by Aaron Chow - University of Toronto T-Space · Aaron Chow Doctor of Philosophy Graduate Department of Mathematics University of Toronto 2015 Let fbe a modular form of even weight

Applications of Fourier coefficients of modular forms

by

Aaron Chow

A thesis submitted in conformity with the requirementsfor the degree of Doctor of PhilosophyGraduate Department of Mathematics

University of Toronto

c© Copyright 2015 by Aaron Chow

Abstract

Applications of Fourier coefficients of modular forms

Aaron Chow

Doctor of Philosophy

Graduate Department of Mathematics

University of Toronto

2015

Let f be a modular form of even weight k and level N which is a normalized eigen-form for the Hecke

operators, and write

f(z) =

∞∑n=1

af (n)e2πinz

for its Fourier expansion at i∞. We assume the Fourier coefficients af (n)∞n=1 are rational integers. A

notable example of such a form is Ramanujan’s cusp form ∆ of weight 12 and level 1. In this case, the

Fourier coefficients are given by the famous Ramanujan τ function:

∆(z) = e2πiz∞∏n=1

(1− e2πinz)24 =

∞∑n=1

τ(n)e2πinz.

In this thesis, we study applications of the Fourier coefficients τ(n), and more generally, af (n).

First, we present a factoring algorithm that uses an oracle that outputs af (n). We show that the

algorithm runs in polynomial time for squarefree integers on average, and quasi-polynomial time for

non-squarefree integers on average. We note that current factoring algorithms have sub-exponential

runtimes. More importantly, when combined with Edixhoven’s (conditional) polynomial time algorithm

for computing af (p), we get an equivalence between factoring and computing af (n).

Secondly, we present an algorithm that uses an oracle that outputs af (n) for testing the squarefree-

ness of an integer. This algorithm exploits the greatest common divisor of sequences of the form

(af (pr))r∈A where p is a prime and A ⊂ N. Thirdly, we present a primality testing algorithm that

uses an oracle that outputs τ(n) and afE (n) where fE is a form of weight 2 corresponding to an elliptic

curve. As a corollary, we use such an oracle to compute in polynomial time the value of the Mobius

function in the squarefree case.

ii

Dedication

For my wife, Alison, and our two children, Alexis and Ashton.

iii

Acknowledgements

Academic Acknowledgements First and foremost, I am forever indebted to my supervisor, Prof. V.

Kumar Murty. This thesis would not have been possible without him – his teaching, guidance, support,

and patience. I am grateful that he has always provided encouragement and optimism, especially at

times when I want to give up. Most importantly, I thank him for taking me seriously as a researcher.

I thank my external examiner, Dr. Ramarathnam Venkatesan, who travelled to Toronto on the

occasion of my defense, for his helpful comments, career advice, and words of wisdom.

I thank the members of my supervisory committee, Prof. John Friedlander and Prof. Henry Kim,

for their helpful suggestions and critiques, and for keeping me on track.

Big thanks goes out to the members of the GANITA Lab: Catalina Anghel, Robby Burko, Anup

Dixit, Payman Eskandari, William George, Ken Giuliani, Jack Kyls, Nataliya Laptyeva, Meng Fai Lim,

Mariam Mourtada, Gaurav Patil, Hamid Usefi, Nikolajs Volkovs, Ren Zhu, Ying Zong. I thank them for

their questions, comments, help, advice, support, encouragement, and most of all, for their friendship.

Personal Acknowledgements I cannot even begin to thank my wife, Alison, who has unconditionally

supported me throughout my graduate studies. Her love, trust, support, encouragement, quiet patience,

and tolerance were fundamental to the completion of my graduate studies. I thank her for taking care

of everything at home, especially in the last few months before my defense, so that I can focus on this

thesis. I am grateful for her faith in me and allowing me to pursue my ambitions.

I thank my parents, Peter and Ida, for their support and encouragement in my decision to attend

graduate school.

Finally, all praise and honour goes to my saviour Jesus Christ.

iv

Contents

1 Introduction 1

1.1 Main contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

2 Background 5

2.1 Modular forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2.1.1 Ramanujan’s cusp form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.1.2 Congruences for coefficients of modular forms . . . . . . . . . . . . . . . . . . . . . 9

2.2 Algorithms and complexity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.2.1 Computational complexity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.2.2 Oracles and reductions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.3 Polynomial solving . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.3.1 Jenkins-Traub algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.3.2 Homotopy methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

3 A factoring algorithm using Fourier coefficients of modular forms 16

3.1 Introduction and main results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

3.1.1 Main results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

3.2 The squarefree case and proof of theorem 3.3 . . . . . . . . . . . . . . . . . . . . . . . . . 21

3.2.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

3.2.2 Proof of theorem 3.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

3.3 The general case and the proof of theorem 3.4 . . . . . . . . . . . . . . . . . . . . . . . . . 44

3.3.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

3.3.2 Proof of theorem 3.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

3.4 Numerical Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

3.5 Computer Code . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

4 A test for squarefree-ness using Fourier coefficients of modular forms 69


4.1.1 Main results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

4.2 A general recurrence relation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

4.3 The greatest common divisor of a tower . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

4.4 Algorithm SQF — a test for squarefree-ness . . . . . . . . . . . . . . . . . . . . . . . . . . 80

4.4.1 Description of the algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

4.4.2 Proof of theorem 4.1 — correctness of the algorithm . . . . . . . . . . . . . . . . . 81

v

4.4.3 Proof of theorem 4.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

4.4.4 Heuristic analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

4.4.5 Computer code . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

4.4.6 Experimental results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

4.5 Values of the Ramanujuan τ function at odd powers . . . . . . . . . . . . . . . . . . . . . 93

4.5.1 Proof of theorem 4.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

5 A test for primality using Fourier coefficients of modular forms 98


5.1.1 Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

5.2 Proof of theorem 5.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

5.2.1 Description of algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

5.2.2 Proof of correctness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

5.3 Proof of theorem 5.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

A Example 3.16 107

B Example 3.17 116

Bibliography 178

vi

Chapter 1

Introduction

Let f be a modular form of even weight k for Γ0(N) which is a normalized eigen-form for the Hecke

operators. Let us write

f(z) =

∞∑n=1

af (n)e2πinz

for its Fourier expansion at i∞. We assume the Fourier coefficients af (n)∞n=1 are rational integers.

Ramanujan’s famous τ function is defined by the relation

e2πiz∞∏n=1

(1− e2πinz)24 =

∞∑n=1

τ(n)e2πinz.

The right hand side is the Fourier expansion at i∞ of a modular form ∆ of weight 12 and level 1 which

is a normalized eigen-form for the Hecke operators.

In cryptography, elliptic curve cryptography (ECC) is becoming increasing prominent. Briefly, elliptic

curve cryptography is an approach to public-key cryptography based on the group structure of elliptic

curves over finite fields. Let E/Fq be an elliptic curve defined over a finite field and consider the abelian

group E(Fq). Let aE(q) be defined by

#E(Fq) = q + 1− aE(q).

A theorem of Hasse says that |aE(q)| ≤ 2√q. For cryptographic applications, computing #E(Fq), or

equivalently aE(q), efficiently is important. The most well-known algorithm for computing aE(q) is

Schoof’s Algorithm.

We now know, by the modularity theorem (previously the Taniyama-Shimura conjecture), that any

elliptic curve E/Q is modular. That is, there is a cusp form fE of weight 2 and some level N which is a

normalized eigen-form for the Hecke operators and such that for p - NE , the conductor of E,

afE (p) = aE(p).

In [ECd+06], Edixhoven et al. presented an analogue of Schoof’s algorithm (albeit, conditional) to

compute in polynomial time, for a prime p, the p-th Fourier coefficient af (p) for modular forms of

weight k and level 1. An immediate corollary is that τ(p) can be computed in polynomial time. This

1

Chapter 1. Introduction 2

algorithm, together with an oracle for factoring an integer n (i.e. an algorithm that gives the factorization

of n in polynomial time), gives an algorithm for computing af (n) in polynomial time. Thus, we may

interpret this as: given an oracle for factoring an integer n, we may compute af (n) in polynomial time.

We may ask about the converse: given an oracle for f (i.e. an algorithm that outputs af (n) in

polynomial time), can we efficiently factor n? For RSA moduli (i.e. n = pq, a product of two distinct

primes), the answer is “yes”; see example 3.1 and [BC07]. In chapter 3 we generalize the technique to

factor any integer n.

More generally, what other applications are possible given an oracle that outputs af (n)? That is,

given an oracle for f , what problems can we solve? Thus the theme of this thesis is: we assume we are

given an oracle for f and we produce algorithms that solve various problems.

Given a positive integer n, one may ask whether n is squarefree. Of course, given the factorization

of n, the answer is easy. In other words, given an oracle for factoring an integer n, deciding whether

n is squarefree is efficient. If we are not given an oracle for factoring an integer n, then the problem

of deciding whether n is squarefree is believed to be hard. In fact, no unconditional polynomial-time

deterministic algorithm for testing the squarefree-ness of an integer is known.

In chapter 4 we present an algorithm called SQFRf for testing whether an integer is squarefree as-

suming we are given an oracle for some f .

An interesting and related problem is the following. Let p be a prime and r an odd positive integer.

We already know that τ(p) | τ(pr). One may ask the converse: is it possible that τ(pr) | τ(p), or

equivalently, τ(pr) = ±τ(p)? This question has been studied in [MMS87], where it was shown that

τ(pr) 6= ±τ(p) when r is sufficiently large. However, in chapter 4, we are able to prove that for all positive

integers r, τ(pr) 6= ±τ(p) for odd primes p such that τ(p) 6= 0, and more generally af (pr) 6= ±af (p) for

odd primes p such that af (p) 6= 0 and af (p) is even.

A primality test is an algorithm that, given an integer n as input, outputs whether n is a prime or

a composite number. Perhaps the most widely known primality tests are the AKS-class primality tests.

In chapter 5, we present a primality testing algorithm assuming we are given an oracle for ∆ and fE ,

where fE is the modular form of weight 2 corresponding to a certain elliptic curve E with Cremona label

54-B3(B).

We also present an algorithm that, given a modular oracle, computes the parity of the number of

factors of a squarefree integer in polynomial time and thus the value of the Mobius function in the

squarefree case.

1.1 Main contributions

This section lists the main contributions of this thesis. All definitions are given in the relevant chapters.

In chapter 3, we prove the following theorems.

Theorem 3.3. Let f be as above. Let n be a positive squarefree integer such that each prime factor of

n is at least 5. Assume af (n) 6= 0. Suppose we are given an oracle for f . Then:

1. There is an average Las Vegas algorithm which gives the factorization of n. Its runtime complexity

is polynomial on average and its storage complexity is polynomial on average.


2. There is a deterministic algorithm which gives the factorization of n. Its runtime complexity is

quasi-polynomial on average and its storage complexity is polynomial on average.

Theorem 3.4. Let f be as above. Let n be a positive integer such that each prime factor of n is at least

5. Assume af (n) 6= 0. Suppose we are given an oracle for f . Then:


is quasi-polynomial on average and its storage complexity is quasi-polynomial on average.


quasi-polynomial on average and its storage complexity is quasi-polynomial on average.

In chapter 4, we describe the squarefree-ness testing algorithm SQFRf . At its heart, it computes a

number

SQFRf (n) =gcd (af (nr), r = 1, 3, 5, . . . , 2R− 1)

|af (n)|.

The following two theorems together ensure SQFRf (n) = 1 if and only if n is squarefree.

Theorem 4.1. Let n be a squarefree integer. Then SQFRf (n) = 1 for any positive integer R.

Theorem 4.2. Let n be a positive non-squarefree integer. Write n =t∏i=1

pαii . Assume n is such that if

pi | n then pi - af (pi). Let

Iodd = i : αi > 1, αi ≡ 1 (mod 2)

denote the set of odd exponents of n greater than 1. Assume that af (pαii ) 6= af (pi) for all i ∈ Iodd. Also,

assume there exists i0 ∈ Iodd such that gcd(af (p

αi0i0

), af (pαi−2i )

)= 1 for all i 6= i0, i ∈ Iodd, and that

gcd(af (p

αi0i0

), af (pri ))

= 1 for all i such that αi = 1 and all positive odd integers r. Then there exists a

positive integer R such that

SQFRf (n) 6= 1.

In other words, the algorithm SQFRf will conclude n is not squarefree.

We argue heuristically in section 4.4.4 that algorithm SQFRf will correctly recognize squarefree inte-

gers n probabilistically in O(log log n) steps if k (the weight of f) is large enough.

Conjecture 4.3. For weights k ≥ 4, algorithm SQFRf will correctly recognize whether an integer n is

squarefree with probability 1− C > 0 by taking R = O(log(1/C) · log t).

We also prove the following theorem which shows τ(pr) - τ(p) for r odd, which is a converse of the

well known property τ(p) | τ(pr):

Theorem 4.3. Let p be an odd prime such that τ(p) 6= 0. Let r be a positive odd integer. Then

τ(pr) 6= ±τ(p).

More generally, af (pr) - af (p) for r odd:

Theorem 4.4. Let p be an odd prime such that af (p) 6= 0 and af (p) ≡ 0 (mod 2). Let r > 1 be a

positive odd integer. Then af (pr) 6= ±af (p).

In chapter 5 we describe a primality test using an oracle for ∆ and fE .


Theorem 5.2. Let ∆ and fE be as above. Suppose we are given an oracle for ∆ and an oracle for

fE. Let n be a positive squarefree integer such that n 6≡ −1 (mod 54). Then there is a deterministic

algorithm that determines whether n is prime or composite in a bounded (independently of n) number of

steps.

Using the proof of the theorem above we are able to create an algorithm for determining the parity

of the number of factors of a squarefree integer.

Theorem 5.3. Let ∆ be as above. Suppose we are given an oracle for ∆. Let n be a positive squarefree

integer. Then there is a deterministic algorithm that determines the parity of the number of factors of

n in a bounded (independently of n) number of steps.

Chapter 2

Background

2.1 Modular forms

We introduce modular forms in this section. We only present what is relevant to this thesis, and is by

no means a complete overview of the subject. We closely follow [Ste07, DS06].

The modular group

SL2(Z) =

(a b

c c

): ad− bc = 1, a, b, c, d ∈ Z

,

acts on the complex upper half plane

h = z ∈ C : Im(z) > 0

by fractional linear transformations, as follows. For any γ = ( a bc c ) ∈ SL2(Z) and z ∈ h we let

γ(z) =az + b

cz + d∈ h.

Modular forms of level 1

Let q = q(z) = e2πiz, which we view as a holomorphic map on C. Let D′ be the open unit disk in Cwith the origin removed. Then q defines a map h → D′. If f is a holomorphic function on h such that

f(z + 1) = f(z), then there exists a function F : D′ → h such that f(z) = F (q(z)). If for all q in a

neighbourhood of 0 we have the equality

f(z) = F (q) =

∞∑n=0

af (n)qn (2.1)

then f is said to be holomorphic at i∞. Equation (2.1) is called the Fourier expansion f at i∞ or

the q-expansion expansion f about i∞, and the numbers af (n) ∈ C in eq. (2.1) are called the Fourier

coefficients of f .

Let k ∈ Z. A modular form of weight k and level 1 is a function f : h→ C that is holomorphic on h

5

Chapter 2. Background 6

and at i∞, and such that

f(γ(z)) = (cz + d)kf(z)

for all γ = ( a bc c ) ∈ SL2(Z). We denote by Mk(SL2(Z)) the set of modular forms of weight k and level 1.

A cusp form of weight k ∈ Z and level 1 is a modular form f of weight k ∈ Z and level 1 whose

leading Fourier coefficient is 0, i.e af (0) = 0 in eq. (2.1). We denote by Sk(SL2(Z)) the set of cusp forms

of weight k and level 1.

It is easy to see that Mk(SL2(Z)) is a vector space, and Sk(SL2(Z)) is a subspace of Mk(SL2(Z)).

Proposition 2.1.

dim(Mk(SL2(Z))) =

0 if k is odd or k < 0

bk/12c if k ≡ 2 (mod 12)

bk/12c+ 1 if k 6≡ 2 (mod 12)

where bxc denotes the largest integer less than or equal to x.

Modular forms of any level

Let N be a positive integer. The principle congruence subgroup of level N is

Γ(N) = ker(SL2(Z)→ SL2(Z/NZ)) =

(a b

c c

)∈ SL2(Z) :

(a b

c c

)≡

(1 0

0 1

)(mod N)

A congruence subgroup of SL2(Z) is any subgroup Γ of SL2(Z) that contains Γ(N) for some N ∈ Z,

N > 0. The smallest such N is called the level of Γ.

Besides Γ(N), the two most important congruence subgroups are

Γ1(N) =

(a b

c c

)∈ SL2(Z) :

(a b

c c

)≡

(1 ∗0 1

)(mod N)

and

Γ0(N) =

(a b

c c

)∈ SL2(Z) :

(a b

c c

)≡

(∗ ∗0 ∗

)(mod N)

where “∗” means “unspecified”, satisfying

Γ(N) ⊂ Γ1(N) ⊂ Γ0(N) ⊂ SL2(Z).

For any γ = ( a bc c ) ∈ SL2(Z) define the factor of automorphy j(γ, z) for z ∈ h to be

j(γ, z) = (cz + d),

and for any γ = ( a bc c ) ∈ SL2(Z) and any integer k define the weight-k operator [γ]k on functions f : h→ Cby

(f [γ]k)(z) = j(γ, z)−kf(γ(z)), z ∈ h (2.2)


Every congruence subgroup Γ of SL2(Z) contains a translation matrix of the form(1 h

0 1

): z → z + h

for some minimal positive integer h. Every holomorphic function f : h→ C satisfying f [γ]k = f for all

γ ∈ Γ is therefore hZ-periodic and thus there exists a function g : D′ → h such that f(z) = g(qh(z))

where D′ is the open unit disk in C with the origin removed and qh = qh(z) = e2πiz/h. If for all qh in a

neighbourhood of 0 we have the equality

f(z) = g(qh) =

∞∑n=0

af (n)qnh (2.3)

then f is said to be holomorphic at i∞. Equation (2.3) is the Fourier expansion expansion f at i∞.

Let

P1(Q) = Q ∪ i∞.

An element γ = ( a bc c ) ∈ SL2(Z) acts on P1(Q) by

γ(z) =

az+bcz+d if z 6= i∞ac if z = i∞

∞ if cz + d = 0.

A cusp of Γ is a Γ-equivalence class of points in P1(Q). Note that we may write any s ∈ P1(Q)

as s = α(i∞) for some α ∈ SL2(Z), and therefore holomorphy at s is naturally defined in terms of

holomorphy at i∞ via the [α]k operator.

Let Γ be a congruence subgroup of SL2(Z) and let k ∈ Z. A modular form of weight k with respect to

Γ is a function f : h→ C such that f is holomorphic on h, f [α]k is holomorphic at i∞ for all α ∈ SL2(Z),

and f [γ]k = f for all γ ∈ Γ. If in addition af (0) = 0 in the Fourier expansion of f [α]k for all α ∈ SL2(Z),

then f is a cusp form of weight k with respect to Γ.

Hecke Operators

In this section, we define Hecke operators for modular forms on level 1 and highlight some properties

that are relevant to this thesis. We note that there are analogous properties for Hecke operators on

modular forms on higher levels.

Recall the weight-k operator [γ]k for any γ = ( a bc c ) ∈ SL2(Z) on functions f : h→ C from eq. (2.2):

(f [γ]k)(z) = j(γ, z)−kf(γ(z)), z ∈ h.

For any positive integer n, let

Xn =

(a b

0 d

): a, b, c ∈ Z, a ≥ 1, ad = n, 0 ≤ b < d

.


The operator Tn,k defined by

Tn,k(f) =∑γ∈Xn

f [γ]k

is called the n-th Hecke operator of weight k and acts on the set of functions f : h → C. When the

weight k is clear from context, we write Tn,k simply as Tn.

A modular form f of weight k is a Hecke eigen-form if f is a eigenvector for all the Hecke operators

Tn. If, in addition, the Fourier coefficient af (1) = 1 in the Fourier expansion of f , then we say f is a

normalized Hecke eigen-form.

Proposition 2.2. If f is a modular form of weight k, then so is Tn,k(f).

Proposition 2.3. On weight k modular forms, if gcd(m,n) = 1 then

Tmn = TmTn,

and if p is prime then

Tpr = TpTpr−1 − Tpr−2pk−1.

For a normalized Hecke eigen-form f , the eigenvalue of Tn is the Fourier coefficient af (1); that is, if

we write

f(z) =

∞∑n=0

af (n)qn

for its Fourier expansion at i∞, then

Tn(f) = af (n)f.

The following is an immediate corollary to proposition 2.3.

Corollary 2.1. Let f be a normalized Hecke eigen-form of weight k. Write

f(z) =

∞∑n=0

af (n)qn

for its Fourier expansion at i∞. If gcd(m,n) = 1 then

af (m,n) = af (m)a(n),

and if p is prime then

af (pr) = af (p)af (pr−1)− af (pr−2)pk−1.

2.1.1 Ramanujan’s cusp form

Ramanujan’s cusp form ∆ defined by

∆(z) = q

∞∏n=1

(1− qn)24


is a cusp form of weight 12 and level 1 which is a normalized eigen-form for the Hecke operators. Following

Ramanujan, we write

∆(z) =

∞∑n=1

τ(n)e2πinz

for its Fourier expansion at i∞ and define the Ramanujan τ function as the n-th Fourier coefficient of

∆.

Remarkably, Ramanujan conjectured that

1. τ(mn) = τ(m)τ(n) if gcd(m,n) = 1,

2. τ(pr) = τ(p)τ(pr−1)− τ(pr−2)p11 for p prime and r ≥ 2, and

3. |τ(p)| ≤ 2p11/2.

Of course, we now know that the first two properties follow from corollary 2.1. The third property is a

theorem of Deligne [Del74].

Also, τ satisfy several congruence relations [SD73]. We only list a few here:

τ(n) ≡ nσ11(n) (mod 961)

τ(n) ≡ σ11(n) (mod 211) if n ≡ 1 (mod 8)

τ(n) ≡ 1217σ11(n) (mod 213) if n ≡ 3 (mod 8)

τ(n) ≡ 1537σ11(n) (mod 212) if n ≡ 5 (mod 8)

τ(n) ≡ 705σ11(n) (mod 214) if n ≡ 6 (mod 8)

τ(n) ≡ n−610σ1231(n) (mod 36) if n ≡ 1 (mod 3)

τ(n) ≡ n−610σ1231(n) (mod 37) if n ≡ 2 (mod 3)

τ(n) ≡ nσ9(n) (mod 7) if n ≡ 0, 1, 2, 4 (mod 7)

τ(n) ≡ nσ9(n) (mod 72) if n ≡ 3, 5, 6 (mod 7).

Here, σx(n) =∑d|n

dx.

2.1.2 Congruences for coefficients of modular forms

Congruences for coefficients of modular forms, like the ones for the Ramanujan τ function in the previous

section, are studied by H. P. F. Swinnerton-Dyer [SD73].

In this thesis, we use the following corollary found in [SD73, pp. 31-32], which details congruences of

the type

af (n) ≡ nmσk−1−2m(n) (mod `), gcd(n, `) = 1

for several modular forms. We shall omit most of the technical details, and only remark that

Q = 1 + 240

∞∑n=1

σ3(n)qn, R = 1− 504

∞∑n=1

σ5(n)qn

and

1728∆Q3 −R2.


Corollary 2.2 ([SD73]). For the 6 known modular forms which satisfy the conditions of Theorem 1

(Deligne-Serre) in [SD73], the exceptional primes ` and the associated values of m are given in the

following table:

Form k 2 3 5 7 11 13 17 19 23 other `

∆ 12 0 0 1 1 no 691

Q∆ 16 0 0 1 1 1 no 3617

R∆ 18 0 0 2 1 1 1 no 43867

Q2∆ 20 0 0 1 2 1 1 no no 283617

QR∆ 22 0 0 2 1 no 1 1 no 131593

Q2R∆ 26 0 0 2 2 1 no 1 1 no 657931

The first column gives the form and its weight; the middle column gives the value of m for each exceptional

` < k, or “no” if ` is not exceptional; the last column gives the exceptional ` > k for which m = 0

necessarily.

2.2 Algorithms and complexity

2.2.1 Computational complexity

Given an algorithm, one the most important question to ask is: how fast is it?

The runtime complexity of an algorithm quantifies the amount of time taken as a function of its

input size. In this thesis, the time taken is estimated in terms of the number of “operations” the

algorithm takes, where an “operation” is taken to be either a basic arithmetic operation like addition

or multiplication or exponentiation over C, or symbolic manipulation of an expression like rearranging

or expanding. The size of the input is measured by its bit-size. For example, if the input is a positive

integer N then its bit-size is the number of bits needed to store N , which is n = O(logN), where the

logarithm is base 2.

Let T (n) be the runtime complexity of an algorithm with input size n. The runtime complexity of

an algorithm is usually expressed in the form T (n) = O(f(n)), where f is a function of the input size n.

Below are some runtime complexities that will appear in this thesis.

polynomial time T (n) = poly(n) = 2O(logn), for example T (n) = n2.

quasi-polynomial time T (n) = 2poly(logn), for example T (n) = nlogn, nlog logn.

sub-exponential time T (n) = 2o(n), for example T (n) = 2n1/3

.

exponential time T (n) = 2poly(n), for example T (n) = 2n2

.

It is clear that they are in ascending order, that is

polynomial time < quasi-polynomial time < sub-exponential time < exponential time.

Similarly, given an algorithm, another important question one may ask is: how much space does it

take?


The storage complexity of an algorithm quantifies the amount of storage space required as a function

of its input size. It is estimated in terms of the number of bits required to store the intermediate variables

including the input and output. Let S(n) be the storage complexity of an algorithm with input size n.

The storage complexity of an algorithm is usually expressed in the form S(n) = O(f(n)), where f is a

function of the input size n.

For example, storing a polynomial p ∈ Z[x] of degree d, say p(x) = adxd + . . . , a1x, a0, is equivalent

to storing its coefficients (ad, . . . , a1, a0), thus the storage complexity is linear in d and is given byd∑i=0

O(log |ai|).

2.2.2 Oracles and reductions

In what follows below, we give what we believe is an intuitive and sufficient definition of an oracle.

Classically, the definition of an oracle involves Turing machines and can be found in computation theory

texts such as [Ada90].

Let f : A→ B. By an oracle for f we mean the following: there is an algorithm such that on every

input x ∈ A it outputs f(x) ∈ B with runtime complexity polynomial in the input size of x. We may

think of an oracle as a “blackbox” that computes f(x).

Oracles are important in the following sense. Suppose we want to compare the “difficultly” of two

problems Pf and Pg:

Pf : given x, compute f(x) and Pg : given x, compute g(x).

We say Pf ⇒ Pg (i.e. Pf is “harder” than Pg) if there is an algorithm that takes x and an oracle for f as

inputs and outputs g(x) in polynomial time; conversely, Pg ⇒ Pf (i.e. Pg is “harder” than Pf ) if there

is an algorithm that takes x and an oracle for g as inputs and outputs f(x) in polynomial time. We say

Pf is equivalent to Pg if Pf ⇒ Pg and Pg ⇒ Pf ; in this case, Pf and Pg are of the same difficulty.

2.3 Polynomial solving

In this thesis, we will need the complexities of numerically solving polynomial equations, both the

univariate case with one equation and the multi-variate case with a system of equations. For the former,

the Jenkins-Traub algorithm seems to be the standard in the field, and for the latter homotopy methods

seems to be the fastest.

For an iterative root finding algorithm (e.g. Newton’s method and its variants), its runtime complex-

ity depends on the number of iterations that need to be run multiplied by the number of operations per

iteration. The number of iterations needed is in turn directly related to the rate of convergence, which

we define below.

Suppose the (xk) converges to x. We say the sequence converges with order q if there exists a number

µ > 0 such that

limk→∞

|xk+1 − x||xk − x|q

= µ.

Roughly speaking, if (xk) is an approximation to x, then xk+1 has q times the number of correct digits

as that of xk. For example, if q = 2 the convergence is called quadratic convergence, and xk+1 has twice

the number of correct digits as that of xk.


In particular, if (xk) is the result of an iterative root finding algorithm and the algorithm converges

with order q, then it takes approximately

logq n =1

log qlog n

iterations to converge to a root of bit-size n.

2.3.1 Jenkins-Traub algorithm

If we are interested in finding the roots of a univariate polynomial then the Jenkins-Traub algorithm is

one of the most efficient algorithms, and is implemented in most software packages.

The are two variants, “CPOLY” for general polynomials with complex coefficients, and “RPOLY”

for polynomials with real coefficients

Roughly, “CPOLY” works as follows. Given a polynomial f ∈ C[z]:

f(z) =

d∑i=0

aizi, ad = 1, a0 6= 0

where ai ∈ C, it finds the d roots ζ1, . . . , ζd of f one at a time in increasing order of magnitude. Once a

root ζi is found, the linear factor (z−ζi) is removed from f , and the process repeats with the polynomial

f(z)/(z − ζi) of degree d− 1.

For f ∈ R[z], “RPOLY” is similar, but instead finds two roots at a time and avoids complex arith-

metic.

The rate of convergence is q ≈ 2.61 > 2, so the runtime complexity for finding all d roots is

O

(d

log qlog n

)where n is the bit-size of the largest root.

For complete details and references, see [Mata]. Internet resources can be found at [Matb].

2.3.2 Homotopy methods

In a paper [BS12] by C. Beltran and M. Shub, the most successful numerical algorithms for solving

polynomial systems based on homotopy methods are surveyed. In this section, we closely follow [BS12],

but will only highlight what is relevant to this thesis. Towards the end of the section, we will state two

homotopy based algorithms and their complexities for solving polynomial systems.

A polynomial system is f : Cn → Cn defined by f = (f1, . . . , fn) where fi : Cn → C is a polynomial

of degree di:

fi(x1, . . . , xn) =∑

α1+···+αn≤di

a(i)α1,...,αnx

α11 · · · , xαnn , a(i)

α1,...,αn ∈ C.

The key ingredient of homotopy methods is a one-line thought: given a goal system to be solved,

choose some other similar system with a known solution ζ0 and move this system to the goal system,

tracking how the known solution moves. Thus, if for every t ∈ [0, 1] we have a system of equations ft

(f0 is the system with a known solution, f1 is the goal system we want to solve), then we are looking

for a path ζt, t ∈ [0, 1], such that ft(ζt) = 0


Given a polynomial system f , an approximate zero of f with associated (exact) zero ζ is a vector

z0 ∈ Cn such that

‖zk − ζ‖ ≤1

22k−1‖z0 − ζ‖, k ≥ 0

where zk is the result of applying k times the Newton operator z 7→ z −Df(z)−1f(z).

The homogeneous counterpart of f is h : Cn+1 → Cn defined by h = (h1, . . . , hn) where hi : Cn → Cis a homogeneous polynomial of degree di:

hi(x0, x1, . . . , xn) =∑

α1+···+αn≤di

a(i)α1,...,αnx

di−∑ni=1 αi

0 xα11 · · ·xαnn , a(i)

α1,...,αn ∈ C.

Note that the zeros of f and the zeros of the homogeneous counterpart h of f are in correspondence.

Indeed, if ζ is a zero of f then (1, ζ) is a zero of the homogeneous counterpart h of f ; reciprocally, if

(ζ0, ζ1, . . . , ζn) is a zero of h, then (ζ1/ζ0, . . . , ζn/ζ0) is zero of f . Thus, we may solve for the zeros of

h and then recover the zeros of f . Moreover, for any λ ∈ C and x ∈ Cn+1, hi(λx) = λdihi(x), and

therefore

h(λx) = Diag(λd1 , . . . , λdn)h(x).

Thus, the zeros of h lie naturally in the projective space P(Cn+1).

Since we will work with homogeneous systems and projective zeros, we will need a definition of

approximate (projective) zero similar to the one above: given a homogeneous system h, an approximate

zero of h with associated (exact) zero ζ ∈ P(Cn+1) is a vector z0 ∈ P(Cn+1) such that

dR(zk, ζ) ≤ 1

22k−1dR(z0, ζ), k ≥ 0

where zk is the result of applying k times the Newton’s operator z 7→ z −Dh(z)|−1z⊥h(z). Here, dR(x, y)

denotes the Riemannian distance from x to y where x and y are elements in some Riemannian manifold,

Dh(z)|z⊥ denotes the restriction of the derivative of h at z to the (complex) orthogonal subspace z⊥ =

y ∈ Cn+1 : y∗x = 0, and y∗ denotes the complex conjugate transpose of the vector y.

Thus, a (projective) approximate zero of h is a projective point such that successive iterates of the

projective Newton operator quickly approach an exact zero of h. Therefore finding an approximate zero

is an excellent output of a numerical zero-finding algorithm to solve h.

Let

Hs = h ∈ C[x0, . . . , xn] : h is homogeneous of degree s ,

the vector space of homogeneous polynomials of fix degree s ≥ 1. Thus its dimension as a vector space

is

dim(Hs) =

(n+ s

n

).

An Hermitian product on Hs given by

〈h, g〉Hs =

⟨ ∑α1+···+αn≤s

a(i)α1,...,αnx

s−∑ni=1 αi

0 xα11 · · ·xαnn ,

∑α1+···+αn≤s

b(i)α1,...,αnxs−

∑ni=1 αi

0 xα11 · · ·xαnn

⟩

=∑

α1+···+αn≤s

α0! · · ·αn!

s!a(i)α1,...,αnb

(i)α1,...,αn .


Then, given a list of degrees (d) = (d1, . . . , dn), consider the vector space

H(d) =

n⊕i=1

Hdi .

Note that an element h ∈ H(d) can be thought of both as a mapping h : Cn+1 → Cn or as a polynomial

system.

Denote by P(H(d)) the projective space associated to H(d), and by N the complex dimension of

P(H(d)) (the dimension of H(d) is N + 1). Then

N + 1 = dim(H(d)) =

n∑i=1

dim(Hdi) =

n∑i=1

(n+ din

).

An Hermitian structure in H(d) given by:

〈h, g〉 =

n∑i=1

〈hi, gi〉Hdi and ‖h‖ = 〈h, h〉1/2

Denote by S the unit sphere in H(d). The algorithm studied in [BS12] is an algorithm they call linear

homotopy : choose some g ∈ S and ζ ∈ P(C)n+1 such that g(ζ) = 0 (call (g, ζ) a “starting pair”). For

input h ∈ S consider the path contained in the great circle

ht = cos(t)g + sin(t)h− 〈h, g〉g‖h− 〈h, g〉g‖

, t ∈ [0, dR(g, h)],

and use the method in Theorem 9 of [BS12] to track how ζ0 moves to ζdR(g,h), a zero of hdR(g,h) = h.

We now state the results of [BS12] concerning the complexity of linear homotopy.

The first theorem is an Average Las Vegas algorithm, coming from the fact that a random choice is

done. The user of such an algorithm plays the role of a Las Vegas casino, not a gambler, i.e. the chances

of winning (i.e. getting a fast answer) are much higher than those of losing (i.e. waiting a long time

before getting an answer).

Theorem 2.1 (Beltran, Pardo, 2008). The process of choosing a random g ∈ S (with the uniform

distribution) and a random zero ζ of g (with the uniform discrete distribution) can be emulated by a

simple linear algebra procedure.

Then the linear homotopy algorithm with the starting pair (g, ζ) as obtained above has average com-

plexity O(N2), where the notation O(X) = O(X(logX)c) for some fixed constant c.

We will refer to theorem 2.1 as the random homotopy method. Hence the random homotopy method

is a Las Vegas algorithm with an average runtime complexity polynomial in N .

The word “average” in theorem 2.1 is understood as follows. For an input system h, let T (h) be the

running time of the linear homotopy algorithm when (g, ζ) is randomly chosen. Then the expected value

of T (h) for random h is O(N2).

One may also ask for a deterministic algorithm, i.e. an algorithm which does not rely on random

choices.

Consider the solution variety V = (h, ζ) ∈ P(H(d))×P(Cn+1) : h(ζ) = 0 and the natural projection


map π1 : V → P(H(d)) Let Σ′ ⊆ V be the set of critical points of π1 and Σ = π1(Σ′) ⊆ P(H(d)) be the

set of critical values of π1.

Theorem 2.2 (Burgisser, Cucker, 2011). The average (for random h ∈ S) running time of the following

procedure is O(N log logN ): on every input h ∈ P(H(d))\Σ, run simultaneously the algorithms of theorems

18 and 20 in [BS12], stopping the computation whenever one of the two algorithms gives an output.

We will refer to theorem 2.2 as the deterministic homotopy method. Hence the deterministic homotopy

method is a deterministic algorithm with an average runtime complexity quasi-polynomial in N .

Chapter 3

A factoring algorithm using Fourier

coefficients of modular forms

3.1 Introduction and main results

For this chapter, let f be a modular form of even weight k for Γ0(N) which is a normalized eigen-form

for the Hecke operators. Let us write

f(z) =

∞∑n=1

af (n)e2πinz


In cryptography, elliptic curve cryptography (ECC) is becoming increasingly prominent. Briefly,

elliptic curve cryptography is an approach to public-key cryptography based on the group structure of

elliptic curves over finite fields.

Let Fq be a finite field with q elements, and let E/Fq be an elliptic curve defined over a finite field

given by the Weierstrass equation

E : y2 + a1xy + a3y = x3 + a2x2 + a4x+ a6

where a1, a2, a3, a4, a6 ∈ Fq. It is well-known that the points (x, y) ∈ F2q that satisfy E, together with

an identity element O, form an abelian group E(Fq) under the group law ⊕; see, for example, [Sil09].

Thus,

E(Fq) = (x, y) ∈ F2q : y2 + a1xy + a3y = x3 + a2x

2 + a4x+ a6 ∪ O.

One of the most important aspects in the study of elliptic curves is point counting, i.e. #E(Fq) —

the number of points in E(Fq), or equivalently, one more than the number of solutions to the Weierstrass

equation E. Since each x yields at most two values for y, a trivial upper bound is #E(Fq) ≤ 2q + 1.

However, the right order of magnitude is actually q. Hasse proved the following result in the 1930s.

Theorem 3.1 (Hasse). Let E/Fq be an elliptic curve defined over a finite field. Then

|#E(Fq)− q − 1| ≤ 2√q

16

Chapter 3. A factoring algorithm 17

Let aE(q) be defined by

#E(Fq) = q + 1− aE(q).

Then Hasse’s theorem says |aE(q)| ≤ 2√q.

For cryptographic applications, computing #E(Fq), or equivalently, aE(q), efficiently is important.

The most well-known algorithm for computing #E(Fq) is Schoof’s Algorithm [Sch85, Kob94, Was08].

It is a polynomial time algorithm, and computes #E(Fq) in O((log q)8) steps. The idea of Schoof’s

Algorithm is to compute aE(q) (mod l) for enough primes l to determine aE(q). Since |aE(q)| ≤ 2√q

by Hasse, it suffices to use all primes l ≤ L such that∏l≤L

l ≥ 2√q. See [Sil09] for details.



normalized eigen-form for the Hecke operators and such that for p - NE , the conductor of E,

afE (p) = aE(p).

In [ECd+06], Edixhoven et al. presented an analogue of Schoof’s algorithm (albeit, conditional) to

compute in polynomial time, for a prime p, the p-th Fourier coefficient af (p) for modular forms of weight

k and level 1. Since af (m1m2) = af (m1)af (m2) whenever gcd(m1,m2) = 1, this algorithm, together

with an oracle for factoring an integer n, gives an algorithm for computing af (n) in polynomial time.

For any positive integer n, let Tn be the n-th Hecke operator, and let T(N, k) denote the sub-Z-

algebra of End(Sk(Γ1(N))) generated by the Tn’s. The main result of Edixhoven et al. in [ECd+06] is

stated below.

Theorem 3.2 (B. Edixhoven, J. Couveignes, R. de Jong, F. Merkl, J. Bosman, 2006). Assume that

the generalized Riemann hypothesis (GRH) holds. There exists a deterministic algorithm that on input

positive integers n and k, together with the factorization of n into prime factors, computes the element

Tn of T(1, k) in running time polynomial in k and log n.

Recall that the Ramanujan’s cusp form ∆ defined by

∆(z) = e2πiz∞∏n=1

(1− e2πinz)24


Ramanujan, we write

∆(z) =

∞∑n=1

τ(n)e2πinz

for its Fourier expansion at i∞. An immediate corollary to theorem 3.2 is:

Corollary 3.1. There exists a deterministic algorithm that on input a prime number p gives τ(p), in

running time polynomial in log p.

As remarked above, one may interpret theorem 3.2 as follows: Given an oracle for factoring n, we

may compute af (n) efficiently. In this chapter, we study the converse: given an oracle for f (i.e. an

algorithm that outputs af (n) in polynomial time), can we efficiently factor n?

For a RSA modulus (i.e. n = pq where p and q are distinct primes), there is an algorithm that uses

an oracle for f ; see example 3.1 and [BC07].


We remark here that the two fastest known methods for factoring a general integer are number field

sieve methods – the second fastest being quadratic sieve (QS) invented by C. Pomerance in [Pom82],

and the fastest being general number field sieve (GNFS) published by Lenstra and Lenstra in [LL93].

The runtime complexities are sub-exponential; for example, GNFS has a (heuristic) runtime complexity,

roughly, 2O((logn)1/3(log logn)2/3). A comparison of the two sieve methods can be found in [Pom08]. Other

factoring algorithms are described in detail in [CP06, MVOV96].

3.1.1 Main results

Recall that f is a modular form of even weight k for Γ0(N) which is a normalized eigen-form for the

Hecke operators, and we write

f(z) =

∞∑n=1

af (n)e2πinz


For squarefree integers, we have the following theorem.

Theorem 3.3. Let f be as above. Let n be a positive squarefree integer such that each prime factor of

n is at least 5. Assume af (n) 6= 0. Suppose we are given an oracle for f . Then:



Specifically, its runtime complexity is

2O(log logn)

on average, and its storage complexity is

2O(log(k−1)+log logn)

on average.




2O(log logn·log log logn)


2O(log(k−1)+log logn)

on average.

We remark that with the average Las Vegas algorithm in theorem 3.3, the worst case runtime and

storage complexities are exponential; specifically, they are

2O(logn/ log logn) and 2O( lognlog logn+log logn+log(k−1))


respectively. With the deterministic algorithm in theorem 3.3, the worst case runtime and storage

complexities are exponential; specifically, they are

2O(logn) and 2O( lognlog logn+log logn+log(k−1))

respectively.

We want to clarify the two “averages” in theorem 3.3. One “average” is in the sense of the average

runtime complexity of the randomized homotopy method and deterministic homotopy methods as de-

scribed in section 2.3.2. The second “average” (and “worst case”) is in the sense of the average number

of distinct prime factors for a randomly chosen (w.r.t. the uniform distribution) positive integer n.

The idea of the algorithm in theorem 3.3 and its proof is the following. Given n, we form and solve

a system of multivariate polynomial equations whose coefficients depend on n and f . The solution of

the system are the coefficients of a univariate polynomial whose roots are rational numbers and their

denominators are powers of the prime factors of n.

The assumption af (n) 6= 0 is a technical one. The last step of the algorithm extracts the prime

factors of n from the denominators of the roots of a univariate polynomial. This can only be done if

af (n) 6= 0. We remark that Lehmer’s conjecture [Leh43] asserts that τ(n) 6= 0 for any n ≥ 1.


∆(z) = e2πiz∞∏n=1

(1− e2πinz)24


Ramanujan, we write

∆(z) =

∞∑n=1

τ(n)e2πinz

for its Fourier expansion at i∞. It is well known that τ(1) = 1 and τ(m) ∈ Z for all m = 1, 2, 3, . . . .

Therefore, applying theorem 3.3 to f = ∆ immediately yields the following corollary:

Corollary 3.2. Let n be a positive squarefree integer such that each prime factor of n is at least 5.

Assume τ(n) 6= 0. Suppose we are given an oracle for ∆. Then:





For a general (not necessarily squarefree) integer, we have the following theorem.

Theorem 3.4. Let f be as above. Let n be a positive integer such that each prime factor of n is at least

5. Assume af (n) 6= 0. Suppose we are given an oracle for f . Then:








on average.




2O(log logn·(log log logn)2)



on average.

We remark that with the average Las Vegas algorithm in theorem 3.4, the worst case runtime and

storage complexities are exponential; specifically, they are

2O(logn− logn·log log lognlog logn ) and 2O(logn(1− log log logn

log logn ))

respectively. With the deterministic algorithm in theorem 3.4, the worst case runtime and storage

complexities are exponential; specifically, they are

2O(logn(log logn−log log logn)) and 2O(logn(1− log log lognlog logn ))

respectively.

For clarification of the meaning of “average”, see the remark that follows theorem 3.3.

Again, applying theorem 3.4 to f = ∆ immediately yields the following corollary:

Corollary 3.3. Let f be as above. Let n be a positive integer such that each prime factor of n is at

least 5. Assume τ(n) 6= 0. Suppose we are given an oracle for ∆. Then:






3.2 The squarefree case and proof of theorem 3.3

In this section, we assume n is squarefree, and write

n =

t∏i=1

pi,

where each pi ≥ 5 is a distinct prime factor of n and t = ω(n) is the number of distinct prime factors,

and assume af (n) 6= 0.

We will first provide a few examples to illustrate how the factoring algorithm works for 2 ≤ t ≤ 5.

Then we will describe the factoring algorithm and prove theorem 3.3.

3.2.1 Examples

We now present examples to illustrate how one uses an oracle for f to factor such an n with 2 ≤ t ≤ 5.

For concreteness and simplicity, we take f = ∆, and af = τ .

Example 3.1 (2 distinct prime factors). 1 2 Suppose n is a RSA modulus, i.e. n = pq, where p, q ≥ 5

are distinct primes. Assume τ(n) 6= 0.

Then

τ(n2) =(τ(p)2 − p11

) (τ(q)2 − q11

)= τ(n)2 + n11 −

(τ(p)2q11 + τ(q)2p11

).

Since τ(n2), τ(n), and n are known, the sum τ(p)2q11 + τ(q)2p11 is known.

Also,

τ(p)2q11 × τ(q)2p11 = τ(n)2n11.

Since τ(n), and n are known, the product τ(p)2q11 × τ(q)2p11 is known.

Therefore, the individual terms τ(p)2q11 and τ(q)2p11 are known. Indeed they are the roots of the

quadratic polynomial

ϕ2(x) = (x− τ(p)2q11)(x− τ(p)2q11) = x2 − (τ(p)2q11 + τ(q)2p11)x+ τ(p)2q11 × τ(q)2p11.

Dividing τ(p)2q11 and τ(q)2p11 by n11 we obtain the rational numbers

τ(p)2

p11and

τ(q)2

q11.

Notice that τ(n) 6= 0, so τ(p), τ(q) 6= 0. Also, p11 - τ(p)2, otherwise

p11 | τ(p)2 ⇒ p6 | τ(p)⇒ p12 | τ(p)2 ⇒ p12 ≤ τ(p)2,

but |τ(p)2| < 4p11 by Deligne’s bound [Del74]. Similarly q11 - τ(q)2. Hence, τ(p)2

p11 and τ(q)2

q11 are not

integers and their denominators are powers of p and q respectively.

1This example of K. Murty was the motivating example that led to the discovery of the factoring algorithm of thischapter.

2A variant of this example using more general modular forms, independently discovered by E. Bach and D. Charles,can be found in [BC07].


We may now extract p and q from the denominators of τ(p)2

p11 and τ(q)2

q11 .

Example 3.2 (3 distinct prime factors). Suppose n = pqr where p, q ≥ 5. Assume τ(n) 6= 0.

Then

τ(n2) =(τ(p)2 − p11

) (τ(q)2 − q11

) (τ(r)2 − r11

)= τ(n)2 − n11

+(τ(p)2q11r11 + τ(q)2p11r11 + τ(r)2p11q11

)−(τ(p)2τ(q)2r11 + τ(p)2τ(r)2q11 + τ(q)2τ(r)2p11

).

Also,

τ(n3)

τ(n)=(τ(p)2 − 2p11

) (τ(q)2 − 2q11

) (τ(r)2 − 2r11

)= τ(n)2 − 8n11

+ 4(τ(p)2q11r11 + τ(q)2p11r11 + τ(r)2p11q11

)− 2

(τ(p)2τ(q)2r11 + τ(p)2τ(r)2q11 + τ(q)2τ(r)2p11

).

Let us write

a = τ(p)2q11r11 b = τ(q)2p11r11 c = τ(r)2p11q11

and

d = τ(p)2τ(q)2r11 e = τ(p)2τ(r)2q11 f = τ(q)2τ(r)2p11.

Also, write

α = a+ b+ c and β = d+ e+ f.

Then we have the system of (linear) equations

α− β = τ(n2)− τ(n)2 + n11

4α− 2β =τ(n3)

τ(n)− τ(n)2 + 8n11

from which we can solve for α and β. In fact,

α =τ(n3)

2τ(n)− τ(n2) +

τ(n)2

2+ 2n11 and β =

τ(n3)

2τ(n)− 2τ(n2) +

3τ(n)2

2+ n11.

Therefore, we know

a+ b+ c = α

ab+ bc+ ac = n11β

abc = τ(n)2n22.

However, these are exactly the coefficients of the cubic polynomial

ϕ3(x) = (x− a)(x− b)(x− c) = x3 − αx2 + n11βx− τ(n)2n22.


Since we know the ϕ we can solve the roots a, b, and c. Dividing them by n11, we get

τ(p)2

p11and

τ(q)2

q11and

τ(r)2

r11

which are non-zero rational numbers and are not integers by the argument from the example above. We

may now extract p, q, and r from the denominators.

Example 3.3 (4 distinct prime factors). Suppose n = p1p2p3p4 where each pi ≥ 5. Assume τ(n) 6= 0.

Let us write

ri = τ(pi)2n

11

p11i

= τ(pi)2∏j 6=i

p11j i = 1, 2, . . . , 4

and consider the polynomial

ϕ4(x) = (x− r1)(x− r2)(x− r3)(x− r4)

= x4 − S1x3 + S2x

2 − S3x+ S4.

Thus,

S1 = r1 + r2 + r3 + r4

S2 = r1r2 + r1r3 + · · ·+ r3r4

S3 = r1r2r3 + r1r2r4 + r1r3r4 + r2r3r4

S4 = r1r2r3r4.

Now,

τ(n2) =(τ(p1)2 − p11

1

) (τ(p2)2 − p11

2

) (τ(p3)2 − p11

3

) (τ(p4)2 − p11

4

)= τ(p1p2p3p4)2

−(τ(p1p2p3)2p11

4 + · · ·+ τ(p2p3p4)2p111

)+(τ(p1p2)2(p3p4)11 + · · ·+ τ(p3p4)2(p1p2)11

)−(τ(p1)2(p2p3p4)11 + · · ·+ τ(p5)2(p1p2p3)11

)+ (p1p2p3p4)11

= τ(n)2 − 1

n22(r1r2r3 + · · · ) +

1

n11(r1r2 + · · · )− (r1 + · · · ) + n11

= τ(n)2 − 1

n22S3 +

1

n11S2 − S1 + n11,

and

τ(n3)

τ(n)=(τ(p1)2 − 2p11

1

) (τ(p2)2 − 2p11

2

) (τ(p3)2 − 2p11

3

) (τ(p4)2 − 2p11

4

)= τ(p1p2p3p4)2

− 2(τ(p1p2p3)2p11

4 + · · ·+ τ(p2p3p4)2p111

)+ 22

(τ(p1p2)2(p3p4)11 + · · ·+ τ(p3p4)2(p1p2)11

)− 23

(τ(p1)2(p2p3p4)11 + · · ·+ τ(p5)2(p1p2p3)11

)


+ 24(p1p2p3p4)11

= τ(n)2 − 2

n22(r1r2r3 + · · · ) +

22

n11(r1r2 + · · · )− 23 (r1 + · · · ) + 24n11

= τ(n)2 − 2

n22S3 +

22

n11S2 − 23S1 + 24n11.

and

τ(n4) =(τ(p1)4 − 3τ(p1)2p11

1 + p221

)· · ·(τ(p4)4 − 3τ(p4)2p11

4 + p224

)= τ(p1p2p3p4)4

+(τ(p1p2p3)4p22

4 + · · ·+ τ(p2p3p4)4p221

)− 3

(τ(p1p2p3)4τ(p4)2p11

4 + · · ·+ τ(p2p3p4)4τ(p1)2p111

)+(τ(p1p2)4(p3p4)22 + · · ·+ τ(p3p4)4(p1p2)22

)− 3

(τ(p1p2)4τ(p3)2p11

3 p224 + · · ·+ τ(p3p4)4τ(p1)2p11

1 p222

)+ 32

(τ(p1p2)4τ(p3p4)2(p3p4)11 + · · ·+ τ(p3p4)4τ(p1p2)2(p1p2)22

)+(τ(p1)4(p2p3p4)22 + · · ·+ τ(p4)4(p1p2p3)22

)− 3

(τ(p1)4τ(p2)2p11

2 (p3p4)22 + · · ·+ τ(p4)4τ(p1)2p111 (p2p3)22

)+ 32

(τ(p1)4τ(p2p3)2(p2p3)11p22

4 + · · ·+ τ(p4)4τ(p1p2)2(p1p2)11p223

)− 33

(τ(p1)4τ(p2p3p4)2(p2p3p4)11 + · · ·+ τ(p4)4τ(p1p2p3)2(p1p2p3)11

)+ (p1p2p3p4)22

− 3(τ(p1)2p11

1 (p2p3p4)22 + · · · τ(p4)2p115 (p1p2p3)22

)+ 32

(τ(p1p2)2(p1p2)11(p3p4)22 + · · ·+ τ(p3p4)2(p3p4)11(p1p2)22

)− 33

(τ(p1p2p3)2(p1p2p3)11(p4)22 + · · ·+ τ(p2p3p4)2(p2p3p4)11(p1)22

)+ 34τ(p1p2p3p4)2(p1p2p3p4)11

= τ(n)4

+1

n44

(r21r

22r

23 + · · ·

)− 3

n55

(r21r

22r

23r4 + · · ·

)+

1

n22

(r21r

22 + · · ·

)− 3

n33

(r21r

22r3 + · · ·

)+

32

n44

(r21r

22r3r4 + · · ·

)+(r21 + · · ·

)− 3

n11

(r21r2 + · · ·

)+

32

n22

(r21r2r3 + · · ·

)− 33

n33

(r21r2r3r4 + · · ·

)+ n22 − 3n11 (r1 + · · · ) + 32 (r1r2 + · · · )− 33

n11(r1r2r3 + · · · ) + 34τ(n)2n11

= τ(n)4

+1

n44

(S2

3 − 2S4S2

)− 3

n55(S4S3)

+1

n22

(S2

2 − S3S1 + S4

)− 3

n33(S2S3 − 3S1S4) +

32

n44(S3S2 − 3S4S1)

+(S2

1 − 2S2

)− 3

n11(S2S1 − 3S3) +

32

n22(S3S1 − 4S4)− 33

n33(S4S1)

+ n22 − 3n11S1 + 32S2 −33

n11S3 + 34τ(n)2n11.

We now have 3 equations and 3 unknowns. We can then solve for S1, S2, S3, and obtain ϕ4(x).


Solving for the roots ri of ϕ4(x) and dividing each ri by n11, we obtain non-zero rational numbers

τ(pi)2

p11i

i = 1, . . . , 4

from which we may extract the pi’s from the denominators.

Example 3.4 (5 distinct prime factors). Suppose n = p1p2p3p4p5 where each pi ≥ 5. Assume τ(n) 6= 0.

Let us write

ri = τ(pi)2n

11

p11i

= τ(pi)2∏j 6=i

p11j i = 1, 2, . . . , 5

and consider the polynomial

ϕ5(x) = (x− r1)(x− r2)(x− r3)(x− r4)(x− r5)

= x5 − S4x4 + S3x

3 − S2x2 + S1x− S0.

Thus,

S0 = r1 · · · r5

S1 = r1r2r3r4 + r1r2r3r5 + · · ·+ r2r3r4r5

S2 = r1r2r3 + r1r2r5 + · · ·+ r3r4r5

S3 = r1r2 + r1r3 + · · ·+ r4r5

S4 = r1 + r2 + · · ·+ r5.

Now,

τ(n2) =(τ(p1)2 − p11

1

) (τ(p2)2 − p11

2

) (τ(p3)2 − p11

3

) (τ(p4)2 − p11

4

) (τ(p5)2 − p11

5

)= τ(p1p2p3p4p5)2

−(τ(p1p2p3p4)2p11

5 + · · ·+ τ(p2p3p4p5)2p111

)+(τ(p1p2p3)2(p4p5)11 + · · ·+ τ(p3p4p5)2(p1p2)11

)−(τ(p1p2)2(p3p4p5)11 + · · ·+ τ(p4p5)2(p1p2p3)11

)+(τ(p1)2(p2p3p4p5)11 + · · ·+ τ(p5)2(p1p2p3p4)11

)− (p1p2p3p4p5)11

= τ(n)2 − 1

n33(r1r2r3r4 + · · · ) +

1

n22(r1r2r3 + · · · )− 1

n11(r1r2 + · · · ) + (r1 + · · · )− n11

= τ(n)2 − 1

n33S1 +

1

n22S2 −

1

n11S3 + S4 − n11,

and

τ(n3)

τ(n)=(τ(p1)2 − 2p11

1

) (τ(p2)2 − 2p11

2

) (τ(p3)2 − 2p11

3

) (τ(p4)2 − 2p11

4

) (τ(p5)2 − 2p11

5

)= τ(p1p2p3p4p5)2

− 2(τ(p1p2p3p4)2p11

5 + · · ·+ τ(p2p3p4p5)2p111

)+ 22

(τ(p1p2p3)2(p4p5)11 + · · ·+ τ(p3p4p5)2(p1p2)11

)


− 23(τ(p1p2)2(p3p4p5)11 + · · ·+ τ(p4p5)2(p1p2p3)11

)+ 24

(τ(p1)2(p2p3p4p5)11 + · · ·+ τ(p5)2(p1p2p3p4)11

)− 25(p1p2p3p4p5)11

= τ(n)2 − 2

n33(r1r2r3r4 + · · · ) +

22

n22(r1r2r3 + · · · )− 23

n11(r1r2 + · · · ) + 24 (r1 + · · · )− 25n11

= τ(n)2 − 2

n33S1 +

22

n22S2 −

23

n11S3 + 24S4 − 25n11,

and

τ(n4) =(τ(p1)4 − 3τ(p1)2p11

1 + p221

)· · ·(τ(p5)4 − 3τ(p5)2p11

5 + p225

)= τ(p1p2p3p4p5)4

+(τ(p1p2p3p4)4p22

5 + · · ·+ τ(p2p3p4p5)4p221

)− 3

(τ(p1p2p3p4)4τ(p5)2p11

5 + · · ·+ τ(p2p3p4p5)4τ(p1)2p111

)+(τ(p1p2p3)4(p4p5)22 + · · ·+ τ(p3p4p5)4(p1p2)22

)− 3

(τ(p1p2p3)4τ(p4)2p11

4 p225 + · · ·+ τ(p3p4p5)4τ(p1)2p11

1 p222

)+ 32

(τ(p1p2p3)4τ(p4p5)2(p4p5)11 + · · ·+ τ(p3p4p5)4τ(p1p2)2(p1p2)22

)+(τ(p1p2)4(p3p4p5)22 + · · ·+ τ(p4p5)4(p1p2p3)22

)− 3

(τ(p1p2)4τ(p3)2p11

3 (p4p5)22 + · · ·+ τ(p4p5)4τ(p1)2p111 (p2p3)22

)+ 32

(τ(p1p2)4τ(p3p4)2(p3p4)11p22

5 + · · ·+ τ(p4p5)4τ(p1p2)2(p1p2)11p223

)− 33

(τ(p1p2)4τ(p3p4p5)2(p3p4p5)11 + · · ·+ τ(p4p5)4τ(p1p2p3)2(p1p2p3)11

)+(τ(p1)4(p2p3p4p5)22 + · · ·+ τ(p5)4(p1p2p3p4)22

)− 3

(τ(p1)4τ(p2)2p11

2 (p3p4p5)22 + · · ·+ τ(p5)4τ(p4)2p114 (p1p2p3)22

)+ 32

(τ(p1)4τ(p2p3)2(p2p3)11(p4p5)22 + · · ·+ τ(p5)4τ(p3p4)2(p3p4)11(p1p2)22

)− 33

(τ(p1)4τ(p2p3p4)2(p2p3p4)11p22

5 + · · ·+ τ(p5)4τ(p2p3p4)2(p2p3p4)11p221

)+ 34

(τ(p1)4τ(p2p3p4p5)2(p2p3p4p5)11 + · · ·+ τ(p5)4τ(p1p2p3p4)2(p1p2p3p4)11

)+((p1p2p3p4p5)22

)− 3

(τ(p1)2p11

1 (p2p3p4p5)22 + · · · τ(p5)2p115 (p1p2p3p4)22

)+ 32

(τ(p1p2)2(p1p2)11(p3p4p5)22 + · · ·+ τ(p4p5)2(p4p5)11(p1p2p3)22

)− 33

(τ(p1p2p3)2(p1p2p3)11(p4p5)22 + · · ·+ τ(p3p4p5)2(p3p4p5)11(p1p2)22

)+ 34

(τ(p1p2p3p4)2(p1p2p3p4)11p22

5 + · · ·+ τ(p2p3p4p5)2(p2p3p4p5)11p221

)− 35

(τ(p1p2p3p4p5)2(p1p2p3p4p5)11

)= τ(n)4 +

1

n66

(r21r

22r

23r

24 + · · ·

)− 3

n77

(r21r

22r

23r

24r5 + · · ·

)+

1

n44

(r21r

22r

23 + · · ·

)− 3

n55

(r21r

22r

23r4 + · · ·

)+

32

n66

(r21r

22r

23r4r5 + · · ·

)+

1

n22

(r21r

22 + · · ·

)− 3

n33

(r21r

22r3 + · · ·

)+

32

n44

(r21r

22r3r4 + · · ·

)− 33

n55

(r21r

22r3r4r5 + · · ·

)+(r21 + · · ·

)− 3

n11

(r21r2 + · · ·

)+

32

n22

(r21r2r3 + · · ·

)− 33

n33

(r21r2r3r4 + · · ·

)+

34

n44

(r21r2r3r4r5 + · · ·

)


+ n22 − 3n11 (r1 + · · · ) + 32 (r1r2 + · · · )− 33

n11(r1r2r3 + · · · ) +

34

n22(r1r2r3r4 + · · · )− 35τ(n)2n11

= τ(n)4 +1

n66

(S2

1 − 2S0S2 + · · ·)− 3

n77(S0S1)

+1

n44

(S2

2 − 2S1S3 + 2S0S4

)− 3

n55(S1S2 − 3S0) +

32

n66(S0S2)

+1

n22

(S2

3 − 2S2S4 + S1

)− 3

n33(S2S3 − 3S1S4 + 5S0) +

32

n44(S1S3 − 4S0S4)− 33

n55(S0S3)

+(S2

4 − 2S3

)− 3

n11(S3S4 − 3S2) +

32

n22(S2S4 − 4S1)− 33

n33(S1S4 − 5S0) +

34

n44(S0S4)

+ n22 − 3n11S4 + 32S3 −33

n11S2 +

34

n22S1 − 35τ(n)2n11,

and

τ(n5)

τ(n)=(τ(p1)4 − 4τ(p1)2p11

1 + 3p221

)· · ·(τ(p5)4 − 4τ(p5)2p11

5 + 3p225

)= τ(p1p2p3p4p5)4

+ 3(τ(p1p2p3p4)4p22

5 + · · ·+ τ(p2p3p4p5)4p221

)− 4

(τ(p1p2p3p4)4τ(p5)2p11

5 + · · ·+ τ(p2p3p4p5)4τ(p1)2p111

)+ 32

(τ(p1p2p3)4(p4p5)22 + · · ·+ τ(p3p4p5)4(p1p2)22

)− 4 · 3

(τ(p1p2p3)4τ(p4)2p11

4 p225 + · · ·+ τ(p3p4p5)4τ(p1)2p11

1 p222

)+ 42

(τ(p1p2p3)4τ(p4p5)2(p4p5)11 + · · ·+ τ(p3p4p5)4τ(p1p2)2(p1p2)22

)+ 33

(τ(p1p2)4(p3p4p5)22 + · · ·+ τ(p4p5)4(p1p2p3)22

)− 4 · 32

(τ(p1p2)4τ(p3)2p11

3 (p4p5)22 + · · ·+ τ(p4p5)4τ(p1)2p111 (p2p3)22

)+ 42 · 3

(τ(p1p2)4τ(p3p4)2(p3p4)11p22

5 + · · ·+ τ(p4p5)4τ(p1p2)2(p1p2)11p223

)− 43

(τ(p1p2)4τ(p3p4p5)2(p3p4p5)11 + · · ·+ τ(p4p5)4τ(p1p2p3)2(p1p2p3)11

)+ 34

(τ(p1)4(p2p3p4p5)22 + · · ·+ τ(p5)4(p1p2p3p4)22

)− 4 · 33

(τ(p1)4τ(p2)2p11

2 (p3p4p5)22 + · · ·+ τ(p5)4τ(p4)2p114 (p1p2p3)22

)+ 42 · 32

(τ(p1)4τ(p2p3)2(p2p3)11(p4p5)22 + · · ·+ τ(p5)4τ(p3p4)2(p3p4)11(p1p2)22

)− 43 · 3

(τ(p1)4τ(p2p3p4)2(p2p3p4)11p22

5 + · · ·+ τ(p5)4τ(p2p3p4)2(p2p3p4)11p221

)+ 44

(τ(p1)4τ(p2p3p4p5)2(p2p3p4p5)11 + · · ·+ τ(p5)4τ(p1p2p3p4)2(p1p2p3p4)11

)+ 35

((p1p2p3p4p5)22

)− 41 · 34

(τ(p1)2p11

1 (p2p3p4p5)22 + · · · τ(p5)2p115 (p1p2p3p4)22

)+ 42 · 33

(τ(p1p2)2(p1p2)11(p3p4p5)22 + · · ·+ τ(p4p5)2(p4p5)11(p1p2p3)22

)− 43 · 32

(τ(p1p2p3)2(p1p2p3)11(p4p5)22 + · · ·+ τ(p3p4p5)2(p3p4p5)11(p1p2)22

)+ 44 · 3

(τ(p1p2p3p4)2(p1p2p3p4)11p22

5 + · · ·+ τ(p2p3p4p5)2(p2p3p4p5)11p221

)− 45

(τ(p1p2p3p4p5)2(p1p2p3p4p5)11

)= τ(n)4 +

3

n66

(r21r

22r

23r

24 + · · ·

)− 4

n77

(r21r

22r

23r

24r5 + · · ·

)+

32

n44

(r21r

22r

23 + · · ·

)− 4 · 3n55

(r21r

22r

23r4 + · · ·

)+

42

n66

(r21r

22r

23r4r5 + · · ·

)


+33

n22

(r21r

22 + · · ·

)− 4 · 32

n33

(r21r

22r3 + · · ·

)+

42 · 31

n44

(r21r

22r3r4 + · · ·

)− 43

n55

(r21r

22r3r4r5 + · · ·

)+ 34

(r21 + · · ·

)− 41 · 33

n11

(r21r2 + · · ·

)+

42 · 32

n22

(r21r2r3 + · · ·

)− 43 · 3

n33

(r21r2r3r4 + · · ·

)+

44

n44

(r21r2r3r4r5 + · · ·

)+ 35n22 − 4 · 34n11 (r1 + · · · ) + 42 · 33 (r1r2 + · · · )− 43 · 32

n11(r1r2r3 + · · · )

+44 · 3n22

(r1r2r3r4 + · · · )− 45τ(n)2n11

= τ(n)4 +3

n66

(S2

1 − 2S0S2 + · · ·)− 4

n77(S0S1)

+32

n44

(S2

2 − 2S1S3 + 2S0S4

)− 4 · 3n55

(S1S2 − 3S0) +42

n66(S0S2)

+33

n22

(S2

3 − 2S2S4 + S1

)− 4 · 32

n33(S2S3 − 3S1S4 + 5S0) +

42 · 3n44

(S1S3 − 4S0S4)− 43

n55(S0S3)

+ 34(S2

4 − 2S3

)− 4 · 33

n11(S3S4 − 3S2) +

42 · 32

n22(S2S4 − 4S1)− 43 · 3

n33(S1S4 − 5S0) +

44

n44(S0S4)

+ 35n22 − 4 · 34n11S4 + 42 · 33S3 −43 · 32

n11S2 +

44 · 3n22

S1 − 45τ(n)2n11

We now have 4 equations and 4 unknowns. We can then solve for S1, S2, S3, and S4 and obtain

ϕ5(x). Solving for the roots ri of ϕ5(x) and dividing each ri by n11, we obtain non-zero rational numbers

τ(pi)2

p11i

i = 1, . . . , 4

from which we may extract the pi’s from the denominators.

3.2.2 Proof of theorem 3.3

In this section, we assume n is squarefree, and write

n =

t∏i=1

pi,



Let

ri = af (pi)2∏j 6=i

pk−1j , i = 1, . . . , t. (3.1)

Define

ϕt(x) =

t∏i=1

(x− ri).

Let Sj be defined by the expansion of ϕt(x):

ϕt(x) = xt − S1xt−1 + S2x

t−2 + · · ·+ (−1)tSt. (3.2)

Note that, by definition, ri > 0, so Sj > 0. Also, note that St =t∏i=1

ri = af (n)2n(k−1)(t−1).


It is well-known that af (pm) can be written as a polynomial expression in af (p)2 and pk−1 with

coefficients that depend only on m. In the following lemma, we make this more explicit.

Lemma 3.1. For even powers m = 2l

af (p2l) =

l∑j=0

c2l,2(l−j)af (p)2(l−j)p(k−1)j

where c2,2 = 1, c2,0 = −1, and the coefficients c2l,2(l−j) are given by the recurrence relation

c2(l+1),2(l+1−j) =

1, j = 0

c2l+1,2(l−j) − c2l,2(l+1−j), j = 1, . . . , l

c2l,0, j = l + 1

.

For odd powers m = 2l + 1

af (p2l+1)

af (p)=

l∑j=0

c2l+1,2(l−j)af (p)2(l−j)p(k−1)j

where c3,2 = 1, c3,0 = −2, and the coefficients c2l+1,2(l−j) are given by the recurrence relation

c2(l+1)+1,2(l+1−j) =

1, j = 0

c2(l+1),2(l+1−j) − c2l+1,2(l+1−j), j = 1, . . . , l + 1.

Proof. By induction. For l = 1,

af (p2) = af (p)2 − pk−1

andaf (p3)

af (p)= af (p)2 − 2pk−1.

For the case m = 2(l + 1),

af (p2(l+1)) = af (p)af (p2l+1)− af (p2l)pk−1

= af (p)2 af (p2l+1)

af (p)− af (p2l)pk−1

= af (p)2l∑

j=0

c2l+1,2(l−j)af (p)2(l−j)p(k−1)j − pk−1l∑

j=0

c2l,2(l−j)af (p)2(l−j)p(k−1)j

=

l∑j=0

c2l+1,2(l−j)af (p)2(l+1−j)p(k−1)j −l∑

j=0

c2l,2(l−j)af (p)2(l−j)p(k−1)(j+1)

=

l∑j=0

c2l+1,2(l−j)af (p)2(l+1−j)p(k−1)j −l+1∑j=1

c2l,2(l+1−j)af (p)2(l+1−j)p(k−1)j

= c2l+1,2laf (p)2(l+1) +

l∑j=1

c2l+1,2(l−j)af (p)2(l+1−j)p(k−1)j


−l∑

j=1

c2l,2(l+1−j)af (p)2(l+1−j)p(k−1)j − c2l,0p(k−1)(l+1)

= af (p)2(l+1) +

l∑j=1

(c2l+1,2(l−j) − c2l,2(l+1−j))af (p)2(l+1−j)p(k−1)j − c2l,0p(k−1)(l+1)

=

l+1∑j=0

c2(l+1),2(l+1−j)af (p)2(l+1−j)p(k−1)j

where we define

c2(l+1),2(l+1−j) =

1, j = 0

c2l+1,2(l−j) − c2l,2(l+1−j), j = 1, . . . , l

c2l,0, j = l + 1

.

For the case m = 2(l + 1) + 1,

af (p2(l+1)+1) = af (p)af (p2(l+1))− af (p2l+1)pk−1

af (p2(l+1)+1)

af (p)= af (p2(l+1))− af (p2l+1)

af (p)pk−1

=

l+1∑j=0

c2(l+1),2(l+1−j)af (p)2(l+1−j)p(k−1)j − pk−1l∑

j=0

c2l+1,2(l−j)af (p)2(l−j)p(k−1)j

=

l+1∑j=0

c2(l+1),2(l+1−j)af (p)2(l+1−j)p(k−1)j −l∑

j=0

c2l+1,2(l−j)af (p)2(l−j)p(k−1)(j+1)

=

l+1∑j=0

c2(l+1),2(l+1−j)af (p)2(l+1−j)p(k−1)j −l+1∑j=1

c2l+1,2(l+1−j)af (p)2(l+1−j)p(k−1)j

= af (p)2(l+1) +

l+1∑j=1

c2(l+1),2(l+1−j)af (p)2(l+1−j)p(k−1)j −l+1∑j=1

c2l+1,2(l+1−j)af (p)2(l+1−j)p(k−1)j

= af (p)2(l+1) +

l+1∑j=1

(c2(l+1),2(l+1−j) − c2l+1,2(l+1−j))af (p)2(l+1−j)p(k−1)j

=

l+1∑j=0

c2(l+1)+1,2(l+1−j)af (p)2(l+1−j)p(k−1)j

where we define

c2(l+1)+1,2(l+1−j) =

1, j = 0

c2(l+1),2(l+1−j) − c2l+1,2(l+1−j), j = 1, . . . , l + 1.

Lemma 3.2. For each 2 ≤ m ≤ t, there exists a symmetric polynomial gf,n,t,m ∈ Q[x1, . . . , xt] whose

coefficients depend on f , n, t, and m such that

af (nm) = gf,n,t,m(r1, . . . , rt) if m is even


af (nm)

af (n)= gf,n,t,m(r1, . . . , rt) if m is odd

where r1, . . . , rt are defined in eq. (3.1). That is, af (nm) (if m is even) oraf (nm)af (n) (if m is odd) can be

written as a symmetric polynomial expression in r1, . . . , rt with rational coefficients that depend on f ,

n, t, and m.

Proof. Recall

ri = af (pi)2∏j 6=i

pk−1j , i = 1, . . . , t.

Since n =t∏i=1

pi, we rewrite ri as

ri =af (pi)

2nk−1

pk−1i

, i = 1, . . . , t.

We will first prove the case when m is even. Let l = m/2, so 1 ≤ l ≤ b t2c. Applying lemma 3.1 we

see that

af (n2l) =

t∏i=1

af (p2li ) =

t∏i=1

l∑j=0

c2l,2(l−j)af (pi)2(l−j)p(k−1)j

.Now,

af (pi)2(l−j)p

(k−1)ji =

(af (pi)

2nk−1

pk−1i

)l−jp

(k−1)(l−j)i

n(k−1)(l−j) p(k−1)ji =

rl−ji

n(k−1)(l−j) p(k−1)li

therefore

af (n2l) =

t∏i=1

l∑j=0

c2l,2(l−j)af (pi)2(l−j)p(k−1)j

=

t∏i=1

l∑j=0

c2l,2(l−j)rl−ji


=

t∏i=1

p(k−1)li

l∑j=0

c2l,2(l−j)

n(k−1)(l−j) rl−ji

= n(k−1)l

t∏i=1

l∑j=0

c2l,2(l−j)


.Hence, if we let

gf,n,2l,t(x1, . . . , xt) = n(k−1)lt∏i=1

l∑j=0

c2l,2(l−j)

n(k−1)(l−j)xl−ji

then af (n2l) = gf,n,2l,t(r1, . . . , rt).

Now suppose m is odd. Let l = (m+ 1)/2, so 1 ≤ l ≤ b t−12 c. Applying lemma 3.1 we see that

af (n2l+1)

af (n)=

t∏i=1

af (p2l+1i )

af (pi)=

t∏i=1

l∑j=0

c2l+1,2(l−j)af (p)2(l−j)p(k−1)j

.


Now,

af (pi)2(l−j)p

(k−1)ji =

(af (pi)

2nk−1

pk−1i

)l−jp

(k−1)(l−j)i

n(k−1)(l−j) p(k−1)ji =

rl−ji


therefore

af (n2l+1)

af (n)=

t∏i=1

l∑j=0

c2l+1,2(l−j)af (pi)2(l−j)p(k−1)j

=

t∏i=1

l∑j=0

c2l+1,2(l−j)rl−ji


=

t∏i=1

p(k−1)li

l∑j=0

c2l+1,2(l−j)


= n(k−1)l

t∏i=1

l∑j=0

c2l+1,2(l−j)


and hence

af (n2l+1) = af (n)n(k−1)lt∏i=1

l∑j=0

c2l+1,2(l−j)


.Hence, if we let

gf,n,2l+1,t(x1, . . . , xt) = af (n)n(k−1)lt∏i=1

l∑j=0

c2l+1,2(l−j)

n(k−1)(l−j) xl−ji

then

af (n2l+1)af (n) = gf,n,2l+1,t(r1, . . . , rt).

Notice that in both cases (m even or odd), gf,n,t,m ∈ Q[x1, . . . , xt] is a polynomial with rational

coefficients that depend on f , m, n, and t.

Example 3.5. For example, take f = ∆ (hence k = 12) and t = 2. Then

g∆,n,2,2(x1, x2) =x1x2

n11+ n11 − x1 − x2.


g∆,n,4,2(x1, x2, x3, x4) =x1x2x3x4

n33− x1x2x3

n22− x1x2x4

n22− x1x3x4

n22− x2x3x4

n22

+x1x2

n11+x1x3

n11+x2x3

n11+x1x4

n11+x2x4

n11+x3x4

n11

+ n11 − x1 − x2 − x3 − x4

g∆,n,4,3(x1, x2, x3, x4) =x1x2x3x4

n33− 2x1x2x3

n22− 2x1x2x4

n22− 2x1x3x4

n22− 2x2x3x4

n22

+4x1x2

n11+

4x1x3

n11+

4x2x3

n11+

4x1x4

n11+

4x2x4

n11+

4x3x4

n11

+ 16n11 − 8x1 − 8x2 − 8x3 − 8x4

g∆,n,4,4(x1, x2, x3, x4) = n22 − 3x1n11 − 3x2n

11 − 3x3n11 − 3x4n

11 + x21 + x2

2 + x23 + x2

4

+ 9x1x2 + 9x1x3 + 9x2x3 + 9x1x4 + 9x2x4 + 9x3x4


− 3x1x22

n11− 3x1x

23

n11− 3x2x

23

n11− 3x1x

24

n11− 3x2x

24

n11− 3x3x

24

n11− 3x2

1x2

n11− 3x2

1x3

n11− 3x2

2x3

n11

− 27x1x2x3

n11− 3x2

1x4

n11− 3x2

2x4

n11− 3x2

3x4

n11− 27x1x2x4

n11− 27x1x3x4

n11− 27x2x3x4

n11

+x2

1x22

n22+x2

1x23

n22+x2

2x23

n22+

9x1x2x23

n22+x2

1x24

n22+x2

2x24

n22+x2

3x24

n22+

9x1x2x24

n22+

9x1x3x24

n22

+9x2x3x

24

n22+

9x1x22x3

n22+

9x21x2x3

n22+

9x1x22x4

n22+

9x1x23x4

n22+

9x2x23x4

n22+

9x21x2x4

n22

+9x2

1x3x4

n22+

9x22x3x4

n22+

81x1x2x3x4

n22− 3x1x

22x

23

n33− 3x2

1x2x23

n33− 3x1x

22x

24

n33− 3x1x

23x

24

n33

− 3x2x23x

24

n33− 3x2

1x2x24

n33− 3x2

1x3x24

n33− 3x2

2x3x24

n33− 27x1x2x3x

24

n33− 3x2

1x22x3

n33− 3x2

1x22x4

n33

− 3x21x

23x4

n33− 3x2

2x23x4

n33− 27x1x2x

23x4

n33− 27x1x

22x3x4

n33− 27x2

1x2x3x4

n33

+x2

1x22x

23

n44+x2

1x22x

24

n44+x2

1x23x

24

n44+x2

2x23x

24

n44+

9x1x2x23x

24

n44+

9x1x22x3x

24

n44+

9x21x2x3x

24

n44

+9x1x

22x

23x4

n44+

9x21x2x

23x4

n44+

9x21x

22x3x4

n44− 3x1x

22x

23x

24

n55− 3x2

1x2x23x

24

n55

− 3x21x

22x3x

24

n55− 3x2

1x22x

23x4

n55+x2

1x22x

23x

24

n66.

Lemma 3.3. For each 2 ≤ m ≤ t, there exists a polynomial Gf,n,t,m ∈ Q[x1, . . . , xt] whose coefficients

depend on f , n, t, and m such that

af (nm) = Gf,n,t,m(S1, . . . , St) if m is even

af (nm)

af (n)= Gf,n,t,m(S1, . . . , St) if m is odd

where S1, . . . , St are defined by eq. (3.2). That is, af (nm) (if m is even) oraf (nm)af (n) (if m is odd) can be

written as a polynomial expression in S1, . . . , St with rational coefficients that depend on f , n, t, and m.

Proof. Recall S1, . . . , St are defined by the expansion of ϕt(x) in eq. (3.2):

ϕt(x) =

t∏i=1

(x− ri) = xt − S1xt−1 + S2x

t−2 + · · ·+ (−1)tSt.

Recall the j-th elementary symmetric polynomial ej in t variables is a degree j homogeneous polynomial

defined by

ej(x1, . . . , xt) =∑

1≤i1<···<ij≤t

xi1 · · ·xij

Thus we see that,

Sj = ej(r1, . . . , rt).

Recall from lemma 3.2 that gf,n,t,m is a symmetric polynomial. Hence, by the Fundamental Theorem

of Symmetric Polynomials, gf,n,t,m has a unique representation

gf,n,t,m(x1, . . . , xt) = Gf,n,t,m(e1(x1, . . . , xt), . . . , et(x1, . . . , xt))


for some Gf,n,t,m ∈ Q[x1, . . . , xt]. Thus specializing xi to ri we see that

gf,n,t,m(r1, . . . , rt) = Gf,n,t,m(e1(r1, . . . , rt), . . . , et(r1, . . . , rt))

and therefore by lemma 3.2

af (nm) = Gf,n,t,m(S1, . . . , St) if m is even

af (nm)

af (n)= Gf,n,t,m(S1, . . . , St) if m is odd.


G∆,n,2,2(x1, x2) =1

n11x2 + n11 − x1.


G∆,n,4,2(x1, x2, x3, x4) =x4

n33− x3

n22+

x2

n11+ n11 − x1

G∆,n,4,3(x1, x2, x3, x4) =x4

n33− 2x3

n22+

4x2

n11+ 16n11 − 8x1

G∆,n,4,4(x1, x2, x3, x4) =x2

4

n66− 3x3x4

n55+

x23

n44+

7x2x4

n44− 3x2x3

n33− 18x1x4

n33+

x22

n22+

7x1x3

n22+

47x4

n22

+ n22 − 3n11x1 −3x1x2

n11− 18x3

n11+ x2

1 + 7x2

Lemma 3.4. For each 2 ≤ m ≤ t, there exists a homogeneous polynomial Hf,n,t,m ∈ Z[x0, x1, . . . , xt−1]

whose coefficients depend on f , n, t, and m such that

Hf,n,t,m(1, S1, . . . , St−1) = 0.

Proof. Consider Gf,n,t,m ∈ Q[x1, . . . , xt] as in lemma 3.3. Recall St = af (n)2n(k−1)(t−1). Let Cm =

Gf,n,t,m(S1, . . . , St) ∈ Z. Consider the polynomial in the variables x1, . . . , xt−1

(Gf,n,t,m − Cm)(x1, . . . , xt−1) = Gf,n,t,m(x1, . . . , xt−1, St)− Cm

and let Dm be the lowest common multiple of the denominators of the coefficients of Gf,n,t,m−Cm. Let

Hf,n,t,m ∈ Z[x0, . . . , xt−1] be the homogenization of Dm(Gf,n,t,m − Cm). Then substituting x0 = 1 into

Hf,n,t,m is just its de-homogenization, so

Hf,n,t,m(1, x1, . . . , xt−1) = (Dm(Gf,n,t,m − Cm))(x1, . . . , xt−1)

= Dm(Gf,n,t,m(x1, . . . , xt−1, St)− Cm)

and therefore

Hf,n,t,m(1, S1, . . . , St−1) = Dm(Gf,n,t,m(S1, . . . , St−1, St)− Cm) = 0.


Lemma 3.5 (Algorithm to generate Hf,n,t,mtm=2). There is an algorithm to generate Hf,n,t,mtm=2.

Proof. We generate two lists containing the symbolic expressions:

Ta = (af (pm))tm=2 and TH = (Hf,n,t,m(x0, x1, . . . , xt−1))

tm=2 .

1. For each 2 ≤ m ≤ t, expand af (pm) symbolically in powers of af (p)2 and pk−1 as in lemma 3.1 and

add it to Ta. Note that we may refer to af (pm−2) and af (pm−1) in Ta when expanding af (pm).

2. For each 2 ≤ m ≤ t,

(a) Expand af (nm) =t∏i=1

af (pmi ) symbolically in powers of af (pi)2 and pk−1

i by referring to the

expansion of af (pmi ) from T1. Substitute symbolically af (pi)2(l−j)p

(k−1)ji =

xl−ji


as in lemma 3.2. This will give the expression

af (nm) = gf,n,t,m(x1, . . . , xt).

(b) Rearrange gf,n,t,m(x1, . . . , xt) in terms of the elementary symmetric polynomials ej(x1, . . . , xt)tj=1.

This will give the expression

af (nm) = Gf,n,t,m(x1, . . . , xt).

(c) Substitute xt = St = af (n)2n(k−1)(t−1), rearrange, clear denominators, and homogenize the

expression af (nm) = Gf,n,t,m(x1, . . . , xt) as in lemma 3.4. This will give the expression

Hf,n,t,m(x0, x1, . . . , xt−1) = 0.

Add Hf,n,t,m(x0, x1, . . . , xt−1) to TH .

Lemma 3.6. Let k be a positive even integer and α be any positive integer. For sufficiently large prime

p,

p(k−1)α -α∑j=0

cjτ(p)2α−2jp(k−1)j .

where cj ∈ Z for j = 0, . . . , α and c0 = 1,

Proof. By (strong) induction on α.

For α = 1, we see that pk−1 - af (p)2 + c1pk−1, since if otherwise, we have that pk−1 | af (p)2 ⇒ pk |

af (p)2 ⇒ pk ≤ af (p)2, but af (p)2 < 4pk−1 by Deligne’s bound [Del74].

Now, assume the lemma holds for 1, . . . , α. Suppose p(k−1)(α+1) |α+1∑j=0

cjaf (p)2(α+1)−2jp(k−1)j . Now

α+1∑j=0

cjaf (p)2(α+1)−2jp(k−1)j =

α∑j=0

cjaf (p)2(α+1)−2jp(k−1)j + cα+1p(k−1)(α+1)


so

p(k−1)(α+1) |α∑j=0

cjaf (p)2(α+1)−2jp(k−1)j .

Writingα∑j=0

cjaf (p)2(α+1)−2jp(k−1)j = af (p)2α∑j=0

cjaf (p)2α−2jp(k−1)j

we see that since p(k−1) - af (p)2,

p(k−1)α+l|α∑j=0

cjaf (p)2α−2jp(k−1)j

for some 1 ≤ l ≤ (k − 1), so in particular,

p(k−1)α|α∑j=0

cjaf (p)2α−2jp(k−1)j .

This is a contraction.

Lemma 3.7. Let k be a positive even integer and α be any positive integer. For sufficiently large prime

p,

p(k−1)α - af (pα)2.

Proof. First, suppose α is even. By lemma 3.1 we may write

af (pα)2 =

α/2∑j=0

c′jaf (p)α−2jp(k−1)j

2

where c′0 = 1

=

α∑j=0

cjaf (p)2α−2jp(k−1)j where c0 = 1.

By the lemma above, it is not divisible by p(k−1)α.

Next, suppose α is odd. By lemma 3.1 we may write

af (pα)2 =

(α−1)/2∑j=0

c′jaf (p)α−2jp(k−1)j

2

where c′0 = 1

=

α−1∑j=0

cjaf (p)2α−2jp(k−1)j where c0 = 1

=

α∑j=0

cjaf (p)2α−2jp(k−1)j where c0 = 1 and cα = 0.

By the lemma above, it is not divisible by p(k−1)α

Description of the algorithm

Let n =t∏i=1

pi be given.


1. Using lemma 3.5, form a system of t − 1 homogeneous polynomial equations in t variables with

integer coefficients:

Hf,n,t,2(x0, . . . , xt−1) = 0

Hf,n,t,3(x0, . . . , xt−1) = 0

...

Hf,n,t,t(x0, . . . , xt−1) = 0.

2. Solve the homogeneous system (Hf,2,n,t(x0, . . . , xt) = 0)tm=2 above using homotopy methods de-

scribed in section section 2.3.2 . By construction and lemma 3.4, (1, S1, . . . , St−1) is a solution.

3. Form the single variate polynomial

ϕt(x) = xt − S1xt−1 + S2x

t−2 + · · ·+ (−1)tSt

and solve the roots. By construction, r1, . . . , rt are the roots.

4. Divide each ri by nk−1 and obtain

rink−1

=af (pi)

2

pk−1i

, i = 1, . . . , t.

By assumption,ri

nk−16= 0 and are not integers by lemma 3.7, i.e. the denominators are powers of

pi.

5. Extract each pi from the denominator ofri

nk−1.

Complexity Analysis

We provide an estimate the runtime and storage complexity of the algorithm.

Recall that each Hf,n,t,m ∈ Hbm2 c, where Hd is the space of homogeneous polynomial of degree d in

the variables x0, . . . , xt−1. Therefore,

(Hf,n,t,m)tm=2 ∈t⊕

m=2

Hbm2 c = H,

and we define N + 1 to be the dimension of this vector space. Now, note that

N + 1 = dim

(t⊕

m=2

Hbm2 c

)=

t∑m=2

dim(Hbm2 c

)= 2

b t2 c∑d=1

dim (Hd) .

It is well known that dim (Hd) =(t−1+dd

), so

N + 1 = 2

b t2 c∑d=1

(t− 1 + d

d

).


If we apply elementary formula for the “diagonal sum” of binomial coefficientsn∑d=0

(r+dd

)=(r+1+nn

)(r+1+nr+1

)and divide by 2 we see that

N + 1

2=

(t+ b t2c

t

)

Recall that ϕt(x) =∏ti=1(x− ri) is an univariate polynomial of degree t. Denote by Φt its homoge-

neous counterpart. Then

Φt ∈ h ∈ C[x0, x1] : h is homogeneous of degree t = H′.

Let N ′ + 1 = dim(H′). Then we see that

N ′ + 1 =

(1 + t

t

)= t+ 1

and so

N ′ = t

Asymptotic analysis of N We now present an asymptotic analysis of N .

For simplicity, assume t is even so that b t2c = t2 . Then

N + 1

2=

(t+ t

2

t

)=

(3t/2

t

).

We now estimate the size of the binomial coefficient(

3t/2t

).

Now, (3t/2

t

)=

(3t/2)!

t!(t/2)!

=(3t/2)(3t/2− 1) · · · (t+ 1)

(t/2)(t/2− 1) . . . 1.

Taking the natural logarithms of both sides we have

ln

(3t/2

t

)=

t/2∑i=1

ln(t+ i)−t/2∑i=1

ln i

=

t/2∑i=1

(ln(t+ i)− ln i) .

We estimate ln(

3t/2t

)using the trapezoid rule. Let f(x) = ln(t+ x)− lnx. On any interval [i, i+ 1],

f ′′(x) =1

x2− 1

(t+ x)2= O

(1

i2

)


which leads to the approximation∫ i+1

i

f(x) dx =f(i) + f(i+ 1)

2− εi

for some error εi = O(

1i2

). Note that f is decreasing and concave up, so this is an over-approximation.

The error is absolutely convergent; by the integral test, we have

t/2∑i=1

εi = C0 +O

(1

t

)

for some absolute constant C0 :=∞∑i=1

εi.

Therefore, performing this sum we conclude that

∫ t/2

1

f(x) dx =

t/2∑1

f(i)− f(1)

2− f(t/2)

2−(C0 +O

(1

t

)).

Rearranging we get

t/2∑1

f(i) =

∫ t/2

1

f(x) dx+f(1)

2+f(t/2)

2+ C0 +O

(1

t

)=[(t+ x) ln(t+ x)− (t+ x)− x lnx+ x

]t/21

+ln(t+ 1)

2+

ln(3t/2)− ln(t/2)

2+ C0 +O

(1

t

)=

3t

2ln

(3t

2

)− t

2ln

(t

2

)− (t+ 1) ln(t+ 1) +

ln(t+ 1)

2+

ln 3

2+ C0 +O

(1

t

)= ln

(27

4

)t

2+ t ln

(t

t+ 1

)− 1

2ln(t+ 1) + C1 +O

(1

t

)< ln

(27

4

)t

2− 1

2ln(t+ 1) + C1 +O

(1

t

)

where C1 = ln 32 + C0 and since ln

(tt+1

)< 0. Hence,

(3t/2

t

)<

(27

4

)t/2(t+ 1)−1/2eC1+O( 1

t )

= C

(27

4

)t/2(t+ 1)−1/2

(1 +O

(1

t

))= C

(27/4)t/2

(t+ 1)1/2

(1 +O

(1

t

))where C = eC1 .

Recall N+12 =

(3t/2t

). Hence

N < 2C(27/4)t/2

(t+ 1)1/2

(1 +O

(1

t

))+ 1


and therefore

N = O

((27/4)t/2

(t+ 1)1/2

).

By a celebrated result of Hardy-Ramanujan in [Har40],

t = ω(n) ∼ log log n

on average. Whenever we use this estimate in a complexity analysis, we will refer to it as the average

complexity. Therefore, on average,

N = O

((27/4)(log logn)/2

(log log n+ 1)1/2

).

For convenience, also note that in this case

logN = O(t log(27/4)1/2 − 1/2 log t

)= O(t) = O(log log n)

log logN = log(O(t)) = O (log t) = O(log log log n)

It is also well known that

t = ω(n) .log n

log log n

is the upper bound. Whenever we use this estimate in a complexity analysis, we will refer to it as the

worst case complexity. Therefore, in the worst case,

N = O

((27/4)logn/2 log logn

(log n/ log log n+ 1)1/2

).


logN = O(t log(27/4)1/2 − 1/2 log t

)= O(t) = O

(log n

log log n

)log logN = O(log log n− log log log n)

Generating the system of polynomial equations We assume we are given an oracle to f , so it

takes t steps to compute af (n), af (n2), . . . , af (nt).

To generate the polynomials Hf,n,t,mtm=2, we use the algorithm described in lemma 3.5:

1. The (m−j)-th entry in Ta contains bm−j+12 c terms, so them-th entry takesO(bm−1

2 cbm2 c) = O(m2)

multiplications to compute. Therefore, runtime complexity to generate Ta is

t∑m=2

O(m2) = O(t3).

2. The generation of TH from Ta requires the manipulation of N+1 terms, so the runtime complexity

is O(N).


Hence, the total runtime complexity to generate Ta and TH is

O(t3) +O(N).

Note that by the definition of Hf,n,t,m, its coefficients are integers that depend on t, n, k, af (n), and

af (nm). Therefore, we may perform a time-memory trade off by pre-computing Hf,n,t,mtm=2 and store

the coefficients in terms of t, n, k, af (n), and af (nm). Also note that the coefficients are essentially the

products of powers of nk−1 and af (nm), and since

|af (nm)| =

∣∣∣∣∣t∏i=1

af (pmi )

∣∣∣∣∣ <t∏i=1

(m+ 1)p(k−1)/2i = t!n(k−1)/2

log |af (nm)| < log(t!) +k − 1

2log n ≤< t log(t) +

k − 1

2log n

so the bit-size of each coefficient is O(t + k−12 log n + k−1

2 log n) = O(t log t + (k − 1) log n). Since

(Hf,n,t,(α),m)2tm=1 ∈ H there are N + 1 coefficients to store. The storage complexity is therefore

O(N(t log t+ (k − 1) log n)).

Solving the system of polynomial equations The runtime complexity in solving

Hf,2,n,t(x0, x2, . . . , xt−1) = 0

Hf,3,n,t(x0, x2, . . . , xt−1) = 0

...

Hf,t,n,t(x0, x2, . . . , xt−1) = 0

using homotopy methods depends on the N .

If we use the randomized homotopy method with random starting points, the runtime complexity is,

on average,

O(N2).

If we use the deterministic homotopy method, the runtime complexity is

O(N log logN ).

We have to store the roots Sjt−1j=1. Recall by the proof of lemma 3.3, Sj = ej(r1, . . . , rt) where ej

is the j-th symmetric polynomial, so

|Sj | = |ej(r1, . . . , rt)| ≤(t

j

)( max1≤i≤t

|ri|)j ≤ 2t( max1≤i≤t

|ri|)j

log |Sj | ≤ t+ j log max1≤i≤t

|ri| = t+ jO(t+ (k − 1) log n) = jO((k − 1) log n).


Therefore the storage complexity is

t−1∑j=1

jO((k − 1) log n) = O((k − 1)2(log n)2).

Finding the factors Recall that to find ri, we solve

ϕt(x) = 0.

We may apply the “RPOLY” variant of the Jenkins-Traub algorithm to solve for the roots ri. Each

root takes O(log ri) = O(log n) iterations, and there are t roots, so the runtime complexity is

O(t log n).

Finally, extracting the pi’s from the ri’s has runtime complexity O(t).

We only need to store the ri’s and the pis, so storage complexity is

O(t log n).

Total Runtime and Storage Complexity Combining the analysis above,

1. The generation of TH = Hf,n,t,mtm=1 has runtime complexity O(t3) +O(N).

2. Solving the system (Hf,n,t,m(x0, . . . , xt−1))tm=1 = 0 has runtime complexity

• O(N2) if we use the randomized homotopy method, and

• O(N log logN ) if we use the deterministic homotopy method.

3. Find the roots of ϕt(x) and extracting the factors pi’s has runtime complexity O(t log n+ t).

Since we are not given t, we simply run the algorithm ω(n)−1 times, each time assuming t = 2, . . . , ω(n).

Therefore, we see that the runtime complexity is dominated by solving the system (Hf,n,t,m(x0, . . . , xt−1))tm=1 =

0.

If use the average estimate t ∼ log log n, and the randomized homotopy method, the runtime com-

plexity is

O(N2) = O

((27/4)t

(t+ 1)

)= O

((27/4)log logn

(log log n+ 1)

)= O

((27/4)log logn

(log log n+ 1)

)= O

((27/4)log logn

(log log n+ 1)

)= 2O(log logn)

which is polynomial in log n.

With the deterministic homotopy method, the runtime complexity is

O(N log logN ) = O(2logN ·log logN

)= 2O(log logn·log log logn)

which is quasi-polynomial in log n.

If use the worst case estimate t . lognlog logn , and the randomized homotopy method, the runtime

complexity is

O(N2) = O

((27/4)logn/ log logn

log n/ log log n

)


= O(

2log(27/4) lognlog logn+log log logn−log logn

)= O

(2logn/ log logn

)= 2O(logn/ log logn)

which is exponential in log n.



)= 2O((logn/2 log logn)(log logn−2 log log logn)) = 2O(logn)


We summarize the runtime complexity below in a table.

randomized homotopy deterministic homotopy

average polynomial quasi-polynomial

worst case exponential exponential

Similarly,

1. The generation of TH = Hf,n,t,mtm=1 has storage complexity O(N(t+ (k − 1) log n)).

2. Solving the system (Hf,n,t,m(x0, . . . , xt−1))tm=1 = 0 has storage complexity O((k − 1)2(log n)2).

3. Find the roots of ϕt(x) and extracting the factors pi’s has storage complexity O(t).

Therefore, we see that the storage complexity is dominated by the generation of TH = Hf,n,t,mtm=1.

If use the average estimate t ∼ log log n, the storage complexity is therefore

O(N(t log t+ (k − 1) log n)) = O

((27/4)(log logn)/2

(log log n+ 1)1/2(log log n · log log log n+ (k − 1) log n)

)= O

((27/4)(log logn)/2+(log(k−1)+log logn)/ log(27/4)

(log log n+ 1)1/2

)= O

((27/4)(1/2+1/ log(27/4)) log logn+log(27/4) log(k−1)

(log log n+ 1)1/2

)= 2O(log(k−1)+log logn−log log logn)

= 2O(log(k−1)+log logn)

which is polynomial in log n and k.

If use the worst case estimate t . lognlog logn , the storage complexity is therefore

O(N(t log t+ (k − 1) log n)) = O

((27/4)logn/2 log logn

(log n/ log log n+ 1)1/2

(log n

log log n(log log n− log log log n) + (k − 1) log n

))= O

((27/4)logn/2 log logn(k − 1)(log n)

)= 2O( logn

log logn+log logn+log(k−1))



3.3 The general case and the proof of theorem 3.4

In this section, we assume n a positive integer, and write

n =

t∏i=1

pαii ,



3.3.1 Examples

Before proving theorem 3.4, we illustrate how one can factor squarefree integers in some “simple” cases,

i.e. when the number of factors are between 2 and 5. These examples form the basis for extending to

the general case, and illustrate the main idea behind the theorem.

In the examples, for concreteness (and historical reasons), we will use the Ramanujan tau function,

which comes from the Fourier coefficients of the weight 11 modular form

∆ = q

∞∏n=1

(1− qn)24 =

∞∑n=1

τ(n)qn

where q = e2πiz.

Example 3.9 (Factoring n = p2q). Suppose n = p2q. Let

a =√τ(p2)2q11 = τ(p2)q11/2 and b =

√τ(q)2p22 = τ(q)p11.

Note

τ(q2) = τ(q)2 − q11

and

τ(p4) = τ(p)τ(p3)− τ(p2)p11

= τ(p)(τ(p)τ(p2)− τ(p)p11

)− τ(p2)p11

= τ(p)2(τ(p2)− p11

)− τ(p2)p11

=(τ(p2) + p11

) (τ(p2)− p11

)− τ(p2)p11

= τ(p2)2 − p22 − τ(p2)p11.

Thus

τ(n2) = τ(p4)τ(q2)

=(τ(p2)2 − τ(p2)p11 − p22

) (τ(q)2 − q11

)= τ(p2)2τ(q)2 − τ(p2)2q11 − τ(p2)τ(q)2p11 + τ(p2)p11q11

− τ(q)2p22 + p22q11


= τ(n)2 − τ(p2)2q11 − τ(n)τ(q)p11 + τ(p2)p11q11 − τ(q)2p22 + n11

= τ(n)2 − a2 − τ(n)b+ n11/2a− b2 + n11

a2 − n11/2a+ b2 + τ(n)b = τ(n)2 + n11 − τ(n2).

Completing the square, we see

(a− n11/2

2

)2

+

(b+

τ(n)

2

)2

=5

4τ(n)2 +

5

4n11 − τ(n2).

Also, we know

ab = τ(n)√n.

Therefore, we get the following system of equations(a− n11/2

2

)2

+(b+ τ(n)

2

)2

= 54τ(n)2 + 5

4n11 − τ(n2)

ab = τ(n)√n

and solve for a and b. We can divide a2 and b2 by n11

a2

n11=τ(p2)2

p22and

b2

n11=τ(q)2

q11.

and get p and q from the denominators.

Example 3.10 (Factoring n = p2qr). Suppose n = p2qr. Let

a = τ(p2)q11/2r11/2 and b = τ(q)p11r11/2 and c = τ(r)p11q11/2.

Note

τ(q2) = τ(q)2 − q11

τ(q3) = τ(q)3 − 2τ(q)q11

and

τ(p4) = τ(p2)2 − τ(p2)p11 − p22

τ(p6) = τ(p2)3 − 2τ(p2)2p11 − τ(p2)p22 + p33

Now,

τ(n) = τ(p2)τ(q)τ(r)

=abc

n11


and

τ(n2) = τ(p4)τ(q2)τ(r2)

=(τ(p2)2 − τ(p2)p11 − p22

) (τ(q)2 − q11

) (τ(r)2 − r11

)...

= a2 + b2 + c2 + τ(n)2 − τ(n)

n11/2bc− 1

n11(a2b2 + a2c2 + b2c2) +

1

n11/2(ab2 + ac2)− n11/2a− n11

and

τ(n3) = τ(p6)τ(q3)τ(r3)

=(τ(p2)3 − 2τ(p2)2p11 − τ(p2)p22 + p33

) (τ(q)3 − 2τ(q)q11

) (τ(r)3 − 2τ(r)r11

)...

= −4τ(n)n11 + τ(n)3 − 2τ(n)2

n11/2bc− τ(n)

n11(2a2b2 + 2a2c2 + b2c2) +

4τ(n)

n11/2(ab2 + ac2)

+1

n33/2b3c3 + 2τ(n)(2a2 + b2 + c2)− 8τ(n)n11/2a− 2

n11/2(b3c+ bc3) + 4n11/2bc.

We have a system of 3 equations in the 3 variables. Once we solve for a, b, and c, we divide a2, b2, and

c2 by n11

a2

n11=τ(p2)2

p22and

b2

n11=τ(q)2

q11and

c2

n11=τ(r)2

r11.

and get p, q, and r from the denominators.


a = τ(p3)2q11 b = τ(q)2p33 c =p11

τ(p)2 − 2p11.

Now,

τ(q2) = τ(q)2 − q11

τ(q3) = τ(q)3 − 2τ(q)q11

and

τ(p3) = τ(p)3 − 2τ(p)p11

τ(p3)2 = τ(p)6 − 4τ(p)4p11 + 4τ(p)2p22

τ(p3)3 = τ(p)9 − 6τ(p)7p11 + 12τ(p)5p22 − 8τ(p)3p33.

First,

ab = τ(p3)2q11τ(q)2p33 = τ(n)2n11.


Since τ(p) =τ(p3)

τ(p)2 − 2p11⇔ τ(p)p11 =

τ(p3)p11

τ(p)2 − 2p11= τ(p3)c, write

τ(p6) = τ(p)6 − 5τ(p)4p11 + 6τ(p)2p22 − p33

= τ(p3)2 − τ(p)p11τ(p3)− p33

= τ(p3)2 − cτ(p3)2 − p33

we have

τ(n2) = τ(p6)τ(q2)

=(τ(p3)2 − cτ(p3)2 − p33

) (τ(q)2 − q11

)= τ(p3)2τ(q)2 − cτ(p3)2τ(q)2 − τ(q)2p33 − τ(p3)2q11 + cτ(p3)2q11 + p33q11

= τ(n)2 − τ(n)2c− b− a+ ac+ n11.

Similarly, using τ(p)p11 = τ(p3)c and writing

τ(p9) = τ(p)9 − 8τ(p)7p11 + 21τ(p)5p22 − 20τ(p)3p33 + 5τ(p)p44

= τ(p3)3 − 2τ(p)p11τ(p3)2 + τ(p)2p22τ(p3)− 2p33τ(p3) + τ(p)p44

= τ(p3)3 − 2cτ(p3)3 + c2τ(p3)3 − 2p33τ(p3) + cτ(p3)p33,

we have

τ(n3) = τ(p9)τ(q3)

=(τ(p3)3 − 2cτ(p3)3 + c2τ(p3)3 − 2p33τ(p3) + cτ(p3)p33

) (τ(q)3 − 2τ(q)q11

)= τ(p3)3τ(q)3 − 2cτ(p3)3τ(q)3 + c2τ(p3)3τ(q)3 − 2τ(p3)τ(q)3p33 + cτ(p3)τ(q)3p33

− 2τ(p3)3τ(q)q11 + 4cτ(p3)3τ(q)q11 − 2c2τ(p3)3τ(q)q11 + 4τ(p3)τ(q)p33q11 − 2cτ(p3)τ(q)p33q11

= τ(n)3 − 2τ(n)3c+ τ(n)2c2 − 2τ(n)τ(q)2p33 + τ(n)τ(q)2p33c

− 2τ(n)τ(p3)2q11 + 4τ(n)τ(p3)2q11c− 2τ(n)τ(p3)2q11c2 + 4τ(n)n11 − 2τ(n)n11c

= τ(n)3 − 2τ(n)3c+ τ(n)2c2 − 2τ(n)b+ τ(n)bc

− 2τ(n)a+ 4τ(n)ac− 2τ(n)ac2 + 4τ(n)n11 − 2τ(n)n11c

Therefore, we obtain a system of 3 equations in the variables a, b, and c:

ab = τ(n)n11

a+ b+ τ(n)2c− ac = τ(n)2 − τ(n2) + n11

2τ(n)a+ 2τ(n)b+ 2τ(n)(τ(n)2 + n11)c = τ(n)3 + 4τ(n)n11 − τ(n3)

−4τ(n)ac− τ(n)bc− τ(n)2c2 + 2τ(n)ac2.

By construction, a solution exists. Once we solve for a and/or b, divide by n11

a

n11=τ(p3)2

p33and

b

n11=τ(q)2

q11.


and extract p and q from the denominators.

Interestingly, if we solve for c we may extract p from the denominator of 1c + 2, since

c =p11

τ(p)2 − 2p11

1

c=τ(p)2 − 2p11

p11=τ(p)2

p11− 2

1

c+ 2 =

τ(p)2

p11.

Example 3.12 (Factoring n = p2q2). Suppose n = p2q2. Let

a = τ(p2)q11 and b = τ(q2)p11.

Now,

ab = τ(n)n11/2

and

τ(n2) = τ(p4)τ(q4)

=(τ(p2)2 − τ(p2)p11 − p22

) (τ(q2)2 − τ(q2)q11 − q22

)= τ(p2)2τ(q2)2 − τ(p2)2τ(q2)q11 − τ(p2)2q22

− τ(p2)τ(q2)2p11 + τ(p2)τ(q2)p11q11 + τ(p2)p11q22

− τ(q2)2p22 + τ(q2)p22q11 + p22q22

= τ(n)2 − τ(n)a− a2

− τ(n)b+ τ(n)n11/2 + n11/2a

− b2 + n11/2b+ n11

a2 + b2 + (τ(n)− n11/2)(a+ b) = τ(n)2 − τ(n)n11/2 + n11 − τ(n2).

Thus, we have a system two equations in variables a and b. By construction a solution exists. Once we

solve for a and/or b, divide by n11/2 to obtain

a

n11/2=τ(p2)

p11and

b

n11/2=τ(q2)

q11

from which we may extract p and q from the denominator(s).


a = τ(p4)q11/2 b = τ(q)p22 c =p11

τ(p)2 − 3p11.


Then

τ(p4) = τ(p)4 − 3τ(p)2p11 + p22

τ(p4)− p22 = τ(p)2(τ(p)2 − 3p11)

τ(p)2p11 = c(τ(p4)− p22)

=ac

q11/2− cp22

First, we have

ab = τ(n)n11/2.

Secondly,

τ(p8) = τ(p4)2 − τ(p4)τ(p)2p11 + τ(p4)p22 − p44

=a2

q11− a

q11/2

(ac

q11/2− cp22

)+

a

q11/2p22 − p44

and

τ(q2) = τ(q)2 − q11

=b2

p44− q11.

Hence,

τ(n2) = τ(p8)τ(q2)

= −a2 − b2 + a2c+a2b2

n11− a2b2c2

n11+

ab2

n11/2+

ab2c

n11/2− n11/2a− n11/2ac+ n11.

Thirdly,

τ(p12) = τ(p4)3 − 2τ(p4)2τ(p)2p11 − 3τ(p4)2p22 − τ(p4)τ(p)2p33 − 2τ(p4)p44 + τ(p)2p55 − p66

=a3

q33/2− 2

a2

q11

(ac

q11/2− cp22

)− 3

a2

q11p22 − a

q11/2

(ac

q11/2− cp22

)p22

− 2a

q11/2p44 +

(ac

q11/2− cp22

)p44 − p66

and

τ(q3) = τ(q)(τ(q)2 − 2q11)

=b

p22

(b2

p44− 2q11

).

Hence,

τ(n3) = τ(p12)τ(q3)


= 6a2b− b3 − 2a2bc− b3c+a3b3

n33/2− 2a3b3c

n33/2− 3a2b3

n11+a2b3c

n11− 2a3b

n11/2− 2ab3

n11/2

+4a3bc

n11/2+

2ab3c

n11/2+ 4n11/2ab− 4n11/2abc+ 2n11b+ 2n11bc.

Therefore, we obtain a system of 3 equations in the variables a, b, and c By construction, a solution

exists. Once we solve for a and/or b, divide by n11

a2

n11=τ(p4)2

p44and

b2

n11=τ(q)2

q11.

and extract p and q from the denominators.


In this section, we assume n a positive integer, and write

n =

t∏i=1

pαii ,



For each i = 1, . . . , t define

ri =af (pαii )n(k−1)/2

p(k−1)αi/2i

and si =

p(k−1)(αi−1)/2

i af (pi)

af (pαii )

if αi is odd

p(k−1)(αi−2)/2

i af (pi)2

af (pαi )−p(k−1)αi/2if αi is even

. (3.3)

Note that ri, si ∈ R and are generally irrational, however it is easy to see that 0 < r2i ∈ Z.

Lemma 3.8. Let m be a positive integer. Then for each i = 1, . . . , t, there exists a polynomial gf,n,αi,m ∈R[x, y] whose coefficients depend on f , n, m, and αi such that

af (pαimi ) = p(k−1)αim/2i gf,n,αi,m(ri, si)

where ri and si are defined in eq. (3.3). That is, af (pαimi ) can be written as a polynomial expression in

ri and si with real coefficients that depend on f , n, m, and αi.

Proof. First, suppose αi is odd. Recall, in this case,


p(k−1)αi/2i

and si =p

(k−1)(αi−1)/2i af (pi)

af (pαii ).

Rearranging the expression for ri we see that

af (pαii ) =rip

11αi/2i

n(k−1)/2


and substituting this into the expression for si and rearranging we see that

af (pi) =af (pαii )si

p(k−1)(αi−1)/2i

=risip

(k−1)αi/2i

p(k−1)(αi−1)/2i n(k−1)/2

=risip

(k−1)/2i

n(k−1)/2.

Hence, by lemma 3.1,

af (pαimi ) =

bαim/2c∑j=0

cjaf (pi)αim−2jp

(k−1)ji

=

bαim/2c∑j=0

cj

(risip

(k−1)/2i

n(k−1)/2

)αim−2j

p(k−1)ji

=

bαim/2c∑j=0

cjn(k−1)(αim−2j)/2

(risi)αim−2jp

(k−1)(αim−2j)/2+(k−1)ji

= p(k−1)αim/2i

bαim/2c∑j=0


(risi)αim−2j .

So, if we let

gf,n,αi,m(x, y) =

bαim/2c∑j=0


(xy)αim−2j

then af (pαimi ) = p(k−1)αim/2i gf,n,αi,m(ri, si).

Now, suppose αi is even. Recall, in this case,


p(k−1)αi/2i

and si =p

(k−1)(αi−2)/2i af (pi)

2

af (pαi)− p(k−1)αi/2.

Rearranging the expression for ri we see that

af (pαii ) =rip

(k−1)αi/2i

n(k−1)/2

and substituting this into the expression for si and rearranging we see that

af (p)2 = siaf (pαii )− p(k−1)αi/2

i

p(k−1)(αi−2)/2i

= si

(rip

(k−1)αi/2i

n(k−1)/2− p(k−1)αi/2

i

)1

p(k−1)(αi−2)/2i

= si

( rin(k−1)/2

− 1) p

(k−1)αi/2i

p(k−1)(αi−2)/2i

=( risin(k−1)/2

− si)p

(k−1)i


Hence, by lemma 3.1,

af (pαimi ) =

αim/2∑j=0

cjaf (p)αim−2jp(k−1)ji

=

αim/2∑j=0

cj(af (p)2

)(αim−2j)/2p

(k−1)ji

=

αim/2∑j=0

cj

(( risin(k−1)/2

− si)p

(k−1)i

)(αim−2j)/2

p(k−1)ji

=

αim/2∑j=0

cj

( risin(k−1)/2

− si)(αim−2j)/2

p(k−1)(αim−2j)/2i p

(k−1)ji

= p(k−1)αim/2i

αim/2∑j=0

cj

( risin(k−1)/2

− si)αim/2−j

.

So, if we let

gf,n,αi,m(x, y) =

αim/2∑j=0

cj

( xy

n(k−1)/2− y)αim/2−j

then af (pαimi ) = p(k−1)αim/2i gf,n,αi,m(ri, si).

Example 3.14. For example, take f = ∆ (hence k = 12), and n = p21p

32 (so t = 2 and (α) = (2, 3)).

Then

g∆,n,2,1(x, y) =xy

n11/2− y − 1

g∆,n,2,2(x, y) = − 2xy2

n11/2− 3xy

n11/2+x2y2

n11+ y2 + 3y + 1

g∆,n,3,1(x, y) =x3y3

n33/2− 2xy

n11/2

g∆,n,3,2(x, y) =x6y6

n33− 5x4y4

n22+

6x2y2

n11− 1

Lemma 3.9. Let m be a positive integer, and let (α) = (α1, . . . , αt). Then there exists a polynomial

Gf,n,t,(α),m ∈ R[x1, . . . , xt, y1, . . . , yt] whose coefficients depend on f , n, t, m, and (α) such that

af (nm) = n(k−1)m/2Gf,n,t,(α),m(r1, . . . , rt, s1, . . . , st)

where the ri’s and si’s are defined in eq. (3.3). That is, af (nm) can be written as a polynomial expression

in r1, . . . , rt and s1, . . . , st with real coefficients that depend on f , n, t, m, and αi.

Proof. By lemma 3.8, for each αi ∈ (α) there exists gf,n,αi,m ∈ R[x, y] such that

af (pαimi ) = p(k−1)αim/2i gf,n,αi,m(ri, si).


Therefore,

af (nm) =

t∏i=1

af (pαimi )

=

t∏i=1

p(k−1)αim/2i gf,n,αi,m(ri, si)

=

t∏i=1

p(k−1)αim/2i

t∏i=1

gf,n,αi,m(ri, si)

= n(k−1)m/2t∏i=1

gf,n,αi,m(ri, si)

So if we let

Gf,n,t,(α),m(x1, . . . , xt, y1, . . . , yt) =

t∏i=1

gf,n,αi,m(xi, yi)

then af (nm) = n(k−1)m/2Gf,n,t,(α),m(r1, . . . , rt, s1, . . . , st).

Example 3.15. For example, take f = ∆ (hence k = 12), and n = p21p

32 (so t = 2 and (α) = (2, 3)).

Then

G∆,n,(α),1(x1, x2, y1, y2) = − x32y

32

n33/2− x3

2y1y32

n33/2+

2x2y2

n11/2+

2x2y1y2

n11/2+x1x

32y1y

32

n22− 2x1x2y1y2

n11

G∆,n,(α),1(x1, x2, y1, y2) = −2x1x62y

21y

62

n77/2− 3x1x

62y1y

62

n77/2+

10x1x42y

21y

42

n55/2+

15x1x42y1y

42

n55/2− 12x1x

22y

21y

22

n33/2

− 18x1x22y1y

22

n33/2+

2x1y21

n11/2+

3x1y1

n11/2+x2

1x62y

21y

62

n44+x6

2y62

n33

+x6

2y21y

62

n33+

3x62y1y

62

n33− 5x2

1x42y

21y

42

n33− 5x4

2y42

n22− 5x4

2y21y

42

n22

− 15x42y1y

42

n22+

6x21x

22y

21y

22

n22+

6x22y

22

n11+

6x22y

21y

22

n11+

18x22y1y

22

n11

− x21y

21

n11− y2

1 − 3y1 − 1

G∆,n,(α),3(x1, x2, y1, y2) = − x92y

92

n99/2+x3

1x92y

31y

92

n66− 3x2

1x92y

31y

92

n121/2+

3x1x92y

31y

92

n55− x9

2y31y

92

n99/2

− 5x21x

92y

21y

92

n121/2+

10x1x92y

21y

92

n55− 5x9

2y21y

92

n99/2+

6x1x92y1y

92

n55− 6x9

2y1y92

n99/2

+8x7

2y72

n77/2− 8x3

1x72y

31y

72

n55+

24x21x

72y

31y

72

n99/2− 24x1x

72y

31y

72

n44+

8x72y

31y

72

n77/2

+40x2

1x72y

21y

72

n99/2− 80x1x

72y

21y

72

n44+

40x72y

21y

72

n77/2− 48x1x

72y1y

72

n44+

48x72y1y

72

n77/2

− 21x52y

52

n55/2+

21x31x

52y

31y

52

n44− 63x2

1x52y

31y

52

n77/2+

63x1x52y

31y

52

n33− 21x5

2y31y

52

n55/2

− 105x21x

52y

21y

52

n77/2+

210x1x52y

21y

52

n33− 105x5

2y21y

52

n55/2+

126x1x52y1y

52

n33− 126x5

2y1y52

n55/2

+20x3

2y32

n33/2− 20x3

1x32y

31y

32

n33+

60x21x

32y

31y

32

n55/2− 60x1x

32y

31y

32

n22+

20x32y

31y

32

n33/2


+100x2

1x32y

21y

32

n55/2− 200x1x

32y

21y

32

n22+

100x32y

21y

32

n33/2− 120x1x

32y1y

32

n22+

120x32y1y

32

n33/2

+5x3

1x2y31y2

n22− 15x2

1x2y31y2

n33/2+

15x1x2y31y2

n11− 5x2y

31y2

n11/2− 25x2

1x2y21y2

n33/2

+50x1x2y

21y2

n11− 25x2y

21y2

n11/2− 5x2y2

n11/2+

30x1x2y1y2

n11− 30x2y1y2

n11/2

G∆,n,(α),4(x1, x2, y1, y2) =x12

2 y122

n66+x4

1x122 y

41y

122

n88− 4x3

1x122 y

41y

122

n165/2+

6x21x

122 y

41y

122

n77− 4x1x

122 y

41y

122

n143/2

+x12

2 y41y

122

n66− 7x3

1x122 y

31y

122

n165/2+

21x21x

122 y

31y

122

n77− 21x1x

122 y

31y

122

n143/2+

7x122 y

31y

122

n66

+15x2

1x122 y

21y

122

n77− 30x1x

122 y

21y

122

n143/2+

15x122 y

21y

122

n66− 10x1x

122 y1y

122

n143/2+

10x122 y1y

122

n66

− 11x102 y

102

n55− 11x4

1x102 y

41y

102

n77+

44x31x

102 y

41y

102

n143/2− 66x2

1x102 y

41y

102

n66+

44x1x102 y

41y

102

n121/2

− 11x102 y

41y

102

n55+

77x31x

102 y

31y

102

n143/2− 231x2

1x102 y

31y

102

n66+

231x1x102 y

31y

102

n121/2− 77x10

2 y31y

102

n55

− 165x21x

102 y

21y

102

n66+

330x1x102 y

21y

102

n121/2− 165x10

2 y21y

102

n55+

110x1x102 y1y

102

n121/2− 110x10

2 y1y102

n55

+45x8

2y82

n44+

45x41x

82y

41y

82

n66− 180x3

1x82y

41y

82

n121/2+

270x21x

82y

41y

82

n55− 180x1x

82y

41y

82

n99/2

+45x8

2y41y

82

n44− 315x3

1x82y

31y

82

n121/2+

945x21x

82y

31y

82

n55− 945x1x

82y

31y

82

n99/2+

315x82y

31y

82

n44

+675x2

1x82y

21y

82

n55− 1350x1x

82y

21y

82

n99/2+

675x82y

21y

82

n44− 450x1x

82y1y

82

n99/2+

450x82y1y

82

n44

− 84x62y

62

n33− 84x4

1x62y

41y

62

n55+

336x31x

62y

41y

62

n99/2− 504x2

1x62y

41y

62

n44+

336x1x62y

41y

62

n77/2

− 84x62y

41y

62

n33+

588x31x

62y

31y

62

n99/2− 1764x2

1x62y

31y

62

n44+

1764x1x62y

31y

62

n77/2− 588x6

2y31y

62

n33

− 1260x21x

62y

21y

62

n44+

2520x1x62y

21y

62

n77/2− 1260x6

2y21y

62

n33+

840x1x62y1y

62

n77/2− 840x6

2y1y62

n33

+70x4

2y42

n22+

70x41x

42y

41y

42

n44− 280x3

1x42y

41y

42

n77/2+

420x21x

42y

41y

42

n33− 280x1x

42y

41y

42

n55/2

+70x4

2y41y

42

n22− 490x3

1x42y

31y

42

n77/2+

1470x21x

42y

31y

42

n33− 1470x1x

42y

31y

42

n55/2+

490x42y

31y

42

n22

+1050x2

1x42y

21y

42

n33− 2100x1x

42y

21y

42

n55/2+

1050x42y

21y

42

n22− 700x1x

42y1y

42

n55/2+

700x42y1y

42

n22

− 21x41x

22y

41y

22

n33+

84x31x

22y

41y

22

n55/2− 126x2

1x22y

41y

22

n22+

84x1x22y

41y

22

n33/2− 21x2

2y41y

22

n11

+147x3

1x22y

31y

22

n55/2− 441x2

1x22y

31y

22

n22+

441x1x22y

31y

22

n33/2− 147x2

2y31y

22

n11− 21x2

2y22

n11

− 315x21x

22y

21y

22

n22+

630x1x22y

21y

22

n33/2− 315x2

2y21y

22

n11+

210x1x22y1y

22

n33/2− 210x2

2y1y22

n11

+x4

1y41

n22− 4x3

1y41

n33/2+

6x21y

41

n11− 4x1y

41

n11/2+ y4

1

− 7x31y

31

n33/2+

21x21y

31

n11− 21x1y

31

n11/2+ 7y3

1 +15x2

1y21

n11

− 30x1y21

n11/2+ 15y2

1 −10x1y1

n11/2+ 10y1 + 1

Lemma 3.10. Let m be a positive integer, and let (α) = (α1, . . . , αt). Then there exists a homogeneous


polynomial Hf,n,t,(α),m ∈ R[x1, . . . , xt, y1, . . . , yt, z] whose coefficients depend on f , n, t, m, and (α) such

that

Hf,n,t,(α),m(r1, . . . , rt, s1, . . . , st, 1) = 0

where the ri’s and si’s are defined in eq. (3.3).

Proof. By lemma 3.9, there exists Gf,n,t,(α),m ∈ R[x1, . . . , xt, y1, . . . , yt] such that

af (nm) = n(k−1)m/2Gf,n,t,(α),m(r1, . . . , rt, s1, . . . , st).

Let Hf,n,t,(α),m ∈ R[x1, . . . , xt, y1, . . . , yt, z] be the homogenization of n(k−1)m/2Gf,n,t,(α),m − af (nm).

The substituting z = 1 is just its de-homogenization, so

Hf,n,t,(α),m(x1, . . . , xt, y1, . . . , yt, 1) = n(k−1)m/2Gf,n,t,(α),m(x1, . . . , xt, y1, . . . , yt)− af (nm)

and therefore

Hf,n,t,(α),m(r1, . . . , rt, s1, . . . , st, 1) = n(k−1)m/2Gf,n,t,(α),m(r1, . . . , rt, s1, . . . , st)− af (nm) = 0

Lemma 3.11 (Algorithm to generate Hf,n,t,(α),m2tm=1). There is an algorithm to generate Hf,n,t,(α),m2tm=1.

Proof. We generate a list of symbolic expressions

Ta = (af (pd))2tαmax

d=1

where αmax = max1≤i≤t

αi, a t× 2t matrix of symbolic expressions

[Tg]i,m = gf,n,αi,m(x, y),

and a list of symbolic expressions

TH =(Hf,n,t,(α),m(x1, . . . , xt, y1, . . . , yt, z)

)2tm=1

.

1. For each 1 ≤ d ≤ 2tαmax, expand af (pd) symbolically in powers of af (p)2 and pk−1 as in lemma 3.1

and add it to Ta. Note that we may refer to af (pd−2) and af (pd−1) in Ta when expanding af (pd).

2. For each 1 ≤ i ≤ t and 1 ≤ m ≤ 2t, look up the expression for af (pαim) in powers of af (p)2 and

pk−1 from Ta, then substitute

af (pi) =

xyp

(k−1)/2i

n(k−1)/2 if αi is odd((xy

n(k−1)/2 − y)p

(k−1)i

)1/2

if αi is even

as in lemma 3.8. This gives the expression for gf,n,αi,m(x, y). Add this to [Tg]im.

3. For each 1 ≤ m ≤ 2t,


(a) Symbolically compute the product∏ti=1 gf,n,αi,m(xi, yi) by referring to [Tg] for gf,n,αi,m. This

gives Gf,n,t,(α),m(x1, . . . , xt, y1, . . . , yt).

(b) Homogenize the expression n(k−1)m/2Gf,n,t,(α),m(x1, . . . , xt, y1, . . . , yt)−af (nm) as in lemma 3.10.

This will give the expression

Hf,n,t,(α),m(x1, . . . , xt, y1, . . . , yt, z) = 0

Add Hf,n,t,(α),m(x1, . . . , xt, y1, . . . , yt, z) to TH .

Description of the algorithm

Let n =∏ti=1 p

αii and (α) = (α1, . . . , αt) be given.

1. Using lemma 3.11, form a system of 2t homogeneous polynomial equations in 2t+ 1 variables with

real coefficients:

Hf,n,t,(α),1(x1, . . . , xt, y1, . . . , yt, z) = 0

Hf,n,t,(α),2(x1, . . . , xt, y1, . . . , yt, z) = 0

...

Hf,n,t,(α),2t(x1, . . . , xt, y1, . . . , yt, z) = 0.

2. Solve the homogeneous system (Hf,n,t,(α),m(x1, . . . , xt, y1, . . . , yt, z) = 0)2tm=1 above using homo-

topy methods described in section 2.3.2 . By construction and lemma 3.10, (r1, . . . , rt, s1, . . . , st, 1)

is a solution.

3. Divide each ri by nk−1 and obtain

r2i

nk−1=af (pαii )2nk−1

p(k−1)αii nk−1

=af (pαii )2

p(k−1)αii

, i = 1, . . . , t.

By assumption,r2i

nk−16= 0, and

r2i

nk−1∈ Q but are not integers lemma 3.7, i.e. the denominators

are powers of pi.

4. Extract each pi from the denominator ofr2i

nk−1.

Complexity Analysis

We provide an estimate the runtime and storage complexity of the algorithm. Recall that n =t∏i=1

pαii

where t = ω(n) andt∑i=1

αi = Ω(n).

From lemma 3.8 we see that gf,n,αi,m is a polynomial of degree 2αim in two variables, and from

lemma 3.9 Gf,n,t,(α),m is a polynomial of degreet∑i=1

2αim = 2mt∑i=1

αi = 2mΩ(n) in 2t variables. There-

fore, Hf,n,t,(α),m ∈ H2mΩ(n), where Hd is the space of homogeneous polynomial of degree d in the


variables x1, . . . , xt, y1, . . . , yt, z, and so

(Hf,n,t,(α),m)2tm=1 ∈

2t⊕m=1

H2mΩ(n) = H.

We define N + 1 to be the dimension of this vector space H. Now, note that

N + 1 = dim

(2t⊕m=1

H2mΩ(n)

)=

2t∑m=1

dim(H2mΩ(n)

).

It is well known that dim (Hd) =(

2t+dd

)=(

2t+d2t

), so

N + 1 =

2t∑m=1

(2t+ 2mΩ(n)

2t

).

Asymptotic analysis of N We now present an asymptotic analysis of N . We estimate the size of

the sum of binomial coefficients2t∑m=1

(2t+2mΩ(n)

2mΩ(n)

).

We apply the following well-known bounds on binomial coefficients:

(nr

)r≤(n

r

)≤(ner

)r.

An upper bound for N is thus given by

N + 1 =

2t∑m=1

(2mΩ(n) + 2t

2t

)

≤2t∑m=1

(2mΩ(n) + 2t

2t

)2t

e2t

= e2t2t∑m=1

(Ω(n)

tm+ 1

)2t

≤ e2t2t∑m=1

(2Ω(n)

tm

)2t

= e2t

(2Ω(n)

t

)2t 2t∑m=1

m2t

= e2t

(2Ω(n)

t

)2t

O((2t)2t+1

)= 22t log e22t(log(2Ω(n))−log t)2O((2t+1) log(2t))

= 2O(t log t+t log Ω(n)).

Similarly, a lower bound for N is given by

N + 1 =

2t∑m=1

(2mΩ(n) + 2t

2t

)


≥2t∑m=1

(2mΩ(n) + 2t

2t

)2t

=

2t∑m=1

(Ω(n)

tm+ 1

)2t

≥2t∑m=1

(Ω(n)

tm

)2t

=

(2Ω(n)

t

)2t 2t∑m=1

m2t

≥(

2Ω(n)

t

)2t

(2t)2t

= 22t log 2+2t log Ω(n)−2t log t+2t log t+2t log 2

= 24t log 2+2t log Ω(n).

Hence,

N ∼ 2O(t log t+t log Ω(n))

By a celebrated result of Hardy-Ramanujan in [Har40],

t = ω(n) ∼ log log n,

and

Ω(n) ∼ log log n

on average. Whenever we use this estimate in a complexity analysis, we will refer to it as the average

complexity. Therefore, on average,

N = 2O(log logn·log log logn).


logN = O(log log n · log log log n)

log logN = O(log log log n+ log log log log n)

It is also well known that

t = ω(n) .log n

log log n

and

Ω(n) .log n

log log n

are the upper bounds. Whenever we use this estimate in a complexity analysis, we will refer to it as the

worst case complexity. Therefore, in the worst case,

N = 2O( lognlog logn (log logn−log log logn)) = 2O(logn− logn·log log logn

log logn ).



logN = O

(log n− log n · log log log n

log log n

)log logN = log

[O

(log n− log n · log log log n

log log n

)]= O(log log n)

Generating the system of polynomial equations To generate the polynomials

Hf,n,t,(α),m(x1, . . . , xt, y1, . . . , yt, z)

2t

m=1,

we use the algorithm described in lemma 3.11:

1. The (d− j)-th entry in Ta contains bd−j+12 c terms, so the d-th entry takes O(bd−1

2 cbd2c) = O(d2)

multiplications to compute. Therefore, runtime complexity to generate Ta is

2tαmax∑d=1

O(d2) = O((2tαmax)3) = O(t3α3max) = O(t3(log n)3).

2. Computing (i,m)-th entry of Tg requires substituting and manipulating bαim2 c terms, so the run-

time complexity to generate Ta is

2t∑m=1

t∑i=1

O(⌊αim

2

⌋)= O(t2Ω(n)).

3. The generation of TH from Tg requires the manipulation of N+1 terms, so the runtime complexity

is O(N).

Hence, the total runtime complexity to generate Ta, Tg and TH is

O(t3(log n)3) +O(t2Ω(n)) +O(N).

Note that the coefficients of Hf,n,t,(α),m are essentially the products of powers of nk−1 and af (nm),

and since

|af (nm)| =

∣∣∣∣∣t∏i=1

af (pmi )

∣∣∣∣∣ <t∏i=1

(m+ 1)p(k−1)/2i = t!n(k−1)/2

log |af (nm)| < log(t!) +k − 1

2log n ≤< t log(t) +

k − 1

2log n

so the bit-size of each coefficient is O(t + k−12 log n + k−1

2 log n) = O(t log t + (k − 1) log n). Since

(Hf,n,t,(α),m)2tm=1 ∈ H there are N + 1 coefficients to store. The storage complexity is therefore

O(N(t log t+ (k − 1) log n)).

Solving the system of polynomial equations The runtime complexity in solving

Hf,n,t,(α),1(x1, . . . , xt, y1, . . . , yt, z) = 0


Hf,n,t,(α),2(x1, . . . , xt, y1, . . . , yt, z) = 0

...

Hf,n,t,(α),2t(x1, . . . , xt, y1, . . . , yt, z) = 0.

using homotopy methods depends on the N .

If we use the randomized homotopy method with random starting points, the runtime complexity is,

on average,

O(N2).

If we use the deterministic homotopy method, the runtime complexity is

O(N log logN ).

We have to store the square of roots (r2i )ti=1. By the definition of ri in eq. (3.3)

|r2i | =

∣∣∣∣∣af (pαii )nk−1

p(k−1)αii

∣∣∣∣∣ ≤ ∣∣af (pαii )nk−1∣∣

log |r2i | ≤ log |af (pαii )|+ log

∣∣nk−1∣∣

< log∣∣∣(αi + 1)p

αi(k−1)/2i

∣∣∣+ log∣∣nk−1

∣∣=αi(k − 1)

2(log(αi + 1) + log pi) + (k − 1) log n

<log n · (k − 1)

2(log(log n+ 1) + log n) + (k − 1) log n

= O((k − 1)(log n)2)

Therefore the storage complexity is

tO((k − 1)(log n)2) = O(t(k − 1)(log n)2).

Finding the factors Extracting the pi’s from the r2i /n

k−1’s has runtime complexity O(t).

We only need to store the pi’s, so storage complexity is

O(t log n).

Total Runtime and Storage Complexity Combining the analysis above,

1. The generation of TH = Hf,n,t,(α),m2tm=1 has runtime complexity O(t3(log n)3) + O(t2Ω(n)) +

O(N).

2. Solving the system (Hf,n,t,(α),m(x1, . . . , xt, y1, . . . , yt, z))2tm=1 = 0 has runtime complexity

• O(N2) if we use the randomized homotopy method, and

• O(N log logN ) if we use the deterministic homotopy method.

3. Extracting the factors pi’s has runtime complexity O(t).


Since we are not given Ω(n), we run simply the algorithm P (Ω(n)) times where P (·) is the partition

function. It is shown in [Har40] that asymptotically

P (x) ≈ 1

4x√

3eπ√

2x/3,

and therefore

P (Ω(n)) ≈= 2O(Ω(n)1/2).

Therefore, we see that the runtime complexity is dominated by solving the system

(Hf,n,t,(α),m(x1, . . . , xt, y1, . . . , yt, z)

)2tm=1

= 0.

If use the average estimate t ∼ log log n, and the randomized homotopy method, the runtime com-

plexity is

O(N2) = O(

2O(log logn·log log logn))

= 2O(log logn·log log logn)



O(N log logN ) = 2O((log logn·log log logn)·(log log logn+log log log logn)) = 2O(log logn·(log log logn)2)


If use the worst case estimate t . lognlog logn , and the randomized homotopy method, the runtime

complexity is

O(N2) = O(

2O(logn− logn·log log lognlog logn )

)= 2O(logn− logn·log log logn

log logn )




)= 2O((logn− logn·log log logn

log logn ) log logn) = 2O(logn(log logn−log log logn))


We summarize the runtime complexity below in a table.

randomized homotopy deterministic homotopy

average quasi-polynomial quasi-polynomial

worst case exponential exponential

Similarly, summarizing the analysis above,

1. The generation of TH = Hf,n,t,(α),m2tm=1 has storage complexity O(N(t log t+ (k − 1) log n)).

2. Solving the system (Hf,n,t,(α),m(x1, . . . , xt, y1, . . . , yt, z))2tm=1 = 0 has storage complexity O(t(k −

1)(log n)2)

3. Extracting the factors pi’s has storage complexity O(t log n).


Therefore, we see that the storage complexity is dominated by the generation of TH = Hf,n,t,(α),m2tm=1.

If use the average estimate t ∼ log log n, the storage complexity is therefore

O(N(t log t+ (k − 1) log n)) = 2O(log logn·log log logn)O(log log n · log log log n+ (k − 1) log n)

= 2O(log logn·log log logn)


If use the worst case estimate t . lognlog logn , the storage complexity is therefore

O(N(t log t+ (k − 1) log n)) = 2O(logn− logn·log log lognlog logn )O

(log n

log log n(log log n− log log log n) + (k − 1) log n

)= 2O(logn− logn·log log logn

log logn )O

(k log n− log log log n

log log n

)= 2O(logn(1− log log logn

log logn ))


3.4 Numerical Examples

In this section, we illustrate theorems 3.3 and 3.4 via some numerical examples.

Example 3.16 (Factoring n = 8575). In this example, we let f = ∆ (hence k = 12), and use the

algorithm in theorem 3.4 to factor n = 8575 = 5273. In this case, we let (α) = (2, 3) and t = 2.

One can verify that the system

τ(8575) = 857511/2G∆,8575,t,(α),1(x1, x2, y1, y2)

τ(85752) = 857511G∆,8575,t,(α),2(x1, x2, y1, y2)

τ(85753) = 857533/2G∆,8575,t,(α),3(x1, x2, y1, y2)

τ(85754) = 857522G∆,8575,t,(α),4(x1, x2, y1, y2)

where G∆,n,t,(α),1(x1, x2, y1, y2), . . . , G∆,n,t,(α),4(x1, x2, y1, y2) are given in example 3.15, and

τ(8575) = −1568772102658481630000

τ(85752) = 933346892568425256627736522049338869983125

τ(85753) = −63747324087651164213662446271685531594267830383031538283845375000

τ(85754) = 286633071832801878332062176280546027855431016943360754958949164999076760969164578515625

(see appendix A) has a solution given by

x1 = r1 = −847413974003634059225√

7 x2 = r2 = 3004020723183593750000

y1 = s1 = − 466578

1486547y2 = s2 = −40353607

74985550.


Thus

r21

857511=

1040336760961

3814697265625=

1040336760961

518

r22

857511=

32171959487977747360000

65712362363534280139543=

32171959487977747360000

727.

and we see that p1 = 5, p2 = 7.

Example 3.17 (Factoring n = 342125). In this example, we let f = ∆ (hence k = 12), and use the

algorithm in theorem 3.4 to factor n = 342125 = 53 · 7 · 17 · 23. In this case, we let (α) = (3, 1, 1, 1) and

t = 4.

One can verify that the system

τ(342125) = 34212511/2G∆,342125,t,(α),1(x1, . . . , x4, y1, . . . , y4)

τ(3421252) = 34212511G∆,342125,t,(α),2(x1, . . . , x4, y1, . . . , y4)

τ(3421253) = 34212533/2G∆,342125,t,(α),3(x1, . . . , x4, y1, . . . , y4)

τ(3421254) = 34212522G∆,342125,t,(α),4(x1, . . . , x4, y1, . . . , y4)

τ(3421255) = 34212555/2G∆,342125,t,(α),5(x1, . . . , x4, y1, . . . , y4)

τ(3421256) = 34212533G∆,342125,t,(α),6(x1, . . . , x4, y1, . . . , y4)

τ(3421257) = 34212577/2G∆,342125,t,(α),7(x1, . . . , x4, y1, . . . , y4)

τ(3421258) = 34212544G∆,342125,t,(α),8(x1, . . . , x4, y1, . . . , y4)

where G∆,n,t,(α),1(x1, . . . , x4, y1, . . . , y4), . . . , G∆,n,t,(α),8(x1, . . . , x4, y1, . . . , y4) are given in appendix B,

and

τ(342125) = −773926209478061930281957056000

τ(3421252) = 1338814814656777847204556177598198929399032443832006734265625

τ(3421253) = 4113694310312566511515614433045797658088479640566568854998957

342844097011053472500000000000

τ(3421254) = 2138071789218539074273394073561778724479877913824244759483067

21346922488881261090887312070171106523102119911240478515625

τ(3421255) = −681257130824909372586086130385764299289920807021787468242182

1448401890106588953679564007172502132637385966577059096939804

8959255383800110062500000000000

τ(3421256) = 2110131107514639884932602763936071469381781185262411127068634

2212080435362778368237518777150368200547368452081986644139963

185227780643105081948269186264266953401744857151031494140625

τ(3421257) = 9781978132084403277808178731893548572501865234710512382559696

6005363122969651965590587743630437701961145055060833727144388

2531157866141287877147888883668606906450245892617489651888518


380134663085937500000000000000

τ(3421258) = −377609679311929301969502438414745934139043807258284255896720

2888737403206997267430377121513742799289706615332136661522903

2027361673489892688978966055652057452489630647140979722378795

5839820979790493410475108020827002622490019738674163818359375

has a solution given by

x1 = r1 = −55140381551268439074311428500√

2737

x2 = r2 = −23348719708931204833984375000√

1955

x3 = r3 = −113991343694610158386230468750√

805

x4 = r4 = 67885498611009510498046875000√

595

y1 = s1 = −1953125

2973094

y2 = s2 = 1

y3 = s3 = 1

y4 = s4 = 1.

Thus

r21

34212511=

8248434570253510416

7450580596923828125=

8248434570253510416

527

r22

34212511=

5721664

40353607=

5721664

79

r23

34212511=

47691924412356

34271896307633=

5721664

1711

r24

34212511=

347571590865984

952809757913927=

5721664

2311

we see that p1 = 5, p2 = 7, p3 = 17, p4 = 23.

3.5 Computer Code

The following Mathematica code was used to generate the examples in this chapter.

Code related to theorem 3.3

(*

* Define a[p^m] recursively.

* Usage: ap[m] expands a[p^m] in powers of w=a[p] and p^(k-1).

*)

ap[0] = 1

ap[1] = w

ap[m_] := Expand[w*ap[m-1] - ap[m-2]*p^(k-1)]


(*

* Computes the polynomial g_f, n, t, m(x1, ..., xt) by substituting

a[pi] with the corresponding xi in the expression for a[p^m].

* Usage: g[n, t, m, x] gives g_f, n, t, m(x1, ..., xt).

*)

gi[n_, m_, x_, i_] := ( r = (Subscript[x, i] * p^(k-1) / n^(k-1))^(1/2);

If[OddQ[m], expr = (ap[m]/((p^((m-1)*(k-1)/2))*ap[1])) /. w -> r, expr =

(ap[m]/p^(m*(k-1)/2)) /. w -> r]; Return[Expand[expr]] )

(*

*

*)

g[n_, t_, m_, x_] := ( expr = 1; Do[expr = expr * gi[n, m, x, i], i, 1,

t]; expr = expr*n^(Floor[m/2]*(k-1)); X = ; Do[X = Append[X,

Subscript[x, i]], i, 1, t]; expr = Expand[expr]; Return[expr] )

(*

* version with x1...xt replaced by a[n]^2 * n^((k-1)*(t-1))).

* g[n_, t_, m_, x_] := ( expr = 1; Do[expr = expr * gi[n, m, x, i], i,

1, t]; expr = expr*n^(Floor[m/2]*(k-1)); X = ; Do[X = Append[X,

Subscript[x, i]], i, 1, t]; expr = Expand[expr]; Return[expr /.

SymmetricPolynomial[t, X] -> (a[n]^2 * n^((k-1)*(t-1)))] )

*)

(*

* Computes the polynomial G_f, n, t, m(s1, ..., st) by evaluating

g_f, n, t, m(s1, ..., st) where si is the i-th symmetric polynomial in

x1, ..., xt.

* Usage: G[n_, m_, t_, x_, s_] gives G_f, n, t, m(s1, ..., st).

*)

G[n_, t_, m_, x_, s_] := ( X = ; S = ; Do[(X = Append[X,

Subscript[x, i]]; S = Append[S, Subscript[s, i]]), i, 1, t]; Return[

SymmetricReduction[g[n, t, m, x], X, S][[1]] ])

(*

* Generates the system of equations G_f, n, t, m(s1, ..., st) =

a_f(n^m) for m = 2, ..., t-1.

* Usage: generateSystem[n, t] returns the system for general t.

generateSystem[n] returns generateSystem[n, PrimeNu[t]].

*)

generateSystem[n_, t_] := ( S = ; Do[ If[OddQ[m], S = Append[S,


Expand[a[n^m]/a[n] == G[n, t, m, x, x]]], S = Append[S, Expand[a[n^m] ==

G[n, m, t, x, x]]]], m, 2, t]; Return[S] )

generateSystem[n_] := generateSystem[n, PrimeNu[n]]

(*

* Generates the solutions ri , i = 1, ..., t.

* Usage: generateSolution[n] returns r1, ..., rt.

*)

generateSolution[n_] := ( factors = FactorInteger[n]; t =

Length[factors]; R=; Do[( p = factors[[i]][[1]]; r = (a[p]^2 *

n^(k-1))/p^(k-1); R = Append[R, r] ), i, 1, t]; Return[R] )

(*

* Generates polynomial phi(x) = (x - r1)...(x - rt).

* Usage: generatePolynomial[n] returns phi(x).

*)

generatePolynomial[n_] := ( S = generateSolution[n]; t = Length[S]; poly

= 1; Do[ poly = poly * (x - S[[i]]) , i, 1, t]; Return[Expand[poly]] )

Code related to theorem 3.4

(*

* Define a[p^m] recursively.

* Usage: ap[m] expands a[p^m] in powers of w=a[p] and p^(k-1).

*)

ap[0] = 1

ap[1] = w

ap[m_] := Expand[w*ap[m-1] - ap[m-2]*p^(k-1)]

(*

* Computes the polynomial g_f, n, a, m(x, y) by substituting a[p] with

the corresponding x and y in the expression for a[p^m].

* Usage: g[a_, m_, x_, y_] gives g_f, n, a, m(x, y).

*)

rs[n_, a_, x_, y_] := If[OddQ[a], x*y*p^((k-1)/2)/n^((k-1)/2),

((x*y)/n^((k-1)/2) - y)^(1/2)*p^((k-1)/2)]

g[n_, a_, m_, x_, y_] := Expand[Simplify[ap[a*m]/(p^((k-1)*a*m/2)) /.

w:>rs[n, a, x, y]]]

(*

* Computes the polynomial G_f, n, t, A, m(x1, ..., xt, y1, ..., yt) by


multiplying g_f, n, a, m(xi, yi) for i = 1, ..., t.

* Usage: G[A, m] gives G_f, n, t, A, m(x1, ..., xt, y1, ..., yt). A =

a1, ..., at.

*)

G[n_, A_, m_] := ( poly = 1; Do[poly = poly * g[n, A[[i]], m,

Subscript[x, i], Subscript[y, i]], i, 1, Length[A]];

Return[Expand[poly]])

(*

* Gets the list of exponents of the factors of n.

* Usage: getA[n] = a1, ..., at

*)

getA[n_] := ( L = FactorInteger[n]; A = ; Do[A = Append[A,

L[[i]][[2]]], i, 1, Length[L]]; Return[A]; )

(*

* Generates the system of equations n^((k-1)m) * G_f, n, t, A, m(x1,

..., xt, y1, ..., yt) = a_f(n^m) for m = 1, ..., 2t.

* Usage: generateSystem[n, A] returns the system for general A=a1, ...,

at. generateSystem[n] returns generateSystem[n, getA[n]].

*)

generateSystem[n_, A_] := ( S = ; t = Length[A]; Do[ (eqn =

Expand[a[n^m] == n^((k-1)*m/2)*G[n, A, m]]; S = Append[S, eqn] ) , m,

1, 2*t]; Return[S] )

generateSystem[n_] := generateSystem[n, getA[n]]

(*

* Generates the solution (r1, ..., rt, s1, ..., st) to system of

equations n^((k-1)m) * G_f, n, t, A, m(x1, ..., xt, y1, ..., yt) =

a_f(n^m) for m = 1, ..., 2t.

* Usage: generateSolutionn_] returns the solution (r1, ..., rt, s1, ...,

st).

*)

generateSolution[n_] := ( factors = FactorInteger[n]; t =

Length[factors]; X=; Y=; Do[ ( p=factors[[i]][[1]];

b=factors[[i]][[2]]; r=a[p^b]*n^((k-1)/2)/p^((k-1)*b/2); If[OddQ[b],

s=(p^((k-1)*(b-1)/2)*a[p])/a[p^b], s=(p^((k-1)*(b-2)/2) *

a[p]^2)/(a[p^b] - p^((k-1)*b/2))]; X = Append[X, r]; Y = Append[Y, s]) ,

i, 1, t]; Return[X, Y] )


(*

* Verifies Sols satisfies Eqns.

* Usage: verifySolution[Eqns_, Sols_] retuns true/false, ...true/false

for each equation in Eqns.

*)

verifySolution[Eqns_, Sols_] := (V = Eqns; X = Sols[[1]]; Y = Sols[[2]];

t = Length[X]; Do[ V = V/.Subscript[x, i]->X[[i]], Subscript[y,

i]->Y[[i]] , i, 1, t ]; Return[V]; )

(*

* Extracts factors pi from the solution, where n = p1â1...ptât.

* Usage:

* computePreFactors[Sols] gives ri^2/n^(k-1)

* extractFactors[Sols] gives pi

* extractDenominators[Sols] gives pi, ai.

*)

computePreFactors[Sols_] := ( PF = ; X = Sols[[1]]; t = Length[X];

Do[( r = X[[i]]; u = r^2/n^(k-1); PF = Append[PF, u]; ), i, 1, t];

Return[PF] )

extractDenominators[Sols_] := ( Denoms = ; X =

computePreFactors[Sols]; t = Length[X]; Do[( d =

FactorInteger[Denominator[X[[i]]]][[1]]; Denoms = Append[Denoms, d]; ) ,

i, 1, t]; Return[Denoms] )

extractFactors[Sols_] := ( Factors = ; X = extractDenominators[Sols];

t = Length[X]; Do[( p = X[[i]][[1]]; Factors = Append[Factors, p]; ),

i, 1, t]; Return[Factors] )

Chapter 4

A test for squarefree-ness using

Fourier coefficients of modular forms


For this chapter, let f be a modular form of even weight k for Γ0(N) which is a normalized eigen-form

for the Hecke operators. Let us write

f(z) =

∞∑n=1

af (n)e2πinz


A positive integer n is squarefree no prime divides n to an exponent larger than 1.

Given a positive integer n, one may ask whether n is squarefree. Of course, given the factorization

of n, the answer is easy. In other words, given an oracle for factoring an integer n, deciding whether n

is squarefree is efficient.

If we are not given an oracle for factoring an integer n, then the problem of deciding whether n

is squarefree is hard. In fact, no unconditional polynomial-time deterministic algorithm for testing the

squarefree-ness of an integer is known.

In section 4.4, we present an algorithm for testing the squarefree-ness of an integer where we assume

we are given an oracle for f (i.e an algorithm that outputs af (n) in polynomial time for any positive

integer n).

We will use the convention that gcd(n1, . . . , nm) > 0. For example, gcd(−2, 4, 6) = 2.

Let p be a prime. It is easy to see that af (p) | af (pr) for any positive odd integer r, and therefore

gcd (af (pr), r = 1, 3, 5, . . . ) = |af (p)|.

If n is squarefree, then

gcd (af (nr), r = 1, 3, 5, . . . ) = |af (n)|.

Motivated by this, our test for squarefree-ness, called SQFRf , is simply as follows. Fix a parameter

69

Chapter 4. A test for squarefree-ness 70

1 ≤ R ∈ Z. Given n, compute the number


|af (n)|(4.1)

and check whether SQFRf (n) = 1.

More generally, we may consider

gcd (af (pr) : r ∈ A)

where A ⊆ N. This is the subject of section 4.2.

Recall the Ramanujuan τ function is defined by the relation

e2πiz∞∏n=1

(1− e2πinz)24 =

∞∑n=1

τ(n)e2πinz.

Let p be a prime and r an odd positive integer. We already know that τ(p) | τ(pr). One may ask the

converse: is it possible that τ(pr) | τ(p), or equivalently,

τ(pr) = ±τ(p)?

This question has been studied in [MMS87], where it was shown that τ(pr) 6= ±τ(p) when r is sufficiently

large. However, in chapter 4, we are able to prove that for all positive integers r > 1, τ(pr) 6= ±τ(p) for

odd primes p such that τ(p) 6= 0.

4.1.1 Main results

To answer the question about the greatest common divisor

gcd (af (pr) : r ∈ A)

where A ⊆ N, we highlight the following 3 lemmas:

Lemma 4.5. Suppose p is a prime such that p - af (p). Let r be any positive integer. Then

gcd(af (pr), af (pr+1)

)= 1.

Lemma 4.14. Suppose p is a prime such that p - af (p). Let r be any positive integer with r ≥ 1. Then


)=

1 if r is even

|af (p)| if r is odd.

Lemma 4.16. Suppose p is a prime such that p - af (p). Let α be a positive integer, and r be an odd

positive integer. Then

gcd(af (pα), af (prα), af (p(r+2)α)

)=

1 if α is even

|af (p)| if α is odd.


The following two theorem state when the algorithm SQFRf will correctly decide on the squarefree-ness

of an integer n. One may think of them as the “necessary” and “sufficient” conditions.





Iodd = i : αi > 1, αi ≡ 1 (mod 2)



αi0i0

), af (pαi−2i )


gcd(af (p

αi0i0

), af (pri ))

= 1 for all i 6∈ Iodd such that αi = 1 and all positive odd integers r. Then there

exists a positive integer R such that

SQFRf (n) 6= 1.


Here R as a parameter — the “cut off” point for how long the algorithm should run. Provably

(theorem 4.2), R is large unfortunately. However, heuristically, algorithm SQFRf will correctly recognize

squarefree integers n probabilistically in O(log log n − log log log n) steps if k (the weight of f) is large

enough:

Conjecture 4.3. For weights k ≥ 4, algorithm SQFRf will correctly recognize whether an integer n is

squarefree with probability 1− C > 0 by taking R = O(log(1/C) · log t).

The following theorem shows τ(pr) - τ(p) for r odd, which is a converse of the well known property

τ(p) | τ(pr):

Theorem 4.3. Let p be an odd prime such that τ(p) 6= 0. Let r > 1 be a positive odd integer. Then

τ(pr) 6= ±τ(p).

More generally, af (pr) - af (p) for r odd:



4.2 A general recurrence relation

It is well-known that the Fourier coefficients af (·) are multiplicative, i.e.

af (n1, n2) = af (n1)af (n2) if gcd(n1, n2) = 1,

and satisfy the recurrence relation

af (pr) = af (p)af (pr−1)− af (pr−2)pk−1 (4.2)

where p is a prime and r ≥ 2 is a positive integer.

We do not know whether the following generalization of eq. (4.2) is well-known or not, but we present

it here nevertheless, since it will be very useful in this chapter.


Lemma 4.1 (General Recurrence Relation). Let p be a prime. Then

af (pr) = af (ps)af (pr−s)− af (ps−1)af (pr−s−1)pk−1

for integers r > s ≥ 1.

Equivalently,

af (pr+s) = af (pr)af (ps)− af (pr−1)af (ps−1)pk−1

for integers r, s ≥ 1.

Proof. By induction on s.

For s = 1, we have the usual relation eq. (4.2):

af (pr) = af (p)af (pr−1)− af (pr−2)pk−1.

Assume af (pr) = af (ps)af (pr−s)−af (ps−1)af (pr−s−1)pk−1. We want to show af (pr) = af (ps+1)af (pr−s−1)−af (ps)af (pr−s−2)pk−1. Now

af (pr) = af (ps)af (pr−s)− af (ps−1)af (pr−s−1)pk−1

= af (ps)[af (p)af (pr−s−1)− af (pr−s−2)pk−1

]− af (ps−1)af (pr−s−1)pk−1

=[af (ps)af (p)− af (ps−1)pk−1

]af (pr−s−1)− af (ps)af (pr−s−2)pk−1

= af (ps+1)af (pr−s−1)− af (ps)af (pr−s−2)pk−1.

The second part of the lemma follows immediately from the first.

4.3 The greatest common divisor of a tower

Let p be a prime number such that p - af (p). We define a tower to mean a sequence

(af (pr))r∈A

where A ⊆ N; we may think of A as a (finite or infinite) subsequence of the sequence 1, 2, 3, . . . Then

the greatest common divisor of a tower (af (pr))r∈A is the integer

gcd (af (pr), r ∈ A) .

It is an elementary exercise to show that af (p) | af (pr) for any odd positive integer r. However, we

present it here for ease of reference and for completeness.

Lemma 4.2. If r is an odd positive integer, then af (pr) is divisible by af (p).

Proof. By induction on r.

For r = 3, we have

af (p3) = af (p)af (p2)− af (p)p11

= af (p)(af (p2)− p11

)


and therefore af (p) | af (p3).

Now, suppose af (p) | af (pr). By the recurrence relation of af we have

af (pr+2) = af (p)af (pr+1)− af (pr)p11

and since af (p) | af (pr) we see af (p) | af (pr+2).

An immediate result we may deduce from lemma 4.2 is that gcd (af (pr), r = 1, 3, 5, . . . ) = |af (p)|.

Lemma 4.3. gcd (af (pr), r = 1, 3, 5, . . . ) = |af (p)|.

Proof. By lemma 4.2, af (p) | af (pr) for all r = 1, 3, 5, . . . .

To simplify the notation for the rest of this chapter, we make the following definition. For a prime p

and positive integer r, define a∗f (pr) by

a∗f (pr) =

a∗f (pr)

af (p) if r is odd

af (p) if r is even. (4.3)

By lemma 4.2, a∗f (pr) ∈ Z. Also, a∗f (p) = 1. The recurrence relation eq. (4.2) then becomes

a∗f (pr) = af (pr−1)− a∗f (pr−2)pk−1 (4.4)

for odd positive integers r ≥ 3.

The definition of a∗f (·) in eq. (4.3) extends to a composite number n =t∏i=1

pαii ; in this case

a∗f (n) =

t∏i=1

a∗f (pαii ).

If we also assume p - af (p), then we are able to show many more results. In particular, for various

towers (i.e for various subsets A ⊆ N) we are able to determine the greatest common divisor.

We remark here that a variant of Lehmer’s conjecture asserts that the number of primes p such that

p | af (p), or equivalently, af (p) ≡ 0 (mod p), is small [Mur07, LM14].

Since we will be assuming p - af (p) for almost the rest of the chapter, the following lemma is useful

and we present it here for ease of reference.

Lemma 4.4. Suppose p is a prime such that p - af (p). Let r be any positive integer. Then p - a∗f (pr).

Proof. Suppose p | a∗f (pr). Now, by lemma 3.1,

a∗f (pr) =

br/2c∑j=0

cjaf (p)2br/2c−2jp(k−1)j , c0 = 1

= af (p)2br/2c + p(k−1)

br/2c∑j=1

cjaf (p)2br/2c−2jp(k−1)(j−1).


Therefore, we see that

p | af (p)2br/2c ⇒ p | af (p)

which is a contradiction.

Lemma 4.5. Suppose p is a prime such that p - af (p). Let r be any positive integer. Then


)= 1.


Suppose r = 1. Suppose d = gcd(af (p), af (p2)

)> 1 and let q > 1 be a prime divisor of d. Since

af (p2) = af (p)2 − pk−1

we see that

q | pk−1

and therefore q = p. However, this implies that

p | af (p).

This is a contradiction unless d = 1.

Now suppose d = gcd(af (pr), af (pr+1)

)> 1 and let q > 1 be a prime divisor of d. Since

af (pr+1) = af (p)af (pr)− af (pr−1)pk−1

we see that

q | af (pr−1)pk−1,

so q | af (pr−1) or q|pk−1 since q is prime.

We may conclude q - pk−1, since otherwise q = p and by lemma 4.4, p | af (p).

Therefore, q | af (pr−1) for all prime factors q of d. So

d | gcd(af (pr−1), af (pr)

)which is a contraction unless d = 1.

This immediately yields the following two lemmas.

Lemma 4.6. Suppose p is a prime such that p - af (p). Then

gcd (af (pr), r = 1, 2, 3, . . . ) = 1.

Proof. It is clear that

gcd (af (pr), r = 1, 2, 3, . . . ) | gcd(af (p), af (p2)

)and gcd

(af (p), af (p2)

)= 1 by lemma 4.5.



gcd (af (pr), r ≥ r0 ≥ 1, r ∈ N) = 1.

Proof. It is clear that

gcd (af (pr), r ≥ r0 ≥ 1, r ∈ N) | gcd(af (pr0), af (pr0+1)

)and gcd

(af (pr0), af (pr0+1)

)= 1 by lemma 4.5.

Recall lemma 4.2 states that gcd (af (pr), r = 1, 3, 5, . . . ) = |af (p)|. That is, the greatest common

divisor of the “full odd tower” is af (p). Its proof relies on the fact that the first element af (p) divides

the rest of the elements af (pr) where r ≥ 1 is odd.

What if we remove the first element af (p), i.e. what is gcd (af (pr), r = 3, 5, 7 . . . )? What about

“even towers”? What is gcd (af (pr), r = 2, 4, 6 . . . )?

For the rest of this section, define

ε = gcd(a∗f (pr), r = 2, 4, 6, . . .

)(4.5)

and

δ = gcd(a∗f (pr), r = 3, 5, 7, . . .

), (4.6)

where a∗f (pr) is as defined in eq. (4.3).

Note that by eq. (4.3),

ε = gcd (af (pr), r = 2, 4, 6, . . . ) ,

and together with lemma 4.2,

δ = gcd

(af (pr)

af (p), r = 3, 5, 7, . . .

)=

gcd (af (pr), r = 3, 5, 7, . . . )

|af (p)|.

Hence δ can be thought of as the “non-trivial” factor of gcd (af (pr), r = 3, 5, 7, . . . ).

Lemma 4.8. Suppose p is a prime such that p - af (p). Then gcd (af (p), δ) = 1.

Proof. Suppose d = gcd (af (p), δ) > 1. Let q be a prime divisor of d.

Now,

δ | a∗f (p3) = af (p2)− pk−1 = af (p)2 − 2pk−1

⇒ d | af (p)2 − 2pk−1

⇒ d | 2pk−1

and

δ | a∗f (p5) = af (p4)− a∗f (p3) = af (p)4 − 4af (p)2pk−1 + 3p2(k−1)

⇒ d | af (p)4 − 4af (p)2pk−1 + 3p2(k−1)


⇒ d | 3p2(k−1)

since d | δ and d | af (p).

Therefore,

d | gcd(

2pk−1, 3p2(k−1))

= pk−1,

which implies q = p and p | af (p). This is a contradiction unless d = 1.

Lemma 4.9. Suppose p is a prime such that p - af (p). Then gcd (p, δ) = 1.

Proof. Suppose gcd (p, δ) > 1. Then in fact δ = pa for some integer a > 1.

By definition of δ in eq. (4.6),

δ | a∗f (p3) = af (p)2 − 2pk−1

and so

p | af (p)2 − 2p(k−1)

⇒ p | af (p)


The following two lemmas show that in the even case

ε = gcd(a∗f (pr), r = 2, 4, 6, . . .

)= 1,

and in the odd case

δ = gcd(a∗f (pr), r = 3, 5, 7, . . .

)= 1

⇒ af (p) = gcd (af (pr), r = 3, 5, 7, . . . ) .

Lemma 4.10. Suppose p is a prime such that p - af (p). Then ε = 1.

Proof. By definition of ε in eq. (4.5),

ε | af (pr−1) and ε | af (pr+1) = af (p)af (pr)− af (pr−1)pk−1

for all odd positive integers r ≥ 3. Therefore,

ε | af (p)af (pr), r = 3, 5, 7, . . . ,

that is,

ε | gcd (af (p)af (pr), r = 3, 5, 7, . . . ) = af (p) · gcd (af (pr), r = 3, 5, 7, . . . ) .

Hence,

ε | af (p)2δ

where δ is defined in eq. (4.6).

Write ε = ab where a | af (p)2 and b | δ. Since gcd (af (p), δ) = 1 by lemma 4.8, gcd(a, b) = 1.


Suppose a > 1. Since ε | af (p2) = af (p)2 − pk−1 and a | ε, we have that a | pk−1 and therefore

p | af (p). Contradiction.

Therefore ε = b and ε | δ. Suppose b = ε > 1. Recall that δ | a∗f (p3) = af (p2)−pk−1 = af (p)2−2pk−1

and ε | af (p2) by definition, we have that ε | pk−1. This implies that p | δ and thus p | af (p).

Contradiction.

Lemma 4.11. Suppose p is a prime such that p - af (p). Then δ = 1.

Proof. We have

δ | a∗f (pr+1) = af (pr)− a∗f (pr−1)pk−1

for all even positive integers r ≥ 2. Therefore,

δ | af (pr), r = 4, 6, 8, . . . .

In particular,

δ | af (p3), af (p4).

By lemma 4.5,

gcd(af (p3), af (p4)

)= 1

so δ = 1.

Lemma 4.12. Suppose p is a prime such that p - af (p). Let r be any odd positive integer. Then

gcd(a∗f (pr), a∗f (pr+2)

)= 1.


For r = 1,

gcd(a∗f (p), a∗f (p3)

)= gcd

(1, a∗f (p3)

)= 1.

We now assume gcd(a∗f (pr), a∗f (pr+2)

)= 1. Let d = gcd

(a∗f (pr+2), a∗f (pr+4)

).

Now,

a∗f (pr+4) = af (pr+3)− a∗f (pr+2)pk−1

so d | af (pr+3). Also,

af (pr+3) = af (p)af (pr+2)− af (pr+1)pk−1

so d | af (pr+1)pk−1. We claim gcd(d, pk−1

)= 1. Indeed, suppose gcd

(d, pk−1

)> 1. Then p | d. By

lemma 3.1, we see that

a∗f (pr+2) =

(r+1)/2∑j=0

cjaf (p)(r+1)−2jp(k−1)j , c0 = 1

and therefore p | af (p)r+1 ⇒ p | af (p) which is a contradiction. Hence d | af (pr+1).

Now,

a∗f (pr+2) = af (pr+1)− a∗f (pr)pk−1

so d | a∗f (pr)pk−1. But by the argument above, gcd(d, pk−1

)= 1, so d | a∗f (pr).


Therefore, d | a∗f (pr) and d | a∗f (pr+2). However, by the induction hypothesis,


)= 1.

Hence d = 1.

.

Lemma 4.13. Suppose p is a prime such that p - af (p). Let r be any even positive integer. Then


)= 1.


For r = 2, we show

gcd(af (p2), af (p4)

)= 1.

Indeed,

af (p2) = af (p)2 − pk−1

and

af (p4) = af (p)af (p3)− af (p2)pk−1

= af (p)(af (p)af (p2)− af (p)pk−1

)− af (p2)pk−1

= af (p)2(af (p2)− pk−1

)− af (p2)pk−1

=(af (p2) + pk−1

) (af (p2)− pk−1

)− af (p2)pk−1

= af (p2)2 − af (p2)pk−1 − p2(k−1).

So if d = gcd(af (p2), af (p4)

)> 1, then

d | p2(k−1)

⇒p | d

⇒p | af (p2) = af (p)2 − pk−1

⇒p | af (p),


We now assume gcd(af (pr), af (pr+2)

)= 1. Let d = gcd

(af (pr+2), af (pr+4)

). Now,

af (pr+4) = af (p)af (pr+3)− af (pr+2)pk−1

so d | af (p)af (pr+3). We claim gcd (d, af (p)) = 1. Otherwise, if e = gcd (d, af (p)) > 1, then by

lemma 3.1, we see that

af (pr+2) =

(r+2)/2∑j=0

cjaf (p)(r+2)−2jp(k−1)j , c0 = 1, c(r+2)/2 = ±1,


and hence

e | af (pr+2)

⇒ d | ±p(k−1)(r+2)/2

⇒ p | d

⇒ p | af (p)

which yields a contradiction. Hence d | af (pr+3).

Therefore, we have that d | af (pr+2), af (pr+3), af (pr+4), so in particular

d | gcd(af (pr+2), af (pr+3)

)= 1

by lemma 4.5. Hence d = 1.

Combining the previous two lemmas, we have:

Lemma 4.14. Suppose p is a prime such that p - af (p). Let r be any positive integer with r ≥ 1. Then


)=

1 if r is even

|af (p)| if r is odd.

or equivalently,


)= 1

We may generalize lemma 4.14 even further from consecutive powers to, in a sense, “arithmetic

progressions” of powers. In particular

gcd(a∗f (pα), a∗f (p3α), a∗f (p5α), . . .

)= 1.

Lemma 4.15. Suppose p is a prime such that p - af (p). Let α and r be positive integers. Then


)| gcd

(af (pα−2), af (pα)

).

Proof. Suppose q > 1 is a prime such that q | af (pα), af (prα), af (p(r+2)α). We will show that this implies

q | af (pα−2), af (pα).

By lemma 4.1,

af (p(r+2)α) = af (prα)af (p2α)− af (prα−1)af (p2α−1)pk−1.

Since q | af (p(r+2)α), af (prα), we see that q | af (prα−1)af (p2α−1)pk−1. We may conclude that q -af (prα−1) (otherwise q | gcd

(af (prα−1), af (prα)

)= 1 by lemma 4.5, which implies q = 1), and q - pk−1

(otherwise q = p | af (p)α ⇒ p | af (p) by lemma 4.4). Thus, q | af (p2α−1).

Now,

af (p2α−1) = af (pα)af (pα−1)− af (pα−1)af (pα−2)pk−1.

Since q | af (p2α−1), af (pα), we see that q | af (pα−1)af (pα−2)pk−1. Again, by a similar argument to the

one above, we may conclude that d | af (pα−2).

Hence, q | gcd(af (pα−2), af (pα)

).


Lemma 4.16. Suppose p is a prime such that p - af (p). Let α be a positive integer, and r be an odd

positive integer. Then


)=

1 if α is even

|af (p)| if α is odd,

or equivalently,

gcd(a∗f (pα), a∗f (prα), a∗f (p(r+2)α)

)= 1

Proof. Let d = gcd(af (pα), af (prα), af (p(r+2)α)

).

By the lemma above,

d | gcd(af (pα−2), af (pα)

)and in particular

d ≤(af (pα−2), af (pα)

).

If α is even, then(af (pα−2), af (pα)

)= 1 by lemma 4.14, and therefore d = 1.

If α is odd, then(af (pα−2), af (pα)

)= af (p) by lemma 4.14, so d ≤ af (p). On the other hand,

af (p) | d, so d ≥ af (p). Therefore d = af (p).


gcd(af (pα), af (p3α), af (p5α), . . .

)=

1 if α is even

|af (p)| if α is odd,

or equivalently,

gcd(a∗f (pα), a∗f (p3α), a∗f (p5α), . . .

)= 1.

Proof. This follows immediately from lemma 4.16.

4.4 Algorithm SQF — a test for squarefree-ness

In this section we present an algorithm for testing whether an integer n is squarefree or not.

Write n =t∏i=1

pαii . We assume that pi - af (pi) for all i = 1, . . . , t. We assume this because we will be

using most of the lemmas in section 4.3.

This assumption is not that unreasonable. For example, taking f = ∆ and af (·) = τ(·), it is

known that the only solutions to the equation τ(p) ≡ 0 (mod p) are p, 2, 3, 5, 7, 2411, 7758337633 for

p ≤ 1010 ≈ 233.

We also assume that, for p prime, |af (pr)| 6= |af (p)| for all odd integers r. For example, f = ∆

satisfies this assumption; see section 4.5.

4.4.1 Description of the algorithm

Given an integer n such that p | n ⇒ p - af (p), an algorithm to determine whether n is squarefree is

described below.


1. Compute


|af (n)|.

2. Conclude n is squarefree if and only if SQFRf (n) = 1.

Here R is a parameter — the “cut off” point for how long the algorithm should run. From eq. (4.1)

we see that computing SQFRf (n) means computing the greatest common divisor of R numbers. Since

computing gcd(m1,m2) can be done in polynomial time (for example by using the Euclidean algorithm),

we may measure the runtime complexity of algorithm SQFRf by R.

It is conceivable that there exists a non-squarefree integer n such SQFRf (n) = 1 for many R’s, however,

as seen in theorem 4.2, the likelihood of this happening should decrease as R increases. Thus a large

value of R (potentially) gives a higher probability of detecting a square-free integer correctly. On the

other hand, section 4.4.6 suggests that, experimentally, a value as small as R = 4 is sufficient when

f = ∆.

The following theorem shows that SQFRf is correct. That is if n is squarefree then it will conclude n

is squarefree.


It is possible that the algorithm SQFRf will fail, that is SQFRf (n) = 1 when n is not squarefree. The

next theorem states that if a non-squarefree integer n satisfies certain conditions, then SQFRf does not

fail, i.e. SQFRf (n) 6= 1.




Iodd = i : αi > 1, αi ≡ 1 (mod 2)



αi0i0

), af (pαi−2i )


gcd(af (p

αi0i0

), af (pri ))



SQFRf (n) 6= 1.


4.4.2 Proof of theorem 4.1 — correctness of the algorithm

Suppose n is squarefree. Write n =t∏i=1

pi where pi’s are distinct primes. Then by the multiplicative

property of af ,

af (nr) = af

(t∏i=1

pri

)=

t∏i=1

af (pri )

for any positive integer r.

Since af (pi) | af (pri ) for 1 ≤ i ≤ t and all odd positive integers r by lemma 4.2,

af (n) | af (nr), r = 1, 3, 5, . . .


and thus

SQFRaf (n) =gcd (af (nr), r = 1, 3, 5, . . . , 2R− 1)

|af (n)|=|af (n)||af (n)|

= 1.

That is, the algorithm will then conclude n is squarefree.


Let n be a non-squarefree integer. Thus we may write n =t∏i=1

pαii with at least one αi > 1. Assume the

algorithm fails, that is

SQFRaf (n) = 1⇔ gcd (af (nr), r = 1, . . . , 2R− 1) = |af (n)| .

Let q > 1 be a prime divisor of af (n), specifically of af (pαi0i0

). We will show q = 1 in lemma 4.23.

Now, if we write the af (nr)’s in terms of the af (prαii )’s, we have the following equations:

af (n) = af (pα11 ) · · · af (pαii ) · · · af (pαtt )

af (n3) = af (p3α11 ) · · · af (p3αi

i ) · · · af (p3αtt )

af (n5) = af (p5α11 ) · · · af (p5αi

i ) · · · af (p5αtt )

...

af (n2R−1) = af (p(2R−1)α1

1 ) · · · af (p(2R−1)αii ) · · · af (p

(2R−1)αtt )

(4.7)

We may regard eq. (4.7) as a matrix with R rows and t columns. Then, since af (n) | af (nr) for

1 ≤ r ≤ 2R − 1, q (recall a > 1 is a non-trivial prime divisor of af (n)) appears in every row. By the

pigeon hole principle, if R is sufficiently large, q must appear in the some column more than once; that

is, q | af (prj1αii ), af (p

rj2αii ) for some 1 ≤ i ≤ t and some rj1 < rj2 .

The assumptions of theorem 4.1 together with the next several lemmas imply that we may assume

αi > 1 and that αi is odd for 1 ≤ i ≤ t.

If αi is even, then rαi is also even. The following two lemmas illustrates what happens when the

exponent αi are even.

Lemma 4.18. Suppose p is a prime such that p - af (p). Let r and s be positive integers. Let q be a

prime such that q | gcd(af (p2r), af (p2r+2s)

).

1. If r = s, then q = 1.

2. If r > s, then q | af (p2r−2s).

3. If s > r, then q | af (p2s−2r−2).

Proof. By lemma 4.1,

af (p2r+2s) = af (p2r)af (p2s)− af (p2r−1)af (p2s−1)pk−1.

We may assume q - pk−1, since otherwise q = p and p | af (p) by lemma 4.4. Also, we may assume

q - af (p2r−1) since otherwise q = 1 by lemma 4.5. Therefore,

q | af (p2s−1).


Suppose r = s. In this case, we see that 2s− 1 = 2r − 1 and this is a contradiction unless q = 1.

Suppose r > s. Then 2r > 2s− 1, and by lemma 4.1

af (p2r) = af (p2r−(2s−1))af (p2s−1)− af (p2r−(2s−1)−1)af (p2s−2)pk−1

= af (p2r−2s+1)af (p2s−1)− af (p2r−2s)af (p2s−2)pk−1.

If q | af (p2s−2) then q = 1 by lemma 4.5. Therefore, q | af (p2r−2s).

Suppose s > r. Then 2s > 2s− 1 > 2r, and by lemma 4.1

af (p2s−1) = af (p2s−1−2r)af (p2r)− af (p2s−1−2r−1)af (p2r)pk−1

= af (p2s−2r−1)af (p2r)− af (p2r−2s−2)af (p2r−1)pk−1.

If q | af (p2r−2) then q = 1 by lemma 4.5. Therefore, q | af (p2r−2s−2).


gcd(af (p2r), af (p2r+2s)

)= 1.

Proof. Let q be a prime such that q | gcd(af (p2r), af (p2r+2s)

).

There are three possibilities: r = s, r > s, or s > r.

Suppose r = s. We are done since lemma 4.18 asserts q = 1.

Suppose r > s. Applying lemma 4.18 repeatedly we see that

q | af (p2r−2s), af (p2r−4s), af (p2r−6s), . . . af (p2r−2j0s)

where j0 is such that

2s > 2r − 2j0s > 0 > 2r − 2(j0 + 1)s.

Note that we may assume 2r − 2j0s > 0, since otherwise 2r − 2j0s = 0 and q | 1. Now let

2r′ = 2r − 2j0s 2s′ = 2s− 2r′ = 2(j0 + 1)s

r′ = r − j0s s′ = (j0 + 1)s− r.

We see that r′ < r and s′ < s, so we have “decreased” r and s. Also s′ > r′, so we are now in the next

case (see below).

Suppose s > r. By the previous lemma, q | af (p2s−2r−2). If 2s− 2r − 2 > 2r, let

2r′ = 2r 2s′ = 2s− 2r − 2− 2r = 2s− 4r − 2

r′ = r s′ = s− 2r − 1.

We see that s′ < s, so we have “decreased” s. Otherwise, if 2r > 2s− 2r − 2, let

2r′ = 2s− 2r − 2 2s′ = 2r − 2r′ = 4r − 2s+ 2

r′ = s− r − 1 s′ = 2r − s+ 1.


We see that r′ < r and s′ < s, so we have “decreased” both r and s.

In either case, s decreases or both r and s decrease. Repeating this, we will end up with s = 0, r = 0

or s = r; in any case, q = 1.

The following lemma is the analogue of lemma 4.18 for odd exponents.

Lemma 4.20. Suppose p is a prime such that p - af (p). Let q be a prime such that

q | gcd(a∗f (p2r+1), a∗f (p2r+1+2s)

).

1. If r = s, then q = 1.

2. If r + 1 > s, then q | a∗f (p2r−2s−1).

3. If s > r + 1, then q | a∗f (p2s−2r−3).

4. If s = r + 1, then gcd(a∗f (p2r+1), a∗f (p2r+1+2s)

)= |a∗f (p2r+1)|.

Proof. By lemma 4.1,

a∗f (p2r+1+2s) = a∗f (p2r+1)af (p2s)− af (p2r)a∗f (p2s−1)pk−1.

We may assume q - pk−1 since otherwise q = p and p | af (p) by lemma 4.4. Also, we may assume

q - af (p2r) since otherwise q = 1 by lemma 4.5. Therefore,

q | a∗f (p2s−1).

Suppose r = s. In this case, 2s− 1 = 2r − 1, and q | a∗f (p2r−1) implies q = 1 by lemma 4.14.

Suppose r + 1 > s. Then 2r + 1 > 2s− 1 and by lemma 4.1

a∗f (p2r+1) = af (p2r+2−2s)a∗f (p2s−1)− a∗f (p2r+1−2s)af (p2s−2)pk−1.

If q | af (p2s−2) then q = 1 by lemma 4.5. Therefore, q | a∗f (p2r+1−2s).

Suppose s > r + 1. Then 2s− 1 > 2r + 1 and by lemma 4.1

a∗f (p2s−1) = af (p2s−2r−2)a∗f (p2r+1)− a∗f (p2s−2r−3)af (p2r)pk−1.

If q | af (p2r) then q = 1 by lemma 4.5. Therefore, q | a∗f (p2r−2s−3).

Suppose s = r+1. In this case, we see that 2r+1+2s = 4r+3 and 2s−1 = 2r+1 and by lemma 4.1

a∗f (p4r+3) = a∗f (p2r+1)af (p2r+2)− af (p2r)a∗f (p2r+1)pk−1

= a∗f (p2r+1)(af (p2r+2)− af (p2r)pk−1

).

Therefore, we see that a∗f (p2r+1) | a∗f (p4r+3), and hence gcd(a∗f (p2r+1), a∗f (p4r+3)

)= |a∗f (p2r+1)|.

The last case, i.e. the case s = r + 1, of lemma 4.20 is generalized in following lemma by noticing

jr + (j − 1) = r + (j − 1)(r + 1).


Lemma 4.21. Let r be any positive integer. Then

gcd(af (pjr+(j−1)), j = 1, 2, 3, . . .

)= |af (pr)|.

Proof. Since for j = 1, af (pjr+(j−1)) = af (pr), therefore it is enough to show af (pr) | af (pjr+(j−1)) for

j = 2, 3, 4, . . . .

We apply induction on j.

The base case j = 1 is trivial, as remarked above.

Suppose af (pr) | af (pjr+(j−1)). For j + 1, we have, by lemma 4.1,

af (p(j+1)r+j) = af (pjr+j)af (pr)− af (pjr+j−1)af (pr−1)pk−1

and thus af (pr) | af (p(j+1)r+j).

An immediate consequence of lemma 4.19 is that for any subset A of the positive even integers,

gcd (af (pr), r ∈ A) = 1.

Recall that q is a prime divisor of af (n) that appears in the some column more than once. Therefore,

by lemma 4.19 we may assume αi > 1 is odd for i = 1, . . . , t.

Lemma 4.22. Suppose p is a prime such that p - af (p). Let α1, . . . , αt be a finite list of odd positive

integers with αi > 1. There exists an infinite increasing sequence of odd numbers (r1, r2, r3, . . . ) such

that

1. r1 = 1,

2. rj+1 > rj > 1 for j > 1, and

3. gcd (a(pαirj1 ), af (pαirj2 )) | af (pαi−2) for all i = 1, . . . , t and any j1 6= j2.

Proof. We construct the sequence (rj)∞j=1 inductively.

Choose r1 = 1.

Suppose (r1, . . . , rJ−1) is chosen. Choose rJ > rJ−1 such that

rJ ≡ −1 (mod αirj + 1)

for all 1 ≤ i ≤ t, 0 ≤ j ≤ J − 1. Such a rJ exists; for example, we may choose

rJ =

∏1≤i≤t

0≤j≤J−1

(αirj + 1)

− 1,

but there may be smaller choices.

Now, let q be a prime dividing gcd (af (pαirj1 ), af (pαirj2 )), i.e. q divides both af (pαirj1 ) and af (pαirj2 ).

For simplicity, write αi = α, rj1 = r1, rj2 = r2, and assume r1 < r2.


By lemma 4.1

af (pr2α) = af (pr1α)af (p(r2−r1)α)− af (pr1α−1)af (p(r2−r1)α−1)pk−1

af (p(r2−r1)α−1) = af (pr1α)af (p(r2−2r1)α−1)− af (pr1α−1)af (p(r2−2r1)α−2)pk−1

af (p(r2−2r1)α−2) = af (pr1α)af (p(r2−3r1)α−2)− af (pr1α−1)af (p(r2−3r1)α−3)pk−1

...

af (p(r2−(c−1)r1)α−(c−1)) = af (pr1α)af (p(r2−cr1)α−(c−1))− af (pr1α−1)af (p(r2−cr1)α−c)pk−1

...

we see that

q | af (p(r2−cr1)α−c) = af (pαr2−c(αr1+1))

for c = 0, 1, 2, . . . . However, by construction,

r2 ≡ −1 (mod αr1 + 1)

αr2 ≡ −α ≡ α(r1 − 1) + 1 (mod αr1 + 1),

so there exists a c such that αr2 − c(αr1 + 1) = α(r1 − 1) + 1. Therefore,

q | af (pα(r1−1)+1).

Since, by lemma 4.1,

af (pr1α) = af (pα(r1−1)+1)af (pα−1)− af (pα(r1−1))af (pα−2)pk−1

we see that

q | af (pα−2).

This shows that gcd (af (pαirj1 ), af (pαirj2 )) | af (pαi−2).

Lemma 4.23. Let n =t∏i=1

pαii where αi > 1 and αi is odd for all i = 1, . . . , t. Suppose there exists an

i0 such that

gcd(af (pαi0 ), af (pαi−2)

)= 1

for all i 6= i0. Then there exists a positive integer S and a finite increasing sequence of odd numbers

(r1, r2, . . . , rS) such that r1 = 1 and

gcd(a∗f (nrj ), j = 1, . . . , S

)< |a∗f (n)|.

In particular, SQFRf (n) 6= 1.

Proof. Let (r1, r2, r3, . . . ) be the (infinite) increasing sequence of odd numbers as in lemma 4.22. Let

S = t+ 1 and consider the finite subsequence (r1, r2, . . . , rS) consisting of the first S terms.


Similar to eq. (4.7), consider the following equations

a∗f (n) = a∗f (pr1α11 ) . . . a∗f (p

r1αi0i0

) · · · a∗f (pr1αtt )

a∗f (nr2) = a∗f (pr2α11 ) · · · a∗f (p

r2αi0i0

) · · · a∗f (pr2αtt )

a∗f (nr3) = a∗f (pr3α11 ) · · · a∗f (p

r3αi0i0

) · · · a∗f (pr3αtt )...

a∗f (nrS ) = a∗f (prSα11 ) · · · a∗f (p

rSαi0i0

) · · · a∗f (prSαtt )

and think of this as a matrix with S rows and t columns.

Now, suppose gcd(a∗f (nrj ), j = 1, . . . , S

)= |a∗f (n)| and let q > 1 be a prime divisor of a∗f (p

αi0i0

). We

will show q = 1.

Then q | a∗f (n) and therefore q | a∗f (nrj ) for all j = 1, . . . , S. That is, q appears in every row.

Therefore q appears in a column, say the i-th, more than once. More precisely,

q | gcd(af (p

rj1αii ), af (p

rj2αii )

)for some j1 < j2. However, by the construction of (rj)

Sj=1,

gcd(af (p

rj1αii ), af (p

rj2αii )

)| af (pαi−2

i )

and hence

q | gcd(af (p

αi0i ), af (pαi−2

i ))

If i 6= i0, then by assumption gcd(af (p

αi0i ), af (pαi−2

i ))

= 1, and so q = 1. If i = i0, then by

lemma 4.14 gcd(af (p

αi0i ), af (p

αi0−2i )

)= 1, and so q = 1.

We now prove the theorem 4.2.




Iodd = i : αi > 1, αi ≡ 1 (mod 2)



αi0i0

), af (pαi−2i )


gcd(af (p

αi0i0

), af (pri ))



SQFRf (n) 6= 1.


Proof. By the remarks above, we may assume without loss of generality that αi > 1 and αi is odd for

all i = 1, . . . , t. Then n satisfies the assumptions of lemma 4.23. Apply lemma 4.23 and let 2R− 1 = rS .

Now, since (rj)Sj=1 is a subsequence of the sequence of odd integers (1, 3, 5, . . . , 2R− 1),

gcd(a∗f (nr), r = 1, 3, . . . , 2R− 1

)≤ gcd

(a∗f (nrj ), j = 1, . . . , S

)< |a∗f (n)|.


Thus,

SQFRf (n) 6= 1.

The value of R in theorem 4.2 is large if we use the construction in lemmas 4.22 and 4.23. However,

we make the following observation: given the values of r1 = 1 and rS , we know (by construction)

the approximate values of r2, . . . , rS−2. Therefore, in practice, we need not compute the full sequence

gcd(a∗f (nr), r = 1, 3, . . . , 2R− 1

), but rather a subsequence gcd

(a∗f (nr), r ∈ AS

), where AS satisfies

r1, r2, . . . , rS ⊆ AS ( 1, 3, 5, . . . , 2R− 1 and has O(S) terms, since

gcd(a∗f (nr), r = 1, 3, . . . , 2R− 1

)≤ gcd

(a∗f (nr), r ∈ AS

)≤ gcd

(a∗f (nrj ), j = 1, . . . , S

)< |a∗f (n)|.

We also note that if a prime factor q appears in two rows of eq. (4.7), then it is almost always the

case that we may “push” the exponent down using the techniques in lemmas 4.16 and 4.20. Therefore,

we believe that SQFRf will work as long as there are t + 1 rows in eq. (4.7). We make the following

conjecture for the value of R.

Conjecture 4.1. We may use R = t+ 1 in the algorithm SQFRf .

If R = t + 1, then by a celebrated result of Hardy-Ramanujan in [Har40], R ∼ log log n on average,

and R . O(log n/ log log n).

Experimentally, we conjecture that R is a constant independent of n if we use f = ∆:

Conjecture 4.2. There exists a constant R0 independent of n such that SQFR0

∆ (n) = 1 if and only if n

is squarefree.

The experiment in section 4.4.6 suggests that the R0 = 2. Also see conjecture 4.4.

4.4.4 Heuristic analysis

In this section, we will give an heuristic argument for the following conjectures.

Conjecture 4.3. Let 0 < C < 1. For weights k ≥ 4, algorithm SQFRf will correctly recognize whether

an integer n is squarefree with probability 1− C > 0 by taking R = O(− logC · log t).

Conjecture 4.4. Let 0 < C < 1. For large enough weight k, algorithm SQFRf will correctly recognize

whether an integer n is squarefree with probability 1− C > 0 by taking R = O(1).

We remark that if conjecture 4.3 is true, then by a celebrated result of Hardy-Ramanujan in [Har40],

log t ∼ log log log n on average, and log t . O(log log n) in the worst case, where t = ω(n). If conjec-

ture 4.4 is true, then R is a constant independent of n.

Recall the definition of a∗f in eq. (4.3).

We assume f satisfy an independence condition as follows. Let p and q be distinct primes, and let α

and β be positive odd integers. Let d be a prime such that d | a∗f (pα). We assume

#q ≤ x : d | a∗f (qβ)#q ≤ x

< 1.


That is, the probability of the set of prime divisors of a∗f (pα) equals the set of prime divisors of a∗f (qβ)

is small if p, q, α, and β are chose randomly. Equivalently, the probability that there exists a prime d

such that d | a∗f (pα) but d - a∗f (pα) is nonzero.

Write n =t∏i=1

pαii with at least one αi > 1. We will estimate the probability of the algorithm failing,

that is

SQFRf (n) = 1⇔ af (n) | af (n), af (n3), . . . , af (n2R−1).

Now, re-write

n = m1m0 =

t1∏i=1

pαii

t0∏i=1

qi

where m1 =t1∏i=1

pαii is the non-squarefree part of n and m0 =t0∏i=1

qi is squarefree part of n. Thus there

are t1 > 0 prime factors dividing n to an exponent larger than 1, and t = t1 + t0.

Consider the following equations, regarded as a R× t matrix as in eq. (4.7):

af (n) = af (pα11 ) · · · af (p

αt1t1 ) af (q1) · · · af (qt0)

af (n3) = af (p3α11 ) · · · af (p

3αt1t1 ) af (q3

1) · · · af (q3t0)

af (n5) = af (p5α11 ) · · · af (p

5αt1t1 ) af (q5

1) · · · af (q5t0)

...

af (n2R−1) = af (p(2R−1)α1

1 ) · · · af (p(2R−1)αt1t1 ) af (q2R−1

1 ) · · · af (q2R−1t0 )

Let dj be a prime divisor of a∗f (pαii ). Then the probability dj divides a∗f (nr) for odd r > 1, i.e. dj

appears in the r-th row, is

P[dj | a∗f (nr)

]≈ t− 1

tεf,dj .

Here, εf,dj is the probability that dj divides a∗f (prα) for randomly chosen p and α, and εd satisfies

0 ≤ εf,dj ≤ 1 and εf,dj → 0 as dj → ∞ if εf,dj < 1. Also, if εf,dj < 1 we expect εf,dj → 0 as k → ∞where k is the weight of f , since af (p) = O(p(k−1)/2) but ω(af (p)) . log af (p)

log log af (p) = O((k− 1) log p). The

factor t−1t appears because there are t− 1 out of t possible columns in which dj may appear.

Let ui = ω(a∗f (pαii )), i.e the number of distinct prime divisors of a∗f (pαii ). Let s =∑t1i=1 ui. Then the

probability that a∗f (n) divides a∗f (nr) for odd r > 1 is

P[a∗f (n) | a∗f (nr)

]≈

s∏j=1

t− 1

tεf,dj =

(t− 1

t

)s s∏j=1

εf,dj .

Since there areR rows, the probability that SQFRf (n) = 1, that is a∗f (n) divides a∗f (n3), a∗f (n5), . . . , a∗f (n2R−1),

is

P[SQFRf (n) = 1

]≈

s∏j=1

t− 1

tεf,dj

R−1

=

(t− 1

t

)s(R−1) s∏j=1

εR−1f,dj

.

Let εf = max1≤j≤s

(εf,dj ). Then 0 ≤ εf ≤ 1 and

s∏j=1

t− 1

tεf,dj

R−1

≤

s∏j=1

t− 1

tεf

R−1

=

(t− 1

tεf

)(R−1)s

.


Also, since a∗f (pαii ) 6= 1 we have s ≥ t1 ≥ 1, so

(t− 1

tεf

)(R−1)s

≤(t− 1

tεf

)(R−1)t1

.

Therefore, for any 0 ≤ C ≤ 1,(t− 1

tεf

)(R−1)t1

≤ C

(R− 1)t1 · log

(t− 1

tεf

)≤ logC

R ≥ logC

t1 log(t−1t εf

) + 1

=− logC

t1· 1

− log εf + log tt−1

+ 1.

For any κ > 0 define

gκ(x) =1

κ+ log xx−1

.

It is an elementary exercise to show that g′(x) > 0 and g′′(x) < 0 for sufficiently large x; hence

gκ(x) = O(x). In fact,

gκ(x) = O(log(x))

where the constant depends on κ. Therefore, if we want P[SQFRf (n) = 1

]≤ C, then we should take

R = O(− logC · g− log εf (t) + 1) = O(− logC · log t).

This is conjecture 4.3.

On the other hand, since t−1t ≤ 1,

(t− 1

tεf

)(R−1)t1

≤ ε(R−1)t1f .

If we assume εf < 1 then for any 0 ≤ C ≤ 1

ε(R−1)t1f ≤ C

(R− 1)t1 log εf ≤ logC

R ≥ logC

log εf· 1

t1+ 1.

In this case, if we want P[SQFRf (n) = 1

]≤ C, then we should take

R = O(1).

This is conjecture 4.4.

We note that the assumption εf < 1 is not likely for small weights k (say k = 2) as the values of


af (pr) may not grow fast enough as p and r grow. 1 Also, for algebraic reasons (see lemma 4.20), εf < 1

is rare if the exponents in the factorization of n are small.

4.4.5 Computer code

The following code in Mathematica can be used to conduct various experiments.

(*

* Computes t(n)/GCD(t(n^r), r=1, 3, ..., R) and check whether it is 1.

*)

SQF[n_, R_] := Module[x, y, (x=RamanujanTau[n]; y=x; Do[y = GCD[y,

RamanujanTau[n^r]], r, 3, R, 2]; Return[Abs[x/y] == 1])]

(*

* Checks whether SQF[n] == SquareFreeQ[n].

*)

SQFcheck[n_, R_] := Return[SQF[n, R] == SquareFreeQ[n]]

(*

* Returns the smallest R <= Rmax until SQFcheck[n, Rmax] becomes true.

If Rmax is reached, returns "No R works!".

*)

SQFminR[n_, Rmax_] := Module[R, (R=3; While[!SQFcheck[n, R] &&

R<=Rmax, R+=2]; If[R <= Rmax, Return[R], Return["No R works!"]] )]

(*

* Generates a random number with at most tmax factors, ie n =

p1â1*...*ptât, with each 11 <= pi <= pmax and 1 <= ai <= amax and 1 <

t <= tmax.

*)

Generate[tmax_, pmax_, amax_] := Module[ x, (x = 1; t =

RandomInteger[2, tmax]; Do[x *= RandomPrime[11,

pmax]^RandomInteger[1, amax], i, t]; Return[x]) ]

(*

* Get a list SQFminR[n, Rmax] for numberOfTrials n’s with t factors, ie

n = p1â1*...*ptât, with each pi <= pmax and ai <= amax.

*)

1J. Tsimerman suggested that SQFRf may work better with higher weight forms, and prompt an experimental study

of low weight (k = 2) forms which partly led to the inclusion of this section.


listSQFminR[tmax_, pmax_, amax_, Rmax_, numberOfTrials_] :=

Module[list, n, (list=; Do[(n=Generate[tmax, pmax, amax];

list=Append[list, SQFminR[n, Rmax]]), i, numberOfTrials];

Return[list])]

(*

* Get the max of listSQFminR[t, pmax, amax, Rmax, numberOfTrials].

*)

maxSQFminR[tmax_, pmax_, amax_, Rmax_, numberOfTrials_] :=

Return[Max[listSQFminR[tmax, pmax, amax, Rmax, numberOfTrials]]]

(*

* Check the odd integers Nstart <= n <= Nend, with sQFcheck[n, R].

Avoids n’s divisible by 2, 3, 5, 7, 2411, 7758337633. Nstart should be

odd.

*)

check[Nstart_, Nend_, R_] := (Do[ (If[ Divisible[n, 3] || Divisible[n,

5] || Divisible[n, 7] || Divisible[n, 2411], Null, If[!SQFcheck[n, R],

Print["n= ", n, " failed, ", "n = ", FactorInteger[n]]] ];

If[Divisible[n+1, 1000], Print["Finished n = ", n, " @",

DateString[]]]), n, Nstart, Nend, 2 ]; Print["Done!"])

(*

* Get the average of listSQFminR[tmax, pmax, amax, Rmax,

numberOfTrials]. Prints milestone every 1000 trials. If SQF fails at n,

prints n.

*)

averageSQFminR[tmax_, pmax_, amax_, Rmax_, numberOfTrials_] := ( count =

0; R = 0; Do[ n = Generate[tmax, pmax, amax]; r = SQFminR[n, Rmax];

If[r=="No R works!", Print["n = ", n, " failed!"], (R += r; count++)];

If[Divisible[i, 1000], Print["avg R = ", N[R/count, 2] , " with ", i , "

trials @", DateString[]]]; , i, numberOfTrials]; Print["Done! avg R =

", N[R/count, 2]]; )

4.4.6 Experimental results

An experiment was run in Mathematica using the code in section 4.4.5 that checked whether SQF5τ (n) = 1

for all integers 11 ≤ n ≤ 1, 000, 000 not divisible by 2, 3, 5, 7 and 2411 using the following command:

check[11,1000000,9]

For such integers n, SQF10τ correctly concludes whether n was squarefree 100% of the time.


Another experiment was run in Mathematica using the code in section 4.4.5 to obtain the average

minimum value of R for which SQFRτ was successful over 500, 000 random integers n where

n =

t∏i=1

pαii

with 2 ≤ t ≤ 70, 11 ≤ pi ≤ 3000, 1 ≤ αi ≤ 70. Note that the pi’s are not necessarily distinct primes.

The following command was used:

averageSQFminR[70,3000,70,9,500000]

The average value of R was 2.

4.5 Values of the Ramanujuan τ function at odd powers


∆(z) = e2πiz∞∏n=1

(1− e2πinz)24

is a modular form of weight 12 and level 1 which is a normalized eigen-form for the Hecke operators.

Following Ramanujan, we write

∆(z) =

∞∑n=1

τ(n)e2πinz

for its Fourier expansion at i∞. It is well known that τ(1) = 1 and τ(m) ∈ Z for all m = 1, 2, 3, . . . .

Let p be a prime and r > 1 an odd positive integer. We already know that τ(p) | τ(pr) from

lemma 4.2. It may happen that this is because τ(pr) = ±τ(p). Thus one may ask: can τ(p) = ±τ(pr),

or equivalently, can τ(pr) | τ(p)?

This question has been studied by M. Murty, K. Murty, and T. N. Shorey in [MMS87], where it was

shown that τ(pr) 6= ±τ(p) when r is sufficiently large.

However, we are able to prove the following theorem for any odd integer r > 1.


τ(pr) 6= ±τ(p).

In fact, the proof of theorem 4.3 will be valid if we replace τ by af as long as af (p) is even. Therefore

we have the following theorem:




Throughout this section, we assume p is an odd prime such that τ(p) 6= 0.

Lemma 4.24. Let p be a prime and s be a positive integer. Then

τ(ps) ≡

1 s ≡ 0 (mod 2)

0 s ≡ 1 (mod 2).


Lemma 4.25. Let p be an odd prime. Then τ(p) 6= ±τ(p3).

Proof. Suppose τ(p) = ±τ(p3). Then by eq. (4.2)

±τ(p) = τ(p)τ(p2)− τ(p)pk−1

±1 = τ(p2)− p11.

Recalling that τ(p2) is odd, we see that the equality above yields a contradiction.

Lemma 4.26. Let r be a positive odd integer. Then

τ∗(pr) ≡

1 (mod 2) if r ≡ 1 (mod 4)

0 (mod 2) if r ≡ 3 (mod 4)

Proof. By induction.

For r = 1, τ∗(p) = 1. For r = 3, τ∗(p3) = τ(p)2 − 2p11 ≡ 0 (mod 2) since τ(p) is even.

Now, let r ≥ 5. If r ≡ 1 (mod 4), then τ∗(pr−2) ≡ 0 (mod 2), and

τ∗(pr) = τ∗(pr−1)− τ∗(pr−2)p11 ≡ 1− 0 ≡ 1 (mod 2).

If r ≡ 3 (mod 4), then τ∗(pr−2) ≡ 1 (mod 2), and

τ∗(pr) = τ∗(pr−1)− τ∗(pr−2)p11 ≡ 1− 1 ≡ 0 (mod 2).

The technique in showing τ(p) 6= ±τ(p3) may be extended to exponents r ≡ 3 (mod 4).

Lemma 4.27. Let r be a positive odd integer such that r ≡ 3 (mod 4). Then τ(pr) 6= ±τ(p).

Proof. Suppose otherwise. Then

τ(pr) = ±τ(p)

τ∗(pr) = ±1

0 ≡ 1 (mod 2)


For r ≡ 1 (mod 4), more work is required. For r = 5 and r = 9, it is fairly easy.

Lemma 4.28. τ(p5) 6= ±τ(p).

Proof. First, note that

τ∗(p5) = τ(p)4 − 4τ(p)2p11 + 3p22.

Suppose τ(p5) = ±τ(p)⇔ τ∗(p5) = ±1. Then

±1 = = τ(p)4 − 4τ(p)2p11 + 3p22


±1 ≡ 0− 0 + 3p2 (mod 8)

±1 ≡ 3 (mod 8)

since a2 ≡ 1 (mod 8) for any odd integer a. This yields a contradiction.

Lemma 4.29. τ(p9) 6= ±τ(p).

Proof. First, note that

τ∗(p9) = τ(p)8 − 8τ(p)6p11 + 21τ(p)4p22 − 20τ(p)2p33 + 5p44.

Suppose τ(p9) = ±τ(p)⇔ τ∗(p9) = ±1. Then

±1 = = τ(p)8 − 8τ(p)6p11 + 21τ(p)4p22 − 20τ(p)2p33 + 5p44

±1 ≡ 5 (mod 8)

since a2 ≡ 1 (mod 8) for any odd integer a. This yields a contradiction.

The technique above doesn’t work when we try to show τ(p13) 6= ±τ(p). This is because

τ∗(p13) = τ(p)12 + · · · − 56τ(p)2p55 + 7p66

and

±1 ≡ 7 (mod 8)

doesn’t immediately yield a contradiction.

However, we are still able to show it.

Lemma 4.30. τ(p13) 6= ±τ(p).

Proof. Suppose τ(p13) = ±τ(p)⇔ τ∗(p13) = ±1. Then

τ(p13) = τ(p7)τ(p6)− τ(p6)τ(p5)p11

±1 = τ(p6)[τ∗(p7)− τ∗(p5)p11

].

Since everything are integers, this implies

τ(p6) = ±1 and τ∗(p7)− τ∗(p5)p11 = ±1.

Now,

τ(p7) = τ(p)τ(p6)− τ(p5)p11

τ∗(p7) = τ(p6)− τ∗(p5)p11

= ±1− τ∗(p5)p11

and therefore

±1 = τ∗(p7)− τ∗(p5)p11


= ±1− τ∗(p5)p11 − τ∗(p5)p11

= ±1− 2τ∗(p5)p11

±1∓ 1 = 2τ∗(p5)p11.

From this we get that

τ∗(p5)p11 = 0 or τ∗(p5)p11 = ±1,

and since p is an odd prime,

τ∗(p5) = 0 or τ∗(p5) = ±1.

The first case is not possible since τ∗(p5) = 0 ⇒ p | τ(p). The second case is also not possible since

τ∗(p5) 6= ±1 by a previous lemma.

We extend this in general, for exponents r ≡ 1 (mod 4).

Lemma 4.31. Let r > 1 be a positive odd integer such that r ≡ 1 (mod 4). Then τ(pr) 6= ±τ(p).

Proof. Suppose τ(pr) = ±τ(p)⇔ τ∗(pr) = ±1. Since r ≡ 1 (mod 4), write r = 4s+ 1.

Then

τ(p4s+1) = τ(p2s+1)τ(p2s)− τ(p2s)τ(p2s−1)p11

±1 = τ(p2s)[τ∗(p2s+1)− τ∗(p2s−1)p11

].

Since everything are integers, this implies

τ(p2s) = ±1 and τ∗(p2s+1)− τ∗(p2s−1)p11 = ±1.

Now,

τ(p2s+1) = τ(p)τ(p2s)− τ(p2s−1)p11

τ∗(p2s+1) = τ(p2s)− τ∗(p2s−1)p11

= ±1− τ∗(p2s−1)p11

and therefore

±1 = τ∗(p2s+1)− τ∗(p2s−1)p11

= ±1− τ∗(p2s−1)p11 − τ∗(p2s−1)p11

= ±1− 2τ∗(p5)p11

±1∓ 1 = 2τ∗(p2s−1)p11.

From this we get that

τ∗(p2s−1)p11 = 0 or τ∗(p2s−1)p11 = ±1.


Since p is an odd prime, the second case (τ∗(p2s−1)p11 = ±1) is not possible.

Therefore, we must have τ∗(p2s−1) = 0. Then, combining this with τ(p2s) = ±1, we have that

τ(p2s) = τ(p)τ(p2s−1)− τ(p2s−2)p11

±1 = τ(p2s−2)p11.

Since p is an odd prime, this is impossible.

Combining lemma 4.27 lemma 4.31 immediately yields theorem 4.3:


τ(pr) 6= ±τ(p).

As remarked in the beginning of section 4.5, the proof of theorem 4.3 will be valid if we replace τ by

af as long as af (p) is even. Therefore we have theorem 4.4:



Chapter 5

A test for primality using Fourier

coefficients of modular forms


A primality test is an algorithm that, given an integer n as input, outputs whether n is a prime or a

composite number.

Perhaps the most widely known primality tests are the AKS-class primality tests (following [CP03]),

which are all variants of the AKS primality test (also known as the Agrawal-Kayal-Saxena primality test

or the cyclotomic AKS test).

The AKS primality test is a deterministic polynomial time algorithm. Roughly, it is based upon the

following theorem (stated as lemma 2.1 in [AKS04]):

Theorem 5.1 (Agrawal-Kayal-Saxena, 2004). Let a ∈ Z, n ∈ N, n ≥ 2, and gcd(a, n) = 1. Then n is

prime if and only if

(X + a)n = Xn + a (mod n). (5.1)

This immediately yields a simple primality test: given an input n, choose an a co-prime to n and test

whether the congruence eq. (5.1) is satisfied. The problem with this is that one will need to evaluate n

coefficients on the left hand side of eq. (5.1), giving a running time exponential in log n. A simple way

to reduce the number of coefficients is to test the whether the following congruence is satisfied:

(X + a)n = Xn + a (mod Xr − 1, n). (5.2)

In [AKS04], it was shown that for appropriately chosen r if eq. (5.2) is satisfied for several a’s then n

must be prime. This reduces the number of coefficients in eq. (5.2) one needs to evaluate and yields a

polynomial time algorithm.

Let f be a modular form of even weight k for Γ0(N) which is a normalized eigen-form for the Hecke

operators. Let us write

f(z) =

∞∑n=1

af (n)e2πinz

98

Chapter 5. A test for primality 99

for its Fourier expansion at i∞. We assume the Fourier coefficients af (n)∞n=1 are rational integers. If

p is prime, then af (p) satisfy the recurrence relation

af (pr) = af (p)af (pr−1)− af (pr−2)pk−1

for any positive integer r ≥ 2.


∆(z) = e2πiz∞∏n=1

(1− e2πinz)24

is a modular form of weight 12 and level 1 which is a normalized eigen-form for the Hecke operators.

Following Ramanujan, we write

∆(z) =

∞∑n=1

τ(n)e2πinz

for its Fourier expansion at i∞ and define the Ramanujan τ function as the n-th Fourier coefficient of

∆. Thus, if p is prime, then τ(p) satisfies the recurrence relation

τ(pr) = τ(p)τ(pr−1)− τ(pr−2)p11

for any positive integer r ≥ 2. It is well-known that the Ramanujan τ function also satisfy remarkable

(and sometimes mysterious) properties. For example, certain congruence relations hold for τ(p) modulo

powers of certain primes l.



normalized eigen-form for the Hecke operators and such the for p - NE , the conductor of E,

afE (p) = aE(p).

For the rest of this chapter , let E be the elliptic curve with Cremona label 54-B3(B) as in [Cre97, Cre],

and fE be the modular form of weight 2 corresponding to E.

In this chapter, we follow the theme of this thesis and explore the following question: given an oracle

for f , can we test whether n is prime?.

It is clear from the recurrence relation satisfied by the Ramanujan τ function that, if n is a prime,

then

τ(nr) = τ(n)τ(nr−1)− τ(nr−2)n11

for any positive integer r ≥ 2. Therefore if the equality above fails then n is composite. This forms the

basis of a primality test. The only question that remains is: if n satisfies the equality above for all r’s

can we conclude n is prime? The answer is given by theorem 5.2.

5.1.1 Main Results

Recall that by an oracle for f we mean an algorithm that computes af (n) in polynomial time.





steps.

The theorem is proved using the idea above that if n is prime then τ(nr) = τ(n)τ(nr−1)−τ(nr−2)n11

for all r’s. It also exploits the congruence properties of the Ramanujan τ function and of fE modulo

powers of certain primes l.

Using the proof of theorem 5.2 we are able to prove the following theorem.




This theorem has an application related to the parity problem and computing the Mobius function.

Recall the Mobius function:

µ(n) =

1 if n is squarefree with an even number of prime factors

−1 if n is squarefree with an odd number of prime factors

0 if n is not squarefree

.

In the case that n is a positive squarefree integer not divisible by 3, we may use theorem 5.3 to compute

the value of Mobius function.

Corollary 5.1. Let ∆ be as above. Suppose we are given an oracle for ∆. Let n be a positive squarefree

integer. Then there is a deterministic algorithm that computes µ(n) in a bounded (independently of n)

number of steps.

For a general (not necessary squarefree) integer n, if we assume conjecture 4.3 or conjecture 4.4, then

we can evaluate the Mobius function efficiently.

Conjecture 5.1. Let ∆ be as above. Suppose we are given an oracle for ∆. Let n ∈ Z.

If conjecture 4.3 is true, then there is a probabilistic algorithm that computes µ(n) (heuristically) in

O(log log log n) steps on average and O(log log n) steps in the worst case.


O(1) steps.

5.2 Proof of theorem 5.2

We first recall theorem 5.2.




steps.


5.2.1 Description of algorithm

Below is a description of the algorithm in theorem 5.2. Given a squarefree integer n 6≡ −1 (mod 54), we

perform the following series of tests:

1. Check if n is divisible by 2 and/or 3.

• If it is then check whether n = 2 or n = 3.

– If it is then conclude n is prime and terminate the algorithm.

– If it is not then conclude n is composite and terminate the algorithm.

• If it is not, then proceed to the next step.

2. Check if n satisfies F6 : τ(n6) = τ(n)τ(n5)− τ(n4)n11.

• If n satisfies F6, then proceed to the next step.

• If n fails to satisfy F6, then conclude n is composite and terminate the algorithm.

3. Check if n satisfies F4τ(n4) = τ(n)τ(n3)− τ(n2)n11.

• If n satisfies F4, then proceed to the next step.

• If n fails to satisfy F4, then conclude n is composite and terminate the algorithm.

4. Compute n mod 6.

(a) If n ≡ 1 (mod 6), compute afE (n2) (mod 9).

• If afE (n2) ≡ 0 (mod 9), then conclude n is composite and terminate the algorithm.

• If afE (n2) 6≡ 0 (mod 9), then conclude n is prime and terminate the algorithm.

(b) If n ≡ −1 (mod 6), compute afE (n) (mod 9).

• If afE (n) ≡ 0 (mod 9), proceed to the next step.

• If afE (n) 6≡ 0 (mod 9), then conclude n is prime and terminate the algorithm.

5. Compute τ(n) (mod 27).

• If τ(n) ≡ 0 (mod 27) then conclude n is composite and terminate the algorithm.

• If τ(n) 6≡ 0 (mod 27) then conclude n is prime and terminate the algorithm.

Lemma 5.1. The algorithm above terminates after a finite number of steps.

Proof. It is clear from the description of the algorithm that at every step the algorithm either terminates

or proceeds to the next step, and the last step always terminates.

5.2.2 Proof of correctness

We now show the output of the algorithm is correct. That is, if we input an integer n 6≡ −1 (mod 54),

the algorithm will always conclude n is prime if and only if n is prime.

Let n be a positive squarefree integer, and write

n =

t∏i=1

pi


where pi’s are distinct primes. By the first step of the algorithm, we may assume pi ≥ 5 for all i = 1, . . . , t.

In particular, n is odd.

For r ∈ Z, r ≥ 2, define the r-th level equation by

Fr : τ(nr) = τ(n)τ(nr−1)− τ(nr−2)n11.

We say n satisfies Fr if n satisfies the r-th level equation, and n fails Fr if n fails to satisfies the r-th

level equation.

As remarked above, if n is prime, then it will satisfy Fr for any integer r ≥ 2. Therefore, if n fails

to satisfy Fr for some r, then n is composite. This leads to the following definition: an positive odd

squarefree integer n is called Ramanujan pseudo-prime 1 if it satisfy the r-th level equation Fr for every

r.

Therefore, for the rest of this section, we assume n is a Ramanujan pseudo-prime.

The following lemma is well-known. We include it here because it is one of the key properties of the

Ramanujan τ function we exploit.

Lemma 5.2 (Ramanujan). Suppose p ≥ 5 is a prime. Then τ(p) ≡ p+ 1 (mod 3). Therefore

• τ(p) ≡ p+ 1 (mod 3),

• τ(p2) ≡ p+ 2 (mod 3),

• τ(p3) ≡ 2p+ 2 (mod 3),

• τ(p4) ≡ 2p (mod 3),

• τ(p5) ≡ 0 (mod 3),

• τ(p6) ≡ 1 (mod 3).

Moreover, τ(pm) ≡ τ(pm mod 6) (mod 3).

Lemma 5.3. A Ramanujan pseudo-prime n has an odd number of factors. That is, t is odd.

Proof. Since n is a Ramanujan pseudo-prime, it satisfies F6:

τ(n6) = τ(n)τ(n5)− τ(n4)n11.

Thus, modulo 3 and applying lemma 5.2 we see that

τ(n6) ≡ τ(n)τ(n5)− τ(n4)n (mod 3)

t∏i=1

τ(p6i ) ≡

t∏i=1

τ(pi)τ(p5i )−

t∏i=1

τ(p4i )pi (mod 3)

t∏i=1

1 ≡t∏i=1

0−t∏i=1

2p2i (mod 3)

1 ≡ −t∏i=1

(−1) (mod 3)

1This excellent terminology was suggested by J. Friedlander.


1 ≡ −(−1)t (mod 3).

This holds if and only if t is odd.

Lemma 5.4. Suppose n =t∏i=1

pi is a Ramanujan pseudo-prime with t odd. Then either

1. pi ≡ 1 (mod 3) for all i = 1, . . . , t, or

2. pi ≡ 2 (mod 3) for all i = 1, . . . , t.

Proof. Since n is a Ramanujan pseudo-prime, it satisfies F4:

τ(n4) = τ(n)τ(n3)− τ(n2)n11.

Thus, modulo 3 and applying lemma 5.2 we see that

τ(n4) ≡ τ(n)τ(n3)− τ(n2)n (mod 3)

t∏i=1

τ(p4i ) ≡

t∏i=1

τ(pi)τ(p3i )−

t∏i=1

τ(p2i )pi (mod 3)

t∏i=1

2pi ≡t∏i=1

(pi + 1)(2pi + 2)−t∏i=1

(pi + 2)pi (mod 3)

2tt∏i=1

pi ≡t∏i=1

(2p2i + 2pi + 2pi + 2)−

t∏i=1

(p2i + 2pi) (mod 3)

2

t∏i=1

pi ≡t∏i=1

(1 + pi)−t∏i=1

(1 + 2pi) (mod 3).

Now suppose pi = 1 (mod 3) and pj = 2 (mod 3) for some i 6= j. Then the right hand side is

t∏i=1

(1 + pi)−t∏i=1

(1 + 2pi) ≡ 0 (mod 3)

but the left hand side is

2

t∏i=1

pi 6≡ 0 (mod 3).

This is a contradiction.


pi is a Ramanujan pseudo-prime with t odd. Then either

1. pi ≡ 1 (mod 6) for all i = 1, . . . , t, or

2. pi ≡ −1 (mod 6) for all i = 1, . . . , t.

Proof. By the lemma above, pi ≡ 1 (mod 3) ∀i = 1, . . . , t or pi ≡ 2 (mod 3)∀i = 1, . . . , t. Recall that

we have assumed n is odd. Therefore, pi ≡ 1 (mod 6)∀i = 1, . . . , t or pi ≡ −1 (mod 6)∀i = 1, . . . , t.

Therefore, at this point, if n is a Ramanujan pseudo-prime, then n has an odd number of prime

factors, and its prime factors are either all congruent to 1 modulo 6, or are all congruent −1 modulo 6.


Recall that E is the elliptic curve with Cremona label 54-B3(B) found in [Cre97], and fE is the

modular form of weight 2 corresponding to E. Then, since 54-B3(B) has conductor NE = 54 has a

torsion of order 9,

afE (p) ≡ 1 + p (mod 9) (5.3)

for any prime p ≥ 5.


pi is a Ramanujan pseudo-prime with t odd and pi ≡ 1 (mod 3) for all

i = 1, . . . , t. Then

afE (n2) ≡ 0 (mod 9)⇐⇒ t ≥ 3.

That is, afE (n2) ≡ 0 (mod 9) if and only if n is composite.

Proof. Let p be a prime, p ≥ 5. If p ≡ 1 (mod 3), then p ≡ 1, 4, 7 (mod 9). The values of afE (p)

(mod 9) and afE (p2) (mod 9) using eq. (5.3) are summarized in the following table:

p (mod 9) afE (p) (mod 9) afE (p2) (mod 9)

1 2 3

4 5 3

7 8 3

Therefore,

afE (n2) =

t∏i=1

afE (p2i ) ≡ 3t (mod 9)

and afE (n2) ≡ 0 (mod 9) if and only if t ≥ 3.


pi is a Ramanujan pseudo-prime with t odd and pi ≡ 2 (mod 3) for all

i = 1, . . . , t. If afE (n) 6≡ 0 (mod 9) then n is prime. If afE (n) ≡ 0 (mod 9) then either

• t ≥ 3, i.e. n is composite, or

• t = 1 and n ≡ −1 (mod 9), i.e. n is a prime congruent to −1 modulo 9.

Proof. Let p be a prime not divisible by 2 or 3. If p ≡ 2 (mod 3), then p ≡ 2, 5, 8 (mod 9). The values

of aE(p) (mod 9) using eq. (5.3) are summarized in the following table:

p (mod 9) aE(p) (mod 9)

2 3

5 6

8 0

We see that with the exception of p ≡ 9 ≡ −1 (mod 9), each aE(p) contains a factor of 3 but is not

divisible by 9. Therefore, afE (n) ≡ 0 (mod 9) when t ≥ 3, or n ≡ −1 (mod 9) when t = 1.

The following congruence is also well-known, and can be found in Ramanujan’s notes and [SD73].

Lemma 5.8 (Ramanujan). Suppose p is a prime. Then

τ(p) ≡ p2(1 + p7) (mod 27). (5.4)


Proof. From [SD73],

τ(n) ≡ n2σ7(n) (mod 33)

where σx(n) is the sum of positive divisors function defined by σx(n) =∑d|n

dx. Therefore, if p is a prime,

then σ7(p) = 1 + p7, and therefore

τ(p) ≡ p2(1 + p7) (mod 27).


pi is a Ramanujan pseudo-prime with t odd and pi ≡ −1 (mod 6) for all

i = 1, . . . , t. If τ(n) 6≡ 0 (mod 27) then n is prime. If τ(n) ≡ 0 (mod 27) then either

• t ≥ 3, i.e. n is composite, or

• t = 1 and n ≡ −1 (mod 54), i.e. n is a prime congruent to −1 modulo 54.

Proof. Let p be a prime, p ≥ 5. If p ≡ −1 (mod 6), then p ≡ 5, 11, 17, 23, 29, 35, 41, 47, 53 (mod 54), so

p ≡ 5, 11, 17, 23, 2, 8, 14, 20, 26 (mod 27). The values of τ(p) (mod 27) using eq. (5.4) are summarized

in the following table:

p (mod 27) τ(p) (mod 27)

5 24

11 12

17 18

23 15

2 3

8 9

14 6

20 21

26 0

We see that with the exception of p ≡ 53 (mod 54), each τ(p) contains a factor of 3 but is not

divisible by 27. Therefore, τ(n) ≡ 0 (mod 27) when t ≥ 3, or n ≡ −1 (mod 54) when t = 1.

Recall the description of the algorithm in theorem 5.2 from section 5.2.1. We now see that every step

of the algorithm imposes a condition that must be satisfied by n.

Write n =t∏i=1

pi . As remarked above, we may assume n is odd and not divisible by 3, and that n is

a Ramanujan-pseudo prime.

If n satisfies F6 then by lemma 5.3 we must have that t is odd.

If n satisfies F4 then by lemma 5.5 we must have that either pi ≡ 1 (mod 6) for all 1 ≤ i ≤ t which

means n ≡ 1 (mod 6), or pi ≡ −1 (mod 6) for all 1 ≤ i ≤ t which means n ≡ −1 (mod 6).

In the case that n ≡ 1 (mod 6), computing afE (n2) will decide whether n is prime or composite by

lemma 5.6.

In the case that n ≡ −1 (mod 6), if afE (n) 6≡ 0 (mod 9) then n is prime by lemma 5.7. Otherwise,

if afE (n) ≡ 0 (mod 9) then n is either a composite or a prime congruent to −1 modulo 9 by lemma 5.7.


In the case that afE (n) ≡ 0 (mod 9), computing τ(n) (mod 27) will decide whether n is prime or

composite by lemma 5.9 since n 6≡ −1 (mod 54).

This completes the proof of theorem 5.2.

5.3 Proof of theorem 5.3

Theorem 5.3 follows from the following lemma. It is actually a stronger result as it computes the parity

of the number of factors congruent to 1 and the number of factors congruent to 2 modulo 3.

Lemma 5.10. Let ∆ be as above. Suppose we are given an oracle for ∆. Let n be a positive squarefree

integer not divisible by 3. Write

n =

r∏i=1

pi

s∏j=1

qj

where pi ≡ 1 (mod 3) for all i = 1 . . . , r and qj ≡ 2 (mod 3) for all j = 1 . . . , s. Then there is a

deterministic algorithm that determines the values of r mod 2 and s mod 2 in a bounded (independently

of n) number of steps.

Proof. The value of n modulo 3 is

n ≡r∏i=1

(pi)

s∏j=1

(qj) ≡ (−1)s (mod 3).

This gives the value of s mod 2.

By lemma 5.2,

τ(n4) ≡r∏i=1

(2pi)

s∏j=1

(2qj) ≡ (−1)r (mod 3).

This gives the value of r mod 2.

The algorithm is clearly deterministic, and the number of steps are independent of n.




Proof. If n is not divisible by 3, then just apply the lemma above. Otherwise, apply the lemma above

to n/3 and then add 1 to the result.

By theorem 5.3 we may compute the µ(n) inO(1) steps if n is squarefree. Therefore, if we combine this

with a squarefree-ness test, we may compute the µ(n) for any integer n. So, if we assume conjecture 4.3

or conjecture 4.4, then we can evaluate the Mobius function efficiently.

Conjecture 5.1. Let ∆ be as above. Suppose we are given an oracle for ∆. Let n ∈ Z.


O(log log log n) steps on average and O(log log n) steps in the worst case.


O(1) steps.

Appendix A

Example 3.16

The system

τ(8575) = 857511/2G∆,8575,t,(α),1(x1, x2, y1, y2)

τ(85752) = 857511G∆,8575,t,(α),2(x1, x2, y1, y2)

τ(85753) = 857533/2G∆,8575,t,(α),3(x1, x2, y1, y2)

τ(85754) = 857522G∆,8575,t,(α),4(x1, x2, y1, y2)

where

τ(8575) = −1568772102658481630000

τ(85752) = 933346892568425256627736522049338869983125

τ(85753) = −63747324087651164213662446271685531594267830383031538283845375000

τ(85754) = 286633071832801878332062176280546027855431016943360754958949164999076760969164578515625

is given by

τ(8575) = − x32y

32

18432125376957522688238845841884613037109375

+x1x

32y1y

32

29909840962041504458595472064827990406379918567836284637451171875√

7

− x32y1y

32

18432125376957522688238845841884613037109375+ 2x2y2

− 2x1x2y1y2

1622701687968798828125√

7+ 2x2y1y2

τ(85752) =x6

2y62

339743245911881497856882784080799065219623283908451194571398445987142622470855712890625

+x2

1x62y

21y

62

6262190104622311082689274825273048379957141284533494695963616315757218564804303857564984035766197933980947709642350673675537109375

− 2x1x62y

21y

62

551301938617208818419265070426787127622871268001090190192645187337041889907851555108209140598773956298828125

√7

107

Appendix A. Example 3.16 108

+x6

2y21y

62

339743245911881497856882784080799065219623283908451194571398445987142622470855712890625

− 3x1x62y1y

62

551301938617208818419265070426787127622871268001090190192645187337041889907851555108209140598773956298828125

√7

+3x6

2y1y62

339743245911881497856882784080799065219623283908451194571398445987142622470855712890625

− x42y

42

3686425075391504537647769168376922607421875

− x21x

42y

21y

42

67948649182376299571376556816159813043924656781690238914279689197428524494171142578125

+2x1x

42y

21y

42

5981968192408300891719094412965598081275983713567256927490234375

√7

− x42y

21y

42

3686425075391504537647769168376922607421875

+3x1x

42y1y

42

5981968192408300891719094412965598081275983713567256927490234375

√7

− 3x42y1y

42

3686425075391504537647769168376922607421875+ 6x2

2y22

+6x2

1x22y

21y

22

18432125376957522688238845841884613037109375

− 12x1x22y

21y

22

1622701687968798828125√

7+ 6x2

2y21y

22

− 18x1x22y1y

22

1622701687968798828125√

7+ 18x2

2y1y22 − x2

1y21

+ 3245403375937597656250√

7x1y21

+ 4868105063906396484375√

7x1y1

− 18432125376957522688238845841884613037109375y21

− 55296376130872568064716537525653839111328125y1

− 18432125376957522688238845841884613037109375

τ(85753) = − x92y

92

6262190104622311082689274825273048379957141284533494695963616315757218564804303857564984035766197933980947709642350673675537109375

+x3

1x92y

31y

92

187301110103323374370040512875166483653899436048528551531452552658991436961486734613230440661344005027975231648045816944378130904137506426035657588669469837683312363196819205768406391143798828125

√7

− 3x21x

92y

31y

92

115425473142741184106243671430202154757864651056629723255120787803425360789805755726167122711232682749672852115524333640340714918757526741277530391016625799238681793212890625

+3x1x

92y

31y

92

10161666453152133126572748878415775266019663553758757065326819134960417901878386012123211254131251174931424119222966861997292653541080653667449951171875

√7

− x92y

31y

92

6262190104622311082689274825273048379957141284533494695963616315757218564804303857564984035766197933980947709642350673675537109375


− x21x

92y

21y

92

23085094628548236821248734286040430951572930211325944651024157560685072157961151145233424542246536549934570423104866728068142983751505348255506078203325159847736358642578125

+2x1x

92y

21y

92

2032333290630426625314549775683155053203932710751751413065363826992083580375677202424642250826250234986284823844593372399458530708216130733489990234375

√7

− x92y

21y

92

1252438020924462216537854965054609675991428256906698939192723263151443712960860771512996807153239586796189541928470134735107421875

+6x1x

92y1y

92

10161666453152133126572748878415775266019663553758757065326819134960417901878386012123211254131251174931424119222966861997292653541080653667449951171875

√7

− 6x92y1y

92

6262190104622311082689274825273048379957141284533494695963616315757218564804303857564984035766197933980947709642350673675537109375

+8x7

2y72

339743245911881497856882784080799065219623283908451194571398445987142622470855712890625

− 8x31x

72y

31y

72

10161666453152133126572748878415775266019663553758757065326819134960417901878386012123211254131251174931424119222966861997292653541080653667449951171875

√7

+24x2

1x72y

31y

72

6262190104622311082689274825273048379957141284533494695963616315757218564804303857564984035766197933980947709642350673675537109375

− 24x1x72y

31y

72

551301938617208818419265070426787127622871268001090190192645187337041889907851555108209140598773956298828125

√7

+8x7

2y31y

72

339743245911881497856882784080799065219623283908451194571398445987142622470855712890625

+8x2

1x72y

21y

72

1252438020924462216537854965054609675991428256906698939192723263151443712960860771512996807153239586796189541928470134735107421875

− 16x1x72y

21y

72

110260387723441763683853014085357425524574253600218038038529037467408377981570311021641828119754791259765625

√7

+8x7

2y21y

72

67948649182376299571376556816159813043924656781690238914279689197428524494171142578125

− 48x1x72y1y

72

551301938617208818419265070426787127622871268001090190192645187337041889907851555108209140598773956298828125

√7

+48x7

2y1y72

339743245911881497856882784080799065219623283908451194571398445987142622470855712890625

− 3x52y

52

2633160768136788955462692263126373291015625

+3x3

1x52y

31y

52

78757419802458402631323581489541018231838752571584312884663598191005984272550222158315591514110565185546875

√7

− 9x21x

52y

31y

52

48534749415983071122411826297257009317089040558350170653056920855306088924407958984375


+9x1x

52y

31y

52

4272834423148786351227924580689712915197131223976612091064453125

√7

− 3x52y

31y

52

2633160768136788955462692263126373291015625

− 3x21x

52y

21y

52

9706949883196614224482365259451401863417808111670034130611384171061217784881591796875

+6x1x

52y

21y

52

854566884629757270245584916137942583039426244795322418212890625√

7

− 3x52y

21y

52

526632153627357791092538452625274658203125

+18x1x

52y1y

52

4272834423148786351227924580689712915197131223976612091064453125√

7

− 18x52y1y

52

2633160768136788955462692263126373291015625

+ 20x32y

32

− 4x31x

32y

31y

32

5981968192408300891719094412965598081275983713567256927490234375√

7

+12x2

1x32y

31y

32

3686425075391504537647769168376922607421875

− 12x1x32y

31y

32

324540337593759765625√

7

+ 20x32y

31y

32

+4x2

1x32y

21y

32

737285015078300907529553833675384521484375

− 8x1x32y

21y

32

64908067518751953125√

7

+ 100x32y

21y

32

− 24x1x32y1y

32

324540337593759765625√

7

+ 120x32y1y

32

+x3

1x2y31y2

324540337593759765625√

7

− 15x21x2y

31y2

+ 24340525319531982421875√

7x1x2y31y2

− 92160626884787613441194229209423065185546875x2y31y2

− 25x21x2y

21y2

+ 81135084398439941406250√

7x1x2y21y2

− 460803134423938067205971146047115325927734375x2y21y2

− 92160626884787613441194229209423065185546875x2y2

+ 48681050639063964843750√

7x1x2y1y2

− 552963761308725680647165375256538391113281250x2y1y2


τ(85754) =x12

2 y122

115425473142741184106243671430202154757864651056629723255120787803425360789805755726167122711232682749672852115524333640340714918757526741277530391016625799238681793212890625

+x4

1x122 y

41y

122

39215024906429591423639876119160322774858847342565140269707051122739763851093961414908401981465835761774891279838750643559644118401178149266162186364357706773897376738531035296586783670240179069022688210598112359129272831703705293193706893362104892730712890625

− 4x31x

122 y

41y

122

3452357544667781813116225998002993234247685520831823092612871584926759994419185998900938331178514356340701321632140679479999288732361909313943687897260663895532956389464319001438110404346682627096769280239385579989175312221050262451171875

√7

+6x2

1x122 y

41y

122

2127536792661648759099860096809456476892930619477294453177171324801336532997206046103959414823627870756635188536673747697614446602507979390153656614839119966597509601540435543974572052405846989131532609462738037109375

− 4x1x122 y

41y

122

187301110103323374370040512875166483653899436048528551531452552658991436961486734613230440661344005027975231648045816944378130904137506426035657588669469837683312363196819205768406391143798828125

√7

+x12

2 y41y

122

115425473142741184106243671430202154757864651056629723255120787803425360789805755726167122711232682749672852115524333640340714918757526741277530391016625799238681793212890625

− x31x

122 y

31y

122

493193934952540259016603714000427604892526502975974727516124512132394284917026571271562618739787765191528760233162954211428469818908844187706241128180094842218993769923474143062587200620954661013824182891340797141310758888721466064453125

√7

+3x2

1x122 y

31y

122

303933827523092679871408585258493782413275802782470636168167332114476647571029435157708487831946838679519312648096249671087778086072568484307665230691302852371072800220062220567796007486549569875933229923248291015625

− 3x1x122 y

31y

122

26757301443331910624291501839309497664842776578361221647350364665570205280212390659032920094477715003996461664006545277768304414876786632290808226952781405383330337599545600824058055877685546875

√7

+x12

2 y31y

122

16489353306105883443749095918600307822552093008089960465017255400489337255686536532309588958747526107096121730789190520048673559822503820182504341573803685605525970458984375

+3x2

1x122 y

21y

122

425507358532329751819972019361891295378586123895458890635434264960267306599441209220791882964725574151327037707334749539522889320501595878030731322967823993319501920308087108794914410481169397826306521892547607421875

− 6x1x122 y

21y

122

37460222020664674874008102575033296730779887209705710306290510531798287392297346922646088132268801005595046329609163388875626180827501285207131517733893967536662472639363841153681278228759765625

√7

+3x12

2 y21y

122

23085094628548236821248734286040430951572930211325944651024157560685072157961151145233424542246536549934570423104866728068142983751505348255506078203325159847736358642578125

− 2x1x122 y1y

122

37460222020664674874008102575033296730779887209705710306290510531798287392297346922646088132268801005595046329609163388875626180827501285207131517733893967536662472639363841153681278228759765625

√7

+2x12

2 y1y122

23085094628548236821248734286040430951572930211325944651024157560685072157961151145233424542246536549934570423104866728068142983751505348255506078203325159847736358642578125

− 11x102 y

102

6262190104622311082689274825273048379957141284533494695963616315757218564804303857564984035766197933980947709642350673675537109375

− 11x41x

102 y

41y

102

2127536792661648759099860096809456476892930619477294453177171324801336532997206046103959414823627870756635188536673747697614446602507979390153656614839119966597509601540435543974572052405846989131532609462738037109375

+44x3

1x102 y

41y

102

187301110103323374370040512875166483653899436048528551531452552658991436961486734613230440661344005027975231648045816944378130904137506426035657588669469837683312363196819205768406391143798828125

√7


− 66x21x

102 y

41y

102

115425473142741184106243671430202154757864651056629723255120787803425360789805755726167122711232682749672852115524333640340714918757526741277530391016625799238681793212890625

+44x1x

102 y

41y

102

10161666453152133126572748878415775266019663553758757065326819134960417901878386012123211254131251174931424119222966861997292653541080653667449951171875

√7

− 11x102 y

41y

102

6262190104622311082689274825273048379957141284533494695963616315757218564804303857564984035766197933980947709642350673675537109375

+11x3

1x102 y

31y

102

26757301443331910624291501839309497664842776578361221647350364665570205280212390659032920094477715003996461664006545277768304414876786632290808226952781405383330337599545600824058055877685546875

√7

− 33x21x

102 y

31y

102

16489353306105883443749095918600307822552093008089960465017255400489337255686536532309588958747526107096121730789190520048673559822503820182504341573803685605525970458984375

+33x1x

102 y

31y

102

1451666636164590446653249839773682180859951936251251009332402733565773985982626573160458750590178739275917731317566694571041807648725807666778564453125

√7

− 11x102 y

31y

102

894598586374615868955610689324721197136734469219070670851945187965316937829186265366426290823742561997278244234621524810791015625

− 33x21x

102 y

21y

102

23085094628548236821248734286040430951572930211325944651024157560685072157961151145233424542246536549934570423104866728068142983751505348255506078203325159847736358642578125

+66x1x

102 y

21y

102

2032333290630426625314549775683155053203932710751751413065363826992083580375677202424642250826250234986284823844593372399458530708216130733489990234375

√7

− 33x102 y

21y

102

1252438020924462216537854965054609675991428256906698939192723263151443712960860771512996807153239586796189541928470134735107421875

+22x1x

102 y1y

102

2032333290630426625314549775683155053203932710751751413065363826992083580375677202424642250826250234986284823844593372399458530708216130733489990234375

√7

− 22x102 y1y

102

1252438020924462216537854965054609675991428256906698939192723263151443712960860771512996807153239586796189541928470134735107421875

+9x8

2y82

67948649182376299571376556816159813043924656781690238914279689197428524494171142578125

+9x4

1x82y

41y

82

23085094628548236821248734286040430951572930211325944651024157560685072157961151145233424542246536549934570423104866728068142983751505348255506078203325159847736358642578125

− 36x31x

82y

41y

82

2032333290630426625314549775683155053203932710751751413065363826992083580375677202424642250826250234986284823844593372399458530708216130733489990234375

√7

+54x2

1x82y

41y

82

1252438020924462216537854965054609675991428256906698939192723263151443712960860771512996807153239586796189541928470134735107421875

− 36x1x82y

41y

82

110260387723441763683853014085357425524574253600218038038529037467408377981570311021641828119754791259765625

√7

+9x8

2y41y

82

67948649182376299571376556816159813043924656781690238914279689197428524494171142578125


− 9x31x

82y

31y

82

290333327232918089330649967954736436171990387250250201866480546713154797196525314632091750118035747855183546263513338914208361529745161533355712890625

√7

+27x2

1x82y

31y

82

178919717274923173791122137864944239427346893843814134170389037593063387565837253073285258164748512399455648846924304962158203125

− 27x1x82y

31y

82

15751483960491680526264716297908203646367750514316862576932719638201196854510044431663118302822113037109375

√7

+9x8

2y31y

82

9706949883196614224482365259451401863417808111670034130611384171061217784881591796875

+27x2

1x82y

21y

82

250487604184892443307570993010921935198285651381339787838544652630288742592172154302599361430647917359237908385694026947021484375

− 54x1x82y

21y

82

22052077544688352736770602817071485104914850720043607607705807493481675596314062204328365623950958251953125

√7

+27x8

2y21y

82

13589729836475259914275311363231962608784931356338047782855937839485704898834228515625

− 18x1x82y1y

82

22052077544688352736770602817071485104914850720043607607705807493481675596314062204328365623950958251953125

√7

+18x8

2y1y82

13589729836475259914275311363231962608784931356338047782855937839485704898834228515625

− 12x62y

62

2633160768136788955462692263126373291015625

− 12x41x

62y

41y

62

894598586374615868955610689324721197136734469219070670851945187965316937829186265366426290823742561997278244234621524810791015625

+48x3

1x62y

41y

62

78757419802458402631323581489541018231838752571584312884663598191005984272550222158315591514110565185546875

√7

− 72x21x

62y

41y

62

48534749415983071122411826297257009317089040558350170653056920855306088924407958984375

+48x1x

62y

41y

62

4272834423148786351227924580689712915197131223976612091064453125√

7

− 12x62y

41y

62

2633160768136788955462692263126373291015625

+12x3

1x62y

31y

62

11251059971779771804474797355648716890262678938797758983523371170143712038935746022616513073444366455078125

√7

− 36x21x

62y

31y

62

6933535630854724446058832328179572759584148651192881521865274407900869846343994140625

+36x1x

62y

31y

62

610404917592683764461132082955673273599590174853801727294921875√

7

− 12x62y

31y

62

376165824019541279351813180446624755859375


− 36x21x

62y

21y

62

9706949883196614224482365259451401863417808111670034130611384171061217784881591796875

+72x1x

62y

21y

62

854566884629757270245584916137942583039426244795322418212890625√

7

− 36x62y

21y

62

526632153627357791092538452625274658203125

+24x1x

62y1y

62

854566884629757270245584916137942583039426244795322418212890625√

7

− 24x62y1y

62

526632153627357791092538452625274658203125

+ 70x42y

42

+2x4

1x42y

41y

42

9706949883196614224482365259451401863417808111670034130611384171061217784881591796875

− 8x31x

42y

41y

42

854566884629757270245584916137942583039426244795322418212890625√

7

+12x2

1x42y

41y

42

526632153627357791092538452625274658203125

− 8x1x42y

41y

42

46362905370537109375√

7

+ 70x42y

41y

42

− 2x31x

42y

31y

42

122080983518536752892226416591134654719918034970760345458984375√

7

+6x2

1x42y

31y

42

75233164803908255870362636089324951171875

− 6x1x42y

31y

42

6623272195791015625√

7

+ 490x42y

31y

42

+6x2

1x42y

21y

42

105326430725471558218507690525054931640625

− 12x1x42y

21y

42

9272581074107421875√

7

+ 1050x42y

21y

42

− 4x1x42y1y

42

9272581074107421875√

7

+ 700x42y1y

42

− 3x41x

22y

41y

22

2633160768136788955462692263126373291015625

+12x3

1x22y

41y

22

231814526852685546875√

7

− 126x21x

22y

41y

22

+ 136306941789379101562500√

7x1x22y

41y

22

− 387074632916107976453015762679576873779296875x22y

41y

22

+3x3

1x22y

31y

22

33116360978955078125√

7


− 441x21x

22y

31y

22

+ 715611444394240283203125√

7x1x22y

31y

22

− 2709522430412755835171110338757038116455078125x22y

31y

22

− 387074632916107976453015762679576873779296875x22y

22

− 315x21x

22y

21y

22

+ 1022302063420343261718750√

7x1x22y

21y

22

− 5806119493741619646795236440193653106689453125x22y

21y

22

+ 340767354473447753906250√

7x1x22y1y

22

− 3870746329161079764530157626795768737792968750x22y1y

22

+ x41y

41

− 6490806751875195312500√

7x31y

41

+ 110592752261745136129433075051307678222656250x21y

41

− 119639363848166017834381888259311961625519674271345138549804687500√

7x1y41

+ 339743245911881497856882784080799065219623283908451194571398445987142622470855712890625y41

− 11358911815781591796875√

7x31y

31

+ 387074632916107976453015762679576873779296875x21y

31

− 628106660202871593630504913361387798533978289924561977386474609375√

7x1y31

+ 2378202721383170484998179488565593456537362987359158361999789121909998357295989990234375y31

+ 276481880654362840323582687628269195556640625x21y

21

− 897295228861245133757864161944839712191397557035088539123535156250√

7x1y21

+ 5096148688678222467853241761211985978294349258626767918570976689807139337062835693359375y21

− 299098409620415044585954720648279904063799185678362846374511718750√

7x1y1

+ 3397432459118814978568827840807990652196232839084511945713984459871426224708557128906250y1

+ 339743245911881497856882784080799065219623283908451194571398445987142622470855712890625

Appendix B

Example 3.17

The polynomials(Gf,n,4,(α),m(x1, . . . , x4, y1, . . . , y4)

)8m=1

where (α) = (3, 1, 1, 1) are given below.

Gf,n,4,(α),1(x1, . . . , y4) =x3

1x2x3x4y31y2y3y4

n33− 2x1x2x3x4y1y2y3y4

n22

Gf,n,4,(α),2(x1, . . . , y4) = −x61y

61

n33+x6

1x22y

22y

61

n44+x6

1x23y

23y

61

n44

− x61x

22x

23y

22y

23y

61

n55+x6

1x24y

24y

61

n44− x6

1x22x

24y

22y

24y

61

n55

− x61x

23x

24y

23y

24y

61

n55+x6

1x22x

23x

24y

22y

23y

24y

61

n66+

5x41y

41

n22

− 5x41x

22y

22y

41

n33− 5x4

1x23y

23y

41

n33+

5x41x

22x

23y

22y

23y

41

n44

− 5x41x

24y

24y

41

n33+

5x41x

22x

24y

22y

24y

41

n44+

5x41x

23x

24y

23y

24y

41

n44

− 5x41x

22x

23x

24y

22y

23y

24y

41

n55− 6x2

1y21

n11+

6x21x

22y

22y

21

n22

+6x2

1x23y

23y

21

n22− 6x2

1x22x

23y

22y

23y

21

n33+

6x21x

24y

24y

21

n22

− 6x21x

22x

24y

22y

24y

21

n33− 6x2

1x23x

24y

23y

24y

21

n33+

6x21x

22x

23x

24y

22y

23y

24y

21

n44

− x22y

22

n11− x2

3y23

n11+x2

2x23y

22y

23

n22

− x24y

24

n11+x2

2x24y

22y

24

n22+x2

3x24y

23y

24

n22

− x22x

23x

24y

22y

23y

24

n33+ 1

Gf,n,4,(α),3(x1, . . . , y4) =x9

1x32x

33x

34y

32y

33y

34y

91

n99− 2x9

1x2x33x

34y2y

33y

34y

91

n88− 2x9

1x32x3x

34y

32y3y

34y

91

n88

+4x9

1x2x3x34y2y3y

34y

91

n77− 2x9

1x32x

33x4y

32y

33y4y

91

n88+

4x91x2x

33x4y2y

33y4y

91

n77

+4x9

1x32x3x4y

32y3y4y

91

n77− 8x9

1x2x3x4y2y3y4y91

n66− 8x7

1x32x

33x

34y

32y

33y

34y

71

n88

+16x7

1x2x33x

34y2y

33y

34y

71

n77+

16x71x

32x3x

34y

32y3y

34y

71

n77− 32x7

1x2x3x34y2y3y

34y

71

n66

116

Appendix B. Example 3.17 117

+16x7

1x32x

33x4y

32y

33y4y

71

n77− 32x7

1x2x33x4y2y

33y4y

71

n66− 32x7

1x32x3x4y

32y3y4y

71

n66

+64x7

1x2x3x4y2y3y4y71

n55+

21x51x

32x

33x

34y

32y

33y

34y

51

n77− 42x5

1x2x33x

34y2y

33y

34y

51

n66

− 42x51x

32x3x

34y

32y3y

34y

51

n66+

84x51x2x3x

34y2y3y

34y

51

n55− 42x5

1x32x

33x4y

32y

33y4y

51

n66

+84x5

1x2x33x4y2y

33y4y

51

n55+

84x51x

32x3x4y

32y3y4y

51

n55− 168x5

1x2x3x4y2y3y4y51

n44

− 20x31x

32x

33x

34y

32y

33y

34y

31

n66+

40x31x2x

33x

34y2y

33y

34y

31

n55+

40x31x

32x3x

34y

32y3y

34y

31

n55

− 80x31x2x3x

34y2y3y

34y

31

n44+

40x31x

32x

33x4y

32y

33y4y

31

n55− 80x3

1x2x33x4y2y

33y4y

31

n44

− 80x31x

32x3x4y

32y3y4y

31

n44+

160x31x2x3x4y2y3y4y

31

n33+

5x1x32x

33x

34y

32y

33y

34y1

n55

− 10x1x2x33x

34y2y

33y

34y1

n44− 10x1x

32x3x

34y

32y3y

34y1

n44+

20x1x2x3x34y2y3y

34y1

n33

− 10x1x32x

33x4y

32y

33y4y1

n44+

20x1x2x33x4y2y

33y4y1

n33+

20x1x32x3x4y

32y3y4y1

n33

− 40x1x2x3x4y2y3y4y1

n22

Gf,n,4,(α),4(x1, . . . , y4) =x12

1 y121

n66+x12

1 x42y

42y

121

n88+x12

1 x43y

43y

121

n88

+x12

1 x42x

43y

42y

43y

121

n110− 3x12

1 x22x

43y

22y

43y

121

n99+x12

1 x44y

44y

121

n88

+x12

1 x42x

44y

42y

44y

121

n110+x12

1 x43x

44y

43y

44y

121

n110+x12

1 x42x

43x

44y

42y

43y

44y

121

n132

− 3x121 x

22x

43x

44y

22y

43y

44y

121

n121− 3x12

1 x22x

44y

22y

44y

121

n99− 3x12

1 x23x

44y

23y

44y

121

n99

− 3x121 x

42x

23x

44y

42y

23y

44y

121

n121+

9x121 x

22x

23x

44y

22y

23y

44y

121

n110− 3x12

1 x22y

22y

121

n77

− 3x121 x

42x

23y

42y

23y

121

n99− 3x12

1 x23y

23y

121

n77+

9x121 x

22x

23y

22y

23y

121

n88

− 3x121 x

42x

24y

42y

24y

121

n99− 3x12

1 x42x

43x

24y

42y

43y

24y

121

n121− 3x12

1 x43x

24y

43y

24y

121

n99

+9x12

1 x22x

43x

24y

22y

43y

24y

121

n110− 3x12

1 x24y

24y

121

n77+

9x121 x

22x

24y

22y

24y

121

n88

+9x12

1 x42x

23x

24y

42y

23y

24y

121

n110+

9x121 x

23x

24y

23y

24y

121

n88− 27x12

1 x22x

23x

24y

22y

23y

24y

121

n99

− 11x101 y

101

n55− 11x10

1 x42y

42y

101

n77− 11x10

1 x43y

43y

101

n77

− 11x101 x

42x

43y

42y

43y

101

n99+

33x101 x

22x

43y

22y

43y

101

n88− 11x10

1 x44y

44y

101

n77

− 11x101 x

42x

44y

42y

44y

101

n99− 11x10

1 x43x

44y

43y

44y

101

n99− 11x10

1 x42x

43x

44y

42y

43y

44y

101

n121

+33x10

1 x22x

43x

44y

22y

43y

44y

101

n110+

33x101 x

22x

44y

22y

44y

101

n88+

33x101 x

23x

44y

23y

44y

101

n88

+33x10

1 x42x

23x

44y

42y

23y

44y

101

n110− 99x10

1 x22x

23x

44y

22y

23y

44y

101

n99+

33x101 x

22y

22y

101

n66

+33x10

1 x42x

23y

42y

23y

101

n88+

33x101 x

23y

23y

101

n66− 99x10

1 x22x

23y

22y

23y

101

n77


+33x10

1 x42x

24y

42y

24y

101

n88+

33x101 x

42x

43x

24y

42y

43y

24y

101

n110+

33x101 x

43x

24y

43y

24y

101

n88

− 99x101 x

22x

43x

24y

22y

43y

24y

101

n99+

33x101 x

24y

24y

101

n66− 99x10

1 x22x

24y

22y

24y

101

n77

− 99x101 x

42x

23x

24y

42y

23y

24y

101

n99− 99x10

1 x23x

24y

23y

24y

101

n77+

297x101 x

22x

23x

24y

22y

23y

24y

101

n88

+45x8

1y81

n44+

45x81x

42y

42y

81

n66+

45x81x

43y

43y

81

n66

+45x8

1x42x

43y

42y

43y

81

n88− 135x8

1x22x

43y

22y

43y

81

n77+

45x81x

44y

44y

81

n66

+45x8

1x42x

44y

42y

44y

81

n88+

45x81x

43x

44y

43y

44y

81

n88+

45x81x

42x

43x

44y

42y

43y

44y

81

n110

− 135x81x

22x

43x

44y

22y

43y

44y

81

n99− 135x8

1x22x

44y

22y

44y

81

n77− 135x8

1x23x

44y

23y

44y

81

n77

− 135x81x

42x

23x

44y

42y

23y

44y

81

n99+

405x81x

22x

23x

44y

22y

23y

44y

81

n88− 135x8

1x22y

22y

81

n55

− 135x81x

42x

23y

42y

23y

81

n77− 135x8

1x23y

23y

81

n55+

405x81x

22x

23y

22y

23y

81

n66

− 135x81x

42x

24y

42y

24y

81

n77− 135x8

1x42x

43x

24y

42y

43y

24y

81

n99− 135x8

1x43x

24y

43y

24y

81

n77

+405x8

1x22x

43x

24y

22y

43y

24y

81

n88− 135x8

1x24y

24y

81

n55+

405x81x

22x

24y

22y

24y

81

n66

+405x8

1x42x

23x

24y

42y

23y

24y

81

n88+

405x81x

23x

24y

23y

24y

81

n66− 1215x8

1x22x

23x

24y

22y

23y

24y

81

n77

− 84x61y

61

n33− 84x6

1x42y

42y

61

n55− 84x6

1x43y

43y

61

n55

− 84x61x

42x

43y

42y

43y

61

n77+

252x61x

22x

43y

22y

43y

61

n66− 84x6

1x44y

44y

61

n55

− 84x61x

42x

44y

42y

44y

61

n77− 84x6

1x43x

44y

43y

44y

61

n77− 84x6

1x42x

43x

44y

42y

43y

44y

61

n99

+252x6

1x22x

43x

44y

22y

43y

44y

61

n88+

252x61x

22x

44y

22y

44y

61

n66+

252x61x

23x

44y

23y

44y

61

n66

+252x6

1x42x

23x

44y

42y

23y

44y

61

n88− 756x6

1x22x

23x

44y

22y

23y

44y

61

n77+

252x61x

22y

22y

61

n44

+252x6

1x42x

23y

42y

23y

61

n66+

252x61x

23y

23y

61

n44− 756x6

1x22x

23y

22y

23y

61

n55

+252x6

1x42x

24y

42y

24y

61

n66+

252x61x

42x

43x

24y

42y

43y

24y

61

n88+

252x61x

43x

24y

43y

24y

61

n66

− 756x61x

22x

43x

24y

22y

43y

24y

61

n77+

252x61x

24y

24y

61

n44− 756x6

1x22x

24y

22y

24y

61

n55

− 756x61x

42x

23x

24y

42y

23y

24y

61

n77− 756x6

1x23x

24y

23y

24y

61

n55+

2268x61x

22x

23x

24y

22y

23y

24y

61

n66

+70x4

1y41

n22+

70x41x

42y

42y

41

n44+

70x41x

43y

43y

41

n44

+70x4

1x42x

43y

42y

43y

41

n66− 210x4

1x22x

43y

22y

43y

41

n55+

70x41x

44y

44y

41

n44

+70x4

1x42x

44y

42y

44y

41

n66+

70x41x

43x

44y

43y

44y

41

n66+

70x41x

42x

43x

44y

42y

43y

44y

41

n88

− 210x41x

22x

43x

44y

22y

43y

44y

41

n77− 210x4

1x22x

44y

22y

44y

41

n55− 210x4

1x23x

44y

23y

44y

41

n55


− 210x41x

42x

23x

44y

42y

23y

44y

41

n77+

630x41x

22x

23x

44y

22y

23y

44y

41

n66− 210x4

1x22y

22y

41

n33

− 210x41x

42x

23y

42y

23y

41

n55− 210x4

1x23y

23y

41

n33+

630x41x

22x

23y

22y

23y

41

n44

− 210x41x

42x

24y

42y

24y

41

n55− 210x4

1x42x

43x

24y

42y

43y

24y

41

n77− 210x4

1x43x

24y

43y

24y

41

n55

+630x4

1x22x

43x

24y

22y

43y

24y

41

n66− 210x4

1x24y

24y

41

n33+

630x41x

22x

24y

22y

24y

41

n44

+630x4

1x42x

23x

24y

42y

23y

24y

41

n66+

630x41x

23x

24y

23y

24y

41

n44− 1890x4

1x22x

23x

24y

22y

23y

24y

41

n55

− 21x21x

42y

42y

21

n33− 21x2

1x43y

43y

21

n33− 21x2

1x42x

43y

42y

43y

21

n55

+63x2

1x22x

43y

22y

43y

21

n44− 21x2

1x44y

44y

21

n33− 21x2

1x42x

44y

42y

44y

21

n55

− 21x21x

43x

44y

43y

44y

21

n55− 21x2

1x42x

43x

44y

42y

43y

44y

21

n77+

63x21x

22x

43x

44y

22y

43y

44y

21

n66

+63x2

1x22x

44y

22y

44y

21

n44+

63x21x

23x

44y

23y

44y

21

n44+

63x21x

42x

23x

44y

42y

23y

44y

21

n66

− 189x21x

22x

23x

44y

22y

23y

44y

21

n55− 21x2

1y21

n11+

63x21x

22y

22y

21

n22

+63x2

1x42x

23y

42y

23y

21

n44+

63x21x

23y

23y

21

n22− 189x2

1x22x

23y

22y

23y

21

n33

+63x2

1x42x

24y

42y

24y

21

n44+

63x21x

42x

43x

24y

42y

43y

24y

21

n66+

63x21x

43x

24y

43y

24y

21

n44

− 189x21x

22x

43x

24y

22y

43y

24y

21

n55+

63x21x

24y

24y

21

n22− 189x2

1x22x

24y

22y

24y

21

n33

− 189x21x

42x

23x

24y

42y

23y

24y

21

n55− 189x2

1x23x

24y

23y

24y

21

n33+

567x21x

22x

23x

24y

22y

23y

24y

21

n44

+x4

2y42

n22+x4

3y43

n22+x4

2x43y

42y

43

n44

− 3x22x

43y

22y

43

n33+x4

4y44

n22+x4

2x44y

42y

44

n44

+x4

3x44y

43y

44

n44+x4

2x43x

44y

42y

43y

44

n66− 3x2

2x43x

44y

22y

43y

44

n55

− 3x22x

44y

22y

44

n33− 3x2

3x44y

23y

44

n33− 3x4

2x23x

44y

42y

23y

44

n55

+9x2

2x23x

44y

22y

23y

44

n44− 3x2

2y22

n11− 3x4

2x23y

42y

23

n33

− 3x23y

23

n11+

9x22x

23y

22y

23

n22− 3x4

2x24y

42y

24

n33

− 3x42x

43x

24y

42y

43y

24

n55− 3x4

3x24y

43y

24

n33+

9x22x

43x

24y

22y

43y

24

n44

− 3x24y

24

n11+

9x22x

24y

22y

24

n22+

9x42x

23x

24y

42y

23y

24

n44

+9x2

3x24y

23y

24

n22− 27x2

2x23x

24y

22y

23y

24

n33+ 1

Gf,n,4,(α),5(x1, . . . , y4) =x15

1 x52x

53x

54y

52y

53y

54y

151

n165− 4x15

1 x32x

53x

54y

32y

53y

54y

151

n154+

3x151 x2x

53x

54y2y

53y

54y

151

n143

− 4x151 x

52x

33x

54y

52y

33y

54y

151

n154+

16x151 x

32x

33x

54y

32y

33y

54y

151

n143− 12x15

1 x2x33x

54y2y

33y

54y

151

n132


+3x15

1 x52x3x

54y

52y3y

54y

151

n143− 12x15

1 x32x3x

54y

32y3y

54y

151

n132+

9x151 x2x3x

54y2y3y

54y

151

n121

− 4x151 x

52x

53x

34y

52y

53y

34y

151

n154+

16x151 x

32x

53x

34y

32y

53y

34y

151

n143− 12x15

1 x2x53x

34y2y

53y

34y

151

n132

+16x15

1 x52x

33x

34y

52y

33y

34y

151

n143− 64x15

1 x32x

33x

34y

32y

33y

34y

151

n132+

48x151 x2x

33x

34y2y

33y

34y

151

n121

− 12x151 x

52x3x

34y

52y3y

34y

151

n132+

48x151 x

32x3x

34y

32y3y

34y

151

n121− 36x15

1 x2x3x34y2y3y

34y

151

n110

+3x15

1 x52x

53x4y

52y

53y4y

151

n143− 12x15

1 x32x

53x4y

32y

53y4y

151

n132+

9x151 x2x

53x4y2y

53y4y

151

n121

− 12x151 x

52x

33x4y

52y

33y4y

151

n132+

48x151 x

32x

33x4y

32y

33y4y

151

n121− 36x15

1 x2x33x4y2y

33y4y

151

n110

+9x15

1 x52x3x4y

52y3y4y

151

n121− 36x15

1 x32x3x4y

32y3y4y

151

n110+

27x151 x2x3x4y2y3y4y

151

n99

− 14x131 x

52x

53x

54y

52y

53y

54y

131

n154+

56x131 x

32x

53x

54y

32y

53y

54y

131

n143− 42x13

1 x2x53x

54y2y

53y

54y

131

n132

+56x13

1 x52x

33x

54y

52y

33y

54y

131

n143− 224x13

1 x32x

33x

54y

32y

33y

54y

131

n132+

168x131 x2x

33x

54y2y

33y

54y

131

n121

− 42x131 x

52x3x

54y

52y3y

54y

131

n132+

168x131 x

32x3x

54y

32y3y

54y

131

n121− 126x13

1 x2x3x54y2y3y

54y

131

n110

+56x13

1 x52x

53x

34y

52y

53y

34y

131

n143− 224x13

1 x32x

53x

34y

32y

53y

34y

131

n132+

168x131 x2x

53x

34y2y

53y

34y

131

n121

− 224x131 x

52x

33x

34y

52y

33y

34y

131

n132+

896x131 x

32x

33x

34y

32y

33y

34y

131

n121− 672x13

1 x2x33x

34y2y

33y

34y

131

n110

+168x13

1 x52x3x

34y

52y3y

34y

131

n121− 672x13

1 x32x3x

34y

32y3y

34y

131

n110+

504x131 x2x3x

34y2y3y

34y

131

n99

− 42x131 x

52x

53x4y

52y

53y4y

131

n132+

168x131 x

32x

53x4y

32y

53y4y

131

n121− 126x13

1 x2x53x4y2y

53y4y

131

n110

+168x13

1 x52x

33x4y

52y

33y4y

131

n121− 672x13

1 x32x

33x4y

32y

33y4y

131

n110+

504x131 x2x

33x4y2y

33y4y

131

n99

− 126x131 x

52x3x4y

52y3y4y

131

n110+

504x131 x

32x3x4y

32y3y4y

131

n99− 378x13

1 x2x3x4y2y3y4y131

n88

+78x11

1 x52x

53x

54y

52y

53y

54y

111

n143− 312x11

1 x32x

53x

54y

32y

53y

54y

111

n132+

234x111 x2x

53x

54y2y

53y

54y

111

n121

− 312x111 x

52x

33x

54y

52y

33y

54y

111

n132+

1248x111 x

32x

33x

54y

32y

33y

54y

111

n121− 936x11

1 x2x33x

54y2y

33y

54y

111

n110

+234x11

1 x52x3x

54y

52y3y

54y

111

n121− 936x11

1 x32x3x

54y

32y3y

54y

111

n110+

702x111 x2x3x

54y2y3y

54y

111

n99

− 312x111 x

52x

53x

34y

52y

53y

34y

111

n132+

1248x111 x

32x

53x

34y

32y

53y

34y

111

n121− 936x11

1 x2x53x

34y2y

53y

34y

111

n110

+1248x11

1 x52x

33x

34y

52y

33y

34y

111

n121− 4992x11

1 x32x

33x

34y

32y

33y

34y

111

n110+

3744x111 x2x

33x

34y2y

33y

34y

111

n99

− 936x111 x

52x3x

34y

52y3y

34y

111

n110+

3744x111 x

32x3x

34y

32y3y

34y

111

n99− 2808x11

1 x2x3x34y2y3y

34y

111

n88

+234x11

1 x52x

53x4y

52y

53y4y

111

n121− 936x11

1 x32x

53x4y

32y

53y4y

111

n110+

702x111 x2x

53x4y2y

53y4y

111

n99

− 936x111 x

52x

33x4y

52y

33y4y

111

n110+

3744x111 x

32x

33x4y

32y

33y4y

111

n99− 2808x11

1 x2x33x4y2y

33y4y

111

n88

+702x11

1 x52x3x4y

52y3y4y

111

n99− 2808x11

1 x32x3x4y

32y3y4y

111

n88+

2106x111 x2x3x4y2y3y4y

111

n77


− 220x91x

52x

53x

54y

52y

53y

54y

91

n132+

880x91x

32x

53x

54y

32y

53y

54y

91

n121− 660x9

1x2x53x

54y2y

53y

54y

91

n110

+880x9

1x52x

33x

54y

52y

33y

54y

91

n121− 3520x9

1x32x

33x

54y

32y

33y

54y

91

n110+

2640x91x2x

33x

54y2y

33y

54y

91

n99

− 660x91x

52x3x

54y

52y3y

54y

91

n110+

2640x91x

32x3x

54y

32y3y

54y

91

n99− 1980x9

1x2x3x54y2y3y

54y

91

n88

+880x9

1x52x

53x

34y

52y

53y

34y

91

n121− 3520x9

1x32x

53x

34y

32y

53y

34y

91

n110+

2640x91x2x

53x

34y2y

53y

34y

91

n99

− 3520x91x

52x

33x

34y

52y

33y

34y

91

n110+

14080x91x

32x

33x

34y

32y

33y

34y

91

n99− 10560x9

1x2x33x

34y2y

33y

34y

91

n88

+2640x9

1x52x3x

34y

52y3y

34y

91

n99− 10560x9

1x32x3x

34y

32y3y

34y

91

n88+

7920x91x2x3x

34y2y3y

34y

91

n77

− 660x91x

52x

53x4y

52y

53y4y

91

n110+

2640x91x

32x

53x4y

32y

53y4y

91

n99− 1980x9

1x2x53x4y2y

53y4y

91

n88

+2640x9

1x52x

33x4y

52y

33y4y

91

n99− 10560x9

1x32x

33x4y

32y

33y4y

91

n88+

7920x91x2x

33x4y2y

33y4y

91

n77

− 1980x91x

52x3x4y

52y3y4y

91

n88+

7920x91x

32x3x4y

32y3y4y

91

n77− 5940x9

1x2x3x4y2y3y4y91

n66

+330x7

1x52x

53x

54y

52y

53y

54y

71

n121− 1320x7

1x32x

53x

54y

32y

53y

54y

71

n110+

990x71x2x

53x

54y2y

53y

54y

71

n99

− 1320x71x

52x

33x

54y

52y

33y

54y

71

n110+

5280x71x

32x

33x

54y

32y

33y

54y

71

n99− 3960x7

1x2x33x

54y2y

33y

54y

71

n88

+990x7

1x52x3x

54y

52y3y

54y

71

n99− 3960x7

1x32x3x

54y

32y3y

54y

71

n88+

2970x71x2x3x

54y2y3y

54y

71

n77

− 1320x71x

52x

53x

34y

52y

53y

34y

71

n110+

5280x71x

32x

53x

34y

32y

53y

34y

71

n99− 3960x7

1x2x53x

34y2y

53y

34y

71

n88

+5280x7

1x52x

33x

34y

52y

33y

34y

71

n99− 21120x7

1x32x

33x

34y

32y

33y

34y

71

n88+

15840x71x2x

33x

34y2y

33y

34y

71

n77

− 3960x71x

52x3x

34y

52y3y

34y

71

n88+

15840x71x

32x3x

34y

32y3y

34y

71

n77− 11880x7

1x2x3x34y2y3y

34y

71

n66

+990x7

1x52x

53x4y

52y

53y4y

71

n99− 3960x7

1x32x

53x4y

32y

53y4y

71

n88+

2970x71x2x

53x4y2y

53y4y

71

n77

− 3960x71x

52x

33x4y

52y

33y4y

71

n88+

15840x71x

32x

33x4y

32y

33y4y

71

n77− 11880x7

1x2x33x4y2y

33y4y

71

n66

+2970x7

1x52x3x4y

52y3y4y

71

n77− 11880x7

1x32x3x4y

32y3y4y

71

n66+

8910x71x2x3x4y2y3y4y

71

n55

− 252x51x

52x

53x

54y

52y

53y

54y

51

n110+

1008x51x

32x

53x

54y

32y

53y

54y

51

n99− 756x5

1x2x53x

54y2y

53y

54y

51

n88

+1008x5

1x52x

33x

54y

52y

33y

54y

51

n99− 4032x5

1x32x

33x

54y

32y

33y

54y

51

n88+

3024x51x2x

33x

54y2y

33y

54y

51

n77

− 756x51x

52x3x

54y

52y3y

54y

51

n88+

3024x51x

32x3x

54y

32y3y

54y

51

n77− 2268x5

1x2x3x54y2y3y

54y

51

n66

+1008x5

1x52x

53x

34y

52y

53y

34y

51

n99− 4032x5

1x32x

53x

34y

32y

53y

34y

51

n88+

3024x51x2x

53x

34y2y

53y

34y

51

n77

− 4032x51x

52x

33x

34y

52y

33y

34y

51

n88+

16128x51x

32x

33x

34y

32y

33y

34y

51

n77− 12096x5

1x2x33x

34y2y

33y

34y

51

n66

+3024x5

1x52x3x

34y

52y3y

34y

51

n77− 12096x5

1x32x3x

34y

32y3y

34y

51

n66+

9072x51x2x3x

34y2y3y

34y

51

n55

− 756x51x

52x

53x4y

52y

53y4y

51

n88+

3024x51x

32x

53x4y

32y

53y4y

51

n77− 2268x5

1x2x53x4y2y

53y4y

51

n66


+3024x5

1x52x

33x4y

52y

33y4y

51

n77− 12096x5

1x32x

33x4y

32y

33y4y

51

n66+

9072x51x2x

33x4y2y

33y4y

51

n55

− 2268x51x

52x3x4y

52y3y4y

51

n66+

9072x51x

32x3x4y

32y3y4y

51

n55− 6804x5

1x2x3x4y2y3y4y51

n44

+84x3

1x52x

53x

54y

52y

53y

54y

31

n99− 336x3

1x32x

53x

54y

32y

53y

54y

31

n88+

252x31x2x

53x

54y2y

53y

54y

31

n77

− 336x31x

52x

33x

54y

52y

33y

54y

31

n88+

1344x31x

32x

33x

54y

32y

33y

54y

31

n77− 1008x3

1x2x33x

54y2y

33y

54y

31

n66

+252x3

1x52x3x

54y

52y3y

54y

31

n77− 1008x3

1x32x3x

54y

32y3y

54y

31

n66+

756x31x2x3x

54y2y3y

54y

31

n55

− 336x31x

52x

53x

34y

52y

53y

34y

31

n88+

1344x31x

32x

53x

34y

32y

53y

34y

31

n77− 1008x3

1x2x53x

34y2y

53y

34y

31

n66

+1344x3

1x52x

33x

34y

52y

33y

34y

31

n77− 5376x3

1x32x

33x

34y

32y

33y

34y

31

n66+

4032x31x2x

33x

34y2y

33y

34y

31

n55

− 1008x31x

52x3x

34y

52y3y

34y

31

n66+

4032x31x

32x3x

34y

32y3y

34y

31

n55− 3024x3

1x2x3x34y2y3y

34y

31

n44

+252x3

1x52x

53x4y

52y

53y4y

31

n77− 1008x3

1x32x

53x4y

32y

53y4y

31

n66+

756x31x2x

53x4y2y

53y4y

31

n55

− 1008x31x

52x

33x4y

52y

33y4y

31

n66+

4032x31x

32x

33x4y

32y

33y4y

31

n55− 3024x3

1x2x33x4y2y

33y4y

31

n44

+756x3

1x52x3x4y

52y3y4y

31

n55− 3024x3

1x32x3x4y

32y3y4y

31

n44+

2268x31x2x3x4y2y3y4y

31

n33

− 8x1x52x

53x

54y

52y

53y

54y1

n88+

32x1x32x

53x

54y

32y

53y

54y1

n77− 24x1x2x

53x

54y2y

53y

54y1

n66

+32x1x

52x

33x

54y

52y

33y

54y1

n77− 128x1x

32x

33x

54y

32y

33y

54y1

n66+

96x1x2x33x

54y2y

33y

54y1

n55

− 24x1x52x3x

54y

52y3y

54y1

n66+

96x1x32x3x

54y

32y3y

54y1

n55− 72x1x2x3x

54y2y3y

54y1

n44

+32x1x

52x

53x

34y

52y

53y

34y1

n77− 128x1x

32x

53x

34y

32y

53y

34y1

n66+

96x1x2x53x

34y2y

53y

34y1

n55

− 128x1x52x

33x

34y

52y

33y

34y1

n66+

512x1x32x

33x

34y

32y

33y

34y1

n55− 384x1x2x

33x

34y2y

33y

34y1

n44

+96x1x

52x3x

34y

52y3y

34y1

n55− 384x1x

32x3x

34y

32y3y

34y1

n44+

288x1x2x3x34y2y3y

34y1

n33

− 24x1x52x

53x4y

52y

53y4y1

n66+

96x1x32x

53x4y

32y

53y4y1

n55− 72x1x2x

53x4y2y

53y4y1

n44

+96x1x

52x

33x4y

52y

33y4y1

n55− 384x1x

32x

33x4y

32y

33y4y1

n44+

288x1x2x33x4y2y

33y4y1

n33

− 72x1x52x3x4y

52y3y4y1

n44+

288x1x32x3x4y

32y3y4y1

n33− 216x1x2x3x4y2y3y4y1

n22

Gf,n,4,(α),6(x1, . . . , y4) = −x181 y

181

n99+x18

1 x62y

62y

181

n132+x18

1 x63y

63y

181

n132

− x181 x

62x

63y

62y

63y

181

n165+

5x181 x

42x

63y

42y

63y

181

n154− 6x18

1 x22x

63y

22y

63y

181

n143

+x18

1 x64y

64y

181

n132− x18

1 x62x

64y

62y

64y

181

n165− x18

1 x63x

64y

63y

64y

181

n165

+x18

1 x62x

63x

64y

62y

63y

64y

181

n198− 5x18

1 x42x

63x

64y

42y

63y

64y

181

n187+

6x181 x

22x

63x

64y

22y

63y

64y

181

n176

+5x18

1 x42x

64y

42y

64y

181

n154+

5x181 x

43x

64y

43y

64y

181

n154− 5x18

1 x62x

43x

64y

62y

43y

64y

181

n187


+25x18

1 x42x

43x

64y

42y

43y

64y

181

n176− 30x18

1 x22x

43x

64y

22y

43y

64y

181

n165− 6x18

1 x22x

64y

22y

64y

181

n143

− 6x181 x

23x

64y

23y

64y

181

n143+

6x181 x

62x

23x

64y

62y

23y

64y

181

n176− 30x18

1 x42x

23x

64y

42y

23y

64y

181

n165

+36x18

1 x22x

23x

64y

22y

23y

64y

181

n154− 5x18

1 x42y

42y

181

n121+

5x181 x

62x

43y

62y

43y

181

n154

− 5x181 x

43y

43y

181

n121− 25x18

1 x42x

43y

42y

43y

181

n143+

30x181 x

22x

43y

22y

43y

181

n132

+5x18

1 x62x

44y

62y

44y

181

n154− 5x18

1 x62x

63x

44y

62y

63y

44y

181

n187+

5x181 x

63x

44y

63y

44y

181

n154

+25x18

1 x42x

63x

44y

42y

63y

44y

181

n176− 30x18

1 x22x

63x

44y

22y

63y

44y

181

n165− 5x18

1 x44y

44y

181

n121

− 25x181 x

42x

44y

42y

44y

181

n143+

25x181 x

62x

43x

44y

62y

43y

44y

181

n176− 25x18

1 x43x

44y

43y

44y

181

n143

− 125x181 x

42x

43x

44y

42y

43y

44y

181

n165+

150x181 x

22x

43x

44y

22y

43y

44y

181

n154+

30x181 x

22x

44y

22y

44y

181

n132

− 30x181 x

62x

23x

44y

62y

23y

44y

181

n165+

30x181 x

23x

44y

23y

44y

181

n132+

150x181 x

42x

23x

44y

42y

23y

44y

181

n154

− 180x181 x

22x

23x

44y

22y

23y

44y

181

n143+

6x181 x

22y

22y

181

n110− 6x18

1 x62x

23y

62y

23y

181

n143

+30x18

1 x42x

23y

42y

23y

181

n132+

6x181 x

23y

23y

181

n110− 36x18

1 x22x

23y

22y

23y

181

n121

− 6x181 x

62x

24y

62y

24y

181

n143+

6x181 x

62x

63x

24y

62y

63y

24y

181

n176− 30x18

1 x42x

63x

24y

42y

63y

24y

181

n165

− 6x181 x

63x

24y

63y

24y

181

n143+

36x181 x

22x

63x

24y

22y

63y

24y

181

n154+

30x181 x

42x

24y

42y

24y

181

n132

− 30x181 x

62x

43x

24y

62y

43y

24y

181

n165+

150x181 x

42x

43x

24y

42y

43y

24y

181

n154+

30x181 x

43x

24y

43y

24y

181

n132

− 180x181 x

22x

43x

24y

22y

43y

24y

181

n143+

6x181 x

24y

24y

181

n110− 36x18

1 x22x

24y

22y

24y

181

n121

+36x18

1 x62x

23x

24y

62y

23y

24y

181

n154− 180x18

1 x42x

23x

24y

42y

23y

24y

181

n143− 36x18

1 x23x

24y

23y

24y

181

n121

+216x18

1 x22x

23x

24y

22y

23y

24y

181

n132+

17x161 y

161

n88− 17x16

1 x62y

62y

161

n121

− 17x161 x

63y

63y

161

n121+

17x161 x

62x

63y

62y

63y

161

n154− 85x16

1 x42x

63y

42y

63y

161

n143

+102x16

1 x22x

63y

22y

63y

161

n132− 17x16

1 x64y

64y

161

n121+

17x161 x

62x

64y

62y

64y

161

n154

+17x16

1 x63x

64y

63y

64y

161

n154− 17x16

1 x62x

63x

64y

62y

63y

64y

161

n187+

85x161 x

42x

63x

64y

42y

63y

64y

161

n176

− 102x161 x

22x

63x

64y

22y

63y

64y

161

n165− 85x16

1 x42x

64y

42y

64y

161

n143− 85x16

1 x43x

64y

43y

64y

161

n143

+85x16

1 x62x

43x

64y

62y

43y

64y

161

n176− 425x16

1 x42x

43x

64y

42y

43y

64y

161

n165+

510x161 x

22x

43x

64y

22y

43y

64y

161

n154

+102x16

1 x22x

64y

22y

64y

161

n132+

102x161 x

23x

64y

23y

64y

161

n132− 102x16

1 x62x

23x

64y

62y

23y

64y

161

n165

+510x16

1 x42x

23x

64y

42y

23y

64y

161

n154− 612x16

1 x22x

23x

64y

22y

23y

64y

161

n143+

85x161 x

42y

42y

161

n110

− 85x161 x

62x

43y

62y

43y

161

n143+

85x161 x

43y

43y

161

n110+

425x161 x

42x

43y

42y

43y

161

n132


− 510x161 x

22x

43y

22y

43y

161

n121− 85x16

1 x62x

44y

62y

44y

161

n143+

85x161 x

62x

63x

44y

62y

63y

44y

161

n176

− 85x161 x

63x

44y

63y

44y

161

n143− 425x16

1 x42x

63x

44y

42y

63y

44y

161

n165+

510x161 x

22x

63x

44y

22y

63y

44y

161

n154

+85x16

1 x44y

44y

161

n110+

425x161 x

42x

44y

42y

44y

161

n132− 425x16

1 x62x

43x

44y

62y

43y

44y

161

n165

+425x16

1 x43x

44y

43y

44y

161

n132+

2125x161 x

42x

43x

44y

42y

43y

44y

161

n154− 2550x16

1 x22x

43x

44y

22y

43y

44y

161

n143

− 510x161 x

22x

44y

22y

44y

161

n121+

510x161 x

62x

23x

44y

62y

23y

44y

161

n154− 510x16

1 x23x

44y

23y

44y

161

n121

− 2550x161 x

42x

23x

44y

42y

23y

44y

161

n143+

3060x161 x

22x

23x

44y

22y

23y

44y

161

n132− 102x16

1 x22y

22y

161

n99

+102x16

1 x62x

23y

62y

23y

161

n132− 510x16

1 x42x

23y

42y

23y

161

n121− 102x16

1 x23y

23y

161

n99

+612x16

1 x22x

23y

22y

23y

161

n110+

102x161 x

62x

24y

62y

24y

161

n132− 102x16

1 x62x

63x

24y

62y

63y

24y

161

n165

+510x16

1 x42x

63x

24y

42y

63y

24y

161

n154+

102x161 x

63x

24y

63y

24y

161

n132− 612x16

1 x22x

63x

24y

22y

63y

24y

161

n143

− 510x161 x

42x

24y

42y

24y

161

n121+

510x161 x

62x

43x

24y

62y

43y

24y

161

n154− 2550x16

1 x42x

43x

24y

42y

43y

24y

161

n143

− 510x161 x

43x

24y

43y

24y

161

n121+

3060x161 x

22x

43x

24y

22y

43y

24y

161

n132− 102x16

1 x24y

24y

161

n99

+612x16

1 x22x

24y

22y

24y

161

n110− 612x16

1 x62x

23x

24y

62y

23y

24y

161

n143+

3060x161 x

42x

23x

24y

42y

23y

24y

161

n132

+612x16

1 x23x

24y

23y

24y

161

n110− 3672x16

1 x22x

23x

24y

22y

23y

24y

161

n121− 120x14

1 y141

n77

+120x14

1 x62y

62y

141

n110+

120x141 x

63y

63y

141

n110− 120x14

1 x62x

63y

62y

63y

141

n143

+600x14

1 x42x

63y

42y

63y

141

n132− 720x14

1 x22x

63y

22y

63y

141

n121+

120x141 x

64y

64y

141

n110

− 120x141 x

62x

64y

62y

64y

141

n143− 120x14

1 x63x

64y

63y

64y

141

n143+

120x141 x

62x

63x

64y

62y

63y

64y

141

n176

− 600x141 x

42x

63x

64y

42y

63y

64y

141

n165+

720x141 x

22x

63x

64y

22y

63y

64y

141

n154+

600x141 x

42x

64y

42y

64y

141

n132

+600x14

1 x43x

64y

43y

64y

141

n132− 600x14

1 x62x

43x

64y

62y

43y

64y

141

n165+

3000x141 x

42x

43x

64y

42y

43y

64y

141

n154

− 3600x141 x

22x

43x

64y

22y

43y

64y

141

n143− 720x14

1 x22x

64y

22y

64y

141

n121− 720x14

1 x23x

64y

23y

64y

141

n121

+720x14

1 x62x

23x

64y

62y

23y

64y

141

n154− 3600x14

1 x42x

23x

64y

42y

23y

64y

141

n143+

4320x141 x

22x

23x

64y

22y

23y

64y

141

n132

− 600x141 x

42y

42y

141

n99+

600x141 x

62x

43y

62y

43y

141

n132− 600x14

1 x43y

43y

141

n99

− 3000x141 x

42x

43y

42y

43y

141

n121+

3600x141 x

22x

43y

22y

43y

141

n110+

600x141 x

62x

44y

62y

44y

141

n132

− 600x141 x

62x

63x

44y

62y

63y

44y

141

n165+

600x141 x

63x

44y

63y

44y

141

n132+

3000x141 x

42x

63x

44y

42y

63y

44y

141

n154

− 3600x141 x

22x

63x

44y

22y

63y

44y

141

n143− 600x14

1 x44y

44y

141

n99− 3000x14

1 x42x

44y

42y

44y

141

n121

+3000x14

1 x62x

43x

44y

62y

43y

44y

141

n154− 3000x14

1 x43x

44y

43y

44y

141

n121− 15000x14

1 x42x

43x

44y

42y

43y

44y

141

n143


+18000x14

1 x22x

43x

44y

22y

43y

44y

141

n132+

3600x141 x

22x

44y

22y

44y

141

n110− 3600x14

1 x62x

23x

44y

62y

23y

44y

141

n143

+3600x14

1 x23x

44y

23y

44y

141

n110+

18000x141 x

42x

23x

44y

42y

23y

44y

141

n132− 21600x14

1 x22x

23x

44y

22y

23y

44y

141

n121

+720x14

1 x22y

22y

141

n88− 720x14

1 x62x

23y

62y

23y

141

n121+

3600x141 x

42x

23y

42y

23y

141

n110

+720x14

1 x23y

23y

141

n88− 4320x14

1 x22x

23y

22y

23y

141

n99− 720x14

1 x62x

24y

62y

24y

141

n121

+720x14

1 x62x

63x

24y

62y

63y

24y

141

n154− 3600x14

1 x42x

63x

24y

42y

63y

24y

141

n143− 720x14

1 x63x

24y

63y

24y

141

n121

+4320x14

1 x22x

63x

24y

22y

63y

24y

141

n132+

3600x141 x

42x

24y

42y

24y

141

n110− 3600x14

1 x62x

43x

24y

62y

43y

24y

141

n143

+18000x14

1 x42x

43x

24y

42y

43y

24y

141

n132+

3600x141 x

43x

24y

43y

24y

141

n110− 21600x14

1 x22x

43x

24y

22y

43y

24y

141

n121

+720x14

1 x24y

24y

141

n88− 4320x14

1 x22x

24y

22y

24y

141

n99+

4320x141 x

62x

23x

24y

62y

23y

24y

141

n132

− 21600x141 x

42x

23x

24y

42y

23y

24y

141

n121− 4320x14

1 x23x

24y

23y

24y

141

n99+

25920x141 x

22x

23x

24y

22y

23y

24y

141

n110

+455x12

1 y121

n66− 455x12

1 x62y

62y

121

n99− 455x12

1 x63y

63y

121

n99

+455x12

1 x62x

63y

62y

63y

121

n132− 2275x12

1 x42x

63y

42y

63y

121

n121+

2730x121 x

22x

63y

22y

63y

121

n110

− 455x121 x

64y

64y

121

n99+

455x121 x

62x

64y

62y

64y

121

n132+

455x121 x

63x

64y

63y

64y

121

n132

− 455x121 x

62x

63x

64y

62y

63y

64y

121

n165+

2275x121 x

42x

63x

64y

42y

63y

64y

121

n154− 2730x12

1 x22x

63x

64y

22y

63y

64y

121

n143

− 2275x121 x

42x

64y

42y

64y

121

n121− 2275x12

1 x43x

64y

43y

64y

121

n121+

2275x121 x

62x

43x

64y

62y

43y

64y

121

n154

− 11375x121 x

42x

43x

64y

42y

43y

64y

121

n143+

13650x121 x

22x

43x

64y

22y

43y

64y

121

n132+

2730x121 x

22x

64y

22y

64y

121

n110

+2730x12

1 x23x

64y

23y

64y

121

n110− 2730x12

1 x62x

23x

64y

62y

23y

64y

121

n143+

13650x121 x

42x

23x

64y

42y

23y

64y

121

n132

− 16380x121 x

22x

23x

64y

22y

23y

64y

121

n121+

2275x121 x

42y

42y

121

n88− 2275x12

1 x62x

43y

62y

43y

121

n121

+2275x12

1 x43y

43y

121

n88+

11375x121 x

42x

43y

42y

43y

121

n110− 13650x12

1 x22x

43y

22y

43y

121

n99

− 2275x121 x

62x

44y

62y

44y

121

n121+

2275x121 x

62x

63x

44y

62y

63y

44y

121

n154− 2275x12

1 x63x

44y

63y

44y

121

n121

− 11375x121 x

42x

63x

44y

42y

63y

44y

121

n143+

13650x121 x

22x

63x

44y

22y

63y

44y

121

n132+

2275x121 x

44y

44y

121

n88

+11375x12

1 x42x

44y

42y

44y

121

n110− 11375x12

1 x62x

43x

44y

62y

43y

44y

121

n143+

11375x121 x

43x

44y

43y

44y

121

n110

+56875x12

1 x42x

43x

44y

42y

43y

44y

121

n132− 68250x12

1 x22x

43x

44y

22y

43y

44y

121

n121− 13650x12

1 x22x

44y

22y

44y

121

n99

+13650x12

1 x62x

23x

44y

62y

23y

44y

121

n132− 13650x12

1 x23x

44y

23y

44y

121

n99− 68250x12

1 x42x

23x

44y

42y

23y

44y

121

n121

+81900x12

1 x22x

23x

44y

22y

23y

44y

121

n110− 2730x12

1 x22y

22y

121

n77+

2730x121 x

62x

23y

62y

23y

121

n110

− 13650x121 x

42x

23y

42y

23y

121

n99− 2730x12

1 x23y

23y

121

n77+

16380x121 x

22x

23y

22y

23y

121

n88


+2730x12

1 x62x

24y

62y

24y

121

n110− 2730x12

1 x62x

63x

24y

62y

63y

24y

121

n143+

13650x121 x

42x

63x

24y

42y

63y

24y

121

n132

+2730x12

1 x63x

24y

63y

24y

121

n110− 16380x12

1 x22x

63x

24y

22y

63y

24y

121

n121− 13650x12

1 x42x

24y

42y

24y

121

n99

+13650x12

1 x62x

43x

24y

62y

43y

24y

121

n132− 68250x12

1 x42x

43x

24y

42y

43y

24y

121

n121− 13650x12

1 x43x

24y

43y

24y

121

n99

+81900x12

1 x22x

43x

24y

22y

43y

24y

121

n110− 2730x12

1 x24y

24y

121

n77+

16380x121 x

22x

24y

22y

24y

121

n88

− 16380x121 x

62x

23x

24y

62y

23y

24y

121

n121+

81900x121 x

42x

23x

24y

42y

23y

24y

121

n110+

16380x121 x

23x

24y

23y

24y

121

n88

− 98280x121 x

22x

23x

24y

22y

23y

24y

121

n99− 1001x10

1 y101

n55+

1001x101 x

62y

62y

101

n88

+1001x10

1 x63y

63y

101

n88− 1001x10

1 x62x

63y

62y

63y

101

n121+

5005x101 x

42x

63y

42y

63y

101

n110

− 6006x101 x

22x

63y

22y

63y

101

n99+

1001x101 x

64y

64y

101

n88− 1001x10

1 x62x

64y

62y

64y

101

n121

− 1001x101 x

63x

64y

63y

64y

101

n121+

1001x101 x

62x

63x

64y

62y

63y

64y

101

n154− 5005x10

1 x42x

63x

64y

42y

63y

64y

101

n143

+6006x10

1 x22x

63x

64y

22y

63y

64y

101

n132+

5005x101 x

42x

64y

42y

64y

101

n110+

5005x101 x

43x

64y

43y

64y

101

n110

− 5005x101 x

62x

43x

64y

62y

43y

64y

101

n143+

25025x101 x

42x

43x

64y

42y

43y

64y

101

n132− 30030x10

1 x22x

43x

64y

22y

43y

64y

101

n121

− 6006x101 x

22x

64y

22y

64y

101

n99− 6006x10

1 x23x

64y

23y

64y

101

n99+

6006x101 x

62x

23x

64y

62y

23y

64y

101

n132

− 30030x101 x

42x

23x

64y

42y

23y

64y

101

n121+

36036x101 x

22x

23x

64y

22y

23y

64y

101

n110− 5005x10

1 x42y

42y

101

n77

+5005x10

1 x62x

43y

62y

43y

101

n110− 5005x10

1 x43y

43y

101

n77− 25025x10

1 x42x

43y

42y

43y

101

n99

+30030x10

1 x22x

43y

22y

43y

101

n88+

5005x101 x

62x

44y

62y

44y

101

n110− 5005x10

1 x62x

63x

44y

62y

63y

44y

101

n143

+5005x10

1 x63x

44y

63y

44y

101

n110+

25025x101 x

42x

63x

44y

42y

63y

44y

101

n132− 30030x10

1 x22x

63x

44y

22y

63y

44y

101

n121

− 5005x101 x

44y

44y

101

n77− 25025x10

1 x42x

44y

42y

44y

101

n99+

25025x101 x

62x

43x

44y

62y

43y

44y

101

n132

− 25025x101 x

43x

44y

43y

44y

101

n99− 125125x10

1 x42x

43x

44y

42y

43y

44y

101

n121+

150150x101 x

22x

43x

44y

22y

43y

44y

101

n110

+30030x10

1 x22x

44y

22y

44y

101

n88− 30030x10

1 x62x

23x

44y

62y

23y

44y

101

n121+

30030x101 x

23x

44y

23y

44y

101

n88

+150150x10

1 x42x

23x

44y

42y

23y

44y

101

n110− 180180x10

1 x22x

23x

44y

22y

23y

44y

101

n99+

6006x101 x

22y

22y

101

n66

− 6006x101 x

62x

23y

62y

23y

101

n99+

30030x101 x

42x

23y

42y

23y

101

n88+

6006x101 x

23y

23y

101

n66

− 36036x101 x

22x

23y

22y

23y

101

n77− 6006x10

1 x62x

24y

62y

24y

101

n99+

6006x101 x

62x

63x

24y

62y

63y

24y

101

n132

− 30030x101 x

42x

63x

24y

42y

63y

24y

101

n121− 6006x10

1 x63x

24y

63y

24y

101

n99+

36036x101 x

22x

63x

24y

22y

63y

24y

101

n110

+30030x10

1 x42x

24y

42y

24y

101

n88− 30030x10

1 x62x

43x

24y

62y

43y

24y

101

n121+

150150x101 x

42x

43x

24y

42y

43y

24y

101

n110

+30030x10

1 x43x

24y

43y

24y

101

n88− 180180x10

1 x22x

43x

24y

22y

43y

24y

101

n99+

6006x101 x

24y

24y

101

n66


− 36036x101 x

22x

24y

22y

24y

101

n77+

36036x101 x

62x

23x

24y

62y

23y

24y

101

n110− 180180x10

1 x42x

23x

24y

42y

23y

24y

101

n99

− 36036x101 x

23x

24y

23y

24y

101

n77+

216216x101 x

22x

23x

24y

22y

23y

24y

101

n88+

1287x81y

81

n44

− 1287x81x

62y

62y

81

n77− 1287x8

1x63y

63y

81

n77+

1287x81x

62x

63y

62y

63y

81

n110

− 6435x81x

42x

63y

42y

63y

81

n99+

7722x81x

22x

63y

22y

63y

81

n88− 1287x8

1x64y

64y

81

n77

+1287x8

1x62x

64y

62y

64y

81

n110+

1287x81x

63x

64y

63y

64y

81

n110− 1287x8

1x62x

63x

64y

62y

63y

64y

81

n143

+6435x8

1x42x

63x

64y

42y

63y

64y

81

n132− 7722x8

1x22x

63x

64y

22y

63y

64y

81

n121− 6435x8

1x42x

64y

42y

64y

81

n99

− 6435x81x

43x

64y

43y

64y

81

n99+

6435x81x

62x

43x

64y

62y

43y

64y

81

n132− 32175x8

1x42x

43x

64y

42y

43y

64y

81

n121

+38610x8

1x22x

43x

64y

22y

43y

64y

81

n110+

7722x81x

22x

64y

22y

64y

81

n88+

7722x81x

23x

64y

23y

64y

81

n88

− 7722x81x

62x

23x

64y

62y

23y

64y

81

n121+

38610x81x

42x

23x

64y

42y

23y

64y

81

n110− 46332x8

1x22x

23x

64y

22y

23y

64y

81

n99

+6435x8

1x42y

42y

81

n66− 6435x8

1x62x

43y

62y

43y

81

n99+

6435x81x

43y

43y

81

n66

+32175x8

1x42x

43y

42y

43y

81

n88− 38610x8

1x22x

43y

22y

43y

81

n77− 6435x8

1x62x

44y

62y

44y

81

n99

+6435x8

1x62x

63x

44y

62y

63y

44y

81

n132− 6435x8

1x63x

44y

63y

44y

81

n99− 32175x8

1x42x

63x

44y

42y

63y

44y

81

n121

+38610x8

1x22x

63x

44y

22y

63y

44y

81

n110+

6435x81x

44y

44y

81

n66+

32175x81x

42x

44y

42y

44y

81

n88

− 32175x81x

62x

43x

44y

62y

43y

44y

81

n121+

32175x81x

43x

44y

43y

44y

81

n88+

160875x81x

42x

43x

44y

42y

43y

44y

81

n110

− 193050x81x

22x

43x

44y

22y

43y

44y

81

n99− 38610x8

1x22x

44y

22y

44y

81

n77+

38610x81x

62x

23x

44y

62y

23y

44y

81

n110

− 38610x81x

23x

44y

23y

44y

81

n77− 193050x8

1x42x

23x

44y

42y

23y

44y

81

n99+

231660x81x

22x

23x

44y

22y

23y

44y

81

n88

− 7722x81x

22y

22y

81

n55+

7722x81x

62x

23y

62y

23y

81

n88− 38610x8

1x42x

23y

42y

23y

81

n77

− 7722x81x

23y

23y

81

n55+

46332x81x

22x

23y

22y

23y

81

n66+

7722x81x

62x

24y

62y

24y

81

n88

− 7722x81x

62x

63x

24y

62y

63y

24y

81

n121+

38610x81x

42x

63x

24y

42y

63y

24y

81

n110+

7722x81x

63x

24y

63y

24y

81

n88

− 46332x81x

22x

63x

24y

22y

63y

24y

81

n99− 38610x8

1x42x

24y

42y

24y

81

n77+

38610x81x

62x

43x

24y

62y

43y

24y

81

n110

− 193050x81x

42x

43x

24y

42y

43y

24y

81

n99− 38610x8

1x43x

24y

43y

24y

81

n77+

231660x81x

22x

43x

24y

22y

43y

24y

81

n88

− 7722x81x

24y

24y

81

n55+

46332x81x

22x

24y

22y

24y

81

n66− 46332x8

1x62x

23x

24y

62y

23y

24y

81

n99

+231660x8

1x42x

23x

24y

42y

23y

24y

81

n88+

46332x81x

23x

24y

23y

24y

81

n66− 277992x8

1x22x

23x

24y

22y

23y

24y

81

n77

− 924x61y

61

n33+

924x61x

62y

62y

61

n66+

924x61x

63y

63y

61

n66

− 924x61x

62x

63y

62y

63y

61

n99+

4620x61x

42x

63y

42y

63y

61

n88− 5544x6

1x22x

63y

22y

63y

61

n77


+924x6

1x64y

64y

61

n66− 924x6

1x62x

64y

62y

64y

61

n99− 924x6

1x63x

64y

63y

64y

61

n99

+924x6

1x62x

63x

64y

62y

63y

64y

61

n132− 4620x6

1x42x

63x

64y

42y

63y

64y

61

n121+

5544x61x

22x

63x

64y

22y

63y

64y

61

n110

+4620x6

1x42x

64y

42y

64y

61

n88+

4620x61x

43x

64y

43y

64y

61

n88− 4620x6

1x62x

43x

64y

62y

43y

64y

61

n121

+23100x6

1x42x

43x

64y

42y

43y

64y

61

n110− 27720x6

1x22x

43x

64y

22y

43y

64y

61

n99− 5544x6

1x22x

64y

22y

64y

61

n77

− 5544x61x

23x

64y

23y

64y

61

n77+

5544x61x

62x

23x

64y

62y

23y

64y

61

n110− 27720x6

1x42x

23x

64y

42y

23y

64y

61

n99

+33264x6

1x22x

23x

64y

22y

23y

64y

61

n88− 4620x6

1x42y

42y

61

n55+

4620x61x

62x

43y

62y

43y

61

n88

− 4620x61x

43y

43y

61

n55− 23100x6

1x42x

43y

42y

43y

61

n77+

27720x61x

22x

43y

22y

43y

61

n66

+4620x6

1x62x

44y

62y

44y

61

n88− 4620x6

1x62x

63x

44y

62y

63y

44y

61

n121+

4620x61x

63x

44y

63y

44y

61

n88

+23100x6

1x42x

63x

44y

42y

63y

44y

61

n110− 27720x6

1x22x

63x

44y

22y

63y

44y

61

n99− 4620x6

1x44y

44y

61

n55

− 23100x61x

42x

44y

42y

44y

61

n77+

23100x61x

62x

43x

44y

62y

43y

44y

61

n110− 23100x6

1x43x

44y

43y

44y

61

n77

− 115500x61x

42x

43x

44y

42y

43y

44y

61

n99+

138600x61x

22x

43x

44y

22y

43y

44y

61

n88+

27720x61x

22x

44y

22y

44y

61

n66

− 27720x61x

62x

23x

44y

62y

23y

44y

61

n99+

27720x61x

23x

44y

23y

44y

61

n66+

138600x61x

42x

23x

44y

42y

23y

44y

61

n88

− 166320x61x

22x

23x

44y

22y

23y

44y

61

n77+

5544x61x

22y

22y

61

n44− 5544x6

1x62x

23y

62y

23y

61

n77

+27720x6

1x42x

23y

42y

23y

61

n66+

5544x61x

23y

23y

61

n44− 33264x6

1x22x

23y

22y

23y

61

n55

− 5544x61x

62x

24y

62y

24y

61

n77+

5544x61x

62x

63x

24y

62y

63y

24y

61

n110− 27720x6

1x42x

63x

24y

42y

63y

24y

61

n99

− 5544x61x

63x

24y

63y

24y

61

n77+

33264x61x

22x

63x

24y

22y

63y

24y

61

n88+

27720x61x

42x

24y

42y

24y

61

n66

− 27720x61x

62x

43x

24y

62y

43y

24y

61

n99+

138600x61x

42x

43x

24y

42y

43y

24y

61

n88+

27720x61x

43x

24y

43y

24y

61

n66

− 166320x61x

22x

43x

24y

22y

43y

24y

61

n77+

5544x61x

24y

24y

61

n44− 33264x6

1x22x

24y

22y

24y

61

n55

+33264x6

1x62x

23x

24y

62y

23y

24y

61

n88− 166320x6

1x42x

23x

24y

42y

23y

24y

61

n77− 33264x6

1x23x

24y

23y

24y

61

n55

+199584x6

1x22x

23x

24y

22y

23y

24y

61

n66− 330x4

1x62y

62y

41

n55− 330x4

1x63y

63y

41

n55

+330x4

1x62x

63y

62y

63y

41

n88− 1650x4

1x42x

63y

42y

63y

41

n77+

1980x41x

22x

63y

22y

63y

41

n66

− 330x41x

64y

64y

41

n55+

330x41x

62x

64y

62y

64y

41

n88+

330x41x

63x

64y

63y

64y

41

n88

− 330x41x

62x

63x

64y

62y

63y

64y

41

n121+

1650x41x

42x

63x

64y

42y

63y

64y

41

n110− 1980x4

1x22x

63x

64y

22y

63y

64y

41

n99

− 1650x41x

42x

64y

42y

64y

41

n77− 1650x4

1x43x

64y

43y

64y

41

n77+

1650x41x

62x

43x

64y

62y

43y

64y

41

n110

− 8250x41x

42x

43x

64y

42y

43y

64y

41

n99+

9900x41x

22x

43x

64y

22y

43y

64y

41

n88+

1980x41x

22x

64y

22y

64y

41

n66


+1980x4

1x23x

64y

23y

64y

41

n66− 1980x4

1x62x

23x

64y

62y

23y

64y

41

n99+

9900x41x

42x

23x

64y

42y

23y

64y

41

n88

− 11880x41x

22x

23x

64y

22y

23y

64y

41

n77+

330x41y

41

n22+

1650x41x

42y

42y

41

n44

− 1650x41x

62x

43y

62y

43y

41

n77+

1650x41x

43y

43y

41

n44+

8250x41x

42x

43y

42y

43y

41

n66

− 9900x41x

22x

43y

22y

43y

41

n55− 1650x4

1x62x

44y

62y

44y

41

n77+

1650x41x

62x

63x

44y

62y

63y

44y

41

n110

− 1650x41x

63x

44y

63y

44y

41

n77− 8250x4

1x42x

63x

44y

42y

63y

44y

41

n99+

9900x41x

22x

63x

44y

22y

63y

44y

41

n88

+1650x4

1x44y

44y

41

n44+

8250x41x

42x

44y

42y

44y

41

n66− 8250x4

1x62x

43x

44y

62y

43y

44y

41

n99

+8250x4

1x43x

44y

43y

44y

41

n66+

41250x41x

42x

43x

44y

42y

43y

44y

41

n88− 49500x4

1x22x

43x

44y

22y

43y

44y

41

n77

− 9900x41x

22x

44y

22y

44y

41

n55+

9900x41x

62x

23x

44y

62y

23y

44y

41

n88− 9900x4

1x23x

44y

23y

44y

41

n55

− 49500x41x

42x

23x

44y

42y

23y

44y

41

n77+

59400x41x

22x

23x

44y

22y

23y

44y

41

n66− 1980x4

1x22y

22y

41

n33

+1980x4

1x62x

23y

62y

23y

41

n66− 9900x4

1x42x

23y

42y

23y

41

n55− 1980x4

1x23y

23y

41

n33

+11880x4

1x22x

23y

22y

23y

41

n44+

1980x41x

62x

24y

62y

24y

41

n66− 1980x4

1x62x

63x

24y

62y

63y

24y

41

n99

+9900x4

1x42x

63x

24y

42y

63y

24y

41

n88+

1980x41x

63x

24y

63y

24y

41

n66− 11880x4

1x22x

63x

24y

22y

63y

24y

41

n77

− 9900x41x

42x

24y

42y

24y

41

n55+

9900x41x

62x

43x

24y

62y

43y

24y

41

n88− 49500x4

1x42x

43x

24y

42y

43y

24y

41

n77

− 9900x41x

43x

24y

43y

24y

41

n55+

59400x41x

22x

43x

24y

22y

43y

24y

41

n66− 1980x4

1x24y

24y

41

n33

+11880x4

1x22x

24y

22y

24y

41

n44− 11880x4

1x62x

23x

24y

62y

23y

24y

41

n77+

59400x41x

42x

23x

24y

42y

23y

24y

41

n66

+11880x4

1x23x

24y

23y

24y

41

n44− 71280x4

1x22x

23x

24y

22y

23y

24y

41

n55+

45x21x

62y

62y

21

n44

+45x2

1x63y

63y

21

n44− 45x2

1x62x

63y

62y

63y

21

n77+

225x21x

42x

63y

42y

63y

21

n66

− 270x21x

22x

63y

22y

63y

21

n55+

45x21x

64y

64y

21

n44− 45x2

1x62x

64y

62y

64y

21

n77

− 45x21x

63x

64y

63y

64y

21

n77+

45x21x

62x

63x

64y

62y

63y

64y

21

n110− 225x2

1x42x

63x

64y

42y

63y

64y

21

n99

+270x2

1x22x

63x

64y

22y

63y

64y

21

n88+

225x21x

42x

64y

42y

64y

21

n66+

225x21x

43x

64y

43y

64y

21

n66

− 225x21x

62x

43x

64y

62y

43y

64y

21

n99+

1125x21x

42x

43x

64y

42y

43y

64y

21

n88− 1350x2

1x22x

43x

64y

22y

43y

64y

21

n77

− 270x21x

22x

64y

22y

64y

21

n55− 270x2

1x23x

64y

23y

64y

21

n55+

270x21x

62x

23x

64y

62y

23y

64y

21

n88

− 1350x21x

42x

23x

64y

42y

23y

64y

21

n77+

1620x21x

22x

23x

64y

22y

23y

64y

21

n66− 225x2

1x42y

42y

21

n33

+225x2

1x62x

43y

62y

43y

21

n66− 225x2

1x43y

43y

21

n33− 1125x2

1x42x

43y

42y

43y

21

n55

+1350x2

1x22x

43y

22y

43y

21

n44+

225x21x

62x

44y

62y

44y

21

n66− 225x2

1x62x

63x

44y

62y

63y

44y

21

n99


+225x2

1x63x

44y

63y

44y

21

n66+

1125x21x

42x

63x

44y

42y

63y

44y

21

n88− 1350x2

1x22x

63x

44y

22y

63y

44y

21

n77

− 225x21x

44y

44y

21

n33− 1125x2

1x42x

44y

42y

44y

21

n55+

1125x21x

62x

43x

44y

62y

43y

44y

21

n88

− 1125x21x

43x

44y

43y

44y

21

n55− 5625x2

1x42x

43x

44y

42y

43y

44y

21

n77+

6750x21x

22x

43x

44y

22y

43y

44y

21

n66

+1350x2

1x22x

44y

22y

44y

21

n44− 1350x2

1x62x

23x

44y

62y

23y

44y

21

n77+

1350x21x

23x

44y

23y

44y

21

n44

+6750x2

1x42x

23x

44y

42y

23y

44y

21

n66− 8100x2

1x22x

23x

44y

22y

23y

44y

21

n55− 45x2

1y21

n11

+270x2

1x22y

22y

21

n22− 270x2

1x62x

23y

62y

23y

21

n55+

1350x21x

42x

23y

42y

23y

21

n44

+270x2

1x23y

23y

21

n22− 1620x2

1x22x

23y

22y

23y

21

n33− 270x2

1x62x

24y

62y

24y

21

n55

+270x2

1x62x

63x

24y

62y

63y

24y

21

n88− 1350x2

1x42x

63x

24y

42y

63y

24y

21

n77− 270x2

1x63x

24y

63y

24y

21

n55

+1620x2

1x22x

63x

24y

22y

63y

24y

21

n66+

1350x21x

42x

24y

42y

24y

21

n44− 1350x2

1x62x

43x

24y

62y

43y

24y

21

n77

+6750x2

1x42x

43x

24y

42y

43y

24y

21

n66+

1350x21x

43x

24y

43y

24y

21

n44− 8100x2

1x22x

43x

24y

22y

43y

24y

21

n55

+270x2

1x24y

24y

21

n22− 1620x2

1x22x

24y

22y

24y

21

n33+

1620x21x

62x

23x

24y

62y

23y

24y

21

n66

− 8100x21x

42x

23x

24y

42y

23y

24y

21

n55− 1620x2

1x23x

24y

23y

24y

21

n33+

9720x21x

22x

23x

24y

22y

23y

24y

21

n44

− x62y

62

n33− x6

3y63

n33+x6

2x63y

62y

63

n66

− 5x42x

63y

42y

63

n55+

6x22x

63y

22y

63

n44− x6

4y64

n33

+x6

2x64y

62y

64

n66+x6

3x64y

63y

64

n66− x6

2x63x

64y

62y

63y

64

n99

+5x4

2x63x

64y

42y

63y

64

n88− 6x2

2x63x

64y

22y

63y

64

n77− 5x4

2x64y

42y

64

n55

− 5x43x

64y

43y

64

n55+

5x62x

43x

64y

62y

43y

64

n88− 25x4

2x43x

64y

42y

43y

64

n77

+30x2

2x43x

64y

22y

43y

64

n66+

6x22x

64y

22y

64

n44+

6x23x

64y

23y

64

n44

− 6x62x

23x

64y

62y

23y

64

n77+

30x42x

23x

64y

42y

23y

64

n66− 36x2

2x23x

64y

22y

23y

64

n55

+5x4

2y42

n22− 5x6

2x43y

62y

43

n55+

5x43y

43

n22

+25x4

2x43y

42y

43

n44− 30x2

2x43y

22y

43

n33− 5x6

2x44y

62y

44

n55

+5x6

2x63x

44y

62y

63y

44

n88− 5x6

3x44y

63y

44

n55− 25x4

2x63x

44y

42y

63y

44

n77

+30x2

2x63x

44y

22y

63y

44

n66+

5x44y

44

n22+

25x42x

44y

42y

44

n44

− 25x62x

43x

44y

62y

43y

44

n77+

25x43x

44y

43y

44

n44+

125x42x

43x

44y

42y

43y

44

n66

− 150x22x

43x

44y

22y

43y

44

n55− 30x2

2x44y

22y

44

n33+

30x62x

23x

44y

62y

23y

44

n66


− 30x23x

44y

23y

44

n33− 150x4

2x23x

44y

42y

23y

44

n55+

180x22x

23x

44y

22y

23y

44

n44

− 6x22y

22

n11+

6x62x

23y

62y

23

n44− 30x4

2x23y

42y

23

n33

− 6x23y

23

n11+

36x22x

23y

22y

23

n22+

6x62x

24y

62y

24

n44

− 6x62x

63x

24y

62y

63y

24

n77+

30x42x

63x

24y

42y

63y

24

n66+

6x63x

24y

63y

24

n44

− 36x22x

63x

24y

22y

63y

24

n55− 30x4

2x24y

42y

24

n33+

30x62x

43x

24y

62y

43y

24

n66

− 150x42x

43x

24y

42y

43y

24

n55− 30x4

3x24y

43y

24

n33+

180x22x

43x

24y

22y

43y

24

n44

− 6x24y

24

n11+

36x22x

24y

22y

24

n22− 36x6

2x23x

24y

62y

23y

24

n55

+180x4

2x23x

24y

42y

23y

24

n44+

36x23x

24y

23y

24

n22− 216x2

2x23x

24y

22y

23y

24

n33

+ 1

Gf,n,4,(α),7(x1, . . . , y4) =x21

1 x72x

73x

74y

72y

73y

74y

211

n231− 6x21

1 x52x

73x

74y

52y

73y

74y

211

n220

+10x21

1 x32x

73x

74y

32y

73y

74y

211

n209− 4x21

1 x2x73x

74y2y

73y

74y

211

n198

− 6x211 x

72x

53x

74y

72y

53y

74y

211

n220+

36x211 x

52x

53x

74y

52y

53y

74y

211

n209

− 60x211 x

32x

53x

74y

32y

53y

74y

211

n198+

24x211 x2x

53x

74y2y

53y

74y

211

n187

+10x21

1 x72x

33x

74y

72y

33y

74y

211

n209− 60x21

1 x52x

33x

74y

52y

33y

74y

211

n198

+100x21

1 x32x

33x

74y

32y

33y

74y

211

n187− 40x21

1 x2x33x

74y2y

33y

74y

211

n176

− 4x211 x

72x3x

74y

72y3y

74y

211

n198+

24x211 x

52x3x

74y

52y3y

74y

211

n187

− 40x211 x

32x3x

74y

32y3y

74y

211

n176+

16x211 x2x3x

74y2y3y

74y

211

n165

− 6x211 x

72x

73x

54y

72y

73y

54y

211

n220+

36x211 x

52x

73x

54y

52y

73y

54y

211

n209

− 60x211 x

32x

73x

54y

32y

73y

54y

211

n198+

24x211 x2x

73x

54y2y

73y

54y

211

n187

+36x21

1 x72x

53x

54y

72y

53y

54y

211

n209− 216x21

1 x52x

53x

54y

52y

53y

54y

211

n198

+360x21

1 x32x

53x

54y

32y

53y

54y

211

n187− 144x21

1 x2x53x

54y2y

53y

54y

211

n176

− 60x211 x

72x

33x

54y

72y

33y

54y

211

n198+

360x211 x

52x

33x

54y

52y

33y

54y

211

n187

− 600x211 x

32x

33x

54y

32y

33y

54y

211

n176+

240x211 x2x

33x

54y2y

33y

54y

211

n165

+24x21

1 x72x3x

54y

72y3y

54y

211

n187− 144x21

1 x52x3x

54y

52y3y

54y

211

n176

+240x21

1 x32x3x

54y

32y3y

54y

211

n165− 96x21

1 x2x3x54y2y3y

54y

211

n154


+10x21

1 x72x

73x

34y

72y

73y

34y

211

n209− 60x21

1 x52x

73x

34y

52y

73y

34y

211

n198

+100x21

1 x32x

73x

34y

32y

73y

34y

211

n187− 40x21

1 x2x73x

34y2y

73y

34y

211

n176

− 60x211 x

72x

53x

34y

72y

53y

34y

211

n198+

360x211 x

52x

53x

34y

52y

53y

34y

211

n187

− 600x211 x

32x

53x

34y

32y

53y

34y

211

n176+

240x211 x2x

53x

34y2y

53y

34y

211

n165

+100x21

1 x72x

33x

34y

72y

33y

34y

211

n187− 600x21

1 x52x

33x

34y

52y

33y

34y

211

n176

+1000x21

1 x32x

33x

34y

32y

33y

34y

211

n165− 400x21

1 x2x33x

34y2y

33y

34y

211

n154

− 40x211 x

72x3x

34y

72y3y

34y

211

n176+

240x211 x

52x3x

34y

52y3y

34y

211

n165

− 400x211 x

32x3x

34y

32y3y

34y

211

n154+

160x211 x2x3x

34y2y3y

34y

211

n143

− 4x211 x

72x

73x4y

72y

73y4y

211

n198+

24x211 x

52x

73x4y

52y

73y4y

211

n187

− 40x211 x

32x

73x4y

32y

73y4y

211

n176+

16x211 x2x

73x4y2y

73y4y

211

n165

+24x21

1 x72x

53x4y

72y

53y4y

211

n187− 144x21

1 x52x

53x4y

52y

53y4y

211

n176

+240x21

1 x32x

53x4y

32y

53y4y

211

n165− 96x21

1 x2x53x4y2y

53y4y

211

n154

− 40x211 x

72x

33x4y

72y

33y4y

211

n176+

240x211 x

52x

33x4y

52y

33y4y

211

n165

− 400x211 x

32x

33x4y

32y

33y4y

211

n154+

160x211 x2x

33x4y2y

33y4y

211

n143

+16x21

1 x72x3x4y

72y3y4y

211

n165− 96x21

1 x52x3x4y

52y3y4y

211

n154

+160x21

1 x32x3x4y

32y3y4y

211

n143− 64x21

1 x2x3x4y2y3y4y211

n132

− 20x191 x

72x

73x

74y

72y

73y

74y

191

n220+

120x191 x

52x

73x

74y

52y

73y

74y

191

n209

− 200x191 x

32x

73x

74y

32y

73y

74y

191

n198+

80x191 x2x

73x

74y2y

73y

74y

191

n187

+120x19

1 x72x

53x

74y

72y

53y

74y

191

n209− 720x19

1 x52x

53x

74y

52y

53y

74y

191

n198

+1200x19

1 x32x

53x

74y

32y

53y

74y

191

n187− 480x19

1 x2x53x

74y2y

53y

74y

191

n176

− 200x191 x

72x

33x

74y

72y

33y

74y

191

n198+

1200x191 x

52x

33x

74y

52y

33y

74y

191

n187

− 2000x191 x

32x

33x

74y

32y

33y

74y

191

n176+

800x191 x2x

33x

74y2y

33y

74y

191

n165

+80x19

1 x72x3x

74y

72y3y

74y

191

n187− 480x19

1 x52x3x

74y

52y3y

74y

191

n176

+800x19

1 x32x3x

74y

32y3y

74y

191

n165− 320x19

1 x2x3x74y2y3y

74y

191

n154

+120x19

1 x72x

73x

54y

72y

73y

54y

191

n209− 720x19

1 x52x

73x

54y

52y

73y

54y

191

n198


+1200x19

1 x32x

73x

54y

32y

73y

54y

191

n187− 480x19

1 x2x73x

54y2y

73y

54y

191

n176

− 720x191 x

72x

53x

54y

72y

53y

54y

191

n198+

4320x191 x

52x

53x

54y

52y

53y

54y

191

n187

− 7200x191 x

32x

53x

54y

32y

53y

54y

191

n176+

2880x191 x2x

53x

54y2y

53y

54y

191

n165

+1200x19

1 x72x

33x

54y

72y

33y

54y

191

n187− 7200x19

1 x52x

33x

54y

52y

33y

54y

191

n176

+12000x19

1 x32x

33x

54y

32y

33y

54y

191

n165− 4800x19

1 x2x33x

54y2y

33y

54y

191

n154

− 480x191 x

72x3x

54y

72y3y

54y

191

n176+

2880x191 x

52x3x

54y

52y3y

54y

191

n165

− 4800x191 x

32x3x

54y

32y3y

54y

191

n154+

1920x191 x2x3x

54y2y3y

54y

191

n143

− 200x191 x

72x

73x

34y

72y

73y

34y

191

n198+

1200x191 x

52x

73x

34y

52y

73y

34y

191

n187

− 2000x191 x

32x

73x

34y

32y

73y

34y

191

n176+

800x191 x2x

73x

34y2y

73y

34y

191

n165

+1200x19

1 x72x

53x

34y

72y

53y

34y

191

n187− 7200x19

1 x52x

53x

34y

52y

53y

34y

191

n176

+12000x19

1 x32x

53x

34y

32y

53y

34y

191

n165− 4800x19

1 x2x53x

34y2y

53y

34y

191

n154

− 2000x191 x

72x

33x

34y

72y

33y

34y

191

n176+

12000x191 x

52x

33x

34y

52y

33y

34y

191

n165

− 20000x191 x

32x

33x

34y

32y

33y

34y

191

n154+

8000x191 x2x

33x

34y2y

33y

34y

191

n143

+800x19

1 x72x3x

34y

72y3y

34y

191

n165− 4800x19

1 x52x3x

34y

52y3y

34y

191

n154

+8000x19

1 x32x3x

34y

32y3y

34y

191

n143− 3200x19

1 x2x3x34y2y3y

34y

191

n132

+80x19

1 x72x

73x4y

72y

73y4y

191

n187− 480x19

1 x52x

73x4y

52y

73y4y

191

n176

+800x19

1 x32x

73x4y

32y

73y4y

191

n165− 320x19

1 x2x73x4y2y

73y4y

191

n154

− 480x191 x

72x

53x4y

72y

53y4y

191

n176+

2880x191 x

52x

53x4y

52y

53y4y

191

n165

− 4800x191 x

32x

53x4y

32y

53y4y

191

n154+

1920x191 x2x

53x4y2y

53y4y

191

n143

+800x19

1 x72x

33x4y

72y

33y4y

191

n165− 4800x19

1 x52x

33x4y

52y

33y4y

191

n154

+8000x19

1 x32x

33x4y

32y

33y4y

191

n143− 3200x19

1 x2x33x4y2y

33y4y

191

n132

− 320x191 x

72x3x4y

72y3y4y

191

n154+

1920x191 x

52x3x4y

52y3y4y

191

n143

− 3200x191 x

32x3x4y

32y3y4y

191

n132+

1280x191 x2x3x4y2y3y4y

191

n121

+171x17

1 x72x

73x

74y

72y

73y

74y

171

n209− 1026x17

1 x52x

73x

74y

52y

73y

74y

171

n198

+1710x17

1 x32x

73x

74y

32y

73y

74y

171

n187− 684x17

1 x2x73x

74y2y

73y

74y

171

n176


− 1026x171 x

72x

53x

74y

72y

53y

74y

171

n198+

6156x171 x

52x

53x

74y

52y

53y

74y

171

n187

− 10260x171 x

32x

53x

74y

32y

53y

74y

171

n176+

4104x171 x2x

53x

74y2y

53y

74y

171

n165

+1710x17

1 x72x

33x

74y

72y

33y

74y

171

n187− 10260x17

1 x52x

33x

74y

52y

33y

74y

171

n176

+17100x17

1 x32x

33x

74y

32y

33y

74y

171

n165− 6840x17

1 x2x33x

74y2y

33y

74y

171

n154

− 684x171 x

72x3x

74y

72y3y

74y

171

n176+

4104x171 x

52x3x

74y

52y3y

74y

171

n165

− 6840x171 x

32x3x

74y

32y3y

74y

171

n154+

2736x171 x2x3x

74y2y3y

74y

171

n143

− 1026x171 x

72x

73x

54y

72y

73y

54y

171

n198+

6156x171 x

52x

73x

54y

52y

73y

54y

171

n187

− 10260x171 x

32x

73x

54y

32y

73y

54y

171

n176+

4104x171 x2x

73x

54y2y

73y

54y

171

n165

+6156x17

1 x72x

53x

54y

72y

53y

54y

171

n187− 36936x17

1 x52x

53x

54y

52y

53y

54y

171

n176

+61560x17

1 x32x

53x

54y

32y

53y

54y

171

n165− 24624x17

1 x2x53x

54y2y

53y

54y

171

n154

− 10260x171 x

72x

33x

54y

72y

33y

54y

171

n176+

61560x171 x

52x

33x

54y

52y

33y

54y

171

n165

− 102600x171 x

32x

33x

54y

32y

33y

54y

171

n154+

41040x171 x2x

33x

54y2y

33y

54y

171

n143

+4104x17

1 x72x3x

54y

72y3y

54y

171

n165− 24624x17

1 x52x3x

54y

52y3y

54y

171

n154

+41040x17

1 x32x3x

54y

32y3y

54y

171

n143− 16416x17

1 x2x3x54y2y3y

54y

171

n132

+1710x17

1 x72x

73x

34y

72y

73y

34y

171

n187− 10260x17

1 x52x

73x

34y

52y

73y

34y

171

n176

+17100x17

1 x32x

73x

34y

32y

73y

34y

171

n165− 6840x17

1 x2x73x

34y2y

73y

34y

171

n154

− 10260x171 x

72x

53x

34y

72y

53y

34y

171

n176+

61560x171 x

52x

53x

34y

52y

53y

34y

171

n165

− 102600x171 x

32x

53x

34y

32y

53y

34y

171

n154+

41040x171 x2x

53x

34y2y

53y

34y

171

n143

+17100x17

1 x72x

33x

34y

72y

33y

34y

171

n165− 102600x17

1 x52x

33x

34y

52y

33y

34y

171

n154

+171000x17

1 x32x

33x

34y

32y

33y

34y

171

n143− 68400x17

1 x2x33x

34y2y

33y

34y

171

n132

− 6840x171 x

72x3x

34y

72y3y

34y

171

n154+

41040x171 x

52x3x

34y

52y3y

34y

171

n143

− 68400x171 x

32x3x

34y

32y3y

34y

171

n132+

27360x171 x2x3x

34y2y3y

34y

171

n121

− 684x171 x

72x

73x4y

72y

73y4y

171

n176+

4104x171 x

52x

73x4y

52y

73y4y

171

n165

− 6840x171 x

32x

73x4y

32y

73y4y

171

n154+

2736x171 x2x

73x4y2y

73y4y

171

n143

+4104x17

1 x72x

53x4y

72y

53y4y

171

n165− 24624x17

1 x52x

53x4y

52y

53y4y

171

n154


+41040x17

1 x32x

53x4y

32y

53y4y

171

n143− 16416x17

1 x2x53x4y2y

53y4y

171

n132

− 6840x171 x

72x

33x4y

72y

33y4y

171

n154+

41040x171 x

52x

33x4y

52y

33y4y

171

n143

− 68400x171 x

32x

33x4y

32y

33y4y

171

n132+

27360x171 x2x

33x4y2y

33y4y

171

n121

+2736x17

1 x72x3x4y

72y3y4y

171

n143− 16416x17

1 x52x3x4y

52y3y4y

171

n132

+27360x17

1 x32x3x4y

32y3y4y

171

n121− 10944x17

1 x2x3x4y2y3y4y171

n110

− 816x151 x

72x

73x

74y

72y

73y

74y

151

n198+

4896x151 x

52x

73x

74y

52y

73y

74y

151

n187

− 8160x151 x

32x

73x

74y

32y

73y

74y

151

n176+

3264x151 x2x

73x

74y2y

73y

74y

151

n165

+4896x15

1 x72x

53x

74y

72y

53y

74y

151

n187− 29376x15

1 x52x

53x

74y

52y

53y

74y

151

n176

+48960x15

1 x32x

53x

74y

32y

53y

74y

151

n165− 19584x15

1 x2x53x

74y2y

53y

74y

151

n154

− 8160x151 x

72x

33x

74y

72y

33y

74y

151

n176+

48960x151 x

52x

33x

74y

52y

33y

74y

151

n165

− 81600x151 x

32x

33x

74y

32y

33y

74y

151

n154+

32640x151 x2x

33x

74y2y

33y

74y

151

n143

+3264x15

1 x72x3x

74y

72y3y

74y

151

n165− 19584x15

1 x52x3x

74y

52y3y

74y

151

n154

+32640x15

1 x32x3x

74y

32y3y

74y

151

n143− 13056x15

1 x2x3x74y2y3y

74y

151

n132

+4896x15

1 x72x

73x

54y

72y

73y

54y

151

n187− 29376x15

1 x52x

73x

54y

52y

73y

54y

151

n176

+48960x15

1 x32x

73x

54y

32y

73y

54y

151

n165− 19584x15

1 x2x73x

54y2y

73y

54y

151

n154

− 29376x151 x

72x

53x

54y

72y

53y

54y

151

n176+

176256x151 x

52x

53x

54y

52y

53y

54y

151

n165

− 293760x151 x

32x

53x

54y

32y

53y

54y

151

n154+

117504x151 x2x

53x

54y2y

53y

54y

151

n143

+48960x15

1 x72x

33x

54y

72y

33y

54y

151

n165− 293760x15

1 x52x

33x

54y

52y

33y

54y

151

n154

+489600x15

1 x32x

33x

54y

32y

33y

54y

151

n143− 195840x15

1 x2x33x

54y2y

33y

54y

151

n132

− 19584x151 x

72x3x

54y

72y3y

54y

151

n154+

117504x151 x

52x3x

54y

52y3y

54y

151

n143

− 195840x151 x

32x3x

54y

32y3y

54y

151

n132+

78336x151 x2x3x

54y2y3y

54y

151

n121

− 8160x151 x

72x

73x

34y

72y

73y

34y

151

n176+

48960x151 x

52x

73x

34y

52y

73y

34y

151

n165

− 81600x151 x

32x

73x

34y

32y

73y

34y

151

n154+

32640x151 x2x

73x

34y2y

73y

34y

151

n143

+48960x15

1 x72x

53x

34y

72y

53y

34y

151

n165− 293760x15

1 x52x

53x

34y

52y

53y

34y

151

n154

+489600x15

1 x32x

53x

34y

32y

53y

34y

151

n143− 195840x15

1 x2x53x

34y2y

53y

34y

151

n132


− 81600x151 x

72x

33x

34y

72y

33y

34y

151

n154+

489600x151 x

52x

33x

34y

52y

33y

34y

151

n143

− 816000x151 x

32x

33x

34y

32y

33y

34y

151

n132+

326400x151 x2x

33x

34y2y

33y

34y

151

n121

+32640x15

1 x72x3x

34y

72y3y

34y

151

n143− 195840x15

1 x52x3x

34y

52y3y

34y

151

n132

+326400x15

1 x32x3x

34y

32y3y

34y

151

n121− 130560x15

1 x2x3x34y2y3y

34y

151

n110

+3264x15

1 x72x

73x4y

72y

73y4y

151

n165− 19584x15

1 x52x

73x4y

52y

73y4y

151

n154

+32640x15

1 x32x

73x4y

32y

73y4y

151

n143− 13056x15

1 x2x73x4y2y

73y4y

151

n132

− 19584x151 x

72x

53x4y

72y

53y4y

151

n154+

117504x151 x

52x

53x4y

52y

53y4y

151

n143

− 195840x151 x

32x

53x4y

32y

53y4y

151

n132+

78336x151 x2x

53x4y2y

53y4y

151

n121

+32640x15

1 x72x

33x4y

72y

33y4y

151

n143− 195840x15

1 x52x

33x4y

52y

33y4y

151

n132

+326400x15

1 x32x

33x4y

32y

33y4y

151

n121− 130560x15

1 x2x33x4y2y

33y4y

151

n110

− 13056x151 x

72x3x4y

72y3y4y

151

n132+

78336x151 x

52x3x4y

52y3y4y

151

n121

− 130560x151 x

32x3x4y

32y3y4y

151

n110+

52224x151 x2x3x4y2y3y4y

151

n99

+2380x13

1 x72x

73x

74y

72y

73y

74y

131

n187− 14280x13

1 x52x

73x

74y

52y

73y

74y

131

n176

+23800x13

1 x32x

73x

74y

32y

73y

74y

131

n165− 9520x13

1 x2x73x

74y2y

73y

74y

131

n154

− 14280x131 x

72x

53x

74y

72y

53y

74y

131

n176+

85680x131 x

52x

53x

74y

52y

53y

74y

131

n165

− 142800x131 x

32x

53x

74y

32y

53y

74y

131

n154+

57120x131 x2x

53x

74y2y

53y

74y

131

n143

+23800x13

1 x72x

33x

74y

72y

33y

74y

131

n165− 142800x13

1 x52x

33x

74y

52y

33y

74y

131

n154

+238000x13

1 x32x

33x

74y

32y

33y

74y

131

n143− 95200x13

1 x2x33x

74y2y

33y

74y

131

n132

− 9520x131 x

72x3x

74y

72y3y

74y

131

n154+

57120x131 x

52x3x

74y

52y3y

74y

131

n143

− 95200x131 x

32x3x

74y

32y3y

74y

131

n132+

38080x131 x2x3x

74y2y3y

74y

131

n121

− 14280x131 x

72x

73x

54y

72y

73y

54y

131

n176+

85680x131 x

52x

73x

54y

52y

73y

54y

131

n165

− 142800x131 x

32x

73x

54y

32y

73y

54y

131

n154+

57120x131 x2x

73x

54y2y

73y

54y

131

n143

+85680x13

1 x72x

53x

54y

72y

53y

54y

131

n165− 514080x13

1 x52x

53x

54y

52y

53y

54y

131

n154

+856800x13

1 x32x

53x

54y

32y

53y

54y

131

n143− 342720x13

1 x2x53x

54y2y

53y

54y

131

n132

− 142800x131 x

72x

33x

54y

72y

33y

54y

131

n154+

856800x131 x

52x

33x

54y

52y

33y

54y

131

n143


− 1428000x131 x

32x

33x

54y

32y

33y

54y

131

n132+

571200x131 x2x

33x

54y2y

33y

54y

131

n121

+57120x13

1 x72x3x

54y

72y3y

54y

131

n143− 342720x13

1 x52x3x

54y

52y3y

54y

131

n132

+571200x13

1 x32x3x

54y

32y3y

54y

131

n121− 228480x13

1 x2x3x54y2y3y

54y

131

n110

+23800x13

1 x72x

73x

34y

72y

73y

34y

131

n165− 142800x13

1 x52x

73x

34y

52y

73y

34y

131

n154

+238000x13

1 x32x

73x

34y

32y

73y

34y

131

n143− 95200x13

1 x2x73x

34y2y

73y

34y

131

n132

− 142800x131 x

72x

53x

34y

72y

53y

34y

131

n154+

856800x131 x

52x

53x

34y

52y

53y

34y

131

n143

− 1428000x131 x

32x

53x

34y

32y

53y

34y

131

n132+

571200x131 x2x

53x

34y2y

53y

34y

131

n121

+238000x13

1 x72x

33x

34y

72y

33y

34y

131

n143− 1428000x13

1 x52x

33x

34y

52y

33y

34y

131

n132

+2380000x13

1 x32x

33x

34y

32y

33y

34y

131

n121− 952000x13

1 x2x33x

34y2y

33y

34y

131

n110

− 95200x131 x

72x3x

34y

72y3y

34y

131

n132+

571200x131 x

52x3x

34y

52y3y

34y

131

n121

− 952000x131 x

32x3x

34y

32y3y

34y

131

n110+

380800x131 x2x3x

34y2y3y

34y

131

n99

− 9520x131 x

72x

73x4y

72y

73y4y

131

n154+

57120x131 x

52x

73x4y

52y

73y4y

131

n143

− 95200x131 x

32x

73x4y

32y

73y4y

131

n132+

38080x131 x2x

73x4y2y

73y4y

131

n121

+57120x13

1 x72x

53x4y

72y

53y4y

131

n143− 342720x13

1 x52x

53x4y

52y

53y4y

131

n132

+571200x13

1 x32x

53x4y

32y

53y4y

131

n121− 228480x13

1 x2x53x4y2y

53y4y

131

n110

− 95200x131 x

72x

33x4y

72y

33y4y

131

n132+

571200x131 x

52x

33x4y

52y

33y4y

131

n121

− 952000x131 x

32x

33x4y

32y

33y4y

131

n110+

380800x131 x2x

33x4y2y

33y4y

131

n99

+38080x13

1 x72x3x4y

72y3y4y

131

n121− 228480x13

1 x52x3x4y

52y3y4y

131

n110

+380800x13

1 x32x3x4y

32y3y4y

131

n99− 152320x13

1 x2x3x4y2y3y4y131

n88

− 4368x111 x

72x

73x

74y

72y

73y

74y

111

n176+

26208x111 x

52x

73x

74y

52y

73y

74y

111

n165

− 43680x111 x

32x

73x

74y

32y

73y

74y

111

n154+

17472x111 x2x

73x

74y2y

73y

74y

111

n143

+26208x11

1 x72x

53x

74y

72y

53y

74y

111

n165− 157248x11

1 x52x

53x

74y

52y

53y

74y

111

n154

+262080x11

1 x32x

53x

74y

32y

53y

74y

111

n143− 104832x11

1 x2x53x

74y2y

53y

74y

111

n132

− 43680x111 x

72x

33x

74y

72y

33y

74y

111

n154+

262080x111 x

52x

33x

74y

52y

33y

74y

111

n143

− 436800x111 x

32x

33x

74y

32y

33y

74y

111

n132+

174720x111 x2x

33x

74y2y

33y

74y

111

n121


+17472x11

1 x72x3x

74y

72y3y

74y

111

n143− 104832x11

1 x52x3x

74y

52y3y

74y

111

n132

+174720x11

1 x32x3x

74y

32y3y

74y

111

n121− 69888x11

1 x2x3x74y2y3y

74y

111

n110

+26208x11

1 x72x

73x

54y

72y

73y

54y

111

n165− 157248x11

1 x52x

73x

54y

52y

73y

54y

111

n154

+262080x11

1 x32x

73x

54y

32y

73y

54y

111

n143− 104832x11

1 x2x73x

54y2y

73y

54y

111

n132

− 157248x111 x

72x

53x

54y

72y

53y

54y

111

n154+

943488x111 x

52x

53x

54y

52y

53y

54y

111

n143

− 1572480x111 x

32x

53x

54y

32y

53y

54y

111

n132+

628992x111 x2x

53x

54y2y

53y

54y

111

n121

+262080x11

1 x72x

33x

54y

72y

33y

54y

111

n143− 1572480x11

1 x52x

33x

54y

52y

33y

54y

111

n132

+2620800x11

1 x32x

33x

54y

32y

33y

54y

111

n121− 1048320x11

1 x2x33x

54y2y

33y

54y

111

n110

− 104832x111 x

72x3x

54y

72y3y

54y

111

n132+

628992x111 x

52x3x

54y

52y3y

54y

111

n121

− 1048320x111 x

32x3x

54y

32y3y

54y

111

n110+

419328x111 x2x3x

54y2y3y

54y

111

n99

− 43680x111 x

72x

73x

34y

72y

73y

34y

111

n154+

262080x111 x

52x

73x

34y

52y

73y

34y

111

n143

− 436800x111 x

32x

73x

34y

32y

73y

34y

111

n132+

174720x111 x2x

73x

34y2y

73y

34y

111

n121

+262080x11

1 x72x

53x

34y

72y

53y

34y

111

n143− 1572480x11

1 x52x

53x

34y

52y

53y

34y

111

n132

+2620800x11

1 x32x

53x

34y

32y

53y

34y

111

n121− 1048320x11

1 x2x53x

34y2y

53y

34y

111

n110

− 436800x111 x

72x

33x

34y

72y

33y

34y

111

n132+

2620800x111 x

52x

33x

34y

52y

33y

34y

111

n121

− 4368000x111 x

32x

33x

34y

32y

33y

34y

111

n110+

1747200x111 x2x

33x

34y2y

33y

34y

111

n99

+174720x11

1 x72x3x

34y

72y3y

34y

111

n121− 1048320x11

1 x52x3x

34y

52y3y

34y

111

n110

+1747200x11

1 x32x3x

34y

32y3y

34y

111

n99− 698880x11

1 x2x3x34y2y3y

34y

111

n88

+17472x11

1 x72x

73x4y

72y

73y4y

111

n143− 104832x11

1 x52x

73x4y

52y

73y4y

111

n132

+174720x11

1 x32x

73x4y

32y

73y4y

111

n121− 69888x11

1 x2x73x4y2y

73y4y

111

n110

− 104832x111 x

72x

53x4y

72y

53y4y

111

n132+

628992x111 x

52x

53x4y

52y

53y4y

111

n121

− 1048320x111 x

32x

53x4y

32y

53y4y

111

n110+

419328x111 x2x

53x4y2y

53y4y

111

n99

+174720x11

1 x72x

33x4y

72y

33y4y

111

n121− 1048320x11

1 x52x

33x4y

52y

33y4y

111

n110

+1747200x11

1 x32x

33x4y

32y

33y4y

111

n99− 698880x11

1 x2x33x4y2y

33y4y

111

n88

− 69888x111 x

72x3x4y

72y3y4y

111

n110+

419328x111 x

52x3x4y

52y3y4y

111

n99


− 698880x111 x

32x3x4y

32y3y4y

111

n88+

279552x111 x2x3x4y2y3y4y

111

n77

+5005x9

1x72x

73x

74y

72y

73y

74y

91

n165− 30030x9

1x52x

73x

74y

52y

73y

74y

91

n154

+50050x9

1x32x

73x

74y

32y

73y

74y

91

n143− 20020x9

1x2x73x

74y2y

73y

74y

91

n132

− 30030x91x

72x

53x

74y

72y

53y

74y

91

n154+

180180x91x

52x

53x

74y

52y

53y

74y

91

n143

− 300300x91x

32x

53x

74y

32y

53y

74y

91

n132+

120120x91x2x

53x

74y2y

53y

74y

91

n121

+50050x9

1x72x

33x

74y

72y

33y

74y

91

n143− 300300x9

1x52x

33x

74y

52y

33y

74y

91

n132

+500500x9

1x32x

33x

74y

32y

33y

74y

91

n121− 200200x9

1x2x33x

74y2y

33y

74y

91

n110

− 20020x91x

72x3x

74y

72y3y

74y

91

n132+

120120x91x

52x3x

74y

52y3y

74y

91

n121

− 200200x91x

32x3x

74y

32y3y

74y

91

n110+

80080x91x2x3x

74y2y3y

74y

91

n99

− 30030x91x

72x

73x

54y

72y

73y

54y

91

n154+

180180x91x

52x

73x

54y

52y

73y

54y

91

n143

− 300300x91x

32x

73x

54y

32y

73y

54y

91

n132+

120120x91x2x

73x

54y2y

73y

54y

91

n121

+180180x9

1x72x

53x

54y

72y

53y

54y

91

n143− 1081080x9

1x52x

53x

54y

52y

53y

54y

91

n132

+1801800x9

1x32x

53x

54y

32y

53y

54y

91

n121− 720720x9

1x2x53x

54y2y

53y

54y

91

n110

− 300300x91x

72x

33x

54y

72y

33y

54y

91

n132+

1801800x91x

52x

33x

54y

52y

33y

54y

91

n121

− 3003000x91x

32x

33x

54y

32y

33y

54y

91

n110+

1201200x91x2x

33x

54y2y

33y

54y

91

n99

+120120x9

1x72x3x

54y

72y3y

54y

91

n121− 720720x9

1x52x3x

54y

52y3y

54y

91

n110

+1201200x9

1x32x3x

54y

32y3y

54y

91

n99− 480480x9

1x2x3x54y2y3y

54y

91

n88

+50050x9

1x72x

73x

34y

72y

73y

34y

91

n143− 300300x9

1x52x

73x

34y

52y

73y

34y

91

n132

+500500x9

1x32x

73x

34y

32y

73y

34y

91

n121− 200200x9

1x2x73x

34y2y

73y

34y

91

n110

− 300300x91x

72x

53x

34y

72y

53y

34y

91

n132+

1801800x91x

52x

53x

34y

52y

53y

34y

91

n121

− 3003000x91x

32x

53x

34y

32y

53y

34y

91

n110+

1201200x91x2x

53x

34y2y

53y

34y

91

n99

+500500x9

1x72x

33x

34y

72y

33y

34y

91

n121− 3003000x9

1x52x

33x

34y

52y

33y

34y

91

n110

+5005000x9

1x32x

33x

34y

32y

33y

34y

91

n99− 2002000x9

1x2x33x

34y2y

33y

34y

91

n88

− 200200x91x

72x3x

34y

72y3y

34y

91

n110+

1201200x91x

52x3x

34y

52y3y

34y

91

n99

− 2002000x91x

32x3x

34y

32y3y

34y

91

n88+

800800x91x2x3x

34y2y3y

34y

91

n77


− 20020x91x

72x

73x4y

72y

73y4y

91

n132+

120120x91x

52x

73x4y

52y

73y4y

91

n121

− 200200x91x

32x

73x4y

32y

73y4y

91

n110+

80080x91x2x

73x4y2y

73y4y

91

n99

+120120x9

1x72x

53x4y

72y

53y4y

91

n121− 720720x9

1x52x

53x4y

52y

53y4y

91

n110

+1201200x9

1x32x

53x4y

32y

53y4y

91

n99− 480480x9

1x2x53x4y2y

53y4y

91

n88

− 200200x91x

72x

33x4y

72y

33y4y

91

n110+

1201200x91x

52x

33x4y

52y

33y4y

91

n99

− 2002000x91x

32x

33x4y

32y

33y4y

91

n88+

800800x91x2x

33x4y2y

33y4y

91

n77

+80080x9

1x72x3x4y

72y3y4y

91

n99− 480480x9

1x52x3x4y

52y3y4y

91

n88

+800800x9

1x32x3x4y

32y3y4y

91

n77− 320320x9

1x2x3x4y2y3y4y91

n66

− 3432x71x

72x

73x

74y

72y

73y

74y

71

n154+

20592x71x

52x

73x

74y

52y

73y

74y

71

n143

− 34320x71x

32x

73x

74y

32y

73y

74y

71

n132+

13728x71x2x

73x

74y2y

73y

74y

71

n121

+20592x7

1x72x

53x

74y

72y

53y

74y

71

n143− 123552x7

1x52x

53x

74y

52y

53y

74y

71

n132

+205920x7

1x32x

53x

74y

32y

53y

74y

71

n121− 82368x7

1x2x53x

74y2y

53y

74y

71

n110

− 34320x71x

72x

33x

74y

72y

33y

74y

71

n132+

205920x71x

52x

33x

74y

52y

33y

74y

71

n121

− 343200x71x

32x

33x

74y

32y

33y

74y

71

n110+

137280x71x2x

33x

74y2y

33y

74y

71

n99

+13728x7

1x72x3x

74y

72y3y

74y

71

n121− 82368x7

1x52x3x

74y

52y3y

74y

71

n110

+137280x7

1x32x3x

74y

32y3y

74y

71

n99− 54912x7

1x2x3x74y2y3y

74y

71

n88

+20592x7

1x72x

73x

54y

72y

73y

54y

71

n143− 123552x7

1x52x

73x

54y

52y

73y

54y

71

n132

+205920x7

1x32x

73x

54y

32y

73y

54y

71

n121− 82368x7

1x2x73x

54y2y

73y

54y

71

n110

− 123552x71x

72x

53x

54y

72y

53y

54y

71

n132+

741312x71x

52x

53x

54y

52y

53y

54y

71

n121

− 1235520x71x

32x

53x

54y

32y

53y

54y

71

n110+

494208x71x2x

53x

54y2y

53y

54y

71

n99

+205920x7

1x72x

33x

54y

72y

33y

54y

71

n121− 1235520x7

1x52x

33x

54y

52y

33y

54y

71

n110

+2059200x7

1x32x

33x

54y

32y

33y

54y

71

n99− 823680x7

1x2x33x

54y2y

33y

54y

71

n88

− 82368x71x

72x3x

54y

72y3y

54y

71

n110+

494208x71x

52x3x

54y

52y3y

54y

71

n99

− 823680x71x

32x3x

54y

32y3y

54y

71

n88+

329472x71x2x3x

54y2y3y

54y

71

n77

− 34320x71x

72x

73x

34y

72y

73y

34y

71

n132+

205920x71x

52x

73x

34y

52y

73y

34y

71

n121


− 343200x71x

32x

73x

34y

32y

73y

34y

71

n110+

137280x71x2x

73x

34y2y

73y

34y

71

n99

+205920x7

1x72x

53x

34y

72y

53y

34y

71

n121− 1235520x7

1x52x

53x

34y

52y

53y

34y

71

n110

+2059200x7

1x32x

53x

34y

32y

53y

34y

71

n99− 823680x7

1x2x53x

34y2y

53y

34y

71

n88

− 343200x71x

72x

33x

34y

72y

33y

34y

71

n110+

2059200x71x

52x

33x

34y

52y

33y

34y

71

n99

− 3432000x71x

32x

33x

34y

32y

33y

34y

71

n88+

1372800x71x2x

33x

34y2y

33y

34y

71

n77

+137280x7

1x72x3x

34y

72y3y

34y

71

n99− 823680x7

1x52x3x

34y

52y3y

34y

71

n88

+1372800x7

1x32x3x

34y

32y3y

34y

71

n77− 549120x7

1x2x3x34y2y3y

34y

71

n66

+13728x7

1x72x

73x4y

72y

73y4y

71

n121− 82368x7

1x52x

73x4y

52y

73y4y

71

n110

+137280x7

1x32x

73x4y

32y

73y4y

71

n99− 54912x7

1x2x73x4y2y

73y4y

71

n88

− 82368x71x

72x

53x4y

72y

53y4y

71

n110+

494208x71x

52x

53x4y

52y

53y4y

71

n99

− 823680x71x

32x

53x4y

32y

53y4y

71

n88+

329472x71x2x

53x4y2y

53y4y

71

n77

+137280x7

1x72x

33x4y

72y

33y4y

71

n99− 823680x7

1x52x

33x4y

52y

33y4y

71

n88

+1372800x7

1x32x

33x4y

32y

33y4y

71

n77− 549120x7

1x2x33x4y2y

33y4y

71

n66

− 54912x71x

72x3x4y

72y3y4y

71

n88+

329472x71x

52x3x4y

52y3y4y

71

n77

− 549120x71x

32x3x4y

32y3y4y

71

n66+

219648x71x2x3x4y2y3y4y

71

n55

+1287x5

1x72x

73x

74y

72y

73y

74y

51

n143− 7722x5

1x52x

73x

74y

52y

73y

74y

51

n132

+12870x5

1x32x

73x

74y

32y

73y

74y

51

n121− 5148x5

1x2x73x

74y2y

73y

74y

51

n110

− 7722x51x

72x

53x

74y

72y

53y

74y

51

n132+

46332x51x

52x

53x

74y

52y

53y

74y

51

n121

− 77220x51x

32x

53x

74y

32y

53y

74y

51

n110+

30888x51x2x

53x

74y2y

53y

74y

51

n99

+12870x5

1x72x

33x

74y

72y

33y

74y

51

n121− 77220x5

1x52x

33x

74y

52y

33y

74y

51

n110

+128700x5

1x32x

33x

74y

32y

33y

74y

51

n99− 51480x5

1x2x33x

74y2y

33y

74y

51

n88

− 5148x51x

72x3x

74y

72y3y

74y

51

n110+

30888x51x

52x3x

74y

52y3y

74y

51

n99

− 51480x51x

32x3x

74y

32y3y

74y

51

n88+

20592x51x2x3x

74y2y3y

74y

51

n77

− 7722x51x

72x

73x

54y

72y

73y

54y

51

n132+

46332x51x

52x

73x

54y

52y

73y

54y

51

n121

− 77220x51x

32x

73x

54y

32y

73y

54y

51

n110+

30888x51x2x

73x

54y2y

73y

54y

51

n99


+46332x5

1x72x

53x

54y

72y

53y

54y

51

n121− 277992x5

1x52x

53x

54y

52y

53y

54y

51

n110

+463320x5

1x32x

53x

54y

32y

53y

54y

51

n99− 185328x5

1x2x53x

54y2y

53y

54y

51

n88

− 77220x51x

72x

33x

54y

72y

33y

54y

51

n110+

463320x51x

52x

33x

54y

52y

33y

54y

51

n99

− 772200x51x

32x

33x

54y

32y

33y

54y

51

n88+

308880x51x2x

33x

54y2y

33y

54y

51

n77

+30888x5

1x72x3x

54y

72y3y

54y

51

n99− 185328x5

1x52x3x

54y

52y3y

54y

51

n88

+308880x5

1x32x3x

54y

32y3y

54y

51

n77− 123552x5

1x2x3x54y2y3y

54y

51

n66

+12870x5

1x72x

73x

34y

72y

73y

34y

51

n121− 77220x5

1x52x

73x

34y

52y

73y

34y

51

n110

+128700x5

1x32x

73x

34y

32y

73y

34y

51

n99− 51480x5

1x2x73x

34y2y

73y

34y

51

n88

− 77220x51x

72x

53x

34y

72y

53y

34y

51

n110+

463320x51x

52x

53x

34y

52y

53y

34y

51

n99

− 772200x51x

32x

53x

34y

32y

53y

34y

51

n88+

308880x51x2x

53x

34y2y

53y

34y

51

n77

+128700x5

1x72x

33x

34y

72y

33y

34y

51

n99− 772200x5

1x52x

33x

34y

52y

33y

34y

51

n88

+1287000x5

1x32x

33x

34y

32y

33y

34y

51

n77− 514800x5

1x2x33x

34y2y

33y

34y

51

n66

− 51480x51x

72x3x

34y

72y3y

34y

51

n88+

308880x51x

52x3x

34y

52y3y

34y

51

n77

− 514800x51x

32x3x

34y

32y3y

34y

51

n66+

205920x51x2x3x

34y2y3y

34y

51

n55

− 5148x51x

72x

73x4y

72y

73y4y

51

n110+

30888x51x

52x

73x4y

52y

73y4y

51

n99

− 51480x51x

32x

73x4y

32y

73y4y

51

n88+

20592x51x2x

73x4y2y

73y4y

51

n77

+30888x5

1x72x

53x4y

72y

53y4y

51

n99− 185328x5

1x52x

53x4y

52y

53y4y

51

n88

+308880x5

1x32x

53x4y

32y

53y4y

51

n77− 123552x5

1x2x53x4y2y

53y4y

51

n66

− 51480x51x

72x

33x4y

72y

33y4y

51

n88+

308880x51x

52x

33x4y

52y

33y4y

51

n77

− 514800x51x

32x

33x4y

32y

33y4y

51

n66+

205920x51x2x

33x4y2y

33y4y

51

n55

+20592x5

1x72x3x4y

72y3y4y

51

n77− 123552x5

1x52x3x4y

52y3y4y

51

n66

+205920x5

1x32x3x4y

32y3y4y

51

n55− 82368x5

1x2x3x4y2y3y4y51

n44

− 220x31x

72x

73x

74y

72y

73y

74y

31

n132+

1320x31x

52x

73x

74y

52y

73y

74y

31

n121

− 2200x31x

32x

73x

74y

32y

73y

74y

31

n110+

880x31x2x

73x

74y2y

73y

74y

31

n99

+1320x3

1x72x

53x

74y

72y

53y

74y

31

n121− 7920x3

1x52x

53x

74y

52y

53y

74y

31

n110


+13200x3

1x32x

53x

74y

32y

53y

74y

31

n99− 5280x3

1x2x53x

74y2y

53y

74y

31

n88

− 2200x31x

72x

33x

74y

72y

33y

74y

31

n110+

13200x31x

52x

33x

74y

52y

33y

74y

31

n99

− 22000x31x

32x

33x

74y

32y

33y

74y

31

n88+

8800x31x2x

33x

74y2y

33y

74y

31

n77

+880x3

1x72x3x

74y

72y3y

74y

31

n99− 5280x3

1x52x3x

74y

52y3y

74y

31

n88

+8800x3

1x32x3x

74y

32y3y

74y

31

n77− 3520x3

1x2x3x74y2y3y

74y

31

n66

+1320x3

1x72x

73x

54y

72y

73y

54y

31

n121− 7920x3

1x52x

73x

54y

52y

73y

54y

31

n110

+13200x3

1x32x

73x

54y

32y

73y

54y

31

n99− 5280x3

1x2x73x

54y2y

73y

54y

31

n88

− 7920x31x

72x

53x

54y

72y

53y

54y

31

n110+

47520x31x

52x

53x

54y

52y

53y

54y

31

n99

− 79200x31x

32x

53x

54y

32y

53y

54y

31

n88+

31680x31x2x

53x

54y2y

53y

54y

31

n77

+13200x3

1x72x

33x

54y

72y

33y

54y

31

n99− 79200x3

1x52x

33x

54y

52y

33y

54y

31

n88

+132000x3

1x32x

33x

54y

32y

33y

54y

31

n77− 52800x3

1x2x33x

54y2y

33y

54y

31

n66

− 5280x31x

72x3x

54y

72y3y

54y

31

n88+

31680x31x

52x3x

54y

52y3y

54y

31

n77

− 52800x31x

32x3x

54y

32y3y

54y

31

n66+

21120x31x2x3x

54y2y3y

54y

31

n55

− 2200x31x

72x

73x

34y

72y

73y

34y

31

n110+

13200x31x

52x

73x

34y

52y

73y

34y

31

n99

− 22000x31x

32x

73x

34y

32y

73y

34y

31

n88+

8800x31x2x

73x

34y2y

73y

34y

31

n77

+13200x3

1x72x

53x

34y

72y

53y

34y

31

n99− 79200x3

1x52x

53x

34y

52y

53y

34y

31

n88

+132000x3

1x32x

53x

34y

32y

53y

34y

31

n77− 52800x3

1x2x53x

34y2y

53y

34y

31

n66

− 22000x31x

72x

33x

34y

72y

33y

34y

31

n88+

132000x31x

52x

33x

34y

52y

33y

34y

31

n77

− 220000x31x

32x

33x

34y

32y

33y

34y

31

n66+

88000x31x2x

33x

34y2y

33y

34y

31

n55

+8800x3

1x72x3x

34y

72y3y

34y

31

n77− 52800x3

1x52x3x

34y

52y3y

34y

31

n66

+88000x3

1x32x3x

34y

32y3y

34y

31

n55− 35200x3

1x2x3x34y2y3y

34y

31

n44

+880x3

1x72x

73x4y

72y

73y4y

31

n99− 5280x3

1x52x

73x4y

52y

73y4y

31

n88

+8800x3

1x32x

73x4y

32y

73y4y

31

n77− 3520x3

1x2x73x4y2y

73y4y

31

n66

− 5280x31x

72x

53x4y

72y

53y4y

31

n88+

31680x31x

52x

53x4y

52y

53y4y

31

n77

− 52800x31x

32x

53x4y

32y

53y4y

31

n66+

21120x31x2x

53x4y2y

53y4y

31

n55


+8800x3

1x72x

33x4y

72y

33y4y

31

n77− 52800x3

1x52x

33x4y

52y

33y4y

31

n66

+88000x3

1x32x

33x4y

32y

33y4y

31

n55− 35200x3

1x2x33x4y2y

33y4y

31

n44

− 3520x31x

72x3x4y

72y3y4y

31

n66+

21120x31x

52x3x4y

52y3y4y

31

n55

− 35200x31x

32x3x4y

32y3y4y

31

n44+

14080x31x2x3x4y2y3y4y

31

n33

+11x1x

72x

73x

74y

72y

73y

74y1

n121− 66x1x

52x

73x

74y

52y

73y

74y1

n110

+110x1x

32x

73x

74y

32y

73y

74y1

n99− 44x1x2x

73x

74y2y

73y

74y1

n88

− 66x1x72x

53x

74y

72y

53y

74y1

n110+

396x1x52x

53x

74y

52y

53y

74y1

n99

− 660x1x32x

53x

74y

32y

53y

74y1

n88+

264x1x2x53x

74y2y

53y

74y1

n77

+110x1x

72x

33x

74y

72y

33y

74y1

n99− 660x1x

52x

33x

74y

52y

33y

74y1

n88

+1100x1x

32x

33x

74y

32y

33y

74y1

n77− 440x1x2x

33x

74y2y

33y

74y1

n66

− 44x1x72x3x

74y

72y3y

74y1

n88+

264x1x52x3x

74y

52y3y

74y1

n77

− 440x1x32x3x

74y

32y3y

74y1

n66+

176x1x2x3x74y2y3y

74y1

n55

− 66x1x72x

73x

54y

72y

73y

54y1

n110+

396x1x52x

73x

54y

52y

73y

54y1

n99

− 660x1x32x

73x

54y

32y

73y

54y1

n88+

264x1x2x73x

54y2y

73y

54y1

n77

+396x1x

72x

53x

54y

72y

53y

54y1

n99− 2376x1x

52x

53x

54y

52y

53y

54y1

n88

+3960x1x

32x

53x

54y

32y

53y

54y1

n77− 1584x1x2x

53x

54y2y

53y

54y1

n66

− 660x1x72x

33x

54y

72y

33y

54y1

n88+

3960x1x52x

33x

54y

52y

33y

54y1

n77

− 6600x1x32x

33x

54y

32y

33y

54y1

n66+

2640x1x2x33x

54y2y

33y

54y1

n55

+264x1x

72x3x

54y

72y3y

54y1

n77− 1584x1x

52x3x

54y

52y3y

54y1

n66

+2640x1x

32x3x

54y

32y3y

54y1

n55− 1056x1x2x3x

54y2y3y

54y1

n44

+110x1x

72x

73x

34y

72y

73y

34y1

n99− 660x1x

52x

73x

34y

52y

73y

34y1

n88

+1100x1x

32x

73x

34y

32y

73y

34y1

n77− 440x1x2x

73x

34y2y

73y

34y1

n66

− 660x1x72x

53x

34y

72y

53y

34y1

n88+

3960x1x52x

53x

34y

52y

53y

34y1

n77

− 6600x1x32x

53x

34y

32y

53y

34y1

n66+

2640x1x2x53x

34y2y

53y

34y1

n55

+1100x1x

72x

33x

34y

72y

33y

34y1

n77− 6600x1x

52x

33x

34y

52y

33y

34y1

n66


+11000x1x

32x

33x

34y

32y

33y

34y1

n55− 4400x1x2x

33x

34y2y

33y

34y1

n44

− 440x1x72x3x

34y

72y3y

34y1

n66+

2640x1x52x3x

34y

52y3y

34y1

n55

− 4400x1x32x3x

34y

32y3y

34y1

n44+

1760x1x2x3x34y2y3y

34y1

n33

− 44x1x72x

73x4y

72y

73y4y1

n88+

264x1x52x

73x4y

52y

73y4y1

n77

− 440x1x32x

73x4y

32y

73y4y1

n66+

176x1x2x73x4y2y

73y4y1

n55

+264x1x

72x

53x4y

72y

53y4y1

n77− 1584x1x

52x

53x4y

52y

53y4y1

n66

+2640x1x

32x

53x4y

32y

53y4y1

n55− 1056x1x2x

53x4y2y

53y4y1

n44

− 440x1x72x

33x4y

72y

33y4y1

n66+

2640x1x52x

33x4y

52y

33y4y1

n55

− 4400x1x32x

33x4y

32y

33y4y1

n44+

1760x1x2x33x4y2y

33y4y1

n33

+176x1x

72x3x4y

72y3y4y1

n55− 1056x1x

52x3x4y

52y3y4y1

n44

+1760x1x

32x3x4y

32y3y4y1

n33− 704x1x2x3x4y2y3y4y1

n22

Gf,n,4,(α),8(x1, . . . , y4) =x24

1 y241

n132+x24

1 x82y

82y

241

n176

+x24

1 x83y

83y

241

n176+x24

1 x82x

83y

82y

83y

241

n220

− 7x241 x

62x

83y

62y

83y

241

n209+

15x241 x

42x

83y

42y

83y

241

n198

− 10x241 x

22x

83y

22y

83y

241

n187+x24

1 x84y

84y

241

n176

+x24

1 x82x

84y

82y

84y

241

n220+x24

1 x83x

84y

83y

84y

241

n220

+x24

1 x82x

83x

84y

82y

83y

84y

241

n264− 7x24

1 x62x

83x

84y

62y

83y

84y

241

n253

+15x24

1 x42x

83x

84y

42y

83y

84y

241

n242− 10x24

1 x22x

83x

84y

22y

83y

84y

241

n231

− 7x241 x

62x

84y

62y

84y

241

n209− 7x24

1 x63x

84y

63y

84y

241

n209

− 7x241 x

82x

63x

84y

82y

63y

84y

241

n253+

49x241 x

62x

63x

84y

62y

63y

84y

241

n242

− 105x241 x

42x

63x

84y

42y

63y

84y

241

n231+

70x241 x

22x

63x

84y

22y

63y

84y

241

n220

+15x24

1 x42x

84y

42y

84y

241

n198+

15x241 x

43x

84y

43y

84y

241

n198

+15x24

1 x82x

43x

84y

82y

43y

84y

241

n242− 105x24

1 x62x

43x

84y

62y

43y

84y

241

n231

+225x24

1 x42x

43x

84y

42y

43y

84y

241

n220− 150x24

1 x22x

43x

84y

22y

43y

84y

241

n209

− 10x241 x

22x

84y

22y

84y

241

n187− 10x24

1 x23x

84y

23y

84y

241

n187


− 10x241 x

82x

23x

84y

82y

23y

84y

241

n231+

70x241 x

62x

23x

84y

62y

23y

84y

241

n220

− 150x241 x

42x

23x

84y

42y

23y

84y

241

n209+

100x241 x

22x

23x

84y

22y

23y

84y

241

n198

− 7x241 x

62y

62y

241

n165− 7x24

1 x82x

63y

82y

63y

241

n209

− 7x241 x

63y

63y

241

n165+

49x241 x

62x

63y

62y

63y

241

n198

− 105x241 x

42x

63y

42y

63y

241

n187+

70x241 x

22x

63y

22y

63y

241

n176

− 7x241 x

82x

64y

82y

64y

241

n209− 7x24

1 x82x

83x

64y

82y

83y

64y

241

n253

− 7x241 x

83x

64y

83y

64y

241

n209+

49x241 x

62x

83x

64y

62y

83y

64y

241

n242

− 105x241 x

42x

83x

64y

42y

83y

64y

241

n231+

70x241 x

22x

83x

64y

22y

83y

64y

241

n220

− 7x241 x

64y

64y

241

n165+

49x241 x

62x

64y

62y

64y

241

n198

+49x24

1 x82x

63x

64y

82y

63y

64y

241

n242+

49x241 x

63x

64y

63y

64y

241

n198

− 343x241 x

62x

63x

64y

62y

63y

64y

241

n231+

735x241 x

42x

63x

64y

42y

63y

64y

241

n220

− 490x241 x

22x

63x

64y

22y

63y

64y

241

n209− 105x24

1 x42x

64y

42y

64y

241

n187

− 105x241 x

82x

43x

64y

82y

43y

64y

241

n231− 105x24

1 x43x

64y

43y

64y

241

n187

+735x24

1 x62x

43x

64y

62y

43y

64y

241

n220− 1575x24

1 x42x

43x

64y

42y

43y

64y

241

n209

+1050x24

1 x22x

43x

64y

22y

43y

64y

241

n198+

70x241 x

22x

64y

22y

64y

241

n176

+70x24

1 x82x

23x

64y

82y

23y

64y

241

n220+

70x241 x

23x

64y

23y

64y

241

n176

− 490x241 x

62x

23x

64y

62y

23y

64y

241

n209+

1050x241 x

42x

23x

64y

42y

23y

64y

241

n198

− 700x241 x

22x

23x

64y

22y

23y

64y

241

n187+

15x241 x

42y

42y

241

n154

+15x24

1 x82x

43y

82y

43y

241

n198− 105x24

1 x62x

43y

62y

43y

241

n187

+15x24

1 x43y

43y

241

n154+

225x241 x

42x

43y

42y

43y

241

n176

− 150x241 x

22x

43y

22y

43y

241

n165+

15x241 x

82x

44y

82y

44y

241

n198

+15x24

1 x82x

83x

44y

82y

83y

44y

241

n242− 105x24

1 x62x

83x

44y

62y

83y

44y

241

n231

+15x24

1 x83x

44y

83y

44y

241

n198+

225x241 x

42x

83x

44y

42y

83y

44y

241

n220

− 150x241 x

22x

83x

44y

22y

83y

44y

241

n209− 105x24

1 x62x

44y

62y

44y

241

n187

− 105x241 x

82x

63x

44y

82y

63y

44y

241

n231+

735x241 x

62x

63x

44y

62y

63y

44y

241

n220


− 105x241 x

63x

44y

63y

44y

241

n187− 1575x24

1 x42x

63x

44y

42y

63y

44y

241

n209

+1050x24

1 x22x

63x

44y

22y

63y

44y

241

n198+

15x241 x

44y

44y

241

n154

+225x24

1 x42x

44y

42y

44y

241

n176+

225x241 x

82x

43x

44y

82y

43y

44y

241

n220

− 1575x241 x

62x

43x

44y

62y

43y

44y

241

n209+

225x241 x

43x

44y

43y

44y

241

n176

+3375x24

1 x42x

43x

44y

42y

43y

44y

241

n198− 2250x24

1 x22x

43x

44y

22y

43y

44y

241

n187

− 150x241 x

22x

44y

22y

44y

241

n165− 150x24

1 x82x

23x

44y

82y

23y

44y

241

n209

+1050x24

1 x62x

23x

44y

62y

23y

44y

241

n198− 150x24

1 x23x

44y

23y

44y

241

n165

− 2250x241 x

42x

23x

44y

42y

23y

44y

241

n187+

1500x241 x

22x

23x

44y

22y

23y

44y

241

n176

− 10x241 x

22y

22y

241

n143− 10x24

1 x82x

23y

82y

23y

241

n187

+70x24

1 x62x

23y

62y

23y

241

n176− 150x24

1 x42x

23y

42y

23y

241

n165

− 10x241 x

23y

23y

241

n143+

100x241 x

22x

23y

22y

23y

241

n154

− 10x241 x

82x

24y

82y

24y

241

n187− 10x24

1 x82x

83x

24y

82y

83y

24y

241

n231

+70x24

1 x62x

83x

24y

62y

83y

24y

241

n220− 150x24

1 x42x

83x

24y

42y

83y

24y

241

n209

− 10x241 x

83x

24y

83y

24y

241

n187+

100x241 x

22x

83x

24y

22y

83y

24y

241

n198

+70x24

1 x62x

24y

62y

24y

241

n176+

70x241 x

82x

63x

24y

82y

63y

24y

241

n220

− 490x241 x

62x

63x

24y

62y

63y

24y

241

n209+

1050x241 x

42x

63x

24y

42y

63y

24y

241

n198

+70x24

1 x63x

24y

63y

24y

241

n176− 700x24

1 x22x

63x

24y

22y

63y

24y

241

n187

− 150x241 x

42x

24y

42y

24y

241

n165− 150x24

1 x82x

43x

24y

82y

43y

24y

241

n209

+1050x24

1 x62x

43x

24y

62y

43y

24y

241

n198− 2250x24

1 x42x

43x

24y

42y

43y

24y

241

n187

− 150x241 x

43x

24y

43y

24y

241

n165+

1500x241 x

22x

43x

24y

22y

43y

24y

241

n176

− 10x241 x

24y

24y

241

n143+

100x241 x

22x

24y

22y

24y

241

n154

+100x24

1 x82x

23x

24y

82y

23y

24y

241

n198− 700x24

1 x62x

23x

24y

62y

23y

24y

241

n187

+1500x24

1 x42x

23x

24y

42y

23y

24y

241

n176+

100x241 x

23x

24y

23y

24y

241

n154

− 1000x241 x

22x

23x

24y

22y

23y

24y

241

n165− 23x22

1 y221

n121

− 23x221 x

82y

82y

221

n165− 23x22

1 x83y

83y

221

n165


− 23x221 x

82x

83y

82y

83y

221

n209+

161x221 x

62x

83y

62y

83y

221

n198

− 345x221 x

42x

83y

42y

83y

221

n187+

230x221 x

22x

83y

22y

83y

221

n176

− 23x221 x

84y

84y

221

n165− 23x22

1 x82x

84y

82y

84y

221

n209

− 23x221 x

83x

84y

83y

84y

221

n209− 23x22

1 x82x

83x

84y

82y

83y

84y

221

n253

+161x22

1 x62x

83x

84y

62y

83y

84y

221

n242− 345x22

1 x42x

83x

84y

42y

83y

84y

221

n231

+230x22

1 x22x

83x

84y

22y

83y

84y

221

n220+

161x221 x

62x

84y

62y

84y

221

n198

+161x22

1 x63x

84y

63y

84y

221

n198+

161x221 x

82x

63x

84y

82y

63y

84y

221

n242

− 1127x221 x

62x

63x

84y

62y

63y

84y

221

n231+

2415x221 x

42x

63x

84y

42y

63y

84y

221

n220

− 1610x221 x

22x

63x

84y

22y

63y

84y

221

n209− 345x22

1 x42x

84y

42y

84y

221

n187

− 345x221 x

43x

84y

43y

84y

221

n187− 345x22

1 x82x

43x

84y

82y

43y

84y

221

n231

+2415x22

1 x62x

43x

84y

62y

43y

84y

221

n220− 5175x22

1 x42x

43x

84y

42y

43y

84y

221

n209

+3450x22

1 x22x

43x

84y

22y

43y

84y

221

n198+

230x221 x

22x

84y

22y

84y

221

n176

+230x22

1 x23x

84y

23y

84y

221

n176+

230x221 x

82x

23x

84y

82y

23y

84y

221

n220

− 1610x221 x

62x

23x

84y

62y

23y

84y

221

n209+

3450x221 x

42x

23x

84y

42y

23y

84y

221

n198

− 2300x221 x

22x

23x

84y

22y

23y

84y

221

n187+

161x221 x

62y

62y

221

n154

+161x22

1 x82x

63y

82y

63y

221

n198+

161x221 x

63y

63y

221

n154

− 1127x221 x

62x

63y

62y

63y

221

n187+

2415x221 x

42x

63y

42y

63y

221

n176

− 1610x221 x

22x

63y

22y

63y

221

n165+

161x221 x

82x

64y

82y

64y

221

n198

+161x22

1 x82x

83x

64y

82y

83y

64y

221

n242+

161x221 x

83x

64y

83y

64y

221

n198

− 1127x221 x

62x

83x

64y

62y

83y

64y

221

n231+

2415x221 x

42x

83x

64y

42y

83y

64y

221

n220

− 1610x221 x

22x

83x

64y

22y

83y

64y

221

n209+

161x221 x

64y

64y

221

n154

− 1127x221 x

62x

64y

62y

64y

221

n187− 1127x22

1 x82x

63x

64y

82y

63y

64y

221

n231

− 1127x221 x

63x

64y

63y

64y

221

n187+

7889x221 x

62x

63x

64y

62y

63y

64y

221

n220

− 16905x221 x

42x

63x

64y

42y

63y

64y

221

n209+

11270x221 x

22x

63x

64y

22y

63y

64y

221

n198

+2415x22

1 x42x

64y

42y

64y

221

n176+

2415x221 x

82x

43x

64y

82y

43y

64y

221

n220


+2415x22

1 x43x

64y

43y

64y

221

n176− 16905x22

1 x62x

43x

64y

62y

43y

64y

221

n209

+36225x22

1 x42x

43x

64y

42y

43y

64y

221

n198− 24150x22

1 x22x

43x

64y

22y

43y

64y

221

n187

− 1610x221 x

22x

64y

22y

64y

221

n165− 1610x22

1 x82x

23x

64y

82y

23y

64y

221

n209

− 1610x221 x

23x

64y

23y

64y

221

n165+

11270x221 x

62x

23x

64y

62y

23y

64y

221

n198

− 24150x221 x

42x

23x

64y

42y

23y

64y

221

n187+

16100x221 x

22x

23x

64y

22y

23y

64y

221

n176

− 345x221 x

42y

42y

221

n143− 345x22

1 x82x

43y

82y

43y

221

n187

+2415x22

1 x62x

43y

62y

43y

221

n176− 345x22

1 x43y

43y

221

n143

− 5175x221 x

42x

43y

42y

43y

221

n165+

3450x221 x

22x

43y

22y

43y

221

n154

− 345x221 x

82x

44y

82y

44y

221

n187− 345x22

1 x82x

83x

44y

82y

83y

44y

221

n231

+2415x22

1 x62x

83x

44y

62y

83y

44y

221

n220− 345x22

1 x83x

44y

83y

44y

221

n187

− 5175x221 x

42x

83x

44y

42y

83y

44y

221

n209+

3450x221 x

22x

83x

44y

22y

83y

44y

221

n198

+2415x22

1 x62x

44y

62y

44y

221

n176+

2415x221 x

82x

63x

44y

82y

63y

44y

221

n220

− 16905x221 x

62x

63x

44y

62y

63y

44y

221

n209+

2415x221 x

63x

44y

63y

44y

221

n176

+36225x22

1 x42x

63x

44y

42y

63y

44y

221

n198− 24150x22

1 x22x

63x

44y

22y

63y

44y

221

n187

− 345x221 x

44y

44y

221

n143− 5175x22

1 x42x

44y

42y

44y

221

n165

− 5175x221 x

82x

43x

44y

82y

43y

44y

221

n209+

36225x221 x

62x

43x

44y

62y

43y

44y

221

n198

− 5175x221 x

43x

44y

43y

44y

221

n165− 77625x22

1 x42x

43x

44y

42y

43y

44y

221

n187

+51750x22

1 x22x

43x

44y

22y

43y

44y

221

n176+

3450x221 x

22x

44y

22y

44y

221

n154

+3450x22

1 x82x

23x

44y

82y

23y

44y

221

n198− 24150x22

1 x62x

23x

44y

62y

23y

44y

221

n187

+3450x22

1 x23x

44y

23y

44y

221

n154+

51750x221 x

42x

23x

44y

42y

23y

44y

221

n176

− 34500x221 x

22x

23x

44y

22y

23y

44y

221

n165+

230x221 x

22y

22y

221

n132

+230x22

1 x82x

23y

82y

23y

221

n176− 1610x22

1 x62x

23y

62y

23y

221

n165

+3450x22

1 x42x

23y

42y

23y

221

n154+

230x221 x

23y

23y

221

n132

− 2300x221 x

22x

23y

22y

23y

221

n143+

230x221 x

82x

24y

82y

24y

221

n176

+230x22

1 x82x

83x

24y

82y

83y

24y

221

n220− 1610x22

1 x62x

83x

24y

62y

83y

24y

221

n209


+3450x22

1 x42x

83x

24y

42y

83y

24y

221

n198+

230x221 x

83x

24y

83y

24y

221

n176

− 2300x221 x

22x

83x

24y

22y

83y

24y

221

n187− 1610x22

1 x62x

24y

62y

24y

221

n165

− 1610x221 x

82x

63x

24y

82y

63y

24y

221

n209+

11270x221 x

62x

63x

24y

62y

63y

24y

221

n198

− 24150x221 x

42x

63x

24y

42y

63y

24y

221

n187− 1610x22

1 x63x

24y

63y

24y

221

n165

+16100x22

1 x22x

63x

24y

22y

63y

24y

221

n176+

3450x221 x

42x

24y

42y

24y

221

n154

+3450x22

1 x82x

43x

24y

82y

43y

24y

221

n198− 24150x22

1 x62x

43x

24y

62y

43y

24y

221

n187

+51750x22

1 x42x

43x

24y

42y

43y

24y

221

n176+

3450x221 x

43x

24y

43y

24y

221

n154

− 34500x221 x

22x

43x

24y

22y

43y

24y

221

n165+

230x221 x

24y

24y

221

n132

− 2300x221 x

22x

24y

22y

24y

221

n143− 2300x22

1 x82x

23x

24y

82y

23y

24y

221

n187

+16100x22

1 x62x

23x

24y

62y

23y

24y

221

n176− 34500x22

1 x42x

23x

24y

42y

23y

24y

221

n165

− 2300x221 x

23x

24y

23y

24y

221

n143+

23000x221 x

22x

23x

24y

22y

23y

24y

221

n154

+231x20

1 y201

n110+

231x201 x

82y

82y

201

n154

+231x20

1 x83y

83y

201

n154+

231x201 x

82x

83y

82y

83y

201

n198

− 1617x201 x

62x

83y

62y

83y

201

n187+

3465x201 x

42x

83y

42y

83y

201

n176

− 2310x201 x

22x

83y

22y

83y

201

n165+

231x201 x

84y

84y

201

n154

+231x20

1 x82x

84y

82y

84y

201

n198+

231x201 x

83x

84y

83y

84y

201

n198

+231x20

1 x82x

83x

84y

82y

83y

84y

201

n242− 1617x20

1 x62x

83x

84y

62y

83y

84y

201

n231

+3465x20

1 x42x

83x

84y

42y

83y

84y

201

n220− 2310x20

1 x22x

83x

84y

22y

83y

84y

201

n209

− 1617x201 x

62x

84y

62y

84y

201

n187− 1617x20

1 x63x

84y

63y

84y

201

n187

− 1617x201 x

82x

63x

84y

82y

63y

84y

201

n231+

11319x201 x

62x

63x

84y

62y

63y

84y

201

n220

− 24255x201 x

42x

63x

84y

42y

63y

84y

201

n209+

16170x201 x

22x

63x

84y

22y

63y

84y

201

n198

+3465x20

1 x42x

84y

42y

84y

201

n176+

3465x201 x

43x

84y

43y

84y

201

n176

+3465x20

1 x82x

43x

84y

82y

43y

84y

201

n220− 24255x20

1 x62x

43x

84y

62y

43y

84y

201

n209

+51975x20

1 x42x

43x

84y

42y

43y

84y

201

n198− 34650x20

1 x22x

43x

84y

22y

43y

84y

201

n187

− 2310x201 x

22x

84y

22y

84y

201

n165− 2310x20

1 x23x

84y

23y

84y

201

n165


− 2310x201 x

82x

23x

84y

82y

23y

84y

201

n209+

16170x201 x

62x

23x

84y

62y

23y

84y

201

n198

− 34650x201 x

42x

23x

84y

42y

23y

84y

201

n187+

23100x201 x

22x

23x

84y

22y

23y

84y

201

n176

− 1617x201 x

62y

62y

201

n143− 1617x20

1 x82x

63y

82y

63y

201

n187

− 1617x201 x

63y

63y

201

n143+

11319x201 x

62x

63y

62y

63y

201

n176

− 24255x201 x

42x

63y

42y

63y

201

n165+

16170x201 x

22x

63y

22y

63y

201

n154

− 1617x201 x

82x

64y

82y

64y

201

n187− 1617x20

1 x82x

83x

64y

82y

83y

64y

201

n231

− 1617x201 x

83x

64y

83y

64y

201

n187+

11319x201 x

62x

83x

64y

62y

83y

64y

201

n220

− 24255x201 x

42x

83x

64y

42y

83y

64y

201

n209+

16170x201 x

22x

83x

64y

22y

83y

64y

201

n198

− 1617x201 x

64y

64y

201

n143+

11319x201 x

62x

64y

62y

64y

201

n176

+11319x20

1 x82x

63x

64y

82y

63y

64y

201

n220+

11319x201 x

63x

64y

63y

64y

201

n176

− 79233x201 x

62x

63x

64y

62y

63y

64y

201

n209+

169785x201 x

42x

63x

64y

42y

63y

64y

201

n198

− 113190x201 x

22x

63x

64y

22y

63y

64y

201

n187− 24255x20

1 x42x

64y

42y

64y

201

n165

− 24255x201 x

82x

43x

64y

82y

43y

64y

201

n209− 24255x20

1 x43x

64y

43y

64y

201

n165

+169785x20

1 x62x

43x

64y

62y

43y

64y

201

n198− 363825x20

1 x42x

43x

64y

42y

43y

64y

201

n187

+242550x20

1 x22x

43x

64y

22y

43y

64y

201

n176+

16170x201 x

22x

64y

22y

64y

201

n154

+16170x20

1 x82x

23x

64y

82y

23y

64y

201

n198+

16170x201 x

23x

64y

23y

64y

201

n154

− 113190x201 x

62x

23x

64y

62y

23y

64y

201

n187+

242550x201 x

42x

23x

64y

42y

23y

64y

201

n176

− 161700x201 x

22x

23x

64y

22y

23y

64y

201

n165+

3465x201 x

42y

42y

201

n132

+3465x20

1 x82x

43y

82y

43y

201

n176− 24255x20

1 x62x

43y

62y

43y

201

n165

+3465x20

1 x43y

43y

201

n132+

51975x201 x

42x

43y

42y

43y

201

n154

− 34650x201 x

22x

43y

22y

43y

201

n143+

3465x201 x

82x

44y

82y

44y

201

n176

+3465x20

1 x82x

83x

44y

82y

83y

44y

201

n220− 24255x20

1 x62x

83x

44y

62y

83y

44y

201

n209

+3465x20

1 x83x

44y

83y

44y

201

n176+

51975x201 x

42x

83x

44y

42y

83y

44y

201

n198

− 34650x201 x

22x

83x

44y

22y

83y

44y

201

n187− 24255x20

1 x62x

44y

62y

44y

201

n165

− 24255x201 x

82x

63x

44y

82y

63y

44y

201

n209+

169785x201 x

62x

63x

44y

62y

63y

44y

201

n198


− 24255x201 x

63x

44y

63y

44y

201

n165− 363825x20

1 x42x

63x

44y

42y

63y

44y

201

n187

+242550x20

1 x22x

63x

44y

22y

63y

44y

201

n176+

3465x201 x

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n132

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44y

42y

44y

201

n154+

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82x

43x

44y

82y

43y

44y

201

n198

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62x

43x

44y

62y

43y

44y

201

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43x

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43y

44y

201

n154

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43x

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42y

43y

44y

201

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43x

44y

22y

43y

44y

201

n165

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22x

44y

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201

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23x

44y

82y

23y

44y

201

n187

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23x

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62y

23y

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201

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44y

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44y

201

n143

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42x

23x

44y

42y

23y

44y

201

n165+

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22x

23x

44y

22y

23y

44y

201

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201

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23y

82y

23y

201

n165

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23y

62y

23y

201

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23y

42y

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201

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23y

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201

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22x

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201

n132

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82x

24y

82y

24y

201

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1 x82x

83x

24y

82y

83y

24y

201

n209

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83x

24y

62y

83y

24y

201

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1 x42x

83x

24y

42y

83y

24y

201

n187

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83x

24y

83y

24y

201

n165+

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83x

24y

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83y

24y

201

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24y

62y

24y

201

n154+

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82x

63x

24y

82y

63y

24y

201

n198

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62x

63x

24y

62y

63y

24y

201

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42x

63x

24y

42y

63y

24y

201

n176

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24y

63y

24y

201

n154− 161700x20

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63x

24y

22y

63y

24y

201

n165

− 34650x201 x

42x

24y

42y

24y

201

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1 x82x

43x

24y

82y

43y

24y

201

n187

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1 x62x

43x

24y

62y

43y

24y

201

n176− 519750x20

1 x42x

43x

24y

42y

43y

24y

201

n165

− 34650x201 x

43x

24y

43y

24y

201

n143+

346500x201 x

22x

43x

24y

22y

43y

24y

201

n154

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24y

24y

201

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22x

24y

22y

24y

201

n132

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1 x82x

23x

24y

82y

23y

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201

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23x

24y

62y

23y

24y

201

n165

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1 x42x

23x

24y

42y

23y

24y

201

n154+

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23x

24y

23y

24y

201

n132

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22x

23x

24y

22y

23y

24y

201

n143− 1330x18

1 y181

n99

− 1330x181 x

82y

82y

181

n143− 1330x18

1 x83y

83y

181

n143


− 1330x181 x

82x

83y

82y

83y

181

n187+

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62x

83y

62y

83y

181

n176

− 19950x181 x

42x

83y

42y

83y

181

n165+

13300x181 x

22x

83y

22y

83y

181

n154

− 1330x181 x

84y

84y

181

n143− 1330x18

1 x82x

84y

82y

84y

181

n187

− 1330x181 x

83x

84y

83y

84y

181

n187− 1330x18

1 x82x

83x

84y

82y

83y

84y

181

n231

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1 x62x

83x

84y

62y

83y

84y

181

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1 x42x

83x

84y

42y

83y

84y

181

n209

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83x

84y

22y

83y

84y

181

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62x

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84y

181

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84y

63y

84y

181

n176+

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82x

63x

84y

82y

63y

84y

181

n220

− 65170x181 x

62x

63x

84y

62y

63y

84y

181

n209+

139650x181 x

42x

63x

84y

42y

63y

84y

181

n198

− 93100x181 x

22x

63x

84y

22y

63y

84y

181

n187− 19950x18

1 x42x

84y

42y

84y

181

n165

− 19950x181 x

43x

84y

43y

84y

181

n165− 19950x18

1 x82x

43x

84y

82y

43y

84y

181

n209

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1 x62x

43x

84y

62y

43y

84y

181

n198− 299250x18

1 x42x

43x

84y

42y

43y

84y

181

n187

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1 x22x

43x

84y

22y

43y

84y

181

n176+

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22x

84y

22y

84y

181

n154

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1 x23x

84y

23y

84y

181

n154+

13300x181 x

82x

23x

84y

82y

23y

84y

181

n198

− 93100x181 x

62x

23x

84y

62y

23y

84y

181

n187+

199500x181 x

42x

23x

84y

42y

23y

84y

181

n176

− 133000x181 x

22x

23x

84y

22y

23y

84y

181

n165+

9310x181 x

62y

62y

181

n132

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1 x82x

63y

82y

63y

181

n176+

9310x181 x

63y

63y

181

n132

− 65170x181 x

62x

63y

62y

63y

181

n165+

139650x181 x

42x

63y

42y

63y

181

n154

− 93100x181 x

22x

63y

22y

63y

181

n143+

9310x181 x

82x

64y

82y

64y

181

n176

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1 x82x

83x

64y

82y

83y

64y

181

n220+

9310x181 x

83x

64y

83y

64y

181

n176

− 65170x181 x

62x

83x

64y

62y

83y

64y

181

n209+

139650x181 x

42x

83x

64y

42y

83y

64y

181

n198

− 93100x181 x

22x

83x

64y

22y

83y

64y

181

n187+

9310x181 x

64y

64y

181

n132

− 65170x181 x

62x

64y

62y

64y

181

n165− 65170x18

1 x82x

63x

64y

82y

63y

64y

181

n209

− 65170x181 x

63x

64y

63y

64y

181

n165+

456190x181 x

62x

63x

64y

62y

63y

64y

181

n198

− 977550x181 x

42x

63x

64y

42y

63y

64y

181

n187+

651700x181 x

22x

63x

64y

22y

63y

64y

181

n176

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1 x42x

64y

42y

64y

181

n154+

139650x181 x

82x

43x

64y

82y

43y

64y

181

n198


+139650x18

1 x43x

64y

43y

64y

181

n154− 977550x18

1 x62x

43x

64y

62y

43y

64y

181

n187

+2094750x18

1 x42x

43x

64y

42y

43y

64y

181

n176− 1396500x18

1 x22x

43x

64y

22y

43y

64y

181

n165

− 93100x181 x

22x

64y

22y

64y

181

n143− 93100x18

1 x82x

23x

64y

82y

23y

64y

181

n187

− 93100x181 x

23x

64y

23y

64y

181

n143+

651700x181 x

62x

23x

64y

62y

23y

64y

181

n176

− 1396500x181 x

42x

23x

64y

42y

23y

64y

181

n165+

931000x181 x

22x

23x

64y

22y

23y

64y

181

n154

− 19950x181 x

42y

42y

181

n121− 19950x18

1 x82x

43y

82y

43y

181

n165

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1 x62x

43y

62y

43y

181

n154− 19950x18

1 x43y

43y

181

n121

− 299250x181 x

42x

43y

42y

43y

181

n143+

199500x181 x

22x

43y

22y

43y

181

n132

− 19950x181 x

82x

44y

82y

44y

181

n165− 19950x18

1 x82x

83x

44y

82y

83y

44y

181

n209

+139650x18

1 x62x

83x

44y

62y

83y

44y

181

n198− 19950x18

1 x83x

44y

83y

44y

181

n165

− 299250x181 x

42x

83x

44y

42y

83y

44y

181

n187+

199500x181 x

22x

83x

44y

22y

83y

44y

181

n176

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1 x62x

44y

62y

44y

181

n154+

139650x181 x

82x

63x

44y

82y

63y

44y

181

n198

− 977550x181 x

62x

63x

44y

62y

63y

44y

181

n187+

139650x181 x

63x

44y

63y

44y

181

n154

+2094750x18

1 x42x

63x

44y

42y

63y

44y

181

n176− 1396500x18

1 x22x

63x

44y

22y

63y

44y

181

n165

− 19950x181 x

44y

44y

181

n121− 299250x18

1 x42x

44y

42y

44y

181

n143

− 299250x181 x

82x

43x

44y

82y

43y

44y

181

n187+

2094750x181 x

62x

43x

44y

62y

43y

44y

181

n176

− 299250x181 x

43x

44y

43y

44y

181

n143− 4488750x18

1 x42x

43x

44y

42y

43y

44y

181

n165

+2992500x18

1 x22x

43x

44y

22y

43y

44y

181

n154+

199500x181 x

22x

44y

22y

44y

181

n132

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1 x82x

23x

44y

82y

23y

44y

181

n176− 1396500x18

1 x62x

23x

44y

62y

23y

44y

181

n165

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1 x23x

44y

23y

44y

181

n132+

2992500x181 x

42x

23x

44y

42y

23y

44y

181

n154

− 1995000x181 x

22x

23x

44y

22y

23y

44y

181

n143+

13300x181 x

22y

22y

181

n110

+13300x18

1 x82x

23y

82y

23y

181

n154− 93100x18

1 x62x

23y

62y

23y

181

n143

+199500x18

1 x42x

23y

42y

23y

181

n132+

13300x181 x

23y

23y

181

n110

− 133000x181 x

22x

23y

22y

23y

181

n121+

13300x181 x

82x

24y

82y

24y

181

n154

+13300x18

1 x82x

83x

24y

82y

83y

24y

181

n198− 93100x18

1 x62x

83x

24y

62y

83y

24y

181

n187


+199500x18

1 x42x

83x

24y

42y

83y

24y

181

n176+

13300x181 x

83x

24y

83y

24y

181

n154

− 133000x181 x

22x

83x

24y

22y

83y

24y

181

n165− 93100x18

1 x62x

24y

62y

24y

181

n143

− 93100x181 x

82x

63x

24y

82y

63y

24y

181

n187+

651700x181 x

62x

63x

24y

62y

63y

24y

181

n176

− 1396500x181 x

42x

63x

24y

42y

63y

24y

181

n165− 93100x18

1 x63x

24y

63y

24y

181

n143

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1 x22x

63x

24y

22y

63y

24y

181

n154+

199500x181 x

42x

24y

42y

24y

181

n132

+199500x18

1 x82x

43x

24y

82y

43y

24y

181

n176− 1396500x18

1 x62x

43x

24y

62y

43y

24y

181

n165

+2992500x18

1 x42x

43x

24y

42y

43y

24y

181

n154+

199500x181 x

43x

24y

43y

24y

181

n132

− 1995000x181 x

22x

43x

24y

22y

43y

24y

181

n143+

13300x181 x

24y

24y

181

n110

− 133000x181 x

22x

24y

22y

24y

181

n121− 133000x18

1 x82x

23x

24y

82y

23y

24y

181

n165

+931000x18

1 x62x

23x

24y

62y

23y

24y

181

n154− 1995000x18

1 x42x

23x

24y

42y

23y

24y

181

n143

− 133000x181 x

23x

24y

23y

24y

181

n121+

1330000x181 x

22x

23x

24y

22y

23y

24y

181

n132

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1 y161

n88+

4845x161 x

82y

82y

161

n132

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1 x83y

83y

161

n132+

4845x161 x

82x

83y

82y

83y

161

n176

− 33915x161 x

62x

83y

62y

83y

161

n165+

72675x161 x

42x

83y

42y

83y

161

n154

− 48450x161 x

22x

83y

22y

83y

161

n143+

4845x161 x

84y

84y

161

n132

+4845x16

1 x82x

84y

82y

84y

161

n176+

4845x161 x

83x

84y

83y

84y

161

n176

+4845x16

1 x82x

83x

84y

82y

83y

84y

161

n220− 33915x16

1 x62x

83x

84y

62y

83y

84y

161

n209

+72675x16

1 x42x

83x

84y

42y

83y

84y

161

n198− 48450x16

1 x22x

83x

84y

22y

83y

84y

161

n187

− 33915x161 x

62x

84y

62y

84y

161

n165− 33915x16

1 x63x

84y

63y

84y

161

n165

− 33915x161 x

82x

63x

84y

82y

63y

84y

161

n209+

237405x161 x

62x

63x

84y

62y

63y

84y

161

n198

− 508725x161 x

42x

63x

84y

42y

63y

84y

161

n187+

339150x161 x

22x

63x

84y

22y

63y

84y

161

n176

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1 x42x

84y

42y

84y

161

n154+

72675x161 x

43x

84y

43y

84y

161

n154

+72675x16

1 x82x

43x

84y

82y

43y

84y

161

n198− 508725x16

1 x62x

43x

84y

62y

43y

84y

161

n187

+1090125x16

1 x42x

43x

84y

42y

43y

84y

161

n176− 726750x16

1 x22x

43x

84y

22y

43y

84y

161

n165

− 48450x161 x

22x

84y

22y

84y

161

n143− 48450x16

1 x23x

84y

23y

84y

161

n143


− 48450x161 x

82x

23x

84y

82y

23y

84y

161

n187+

339150x161 x

62x

23x

84y

62y

23y

84y

161

n176

− 726750x161 x

42x

23x

84y

42y

23y

84y

161

n165+

484500x161 x

22x

23x

84y

22y

23y

84y

161

n154

− 33915x161 x

62y

62y

161

n121− 33915x16

1 x82x

63y

82y

63y

161

n165

− 33915x161 x

63y

63y

161

n121+

237405x161 x

62x

63y

62y

63y

161

n154

− 508725x161 x

42x

63y

42y

63y

161

n143+

339150x161 x

22x

63y

22y

63y

161

n132

− 33915x161 x

82x

64y

82y

64y

161

n165− 33915x16

1 x82x

83x

64y

82y

83y

64y

161

n209

− 33915x161 x

83x

64y

83y

64y

161

n165+

237405x161 x

62x

83x

64y

62y

83y

64y

161

n198

− 508725x161 x

42x

83x

64y

42y

83y

64y

161

n187+

339150x161 x

22x

83x

64y

22y

83y

64y

161

n176

− 33915x161 x

64y

64y

161

n121+

237405x161 x

62x

64y

62y

64y

161

n154

+237405x16

1 x82x

63x

64y

82y

63y

64y

161

n198+

237405x161 x

63x

64y

63y

64y

161

n154

− 1661835x161 x

62x

63x

64y

62y

63y

64y

161

n187+

3561075x161 x

42x

63x

64y

42y

63y

64y

161

n176

− 2374050x161 x

22x

63x

64y

22y

63y

64y

161

n165− 508725x16

1 x42x

64y

42y

64y

161

n143

− 508725x161 x

82x

43x

64y

82y

43y

64y

161

n187− 508725x16

1 x43x

64y

43y

64y

161

n143

+3561075x16

1 x62x

43x

64y

62y

43y

64y

161

n176− 7630875x16

1 x42x

43x

64y

42y

43y

64y

161

n165

+5087250x16

1 x22x

43x

64y

22y

43y

64y

161

n154+

339150x161 x

22x

64y

22y

64y

161

n132

+339150x16

1 x82x

23x

64y

82y

23y

64y

161

n176+

339150x161 x

23x

64y

23y

64y

161

n132

− 2374050x161 x

62x

23x

64y

62y

23y

64y

161

n165+

5087250x161 x

42x

23x

64y

42y

23y

64y

161

n154

− 3391500x161 x

22x

23x

64y

22y

23y

64y

161

n143+

72675x161 x

42y

42y

161

n110

+72675x16

1 x82x

43y

82y

43y

161

n154− 508725x16

1 x62x

43y

62y

43y

161

n143

+72675x16

1 x43y

43y

161

n110+

1090125x161 x

42x

43y

42y

43y

161

n132

− 726750x161 x

22x

43y

22y

43y

161

n121+

72675x161 x

82x

44y

82y

44y

161

n154

+72675x16

1 x82x

83x

44y

82y

83y

44y

161

n198− 508725x16

1 x62x

83x

44y

62y

83y

44y

161

n187

+72675x16

1 x83x

44y

83y

44y

161

n154+

1090125x161 x

42x

83x

44y

42y

83y

44y

161

n176

− 726750x161 x

22x

83x

44y

22y

83y

44y

161

n165− 508725x16

1 x62x

44y

62y

44y

161

n143

− 508725x161 x

82x

63x

44y

82y

63y

44y

161

n187+

3561075x161 x

62x

63x

44y

62y

63y

44y

161

n176


− 508725x161 x

63x

44y

63y

44y

161

n143− 7630875x16

1 x42x

63x

44y

42y

63y

44y

161

n165

+5087250x16

1 x22x

63x

44y

22y

63y

44y

161

n154+

72675x161 x

44y

44y

161

n110

+1090125x16

1 x42x

44y

42y

44y

161

n132+

1090125x161 x

82x

43x

44y

82y

43y

44y

161

n176

− 7630875x161 x

62x

43x

44y

62y

43y

44y

161

n165+

1090125x161 x

43x

44y

43y

44y

161

n132

+16351875x16

1 x42x

43x

44y

42y

43y

44y

161

n154− 10901250x16

1 x22x

43x

44y

22y

43y

44y

161

n143

− 726750x161 x

22x

44y

22y

44y

161

n121− 726750x16

1 x82x

23x

44y

82y

23y

44y

161

n165

+5087250x16

1 x62x

23x

44y

62y

23y

44y

161

n154− 726750x16

1 x23x

44y

23y

44y

161

n121

− 10901250x161 x

42x

23x

44y

42y

23y

44y

161

n143+

7267500x161 x

22x

23x

44y

22y

23y

44y

161

n132

− 48450x161 x

22y

22y

161

n99− 48450x16

1 x82x

23y

82y

23y

161

n143

+339150x16

1 x62x

23y

62y

23y

161

n132− 726750x16

1 x42x

23y

42y

23y

161

n121

− 48450x161 x

23y

23y

161

n99+

484500x161 x

22x

23y

22y

23y

161

n110

− 48450x161 x

82x

24y

82y

24y

161

n143− 48450x16

1 x82x

83x

24y

82y

83y

24y

161

n187

+339150x16

1 x62x

83x

24y

62y

83y

24y

161

n176− 726750x16

1 x42x

83x

24y

42y

83y

24y

161

n165

− 48450x161 x

83x

24y

83y

24y

161

n143+

484500x161 x

22x

83x

24y

22y

83y

24y

161

n154

+339150x16

1 x62x

24y

62y

24y

161

n132+

339150x161 x

82x

63x

24y

82y

63y

24y

161

n176

− 2374050x161 x

62x

63x

24y

62y

63y

24y

161

n165+

5087250x161 x

42x

63x

24y

42y

63y

24y

161

n154

+339150x16

1 x63x

24y

63y

24y

161

n132− 3391500x16

1 x22x

63x

24y

22y

63y

24y

161

n143

− 726750x161 x

42x

24y

42y

24y

161

n121− 726750x16

1 x82x

43x

24y

82y

43y

24y

161

n165

+5087250x16

1 x62x

43x

24y

62y

43y

24y

161

n154− 10901250x16

1 x42x

43x

24y

42y

43y

24y

161

n143

− 726750x161 x

43x

24y

43y

24y

161

n121+

7267500x161 x

22x

43x

24y

22y

43y

24y

161

n132

− 48450x161 x

24y

24y

161

n99+

484500x161 x

22x

24y

22y

24y

161

n110

+484500x16

1 x82x

23x

24y

82y

23y

24y

161

n154− 3391500x16

1 x62x

23x

24y

62y

23y

24y

161

n143

+7267500x16

1 x42x

23x

24y

42y

23y

24y

161

n132+

484500x161 x

23x

24y

23y

24y

161

n110

− 4845000x161 x

22x

23x

24y

22y

23y

24y

161

n121− 11628x14

1 y141

n77

− 11628x141 x

82y

82y

141

n121− 11628x14

1 x83y

83y

141

n121


− 11628x141 x

82x

83y

82y

83y

141

n165+

81396x141 x

62x

83y

62y

83y

141

n154

− 174420x141 x

42x

83y

42y

83y

141

n143+

116280x141 x

22x

83y

22y

83y

141

n132

− 11628x141 x

84y

84y

141

n121− 11628x14

1 x82x

84y

82y

84y

141

n165

− 11628x141 x

83x

84y

83y

84y

141

n165− 11628x14

1 x82x

83x

84y

82y

83y

84y

141

n209

+81396x14

1 x62x

83x

84y

62y

83y

84y

141

n198− 174420x14

1 x42x

83x

84y

42y

83y

84y

141

n187

+116280x14

1 x22x

83x

84y

22y

83y

84y

141

n176+

81396x141 x

62x

84y

62y

84y

141

n154

+81396x14

1 x63x

84y

63y

84y

141

n154+

81396x141 x

82x

63x

84y

82y

63y

84y

141

n198

− 569772x141 x

62x

63x

84y

62y

63y

84y

141

n187+

1220940x141 x

42x

63x

84y

42y

63y

84y

141

n176

− 813960x141 x

22x

63x

84y

22y

63y

84y

141

n165− 174420x14

1 x42x

84y

42y

84y

141

n143

− 174420x141 x

43x

84y

43y

84y

141

n143− 174420x14

1 x82x

43x

84y

82y

43y

84y

141

n187

+1220940x14

1 x62x

43x

84y

62y

43y

84y

141

n176− 2616300x14

1 x42x

43x

84y

42y

43y

84y

141

n165

+1744200x14

1 x22x

43x

84y

22y

43y

84y

141

n154+

116280x141 x

22x

84y

22y

84y

141

n132

+116280x14

1 x23x

84y

23y

84y

141

n132+

116280x141 x

82x

23x

84y

82y

23y

84y

141

n176

− 813960x141 x

62x

23x

84y

62y

23y

84y

141

n165+

1744200x141 x

42x

23x

84y

42y

23y

84y

141

n154

− 1162800x141 x

22x

23x

84y

22y

23y

84y

141

n143+

81396x141 x

62y

62y

141

n110

+81396x14

1 x82x

63y

82y

63y

141

n154+

81396x141 x

63y

63y

141

n110

− 569772x141 x

62x

63y

62y

63y

141

n143+

1220940x141 x

42x

63y

42y

63y

141

n132

− 813960x141 x

22x

63y

22y

63y

141

n121+

81396x141 x

82x

64y

82y

64y

141

n154

+81396x14

1 x82x

83x

64y

82y

83y

64y

141

n198+

81396x141 x

83x

64y

83y

64y

141

n154

− 569772x141 x

62x

83x

64y

62y

83y

64y

141

n187+

1220940x141 x

42x

83x

64y

42y

83y

64y

141

n176

− 813960x141 x

22x

83x

64y

22y

83y

64y

141

n165+

81396x141 x

64y

64y

141

n110

− 569772x141 x

62x

64y

62y

64y

141

n143− 569772x14

1 x82x

63x

64y

82y

63y

64y

141

n187

− 569772x141 x

63x

64y

63y

64y

141

n143+

3988404x141 x

62x

63x

64y

62y

63y

64y

141

n176

− 8546580x141 x

42x

63x

64y

42y

63y

64y

141

n165+

5697720x141 x

22x

63x

64y

22y

63y

64y

141

n154

+1220940x14

1 x42x

64y

42y

64y

141

n132+

1220940x141 x

82x

43x

64y

82y

43y

64y

141

n176


+1220940x14

1 x43x

64y

43y

64y

141

n132− 8546580x14

1 x62x

43x

64y

62y

43y

64y

141

n165

+18314100x14

1 x42x

43x

64y

42y

43y

64y

141

n154− 12209400x14

1 x22x

43x

64y

22y

43y

64y

141

n143

− 813960x141 x

22x

64y

22y

64y

141

n121− 813960x14

1 x82x

23x

64y

82y

23y

64y

141

n165

− 813960x141 x

23x

64y

23y

64y

141

n121+

5697720x141 x

62x

23x

64y

62y

23y

64y

141

n154

− 12209400x141 x

42x

23x

64y

42y

23y

64y

141

n143+

8139600x141 x

22x

23x

64y

22y

23y

64y

141

n132

− 174420x141 x

42y

42y

141

n99− 174420x14

1 x82x

43y

82y

43y

141

n143

+1220940x14

1 x62x

43y

62y

43y

141

n132− 174420x14

1 x43y

43y

141

n99

− 2616300x141 x

42x

43y

42y

43y

141

n121+

1744200x141 x

22x

43y

22y

43y

141

n110

− 174420x141 x

82x

44y

82y

44y

141

n143− 174420x14

1 x82x

83x

44y

82y

83y

44y

141

n187

+1220940x14

1 x62x

83x

44y

62y

83y

44y

141

n176− 174420x14

1 x83x

44y

83y

44y

141

n143

− 2616300x141 x

42x

83x

44y

42y

83y

44y

141

n165+

1744200x141 x

22x

83x

44y

22y

83y

44y

141

n154

+1220940x14

1 x62x

44y

62y

44y

141

n132+

1220940x141 x

82x

63x

44y

82y

63y

44y

141

n176

− 8546580x141 x

62x

63x

44y

62y

63y

44y

141

n165+

1220940x141 x

63x

44y

63y

44y

141

n132

+18314100x14

1 x42x

63x

44y

42y

63y

44y

141

n154− 12209400x14

1 x22x

63x

44y

22y

63y

44y

141

n143

− 174420x141 x

44y

44y

141

n99− 2616300x14

1 x42x

44y

42y

44y

141

n121

− 2616300x141 x

82x

43x

44y

82y

43y

44y

141

n165+

18314100x141 x

62x

43x

44y

62y

43y

44y

141

n154

− 2616300x141 x

43x

44y

43y

44y

141

n121− 39244500x14

1 x42x

43x

44y

42y

43y

44y

141

n143

+26163000x14

1 x22x

43x

44y

22y

43y

44y

141

n132+

1744200x141 x

22x

44y

22y

44y

141

n110

+1744200x14

1 x82x

23x

44y

82y

23y

44y

141

n154− 12209400x14

1 x62x

23x

44y

62y

23y

44y

141

n143

+1744200x14

1 x23x

44y

23y

44y

141

n110+

26163000x141 x

42x

23x

44y

42y

23y

44y

141

n132

− 17442000x141 x

22x

23x

44y

22y

23y

44y

141

n121+

116280x141 x

22y

22y

141

n88

+116280x14

1 x82x

23y

82y

23y

141

n132− 813960x14

1 x62x

23y

62y

23y

141

n121

+1744200x14

1 x42x

23y

42y

23y

141

n110+

116280x141 x

23y

23y

141

n88

− 1162800x141 x

22x

23y

22y

23y

141

n99+

116280x141 x

82x

24y

82y

24y

141

n132

+116280x14

1 x82x

83x

24y

82y

83y

24y

141

n176− 813960x14

1 x62x

83x

24y

62y

83y

24y

141

n165


+1744200x14

1 x42x

83x

24y

42y

83y

24y

141

n154+

116280x141 x

83x

24y

83y

24y

141

n132

− 1162800x141 x

22x

83x

24y

22y

83y

24y

141

n143− 813960x14

1 x62x

24y

62y

24y

141

n121

− 813960x141 x

82x

63x

24y

82y

63y

24y

141

n165+

5697720x141 x

62x

63x

24y

62y

63y

24y

141

n154

− 12209400x141 x

42x

63x

24y

42y

63y

24y

141

n143− 813960x14

1 x63x

24y

63y

24y

141

n121

+8139600x14

1 x22x

63x

24y

22y

63y

24y

141

n132+

1744200x141 x

42x

24y

42y

24y

141

n110

+1744200x14

1 x82x

43x

24y

82y

43y

24y

141

n154− 12209400x14

1 x62x

43x

24y

62y

43y

24y

141

n143

+26163000x14

1 x42x

43x

24y

42y

43y

24y

141

n132+

1744200x141 x

43x

24y

43y

24y

141

n110

− 17442000x141 x

22x

43x

24y

22y

43y

24y

141

n121+

116280x141 x

24y

24y

141

n88

− 1162800x141 x

22x

24y

22y

24y

141

n99− 1162800x14

1 x82x

23x

24y

82y

23y

24y

141

n143

+8139600x14

1 x62x

23x

24y

62y

23y

24y

141

n132− 17442000x14

1 x42x

23x

24y

42y

23y

24y

141

n121

− 1162800x141 x

23x

24y

23y

24y

141

n99+

11628000x141 x

22x

23x

24y

22y

23y

24y

141

n110

+18564x12

1 y121

n66+

18564x121 x

82y

82y

121

n110

+18564x12

1 x83y

83y

121

n110+

18564x121 x

82x

83y

82y

83y

121

n154

− 129948x121 x

62x

83y

62y

83y

121

n143+

278460x121 x

42x

83y

42y

83y

121

n132

− 185640x121 x

22x

83y

22y

83y

121

n121+

18564x121 x

84y

84y

121

n110

+18564x12

1 x82x

84y

82y

84y

121

n154+

18564x121 x

83x

84y

83y

84y

121

n154

+18564x12

1 x82x

83x

84y

82y

83y

84y

121

n198− 129948x12

1 x62x

83x

84y

62y

83y

84y

121

n187

+278460x12

1 x42x

83x

84y

42y

83y

84y

121

n176− 185640x12

1 x22x

83x

84y

22y

83y

84y

121

n165

− 129948x121 x

62x

84y

62y

84y

121

n143− 129948x12

1 x63x

84y

63y

84y

121

n143

− 129948x121 x

82x

63x

84y

82y

63y

84y

121

n187+

909636x121 x

62x

63x

84y

62y

63y

84y

121

n176

− 1949220x121 x

42x

63x

84y

42y

63y

84y

121

n165+

1299480x121 x

22x

63x

84y

22y

63y

84y

121

n154

+278460x12

1 x42x

84y

42y

84y

121

n132+

278460x121 x

43x

84y

43y

84y

121

n132

+278460x12

1 x82x

43x

84y

82y

43y

84y

121

n176− 1949220x12

1 x62x

43x

84y

62y

43y

84y

121

n165

+4176900x12

1 x42x

43x

84y

42y

43y

84y

121

n154− 2784600x12

1 x22x

43x

84y

22y

43y

84y

121

n143

− 185640x121 x

22x

84y

22y

84y

121

n121− 185640x12

1 x23x

84y

23y

84y

121

n121


− 185640x121 x

82x

23x

84y

82y

23y

84y

121

n165+

1299480x121 x

62x

23x

84y

62y

23y

84y

121

n154

− 2784600x121 x

42x

23x

84y

42y

23y

84y

121

n143+

1856400x121 x

22x

23x

84y

22y

23y

84y

121

n132

− 129948x121 x

62y

62y

121

n99− 129948x12

1 x82x

63y

82y

63y

121

n143

− 129948x121 x

63y

63y

121

n99+

909636x121 x

62x

63y

62y

63y

121

n132

− 1949220x121 x

42x

63y

42y

63y

121

n121+

1299480x121 x

22x

63y

22y

63y

121

n110

− 129948x121 x

82x

64y

82y

64y

121

n143− 129948x12

1 x82x

83x

64y

82y

83y

64y

121

n187

− 129948x121 x

83x

64y

83y

64y

121

n143+

909636x121 x

62x

83x

64y

62y

83y

64y

121

n176

− 1949220x121 x

42x

83x

64y

42y

83y

64y

121

n165+

1299480x121 x

22x

83x

64y

22y

83y

64y

121

n154

− 129948x121 x

64y

64y

121

n99+

909636x121 x

62x

64y

62y

64y

121

n132

+909636x12

1 x82x

63x

64y

82y

63y

64y

121

n176+

909636x121 x

63x

64y

63y

64y

121

n132

− 6367452x121 x

62x

63x

64y

62y

63y

64y

121

n165+

13644540x121 x

42x

63x

64y

42y

63y

64y

121

n154

− 9096360x121 x

22x

63x

64y

22y

63y

64y

121

n143− 1949220x12

1 x42x

64y

42y

64y

121

n121

− 1949220x121 x

82x

43x

64y

82y

43y

64y

121

n165− 1949220x12

1 x43x

64y

43y

64y

121

n121

+13644540x12

1 x62x

43x

64y

62y

43y

64y

121

n154− 29238300x12

1 x42x

43x

64y

42y

43y

64y

121

n143

+19492200x12

1 x22x

43x

64y

22y

43y

64y

121

n132+

1299480x121 x

22x

64y

22y

64y

121

n110

+1299480x12

1 x82x

23x

64y

82y

23y

64y

121

n154+

1299480x121 x

23x

64y

23y

64y

121

n110

− 9096360x121 x

62x

23x

64y

62y

23y

64y

121

n143+

19492200x121 x

42x

23x

64y

42y

23y

64y

121

n132

− 12994800x121 x

22x

23x

64y

22y

23y

64y

121

n121+

278460x121 x

42y

42y

121

n88

+278460x12

1 x82x

43y

82y

43y

121

n132− 1949220x12

1 x62x

43y

62y

43y

121

n121

+278460x12

1 x43y

43y

121

n88+

4176900x121 x

42x

43y

42y

43y

121

n110

− 2784600x121 x

22x

43y

22y

43y

121

n99+

278460x121 x

82x

44y

82y

44y

121

n132

+278460x12

1 x82x

83x

44y

82y

83y

44y

121

n176− 1949220x12

1 x62x

83x

44y

62y

83y

44y

121

n165

+278460x12

1 x83x

44y

83y

44y

121

n132+

4176900x121 x

42x

83x

44y

42y

83y

44y

121

n154

− 2784600x121 x

22x

83x

44y

22y

83y

44y

121

n143− 1949220x12

1 x62x

44y

62y

44y

121

n121

− 1949220x121 x

82x

63x

44y

82y

63y

44y

121

n165+

13644540x121 x

62x

63x

44y

62y

63y

44y

121

n154


− 1949220x121 x

63x

44y

63y

44y

121

n121− 29238300x12

1 x42x

63x

44y

42y

63y

44y

121

n143

+19492200x12

1 x22x

63x

44y

22y

63y

44y

121

n132+

278460x121 x

44y

44y

121

n88

+4176900x12

1 x42x

44y

42y

44y

121

n110+

4176900x121 x

82x

43x

44y

82y

43y

44y

121

n154

− 29238300x121 x

62x

43x

44y

62y

43y

44y

121

n143+

4176900x121 x

43x

44y

43y

44y

121

n110

+62653500x12

1 x42x

43x

44y

42y

43y

44y

121

n132− 41769000x12

1 x22x

43x

44y

22y

43y

44y

121

n121

− 2784600x121 x

22x

44y

22y

44y

121

n99− 2784600x12

1 x82x

23x

44y

82y

23y

44y

121

n143

+19492200x12

1 x62x

23x

44y

62y

23y

44y

121

n132− 2784600x12

1 x23x

44y

23y

44y

121

n99

− 41769000x121 x

42x

23x

44y

42y

23y

44y

121

n121+

27846000x121 x

22x

23x

44y

22y

23y

44y

121

n110

− 185640x121 x

22y

22y

121

n77− 185640x12

1 x82x

23y

82y

23y

121

n121

+1299480x12

1 x62x

23y

62y

23y

121

n110− 2784600x12

1 x42x

23y

42y

23y

121

n99

− 185640x121 x

23y

23y

121

n77+

1856400x121 x

22x

23y

22y

23y

121

n88

− 185640x121 x

82x

24y

82y

24y

121

n121− 185640x12

1 x82x

83x

24y

82y

83y

24y

121

n165

+1299480x12

1 x62x

83x

24y

62y

83y

24y

121

n154− 2784600x12

1 x42x

83x

24y

42y

83y

24y

121

n143

− 185640x121 x

83x

24y

83y

24y

121

n121+

1856400x121 x

22x

83x

24y

22y

83y

24y

121

n132

+1299480x12

1 x62x

24y

62y

24y

121

n110+

1299480x121 x

82x

63x

24y

82y

63y

24y

121

n154

− 9096360x121 x

62x

63x

24y

62y

63y

24y

121

n143+

19492200x121 x

42x

63x

24y

42y

63y

24y

121

n132

+1299480x12

1 x63x

24y

63y

24y

121

n110− 12994800x12

1 x22x

63x

24y

22y

63y

24y

121

n121

− 2784600x121 x

42x

24y

42y

24y

121

n99− 2784600x12

1 x82x

43x

24y

82y

43y

24y

121

n143

+19492200x12

1 x62x

43x

24y

62y

43y

24y

121

n132− 41769000x12

1 x42x

43x

24y

42y

43y

24y

121

n121

− 2784600x121 x

43x

24y

43y

24y

121

n99+

27846000x121 x

22x

43x

24y

22y

43y

24y

121

n110

− 185640x121 x

24y

24y

121

n77+

1856400x121 x

22x

24y

22y

24y

121

n88

+1856400x12

1 x82x

23x

24y

82y

23y

24y

121

n132− 12994800x12

1 x62x

23x

24y

62y

23y

24y

121

n121

+27846000x12

1 x42x

23x

24y

42y

23y

24y

121

n110+

1856400x121 x

23x

24y

23y

24y

121

n88

− 18564000x121 x

22x

23x

24y

22y

23y

24y

121

n99− 19448x10

1 y101

n55

− 19448x101 x

82y

82y

101

n99− 19448x10

1 x83y

83y

101

n99


− 19448x101 x

82x

83y

82y

83y

101

n143+

136136x101 x

62x

83y

62y

83y

101

n132

− 291720x101 x

42x

83y

42y

83y

101

n121+

194480x101 x

22x

83y

22y

83y

101

n110

− 19448x101 x

84y

84y

101

n99− 19448x10

1 x82x

84y

82y

84y

101

n143

− 19448x101 x

83x

84y

83y

84y

101

n143− 19448x10

1 x82x

83x

84y

82y

83y

84y

101

n187

+136136x10

1 x62x

83x

84y

62y

83y

84y

101

n176− 291720x10

1 x42x

83x

84y

42y

83y

84y

101

n165

+194480x10

1 x22x

83x

84y

22y

83y

84y

101

n154+

136136x101 x

62x

84y

62y

84y

101

n132

+136136x10

1 x63x

84y

63y

84y

101

n132+

136136x101 x

82x

63x

84y

82y

63y

84y

101

n176

− 952952x101 x

62x

63x

84y

62y

63y

84y

101

n165+

2042040x101 x

42x

63x

84y

42y

63y

84y

101

n154

− 1361360x101 x

22x

63x

84y

22y

63y

84y

101

n143− 291720x10

1 x42x

84y

42y

84y

101

n121

− 291720x101 x

43x

84y

43y

84y

101

n121− 291720x10

1 x82x

43x

84y

82y

43y

84y

101

n165

+2042040x10

1 x62x

43x

84y

62y

43y

84y

101

n154− 4375800x10

1 x42x

43x

84y

42y

43y

84y

101

n143

+2917200x10

1 x22x

43x

84y

22y

43y

84y

101

n132+

194480x101 x

22x

84y

22y

84y

101

n110

+194480x10

1 x23x

84y

23y

84y

101

n110+

194480x101 x

82x

23x

84y

82y

23y

84y

101

n154

− 1361360x101 x

62x

23x

84y

62y

23y

84y

101

n143+

2917200x101 x

42x

23x

84y

42y

23y

84y

101

n132

− 1944800x101 x

22x

23x

84y

22y

23y

84y

101

n121+

136136x101 x

62y

62y

101

n88

+136136x10

1 x82x

63y

82y

63y

101

n132+

136136x101 x

63y

63y

101

n88

− 952952x101 x

62x

63y

62y

63y

101

n121+

2042040x101 x

42x

63y

42y

63y

101

n110

− 1361360x101 x

22x

63y

22y

63y

101

n99+

136136x101 x

82x

64y

82y

64y

101

n132

+136136x10

1 x82x

83x

64y

82y

83y

64y

101

n176+

136136x101 x

83x

64y

83y

64y

101

n132

− 952952x101 x

62x

83x

64y

62y

83y

64y

101

n165+

2042040x101 x

42x

83x

64y

42y

83y

64y

101

n154

− 1361360x101 x

22x

83x

64y

22y

83y

64y

101

n143+

136136x101 x

64y

64y

101

n88

− 952952x101 x

62x

64y

62y

64y

101

n121− 952952x10

1 x82x

63x

64y

82y

63y

64y

101

n165

− 952952x101 x

63x

64y

63y

64y

101

n121+

6670664x101 x

62x

63x

64y

62y

63y

64y

101

n154

− 14294280x101 x

42x

63x

64y

42y

63y

64y

101

n143+

9529520x101 x

22x

63x

64y

22y

63y

64y

101

n132

+2042040x10

1 x42x

64y

42y

64y

101

n110+

2042040x101 x

82x

43x

64y

82y

43y

64y

101

n154


+2042040x10

1 x43x

64y

43y

64y

101

n110− 14294280x10

1 x62x

43x

64y

62y

43y

64y

101

n143

+30630600x10

1 x42x

43x

64y

42y

43y

64y

101

n132− 20420400x10

1 x22x

43x

64y

22y

43y

64y

101

n121

− 1361360x101 x

22x

64y

22y

64y

101

n99− 1361360x10

1 x82x

23x

64y

82y

23y

64y

101

n143

− 1361360x101 x

23x

64y

23y

64y

101

n99+

9529520x101 x

62x

23x

64y

62y

23y

64y

101

n132

− 20420400x101 x

42x

23x

64y

42y

23y

64y

101

n121+

13613600x101 x

22x

23x

64y

22y

23y

64y

101

n110

− 291720x101 x

42y

42y

101

n77− 291720x10

1 x82x

43y

82y

43y

101

n121

+2042040x10

1 x62x

43y

62y

43y

101

n110− 291720x10

1 x43y

43y

101

n77

− 4375800x101 x

42x

43y

42y

43y

101

n99+

2917200x101 x

22x

43y

22y

43y

101

n88

− 291720x101 x

82x

44y

82y

44y

101

n121− 291720x10

1 x82x

83x

44y

82y

83y

44y

101

n165

+2042040x10

1 x62x

83x

44y

62y

83y

44y

101

n154− 291720x10

1 x83x

44y

83y

44y

101

n121

− 4375800x101 x

42x

83x

44y

42y

83y

44y

101

n143+

2917200x101 x

22x

83x

44y

22y

83y

44y

101

n132

+2042040x10

1 x62x

44y

62y

44y

101

n110+

2042040x101 x

82x

63x

44y

82y

63y

44y

101

n154

− 14294280x101 x

62x

63x

44y

62y

63y

44y

101

n143+

2042040x101 x

63x

44y

63y

44y

101

n110

+30630600x10

1 x42x

63x

44y

42y

63y

44y

101

n132− 20420400x10

1 x22x

63x

44y

22y

63y

44y

101

n121

− 291720x101 x

44y

44y

101

n77− 4375800x10

1 x42x

44y

42y

44y

101

n99

− 4375800x101 x

82x

43x

44y

82y

43y

44y

101

n143+

30630600x101 x

62x

43x

44y

62y

43y

44y

101

n132

− 4375800x101 x

43x

44y

43y

44y

101

n99− 65637000x10

1 x42x

43x

44y

42y

43y

44y

101

n121

+43758000x10

1 x22x

43x

44y

22y

43y

44y

101

n110+

2917200x101 x

22x

44y

22y

44y

101

n88

+2917200x10

1 x82x

23x

44y

82y

23y

44y

101

n132− 20420400x10

1 x62x

23x

44y

62y

23y

44y

101

n121

+2917200x10

1 x23x

44y

23y

44y

101

n88+

43758000x101 x

42x

23x

44y

42y

23y

44y

101

n110

− 29172000x101 x

22x

23x

44y

22y

23y

44y

101

n99+

194480x101 x

22y

22y

101

n66

+194480x10

1 x82x

23y

82y

23y

101

n110− 1361360x10

1 x62x

23y

62y

23y

101

n99

+2917200x10

1 x42x

23y

42y

23y

101

n88+

194480x101 x

23y

23y

101

n66

− 1944800x101 x

22x

23y

22y

23y

101

n77+

194480x101 x

82x

24y

82y

24y

101

n110

+194480x10

1 x82x

83x

24y

82y

83y

24y

101

n154− 1361360x10

1 x62x

83x

24y

62y

83y

24y

101

n143


+2917200x10

1 x42x

83x

24y

42y

83y

24y

101

n132+

194480x101 x

83x

24y

83y

24y

101

n110

− 1944800x101 x

22x

83x

24y

22y

83y

24y

101

n121− 1361360x10

1 x62x

24y

62y

24y

101

n99

− 1361360x101 x

82x

63x

24y

82y

63y

24y

101

n143+

9529520x101 x

62x

63x

24y

62y

63y

24y

101

n132

− 20420400x101 x

42x

63x

24y

42y

63y

24y

101

n121− 1361360x10

1 x63x

24y

63y

24y

101

n99

+13613600x10

1 x22x

63x

24y

22y

63y

24y

101

n110+

2917200x101 x

42x

24y

42y

24y

101

n88

+2917200x10

1 x82x

43x

24y

82y

43y

24y

101

n132− 20420400x10

1 x62x

43x

24y

62y

43y

24y

101

n121

+43758000x10

1 x42x

43x

24y

42y

43y

24y

101

n110+

2917200x101 x

43x

24y

43y

24y

101

n88

− 29172000x101 x

22x

43x

24y

22y

43y

24y

101

n99+

194480x101 x

24y

24y

101

n66

− 1944800x101 x

22x

24y

22y

24y

101

n77− 1944800x10

1 x82x

23x

24y

82y

23y

24y

101

n121

+13613600x10

1 x62x

23x

24y

62y

23y

24y

101

n110− 29172000x10

1 x42x

23x

24y

42y

23y

24y

101

n99

− 1944800x101 x

23x

24y

23y

24y

101

n77+

19448000x101 x

22x

23x

24y

22y

23y

24y

101

n88

+12870x8

1y81

n44+

12870x81x

82y

82y

81

n88

+12870x8

1x83y

83y

81

n88+

12870x81x

82x

83y

82y

83y

81

n132

− 90090x81x

62x

83y

62y

83y

81

n121+

193050x81x

42x

83y

42y

83y

81

n110

− 128700x81x

22x

83y

22y

83y

81

n99+

12870x81x

84y

84y

81

n88

+12870x8

1x82x

84y

82y

84y

81

n132+

12870x81x

83x

84y

83y

84y

81

n132

+12870x8

1x82x

83x

84y

82y

83y

84y

81

n176− 90090x8

1x62x

83x

84y

62y

83y

84y

81

n165

+193050x8

1x42x

83x

84y

42y

83y

84y

81

n154− 128700x8

1x22x

83x

84y

22y

83y

84y

81

n143

− 90090x81x

62x

84y

62y

84y

81

n121− 90090x8

1x63x

84y

63y

84y

81

n121

− 90090x81x

82x

63x

84y

82y

63y

84y

81

n165+

630630x81x

62x

63x

84y

62y

63y

84y

81

n154

− 1351350x81x

42x

63x

84y

42y

63y

84y

81

n143+

900900x81x

22x

63x

84y

22y

63y

84y

81

n132

+193050x8

1x42x

84y

42y

84y

81

n110+

193050x81x

43x

84y

43y

84y

81

n110

+193050x8

1x82x

43x

84y

82y

43y

84y

81

n154− 1351350x8

1x62x

43x

84y

62y

43y

84y

81

n143

+2895750x8

1x42x

43x

84y

42y

43y

84y

81

n132− 1930500x8

1x22x

43x

84y

22y

43y

84y

81

n121

− 128700x81x

22x

84y

22y

84y

81

n99− 128700x8

1x23x

84y

23y

84y

81

n99


− 128700x81x

82x

23x

84y

82y

23y

84y

81

n143+

900900x81x

62x

23x

84y

62y

23y

84y

81

n132

− 1930500x81x

42x

23x

84y

42y

23y

84y

81

n121+

1287000x81x

22x

23x

84y

22y

23y

84y

81

n110

− 90090x81x

62y

62y

81

n77− 90090x8

1x82x

63y

82y

63y

81

n121

− 90090x81x

63y

63y

81

n77+

630630x81x

62x

63y

62y

63y

81

n110

− 1351350x81x

42x

63y

42y

63y

81

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n143+

9459450x81x

62x

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81

n132


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61

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61

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61

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61

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− 245245x61x

63x

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61

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61

n132


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61

n99

− 350350x61x

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61

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61

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61

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n99

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− 75075x61x

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61

n99

− 75075x61x

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61

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− 500500x61x

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82x

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n88

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61

n121


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61

n99− 350350x6

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61

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61

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61

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61

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61

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61

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61

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61

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− 500500x61x

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61

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41

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41

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41

n99

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41

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41

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1001x41x

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1001x41x

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− 7007x41x

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84y

41

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15015x41x

42x

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42y

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41

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− 10010x41x

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41

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1x62x

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84y

41

n99

− 7007x41x

63x

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41

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41

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41

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41

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82x

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41

n132

− 105105x41x

62x

43x

84y

62y

43y

84y

41

n121+

225225x41x

42x

43x

84y

42y

43y

84y

41

n110

− 150150x41x

22x

43x

84y

22y

43y

84y

41

n99− 10010x4

1x22x

84y

22y

84y

41

n77

− 10010x41x

23x

84y

23y

84y

41

n77− 10010x4

1x82x

23x

84y

82y

23y

84y

41

n121


+70070x4

1x62x

23x

84y

62y

23y

84y

41

n110− 150150x4

1x42x

23x

84y

42y

23y

84y

41

n99

+100100x4

1x22x

23x

84y

22y

23y

84y

41

n88− 7007x4

1x62y

62y

41

n55

− 7007x41x

82x

63y

82y

63y

41

n99− 7007x4

1x63y

63y

41

n55

+49049x4

1x62x

63y

62y

63y

41

n88− 105105x4

1x42x

63y

42y

63y

41

n77

+70070x4

1x22x

63y

22y

63y

41

n66− 7007x4

1x82x

64y

82y

64y

41

n99

− 7007x41x

82x

83x

64y

82y

83y

64y

41

n143− 7007x4

1x83x

64y

83y

64y

41

n99

+49049x4

1x62x

83x

64y

62y

83y

64y

41

n132− 105105x4

1x42x

83x

64y

42y

83y

64y

41

n121

+70070x4

1x22x

83x

64y

22y

83y

64y

41

n110− 7007x4

1x64y

64y

41

n55

+49049x4

1x62x

64y

62y

64y

41

n88+

49049x41x

82x

63x

64y

82y

63y

64y

41

n132

+49049x4

1x63x

64y

63y

64y

41

n88− 343343x4

1x62x

63x

64y

62y

63y

64y

41

n121

+735735x4

1x42x

63x

64y

42y

63y

64y

41

n110− 490490x4

1x22x

63x

64y

22y

63y

64y

41

n99

− 105105x41x

42x

64y

42y

64y

41

n77− 105105x4

1x82x

43x

64y

82y

43y

64y

41

n121

− 105105x41x

43x

64y

43y

64y

41

n77+

735735x41x

62x

43x

64y

62y

43y

64y

41

n110

− 1576575x41x

42x

43x

64y

42y

43y

64y

41

n99+

1051050x41x

22x

43x

64y

22y

43y

64y

41

n88

+70070x4

1x22x

64y

22y

64y

41

n66+

70070x41x

82x

23x

64y

82y

23y

64y

41

n110

+70070x4

1x23x

64y

23y

64y

41

n66− 490490x4

1x62x

23x

64y

62y

23y

64y

41

n99

+1051050x4

1x42x

23x

64y

42y

23y

64y

41

n88− 700700x4

1x22x

23x

64y

22y

23y

64y

41

n77

+1001x4

1y41

n22+

15015x41x

42y

42y

41

n44

+15015x4

1x82x

43y

82y

43y

41

n88− 105105x4

1x62x

43y

62y

43y

41

n77

+15015x4

1x43y

43y

41

n44+

225225x41x

42x

43y

42y

43y

41

n66

− 150150x41x

22x

43y

22y

43y

41

n55+

15015x41x

82x

44y

82y

44y

41

n88

+15015x4

1x82x

83x

44y

82y

83y

44y

41

n132− 105105x4

1x62x

83x

44y

62y

83y

44y

41

n121

+15015x4

1x83x

44y

83y

44y

41

n88+

225225x41x

42x

83x

44y

42y

83y

44y

41

n110

− 150150x41x

22x

83x

44y

22y

83y

44y

41

n99− 105105x4

1x62x

44y

62y

44y

41

n77

− 105105x41x

82x

63x

44y

82y

63y

44y

41

n121+

735735x41x

62x

63x

44y

62y

63y

44y

41

n110


− 105105x41x

63x

44y

63y

44y

41

n77− 1576575x4

1x42x

63x

44y

42y

63y

44y

41

n99

+1051050x4

1x22x

63x

44y

22y

63y

44y

41

n88+

15015x41x

44y

44y

41

n44

+225225x4

1x42x

44y

42y

44y

41

n66+

225225x41x

82x

43x

44y

82y

43y

44y

41

n110

− 1576575x41x

62x

43x

44y

62y

43y

44y

41

n99+

225225x41x

43x

44y

43y

44y

41

n66

+3378375x4

1x42x

43x

44y

42y

43y

44y

41

n88− 2252250x4

1x22x

43x

44y

22y

43y

44y

41

n77

− 150150x41x

22x

44y

22y

44y

41

n55− 150150x4

1x82x

23x

44y

82y

23y

44y

41

n99

+1051050x4

1x62x

23x

44y

62y

23y

44y

41

n88− 150150x4

1x23x

44y

23y

44y

41

n55

− 2252250x41x

42x

23x

44y

42y

23y

44y

41

n77+

1501500x41x

22x

23x

44y

22y

23y

44y

41

n66

− 10010x41x

22y

22y

41

n33− 10010x4

1x82x

23y

82y

23y

41

n77

+70070x4

1x62x

23y

62y

23y

41

n66− 150150x4

1x42x

23y

42y

23y

41

n55

− 10010x41x

23y

23y

41

n33+

100100x41x

22x

23y

22y

23y

41

n44

− 10010x41x

82x

24y

82y

24y

41

n77− 10010x4

1x82x

83x

24y

82y

83y

24y

41

n121

+70070x4

1x62x

83x

24y

62y

83y

24y

41

n110− 150150x4

1x42x

83x

24y

42y

83y

24y

41

n99

− 10010x41x

83x

24y

83y

24y

41

n77+

100100x41x

22x

83x

24y

22y

83y

24y

41

n88

+70070x4

1x62x

24y

62y

24y

41

n66+

70070x41x

82x

63x

24y

82y

63y

24y

41

n110

− 490490x41x

62x

63x

24y

62y

63y

24y

41

n99+

1051050x41x

42x

63x

24y

42y

63y

24y

41

n88

+70070x4

1x63x

24y

63y

24y

41

n66− 700700x4

1x22x

63x

24y

22y

63y

24y

41

n77

− 150150x41x

42x

24y

42y

24y

41

n55− 150150x4

1x82x

43x

24y

82y

43y

24y

41

n99

+1051050x4

1x62x

43x

24y

62y

43y

24y

41

n88− 2252250x4

1x42x

43x

24y

42y

43y

24y

41

n77

− 150150x41x

43x

24y

43y

24y

41

n55+

1501500x41x

22x

43x

24y

22y

43y

24y

41

n66

− 10010x41x

24y

24y

41

n33+

100100x41x

22x

24y

22y

24y

41

n44

+100100x4

1x82x

23x

24y

82y

23y

24y

41

n88− 700700x4

1x62x

23x

24y

62y

23y

24y

41

n77

+1501500x4

1x42x

23x

24y

42y

23y

24y

41

n66+

100100x41x

23x

24y

23y

24y

41

n44

− 1001000x41x

22x

23x

24y

22y

23y

24y

41

n55− 78x2

1x82y

82y

21

n55

− 78x21x

83y

83y

21

n55− 78x2

1x82x

83y

82y

83y

21

n99


+546x2

1x62x

83y

62y

83y

21

n88− 1170x2

1x42x

83y

42y

83y

21

n77

+780x2

1x22x

83y

22y

83y

21

n66− 78x2

1x84y

84y

21

n55

− 78x21x

82x

84y

82y

84y

21

n99− 78x2

1x83x

84y

83y

84y

21

n99

− 78x21x

82x

83x

84y

82y

83y

84y

21

n143+

546x21x

62x

83x

84y

62y

83y

84y

21

n132

− 1170x21x

42x

83x

84y

42y

83y

84y

21

n121+

780x21x

22x

83x

84y

22y

83y

84y

21

n110

+546x2

1x62x

84y

62y

84y

21

n88+

546x21x

63x

84y

63y

84y

21

n88

+546x2

1x82x

63x

84y

82y

63y

84y

21

n132− 3822x2

1x62x

63x

84y

62y

63y

84y

21

n121

+8190x2

1x42x

63x

84y

42y

63y

84y

21

n110− 5460x2

1x22x

63x

84y

22y

63y

84y

21

n99

− 1170x21x

42x

84y

42y

84y

21

n77− 1170x2

1x43x

84y

43y

84y

21

n77

− 1170x21x

82x

43x

84y

82y

43y

84y

21

n121+

8190x21x

62x

43x

84y

62y

43y

84y

21

n110

− 17550x21x

42x

43x

84y

42y

43y

84y

21

n99+

11700x21x

22x

43x

84y

22y

43y

84y

21

n88

+780x2

1x22x

84y

22y

84y

21

n66+

780x21x

23x

84y

23y

84y

21

n66

+780x2

1x82x

23x

84y

82y

23y

84y

21

n110− 5460x2

1x62x

23x

84y

62y

23y

84y

21

n99

+11700x2

1x42x

23x

84y

42y

23y

84y

21

n88− 7800x2

1x22x

23x

84y

22y

23y

84y

21

n77

+546x2

1x62y

62y

21

n44+

546x21x

82x

63y

82y

63y

21

n88

+546x2

1x63y

63y

21

n44− 3822x2

1x62x

63y

62y

63y

21

n77

+8190x2

1x42x

63y

42y

63y

21

n66− 5460x2

1x22x

63y

22y

63y

21

n55

+546x2

1x82x

64y

82y

64y

21

n88+

546x21x

82x

83x

64y

82y

83y

64y

21

n132

+546x2

1x83x

64y

83y

64y

21

n88− 3822x2

1x62x

83x

64y

62y

83y

64y

21

n121

+8190x2

1x42x

83x

64y

42y

83y

64y

21

n110− 5460x2

1x22x

83x

64y

22y

83y

64y

21

n99

+546x2

1x64y

64y

21

n44− 3822x2

1x62x

64y

62y

64y

21

n77

− 3822x21x

82x

63x

64y

82y

63y

64y

21

n121− 3822x2

1x63x

64y

63y

64y

21

n77

+26754x2

1x62x

63x

64y

62y

63y

64y

21

n110− 57330x2

1x42x

63x

64y

42y

63y

64y

21

n99

+38220x2

1x22x

63x

64y

22y

63y

64y

21

n88+

8190x21x

42x

64y

42y

64y

21

n66

+8190x2

1x82x

43x

64y

82y

43y

64y

21

n110+

8190x21x

43x

64y

43y

64y

21

n66


− 57330x21x

62x

43x

64y

62y

43y

64y

21

n99+

122850x21x

42x

43x

64y

42y

43y

64y

21

n88

− 81900x21x

22x

43x

64y

22y

43y

64y

21

n77− 5460x2

1x22x

64y

22y

64y

21

n55

− 5460x21x

82x

23x

64y

82y

23y

64y

21

n99− 5460x2

1x23x

64y

23y

64y

21

n55

+38220x2

1x62x

23x

64y

62y

23y

64y

21

n88− 81900x2

1x42x

23x

64y

42y

23y

64y

21

n77

+54600x2

1x22x

23x

64y

22y

23y

64y

21

n66− 1170x2

1x42y

42y

21

n33

− 1170x21x

82x

43y

82y

43y

21

n77+

8190x21x

62x

43y

62y

43y

21

n66

− 1170x21x

43y

43y

21

n33− 17550x2

1x42x

43y

42y

43y

21

n55

+11700x2

1x22x

43y

22y

43y

21

n44− 1170x2

1x82x

44y

82y

44y

21

n77

− 1170x21x

82x

83x

44y

82y

83y

44y

21

n121+

8190x21x

62x

83x

44y

62y

83y

44y

21

n110

− 1170x21x

83x

44y

83y

44y

21

n77− 17550x2

1x42x

83x

44y

42y

83y

44y

21

n99

+11700x2

1x22x

83x

44y

22y

83y

44y

21

n88+

8190x21x

62x

44y

62y

44y

21

n66

+8190x2

1x82x

63x

44y

82y

63y

44y

21

n110− 57330x2

1x62x

63x

44y

62y

63y

44y

21

n99

+8190x2

1x63x

44y

63y

44y

21

n66+

122850x21x

42x

63x

44y

42y

63y

44y

21

n88

− 81900x21x

22x

63x

44y

22y

63y

44y

21

n77− 1170x2

1x44y

44y

21

n33

− 17550x21x

42x

44y

42y

44y

21

n55− 17550x2

1x82x

43x

44y

82y

43y

44y

21

n99

+122850x2

1x62x

43x

44y

62y

43y

44y

21

n88− 17550x2

1x43x

44y

43y

44y

21

n55

− 263250x21x

42x

43x

44y

42y

43y

44y

21

n77+

175500x21x

22x

43x

44y

22y

43y

44y

21

n66

+11700x2

1x22x

44y

22y

44y

21

n44+

11700x21x

82x

23x

44y

82y

23y

44y

21

n88

− 81900x21x

62x

23x

44y

62y

23y

44y

21

n77+

11700x21x

23x

44y

23y

44y

21

n44

+175500x2

1x42x

23x

44y

42y

23y

44y

21

n66− 117000x2

1x22x

23x

44y

22y

23y

44y

21

n55

− 78x21y

21

n11+

780x21x

22y

22y

21

n22

+780x2

1x82x

23y

82y

23y

21

n66− 5460x2

1x62x

23y

62y

23y

21

n55

+11700x2

1x42x

23y

42y

23y

21

n44+

780x21x

23y

23y

21

n22

− 7800x21x

22x

23y

22y

23y

21

n33+

780x21x

82x

24y

82y

24y

21

n66

+780x2

1x82x

83x

24y

82y

83y

24y

21

n110− 5460x2

1x62x

83x

24y

62y

83y

24y

21

n99


+11700x2

1x42x

83x

24y

42y

83y

24y

21

n88+

780x21x

83x

24y

83y

24y

21

n66

− 7800x21x

22x

83x

24y

22y

83y

24y

21

n77− 5460x2

1x62x

24y

62y

24y

21

n55

− 5460x21x

82x

63x

24y

82y

63y

24y

21

n99+

38220x21x

62x

63x

24y

62y

63y

24y

21

n88

− 81900x21x

42x

63x

24y

42y

63y

24y

21

n77− 5460x2

1x63x

24y

63y

24y

21

n55

+54600x2

1x22x

63x

24y

22y

63y

24y

21

n66+

11700x21x

42x

24y

42y

24y

21

n44

+11700x2

1x82x

43x

24y

82y

43y

24y

21

n88− 81900x2

1x62x

43x

24y

62y

43y

24y

21

n77

+175500x2

1x42x

43x

24y

42y

43y

24y

21

n66+

11700x21x

43x

24y

43y

24y

21

n44

− 117000x21x

22x

43x

24y

22y

43y

24y

21

n55+

780x21x

24y

24y

21

n22

− 7800x21x

22x

24y

22y

24y

21

n33− 7800x2

1x82x

23x

24y

82y

23y

24y

21

n77

+54600x2

1x62x

23x

24y

62y

23y

24y

21

n66− 117000x2

1x42x

23x

24y

42y

23y

24y

21

n55

− 7800x21x

23x

24y

23y

24y

21

n33+

78000x21x

22x

23x

24y

22y

23y

24y

21

n44

+x8

2y82

n44+x8

3y83

n44

+x8

2x83y

82y

83

n88− 7x6

2x83y

62y

83

n77

+15x4

2x83y

42y

83

n66− 10x2

2x83y

22y

83

n55

+x8

4y84

n44+x8

2x84y

82y

84

n88

+x8

3x84y

83y

84

n88+x8

2x83x

84y

82y

83y

84

n132

− 7x62x

83x

84y

62y

83y

84

n121+

15x42x

83x

84y

42y

83y

84

n110

− 10x22x

83x

84y

22y

83y

84

n99− 7x6

2x84y

62y

84

n77

− 7x63x

84y

63y

84

n77− 7x8

2x63x

84y

82y

63y

84

n121

+49x6

2x63x

84y

62y

63y

84

n110− 105x4

2x63x

84y

42y

63y

84

n99

+70x2

2x63x

84y

22y

63y

84

n88+

15x42x

84y

42y

84

n66

+15x4

3x84y

43y

84

n66+

15x82x

43x

84y

82y

43y

84

n110

− 105x62x

43x

84y

62y

43y

84

n99+

225x42x

43x

84y

42y

43y

84

n88

− 150x22x

43x

84y

22y

43y

84

n77− 10x2

2x84y

22y

84

n55

− 10x23x

84y

23y

84

n55− 10x8

2x23x

84y

82y

23y

84

n99


+70x6

2x23x

84y

62y

23y

84

n88− 150x4

2x23x

84y

42y

23y

84

n77

+100x2

2x23x

84y

22y

23y

84

n66− 7x6

2y62

n33

− 7x82x

63y

82y

63

n77− 7x6

3y63

n33

+49x6

2x63y

62y

63

n66− 105x4

2x63y

42y

63

n55

+70x2

2x63y

22y

63

n44− 7x8

2x64y

82y

64

n77

− 7x82x

83x

64y

82y

83y

64

n121− 7x8

3x64y

83y

64

n77

+49x6

2x83x

64y

62y

83y

64

n110− 105x4

2x83x

64y

42y

83y

64

n99

+70x2

2x83x

64y

22y

83y

64

n88− 7x6

4y64

n33

+49x6

2x64y

62y

64

n66+

49x82x

63x

64y

82y

63y

64

n110

+49x6

3x64y

63y

64

n66− 343x6

2x63x

64y

62y

63y

64

n99

+735x4

2x63x

64y

42y

63y

64

n88− 490x2

2x63x

64y

22y

63y

64

n77

− 105x42x

64y

42y

64

n55− 105x8

2x43x

64y

82y

43y

64

n99

− 105x43x

64y

43y

64

n55+

735x62x

43x

64y

62y

43y

64

n88

− 1575x42x

43x

64y

42y

43y

64

n77+

1050x22x

43x

64y

22y

43y

64

n66

+70x2

2x64y

22y

64

n44+

70x82x

23x

64y

82y

23y

64

n88

+70x2

3x64y

23y

64

n44− 490x6

2x23x

64y

62y

23y

64

n77

+1050x4

2x23x

64y

42y

23y

64

n66− 700x2

2x23x

64y

22y

23y

64

n55

+15x4

2y42

n22+

15x82x

43y

82y

43

n66

− 105x62x

43y

62y

43

n55+

15x43y

43

n22

+225x4

2x43y

42y

43

n44− 150x2

2x43y

22y

43

n33

+15x8

2x44y

82y

44

n66+

15x82x

83x

44y

82y

83y

44

n110

− 105x62x

83x

44y

62y

83y

44

n99+

15x83x

44y

83y

44

n66

+225x4

2x83x

44y

42y

83y

44

n88− 150x2

2x83x

44y

22y

83y

44

n77

− 105x62x

44y

62y

44

n55− 105x8

2x63x

44y

82y

63y

44

n99

+735x6

2x63x

44y

62y

63y

44

n88− 105x6

3x44y

63y

44

n55


− 1575x42x

63x

44y

42y

63y

44

n77+

1050x22x

63x

44y

22y

63y

44

n66

+15x4

4y44

n22+

225x42x

44y

42y

44

n44

+225x8

2x43x

44y

82y

43y

44

n88− 1575x6

2x43x

44y

62y

43y

44

n77

+225x4

3x44y

43y

44

n44+

3375x42x

43x

44y

42y

43y

44

n66

− 2250x22x

43x

44y

22y

43y

44

n55− 150x2

2x44y

22y

44

n33

− 150x82x

23x

44y

82y

23y

44

n77+

1050x62x

23x

44y

62y

23y

44

n66

− 150x23x

44y

23y

44

n33− 2250x4

2x23x

44y

42y

23y

44

n55

+1500x2

2x23x

44y

22y

23y

44

n44− 10x2

2y22

n11

− 10x82x

23y

82y

23

n55+

70x62x

23y

62y

23

n44

− 150x42x

23y

42y

23

n33− 10x2

3y23

n11

+100x2

2x23y

22y

23

n22− 10x8

2x24y

82y

24

n55

− 10x82x

83x

24y

82y

83y

24

n99+

70x62x

83x

24y

62y

83y

24

n88

− 150x42x

83x

24y

42y

83y

24

n77− 10x8

3x24y

83y

24

n55

+100x2

2x83x

24y

22y

83y

24

n66+

70x62x

24y

62y

24

n44

+70x8

2x63x

24y

82y

63y

24

n88− 490x6

2x63x

24y

62y

63y

24

n77

+1050x4

2x63x

24y

42y

63y

24

n66+

70x63x

24y

63y

24

n44

− 700x22x

63x

24y

22y

63y

24

n55− 150x4

2x24y

42y

24

n33

− 150x82x

43x

24y

82y

43y

24

n77+

1050x62x

43x

24y

62y

43y

24

n66

− 2250x42x

43x

24y

42y

43y

24

n55− 150x4

3x24y

43y

24

n33

+1500x2

2x43x

24y

22y

43y

24

n44− 10x2

4y24

n11

+100x2

2x24y

22y

24

n22+

100x82x

23x

24y

82y

23y

24

n66

− 700x62x

23x

24y

62y

23y

24

n55+

1500x42x

23x

24y

42y

23y

24

n44

+100x2

3x24y

23y

24

n22− 1000x2

2x23x

24y

22y

23y

24

n33

+ 1

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by Aaron Chow - University of Toronto T-Space · Aaron Chow Doctor of Philosophy Graduate Department of Mathematics University of Toronto 2015 Let fbe a modular form of even weight