Serre’s Conjecture Mod pq?

B. Mazur  W.A. Stein

January 11, 1999

Contents

1 Introduction 1

2 Letters 2

3 Table of Congruences 5

4 Table of Eigenforms 22

1 Introduction

I haven’t yet written an introduction, but here are some helpful remarks as to how to read the table. Please see Barry Mazur’s letter below which motivated the computation. Basically, when both an f and g occur together in a given row, then “Serre’s Conjecture mod pq” for a certain representation has an easy positive answer.

The rows of the congruence table contain an f whenever the corresponding eigenform at level NM is congruent to the reduction of f modulo the prime P listed in the first column, at least to the precision computed. Similarly for g. In some cases the program did not perform a complete check that the forms were congruent to the precision computed, but only checked, with high probability, that there is a congruence to that precision. In such cases this is indicated by a question mark subscript. In many cases Ribet’s theorem implies that there must be a congruence.

I do not know if Sturm’s result can be applied in this context because the congruence is only checked for those a_{p} for which p is prime to NMpq.

In a few cases it was too much extra work to reduce the prime level ≤ 97 eigenforms modulo 2, and so this is not done, even though 2 might be among the canidate pairs (P, Q). This is indicated above the relevant table.

The eigenforms refered to in the congruence table are uniquely determined, up to conjugacy, by a factor of the characteristic polynomial of some Hecke operator. These factors are given, so that further investigation of particular eigenforms is possible.

To understand why this table was computed, see the letter from Barry Mazur in the last section. Note that the congruence table does not omit primes for which the corresponding representations are reducible.

2 Letters

From: Barry Mazur
Subject: SERRE'S CONJECTURE MOD PQ
To: William Stein

Dear William,

To put perspective on the subject of this e-mail, let me begin with an utterly untestable version of what I was suggesting to you. Let p and q be distinct prime numbers and G the Galois group of the maximal algebraic extension of Q unramified outside a finite set S of primes. Suppose that we have an odd continuous representation r : G → GL_{2}(Z/pqZ) which is irreducible when reduced mod p and mod q. Is r ”modular”?

To begin to give some shape to the above question, we might as well assume the “classical” Serre conjecture for single primes, so we can start out by giving ourselves two modular newforms f and g and then look for a modular newform h that is congruent mod p to f and mod q to g – under the assumption that the associated Galois representations mod p and mod q respectively are irreducible. I should be more careful about what I mean “mod p” and “mod q”: I really mean to take p and q prime ideals in the ring generated by the Fourier coefficients of f and g, and ditto with regard to h. To simplify one’s life, one can ”start out” by working only with pairs f and g of newforms whose ring of Fourier coefficients is Z (i.e., the f and g in question correspond to elliptic curves). But even if one does this, one has to face the possibility (the “probability”, in fact) that h, if it exists at all, no longer has its ring of Fourier coefficients equal to Z.

Of course, phrased this way, our qeustion is still relatively un“falsifiable” so we have better be more specific.

Ken and I thought a bit about how to shape a more specific testable question, and here is what we came up with– very tentatively still. Suppose that N and M are distinct prime numbers, and f and g are newforms of weight two on X_{0}(N) and X_{0}(M) respectively. Now let p be a prime number (or ideal) dividing either 1 + M − a_{M}(f) or 1 + M + a_{M}(f), and let q be a prime number (or ideal) dividing either 1 + N − a_{N}(g) or 1 + N + a_{N}(g). Let us suppose that the residual characteristics of the ideals p and q are distinct (i.e., if we are in a case where p and q are prime numbers, we just want them to be distinct). Suppose also that the corresponding Galois representations associated to f mod p and g mod q are irreducible. Let us say that we have “maximum success” if for any such choice of f, g, p, q there is a modular newform h of weight two on X_{0}(NM) which is congruent to f mod p and to g mod q. Here a_{N}(f) means the Nth Fourier coefficient of f, etc.

The reason why we call the above “maximum success” is that we are putting down the “evident” conditions necessary for there to be such a newform h (and looking for such an h at the minimal conceivable level, and weight). It is probably unreasonable to expect “maximum success”, but initial experiments might give us a sense of how far from the mark “maximal success” actually is, in which case we will weaken our expectations accordingly... I should also say, that the first case is N = 11 and M = 17, where the level of the sought for h is 187, i.e., not too large yet...

Does this sound as if it is a conceivable, fun, etc. computation to make?

Barry

Date: Mon, 26 Oct 1998 12:46:09 -0500

Dear William,

That’s terrific! Actually I had debated about whether to exclude the prime 2 from the search– it being slightly problematic for level-raising, level-lowering issues. I also excluded reducible representations as being significantly more problematic on a number of other grounds. But, while you are systemtically doing the computation, it might make sense not to exclude reducible representations from the search, but just indicate in the data when they are reducible. After all, we may also be getting some “maximal succeses” there too (but if we get a ”failure” for a reducible case, I wouldn’t worry). Keep me posted!

Barry

Date: Tue, 3 Nov 1998 20:32:35 +0000 (GMT)
From: Kevin Buzzard <buzzard@ic.ac.uk>
To: was@bmw.autobahn.org
Subject: Re: congruence table
I looked through your tables a fair bit yesterday and can't find anything
which smacks of contradiction, which is very comforting. It all seems
to fit together very well, which means that I am getting more and more
confident in your programs!
[NB my experience with these programs is that if they "usually" work
then they still might not work 100% because if numbers get too big
then there's weird overflows and so maybe things don't work then---
but I don't have any evidence at all that your programs don't
work 100%]
Anyway, here's some random comments about them. When you say
in the intro that "in _some_cases the program didn't perform
a complete check of congruence"---really, you never performed
a _complete_check. If you mean "complete check up to p=***"
whatever *** is, maybe you should say this. Also it's a bit
strange that you then go on to talk about the ?? cases, and
say that they're congruences with a "high probability", whereas
in reality we're *always* only dealing with probabilities (except,

I guess, in cases where Ken's theorem predicts that a form
exists and only 1 form can possibly work at level NM). In practice
it's very difficult to verify that forms are actually congruent
because the effective Chebatorev bounds aren't good enough.

3 Table of Congruences

f = 11A, g = 17A

P Q 187A 187B 187C 187D 187E 187F
2 3 f f fg f f g
5 2  g
5 3  g  g

f = 11A, g = 19A

P Q 209A 209B 209C 209D
2 3  f  f_{?}  f_{?}g  fg
2 5  f  f_{?}  f_{?}g  f
5 3  f  g  g

f = 11A, g = 23A

P  Q  253A 253B 253C 253D
23  (x + 8, 11)  f
23 (x + 15, 19)  g  f
23 (x^{2} + x + 1, 2)  g  fg
23  (x + 3, 5)  g  f
5  (x + 8, 11)  f
5  (x + 15, 19)  g  f
5 (x^{2} + x + 1, 2)  f  g  g

f = 11A, g = 29A

P  Q  319A 319B 319C 319D 319E
2 (x + 14, 167)  f  f  g
2 (x + 7, 17)  f  g  f
2  (x + 4, 7)  f  f
3 (x + 14, 167)  f  f  g
3 (x + 7, 17)  fg  f
3  (x + 4, 7)  f  f
5 (x + 14, 167)  f  g
5 (x + 7, 17)  g  f
5  (x + 4, 7)  f

f = 17A, g = 19A

P Q 323A 323B 323C 323D 323E 323F
2 3  f  f  f_{?}g  f_{?}
2 5  f  f  f_{?}  f_{?}  g
2 7  f  g  f  f_{?}  f_{?}
3 5  f  g
3 7  fg

f = 11A, g = 31A

P  Q  341A 341B 341C 341D
13 (x^{2} + x + 1, 2)  g_{?}  fg_{?}
13  (x + 2, 5)  g  f
13 (x^{2} + 6x + 6, 7)  g_{?}  f
3 (x^{2} + x + 1, 2)  g_{?}  f_{?}g_{?}
3  (x + 2, 5)  g  f_{?}
3 (x^{2} + 6x + 6, 7)  g_{?}  f_{?}
5 (x^{2} + x + 1, 2)  f  g_{?}  g_{?}
5 (x^{2} + 6x + 6, 7)  f  g_{?}

f = 17A, g = 23A

P  Q  391A 391B 391C 391D 391E
2 (x + 99, 109)  g  f_{?}
2  (x + 8, 11)  f_{?}
2  (x + 3, 5)  g  f_{?}
5 (x + 99, 109)  f  g
5  (x + 8, 11)  f
5 (x^{2} + x + 1, 2)  f  g  g_{?}
7 (x + 99, 109)  g  f_{?}
7  (x + 8, 11)  f_{?}
7 (x^{2} + x + 1, 2)  g  f_{?}g_{?}
7  (x + 3, 5)  g  f_{?}

f = 11A, g = 37A

P Q 407A 407B 407C 407D
41 17  fg
41 7 g f
5 17  g
5 7  g
7 17  f  g

f = 11A, g = 37B

P Q 407A 407B 407C 407D
41 3  g  f  g
41 5  f  g
5 3  g  g
7 3  fg  g
7 5  f  g

f = 19A, g = 23A

P  Q  437A 437B 437C 437D 437E 437F 437G 437H
2  (x + 4, 11)  f  f  g  f  f_{?}  f_{?}  f_{?}
2  (x + 8, 11)  f  f  f  f_{?}  f_{?}  f_{?}
2 (x^{2} + x + 2, 3)  f  f  f  f_{?}  f_{?}g  f_{?}
3  (x + 4, 11)  g  f
3  (x + 8, 11)  f
3 (x^{2} + x + 1, 2)  g_{?}  fg_{?}

f = 11A, g = 41A (char 2 primes for g omitted)

P  Q  451A 451B 451C 451D 451E
17  (x + 2, 5)  g  f
17 (x + 413, 863)  f  g
2  (x + 2, 5)  g  f_{?}  f
2 (x + 413, 863)  f_{?}g  f
5 (x + 413, 863)  f  g

f = 11A, g = 43A

P Q 473A 473B 473C 473D 473E 473F 473G
19 3  fg  g
19 5  g  f
2 3  f_{?}  g  fg
2 5  g  f_{?}  f
5 3  f  g  g

f = 11A, g = 43B

P  Q  473A 473B 473C 473D 473E 473F 473G
19 (x + 62, 113)  g  f
19 (x + 5, 23)  g  f
19 (x + 4, 7)  f
2 (x + 62, 113)  f_{?}g  f
2 (x + 5, 23)  g  f_{?}  f
2  (x + 4, 7)  f_{?}  f
5 (x + 62, 113)  f  g
5 (x + 5, 23)  f  g
5  (x + 4, 7)  f

f = 17A, g = 29A

P  Q  493A 493B 493C 493D 493E 493F 493G 493H
2 (x + 24, 31)  f  f  f_{?}g
2 (x + 4, 7)  g  f  f  f_{?}
3 (x + 1, 2)  fg  g  g  g  f
3 (x + 24, 31)  f  g  f
3 (x + 4, 7)  f  g  f

f = 11A, g = 47A

P  Q  517A 517B 517C 517D 517E 517F 517G 517H 517I 517J 517K
2  (x + 20, 23)  f  f  f_{?}  f_{?}  f_{?}  f_{?}  f
2  (x + 1, 3)  f  fg  f_{?}  f_{?}g  g_{?}  f_{?}  f_{?}  fg
2  (x + 8, 31)  f  f  f_{?}g  f_{?}  f_{?}  f_{?}  f
2  (x + 2, 7)  f  f  f_{?}  g  f_{?}  f_{?}  f_{?}  f
5 (x^{4} + x^{3} + x^{2} + x + 1, 2)  g_{?}  g_{?}
5  (x + 20, 23)
5  (x + 1, 3)  g  g  g_{?}  g
5  (x + 8, 31)  g
5  (x + 2, 7)  g
7 (x^{4} + x^{3} + x^{2} + x + 1, 2)  f  g_{?}  g_{?}
7  (x + 20, 23)  f
7  (x + 1, 3)  g  f  g  g_{?}  g
7  (x + 8, 31)  g  f

f = 17A, g = 31A

P  Q  527A 527B 527C 527D 527E
2 (x + 98, 109)  g  f_{?}
2  (x + 7, 11)  g  f_{?}
2  (x + 2, 5)  f_{?}
3 (x + 98, 109)  g  f_{?}
3  (x + 7, 11)  g  f_{?}
3 (x^{2} + x + 1, 2)  g  f_{?}g_{?}
3  (x + 2, 5)  f_{?}
7 (x + 98, 109)  f  g
7  (x + 7, 11)  fg
7 (x^{2} + x + 1, 2)  f  g  g_{?}
7  (x + 2, 5)  f

f = 19A, g = 29A

P  Q  551A 551B 551C 551D 551E 551F 551G 551H
2 (x^{2} + 2x + 12, 13)  f  f  f  f_{?}  f_{?}g
2  (x + 4, 7)  f  f  f  f_{?}  f_{?}
2  (x + 5, 7)  f  fg  f  f_{?}  f_{?}
3 (x^{2} + 2x + 12, 13)  f_{?}  g
3  (x + 1, 2)  g  g  f_{?}g_{?}  g_{?}  g
3  (x + 4, 7)  f_{?}
3  (x + 5, 7)  g  f_{?}

f = 11A, g = 53A

P Q 583A 583B 583C 583D 583E 583F 583G 583H 583I
2 3  f  fg  f  fg  f_{?}  f_{?}  f_{?}
3 2 fg  g_{?}  f  g_{?}
5 2  g  g_{?}  g_{?}
5 3  g  g

f = 11A, g = 53B

P  Q  583A 583B 583C 583D 583E 583F 583G 583H 583I
2 (x + 10, 13)  f  f  f  f  f_{?}  f_{?}  f_{?}
2 (x + 224, 281)  f  f  f  f  f_{?}  f_{?}  f_{?}g
2  (x + 5, 43)  f  f  f  f  f_{?}g  f_{?}  f_{?}
3 (x + 10, 13)  f  f
3  (x + 1, 2)  fg  g_{?}  f  g_{?}
3 (x + 224, 281)  f  f  g
3  (x + 5, 43)  f  fg
5 (x + 10, 13)
5 (x + 1, 2) g g_{?} g_{?}
5 (x + 224, 281)  g
5  (x + 5, 43)  g

f = 19A, g = 31A

P  Q  589A 589B 589C 589D 589E
2 (x + 2, 5)  f  f  f_{?}g  f
2 (x + 29, 79)  f  f  f_{?}g  f
2 (x + 49, 79)  f  f  f_{?}  fg
3 (x + 2, 5)  g  f
3 (x + 29, 79)  g  f
3 (x + 49, 79)  fg
7 (x + 2, 5)  f  g
7 (x + 29, 79)  f  g
7 (x + 49, 79)  f  g

f = 17A, g = 37A

P Q 629A 629B 629C 629D 629E 629F 629G 629H
2 3  g  f_{?}  f_{?}g  fg
3 2  fg  g_{?}  f  g  g
5 2  f  g  g_{?}  g  g
5 3  g  f  g  g

f = 17A, g = 37B

P Q 629A 629B 629C 629D 629E 629F 629G 629H
2 3  f_{?}g  f_{?}  f
3 2  fg  g_{?}  f  g  g
5 2  f  g  g_{?}  g  g
5 3  f  g

f = 11A, g = 59A (char 2 primes for g omitted)

P  Q  649A 649B 649C 649D 649E 649F
11  (x + 9, 19)  fg
11 (x + 1940, 2671)  f  g
11  (x + 26, 29)  f
11  (x + 1, 7)  g  f
13  (x + 9, 19)  g  f
13 (x + 1940, 2671)  g  f
13  (x + 26, 29)  f
13  (x + 1, 7)  g  f
5  (x + 9, 19)  g  f
5 (x + 1940, 2671)  g  f
5  (x + 26, 29)  f
5  (x + 1, 7)  g  f

f = 23A, g = 29A

P  Q  667A 667B 667C 667D
(x + 4, 11)  (x + 52, 113)  f  g
(x + 4, 11)  (x + 1, 2)  f  g  g_{?}
(x + 4, 11)  (x + 19, 23)  fg
(x + 4, 11)  (x + 4, 7)  f
(x + 8, 11)  (x + 52, 113)  g
(x + 8, 11)  (x + 1, 2)  g  g_{?}
(x + 8, 11)  (x + 19, 23)  g
(x + 8, 11)  (x + 4, 7)
(x^{2} + x + 2, 3) (x + 52, 113)  f  f  g
(x^{2} + x + 2, 3) (x + 1, 2)  f  g  fg_{?}
(x^{2} + x + 2, 3) (x + 19, 23)  fg  f
(x^{2} + x + 2, 3) (x + 4, 7)  f  f

f = 11A, g = 61A

P Q 671A 671B 671C 671D
2 17  f  f  g
2 7  fg  f
37 17  fg
37 7  g  f
5 17  f  g
5 7  f  g

f = 11A, g = 61B

P  Q  671A 671B 671C 671D
2 (x + 3, 13)  f  fg
2 (x + 9, 17)  g  f  f
2 (x + 16, 331)  f  fg
2  (x + 2, 5)  g  f  f
37 (x + 3, 13)  g  f
37 (x + 9, 17)  g  f
37 (x + 16, 331)  g  f
37 (x + 2, 5)  g  f
5 (x + 3, 13)  f  g
5 (x + 9, 17)  fg
5 (x + 16, 331)  f  g

f = 17A, g = 41A (char 2 primes for g omitted)

P  Q  697A 697B 697C 697D 697E 697F
2  (x + 2, 5)  f  f_{?}  f  f_{?}  f_{?}
2 (x^{2} + 4x + 2, 5)  f  f_{?}  f  g  f_{?}  f_{?}
3  (x + 2, 5)  f  f
3 (x^{2} + 4x + 2, 5)  f  g  f

f = 19A, g = 37A

P Q 703A 703B 703C 703D 703E 703F 703G 703H 703I
2 5  f  f_{?}  fg  f_{?}g
3 2  g  g  f  g  g_{?}  g_{?}  g
3 5  f  g  g
5 2  g  g  g  g_{?}  g_{?}  fg

f = 19A, g = 37B

P Q 703A 703B 703C 703D 703E 703F 703G 703H 703I
2 11  f  f_{?}g  f  f_{?}
2 3  f  g  f_{?}  f  f_{?}
3 11  f  g
3 2  g  g  f  g  g_{?}  g_{?}  g
5 11  g  f
5 2  g  g  g  g_{?}  g_{?}  fg
5 3  g  f

f = 23A, g = 31A

P  Q  713A 713B 713C 713D 713E 713F 713G
(x + 4, 11) (x^{2} + x + 1, 2)  g  fg_{?}
(x + 4, 11)  (x + 5, 29)  f  g
(x + 4, 11)  (x + 2, 5)  f
(x + 8, 11) (x^{2} + x + 1, 2)  g  g_{?}
(x + 8, 11)  (x + 5, 29)  g
(x + 8, 11)  (x + 2, 5)
(x + 10, 89)  (x + 7, 11)  g  f
(x + 10, 89) (x^{2} + x + 1, 2)  g  g_{?}  f
(x + 10, 89)  (x + 5, 29)  fg
(x + 10, 89)  (x + 2, 5)  f
(x + 80, 89)  (x + 7, 11)  g  f_{?}
(x + 80, 89) (x^{2} + x + 1, 2)  g  f_{?}g_{?}
(x + 80, 89)  (x + 5, 29)  f_{?}  g
(x + 80, 89)  (x + 2, 5)  f_{?}

f = 17A, g = 43A

P Q 731A 731B 731C 731D 731E 731F
2 3  g  f  f_{?}  f_{?}g_{?}
2 5  f  g  f_{?}  f_{?}
2 7  f  f_{?}  f_{?}g
3 5  g  f_{?}
3 7  f_{?}g
5 3  g  f  g_{?}
5 7  f  g

f = 17A, g = 43B

P  Q  731A 731B 731C 731D 731E 731F
2  (x + 5, 23)  f  g  f_{?}  f_{?}
2 (x + 272, 521)  f  f_{?}g  f_{?}
2  (x + 4, 7)  f  f_{?}  f_{?}
3  (x + 5, 23)  g  f_{?}
3 (x + 272, 521)  g  f_{?}
3  (x + 4, 7)  f_{?}
5  (x + 5, 23)  fg
5 (x + 272, 521)  f  g
5  (x + 4, 7)  f

f = 11A, g = 67A

P Q 737A 737B 737C 737D 737E 737F
3 2 fg  g  g_{?}  g  g_{?}
5 2  g  g  g_{?}  g  g_{?}  f
61 2  g  g  fg_{?}  g  g_{?}

f = 11A, g = 67B

P  Q  737A 737B 737C 737D 737E 737F
3  (x + 4, 11)  f  g
3  (x + 8, 11)  f
3 (x^{2} + x + 12, 13)  f  g
5  (x + 4, 11)  fg
5  (x + 8, 11)  f
5 (x^{2} + x + 12, 13)  g  f
61  (x + 4, 11)  f  g
61  (x + 8, 11)  f
61 (x^{2} + x + 12, 13)  f  g

f = 11A, g = 67C

P  Q  737A 737B 737C 737D 737E 737F
3 (x + 65, 139)  f  g
3 (x + 77, 139)  f  g
5 (x + 65, 139)  g  f
5 (x + 77, 139)  g  f
61 (x + 65, 139)  f  g
61 (x + 77, 139)  fg

f = 19A, g = 41A (char 2 primes for g omitted)

P  Q  779A 779B 779C 779D 779E
2  (x + 2, 5)  f  f  f  f_{?}  f_{?}g_{?}
2 (x + 171, 607)  f  fg  f  f_{?}  f_{?}
2 (x + 299, 929)  f  f  f  f_{?}  f_{?}g
3  (x + 2, 5)  f  g_{?}
3 (x + 171, 607)  g  f
3 (x + 299, 929)  f  g

f = 11A, g = 71A

P  Q  781A 781B 781C 781D 781E 781F 781G
23 (x^{3} + x + 1, 2)  g_{?}  fg_{?}
23  (x, 3)  g  g  g  g  f
23  (x + 2, 5)  g  f
23 (x + 20, 53)  g  f
3 (x^{3} + x + 1, 2)  f  g_{?}  f_{?}g_{?}
3  (x + 2, 5)  f  g  f_{?}
3  (x + 20, 53)  f  g  f_{?}
5 (x^{3} + x + 1, 2)  f  g_{?}  g_{?}
5  (x, 3)  g  g  g  fg
5  (x + 20, 53)  f  g

f = 11A, g = 71B

P  Q  781A 781B 781C 781D 781E 781F 781G
23 (x^{3} + x^{2} + 1, 2)  g_{?}  fg_{?}
23  (x, 3)  g  g  g  g  f
23  (x + 27, 61)  g_{?}  f
23  (x + 4, 7)  f
3 (x^{3} + x^{2} + 1, 2)  f  g_{?}  f_{?}g_{?}
3  (x + 27, 61)  f  g_{?}  f_{?}
3  (x + 4, 7)  f  f_{?}
5 (x^{3} + x^{2} + 1, 2)  f  g_{?}  g_{?}
5  (x, 3)  g  g  g  fg
5  (x + 27, 61)  f  g_{?}
5  (x + 4, 7)  f

f = 17A, g = 47A

P  Q  799A 799B 799C 799D 799E 799F 799G
2  (x + 20, 23)  f  f  f_{?}  f_{?}  f
2  (x + 1, 3)  f  f  f_{?}g_{?}  f_{?}  fg
2 (x + 3495, 43991)  f  f  f_{?}g  f_{?}  f
2  (x + 55, 937)  f  f  f_{?}  g  f_{?}  f
3  (x + 20, 23)  f  f
3 (x + 3495, 43991)  g  f  f
3  (x + 55, 937)  fg  f

f = 11A, g = 73A

P Q 803A 803B 803C 803D 803E 803F
13 2  f  g_{?}
13 5  f  g
13 7  f  g
2 5  f  f_{?}g  f
2 7  f  f  g
3 2  f_{?}  g_{?}
3 5  f_{?}  g
3 7  f_{?}  g
5 2  g_{?}
5 7  g
7 2  g_{?}  f
7 5  g  f

f = 11A, g = 73B

P  Q  803A 803B 803C 803D 803E 803F
13 (x + 14, 23)  f  g
13  (x, 3)  g  f
13 (x + 15, 79)  f  g
2 (x + 14, 23)  f  f  g
2  (x, 3)  fg  f
2 (x + 15, 79)  f  g  f
3 (x + 14, 23)  f_{?}  g
3 (x + 15, 79)  f_{?}  g
5 (x + 14, 23)  g
5 (x, 3) g
5 (x + 15, 79)  g
7 (x + 14, 23)  fg
7  (x, 3)  g  f
7 (x + 15, 79)  g  f

f = 11A, g = 73C

P  Q  803A 803B 803C 803D 803E 803F
13 (x + 100, 109)  f  g
13 (x + 15, 181)  fg
2 (x + 100, 109)  f  fg
2 (x + 15, 181)  f  g  f
3 (x + 100, 109)  f_{?}  g
3 (x + 15, 181)  f_{?}g
5 (x + 100, 109)  g
5 (x + 15, 181)  g
7 (x + 100, 109)  g  f
7 (x + 15, 181)  g  f

f = 19A, g = 43A

P Q 817A 817B 817C 817D 817E 817F
3 11  f_{?}  g_{?}
3 2  g  g  f_{?}  g_{?}  g_{?}
43 11  fg_{?}
43 2  g  g  g_{?}  fg_{?}
43 3  g  f
5 11  f  g_{?}
5 2  fg  g  g_{?}  g_{?}
5 3  f  g

f = 19A, g = 43B

P  Q  817A 817B 817C 817D 817E 817F
3 (x + 11, 17)  f_{?}g_{?}
3 (x, 2) g g f_{?} g_{?} g_{?}
3 (x + 4, 7)  f_{?}
3 (x + 70, 79)  f_{?}  g
43 (x + 11, 17)  g_{?}  f
43  (x, 2)  g  g  g_{?}  fg_{?}
43 (x + 4, 7)  f
43 (x + 70, 79)  g  f
5 (x + 11, 17)  f  g_{?}
5  (x, 2)  fg  g  g_{?}  g_{?}
5 (x + 4, 7)  f
5 (x + 70, 79)  f  g

f = 23A, g = 37A

P  Q 851A 851B 851C 851D 851E 851F 851G 851H
(x + 8, 11)  13  g
(x + 8, 11)  2  g  g  g_{?}  g
(x^{2} + x + 1, 2) 11  g  f_{?}  f_{?}
(x^{2} + x + 1, 2) 13  f_{?}g  f_{?}
(x + 19, 31) 11  f  g
(x + 19, 31) 13  f  g
(x + 19, 31)  2  g  fg  g_{?}  g
(x + 360, 379) 11  fg
(x + 360, 379) 13  f  g
(x + 360, 379) 2  g  g  fg_{?}  g

f = 23A, g = 37B

P  Q 851A 851B 851C 851D 851E 851F 851G 851H
(x + 8, 11)  2  g  g  g_{?}  g
(x + 8, 11)  3  g_{?}  g
(x + 8, 11)  5  g
(x^{2} + x + 1, 2) 3  g_{?}  f_{?}  f_{?}g
(x^{2} + x + 1, 2) 5  g  f_{?}  f_{?}
(x + 19, 31) 2  g  fg  g_{?}  g
(x + 19, 31) 3  g_{?}  f  g
(x + 19, 31) 5  g  f
(x + 360, 379) 2  g  g  fg_{?}  g
(x + 360, 379) 3  g_{?}  f  g
(x + 360, 379) 5  g  f

f = 11A, g = 79A

P Q 869A 869B 869C 869D 869E 869F 869G 869H
2 5  f  g_{?}  f_{?}
2 7  f  g  f_{?}
3 2 fg  g  f  g_{?}  g_{?}
3 5  f  g_{?}  f
3 7  f  f  g
5 2  g  g  fg_{?}  g_{?}
5 7  fg
7 2  g  g  fg_{?}  g_{?}
7 5  g_{?}  f

f = 11A, g = 79B

P  Q  869A 869B 869C 869D 869E 869F 869G 869H
2  (x + 10, 13)  f  f_{?}
2 (x + 1637, 1861)  f  f_{?}g
2 (x + 3680, 5927)  f  g  f_{?}
2  (x + 55, 61)  f  f_{?}g
3  (x + 10, 13)  f  f
3 (x + 1637, 1861)  f  f  g
3  (x + 1, 2)  fg  g  f  g_{?}  g_{?}
3 (x + 3680, 5927)  f  fg
3  (x + 55, 61)  f  f  g
5  (x + 10, 13)  f
5 (x + 1637, 1861)  f  g
5  (x + 1, 2)  g  g  fg_{?}  g_{?}
5 (x + 3680, 5927)  g  f
5  (x + 55, 61)  f  g
7  (x + 10, 13)  f
7 (x + 1637, 1861)  f  g
7  (x + 1, 2)  g  g  fg_{?}  g_{?}
7 (x + 3680, 5927)  g  f
7  (x + 55, 61)  f  g

f = 19A, g = 47A

P  Q  893A 893B 893C 893D
17  (x + 304, 1373)  g  f
17  (x + 37, 139)  fg
17 (x^{4} + x^{3} + x^{2} + x + 1, 2)  g_{?}  g  f
17  (x + 20, 23)  f
17  (x + 1, 3)  fg
17  (x + 2, 7)  g_{?}  f
3  (x + 304, 1373)  g  f
3  (x + 37, 139)  f  g
3 (x^{4} + x^{3} + x^{2} + x + 1, 2)  fg_{?}  g
3  (x + 20, 23)  f
3  (x + 2, 7)  g_{?}  f
5  (x + 304, 1373)  g  f
5  (x + 37, 139)  f  g
5 (x^{4} + x^{3} + x^{2} + x + 1, 2)  fg_{?}  g
5  (x + 20, 23)  f
5  (x + 1, 3)  f  g
5  (x + 2, 7)  g_{?}  f

f = 29A, g = 31A

P  Q  899A 899B 899C 899D 899E 899F
(x + 52, 113) (x^{2} + x + 1, 2)  g  g_{?}  f
(x + 52, 113)  (x + 12, 31)  fg
(x + 52, 113)  (x + 2, 5)  g  f
(x + 52, 113)  (x + 43, 61)  g  f
(x + 41, 47)  (x^{2} + x + 1, 2)  g  fg_{?}
(x + 41, 47)  (x + 12, 31)  f  g
(x + 41, 47)  (x + 2, 5)  g  f
(x + 41, 47)  (x + 43, 61)  g  f
(x^{2} + 2x + 4, 5) (x^{2} + x + 1, 2)  g  fg_{?}
(x^{2} + 2x + 4, 5) (x + 12, 31)  f  g
(x^{2} + 2x + 4, 5) (x + 43, 61)  g  f
(x + 4, 7)  (x^{2} + x + 1, 2)  g  g_{?}
(x + 4, 7)  (x + 12, 31)  g
(x + 4, 7)  (x + 2, 5)  g
(x + 4, 7)  (x + 43, 61)  g

f = 17A, g = 53A

P Q 901A 901B 901C 901D 901E 901F 901G 901H 901I 901J
2 3  f  fg  g  f_{?}  f_{?}  f_{?}
2 5 fg  f  f_{?}  f_{?}  f_{?}
2 7  f  f  f_{?}  f_{?}  f_{?}g
3 5  g  f  f_{?}
3 7  f  f_{?}g
5 3  g  fg
5 7  f  g

f = 17A, g = 53B

P  Q  901A 901B 901C 901D 901E 901F 901G 901H 901I 901J
2  (x + 47, 113)  f  f  f_{?}g  f_{?}  f_{?}
2  (x + 10, 13)  f  f  f_{?}  f_{?}  f_{?}
2 (x + 2078, 4139)  f  f  f_{?}  f_{?}g  f_{?}
2  (x + 3, 5)  f  f  g  f_{?}  f_{?}  f_{?}
3  (x + 47, 113)  f  g  f_{?}
3  (x + 10, 13)  f  f_{?}
3 (x + 2078, 4139)  f  g  f_{?}
3  (x + 3, 5)  fg  f_{?}
5  (x + 47, 113)  f  g
5  (x + 10, 13)  f
5 (x + 2078, 4139)  f  g

f = 11A, g = 83A

P Q 913A 913B 913C 913D 913E 913F 913G
13 3  g  fg
13 5  g  f
2 3  f  fg  g
2 5  f  fg
3 5  g  f  f_{?}
5 3  g  g

f = 11A, g = 83B

P  Q  913A 913B 913C 913D 913E 913F 913G
13 (x + 5727, 10007)  g  f
13  (x + 31, 197)  f  g
13  (x + 5, 37)  f  g
13  (x + 69, 379)  f  g
13  (x + 38, 41)  f
13  (x + 1, 5)  g  f
2 (x + 5727, 10007)  f  g  f
2  (x + 31, 197)  f  f  g
2  (x + 5, 37)  f  f  g
2  (x + 69, 379)  f  f  g
2  (x + 38, 41)  f  f
2  (x + 1, 5)  g  f  f
3 (x + 5727, 10007)  g  f  f_{?}
3  (x + 31, 197)  f  f_{?}g
3  (x + 5, 37)  f  g  f_{?}
3  (x + 69, 379)  f  f_{?}g
3  (x + 38, 41)  f  f_{?}
3  (x + 1, 5)  g  f  f_{?}
5 (x + 5727, 10007)  g
5  (x + 31, 197)  g
5  (x + 5, 37)  g
5  (x + 69, 379)  g
5  (x + 38, 41)

f = 23A, g = 41A (char 2 primes for g omitted

P  Q  943A 943B 943C 943D 943E
(x + 8, 11)  (x + 7, 13)  g
(x + 8, 11) (x + 10, 37)  g
(x + 8, 11)  (x + 2, 5)
(x + 8, 11) (x + 31, 67)  g
(x + 124, 151) (x + 7, 13)  g  f
(x + 124, 151) (x + 10, 37)  g  f
(x + 124, 151) (x + 2, 5)  f
(x + 124, 151) (x + 31, 67)  fg
(x + 13, 31) (x + 7, 13)  g  f_{?}
(x + 13, 31) (x + 10, 37)  g  f_{?}
(x + 13, 31)  (x + 2, 5)  f_{?}
(x + 13, 31) (x + 31, 67)  f_{?}  g
(x + 34, 59) (x + 7, 13)  g  f
(x + 34, 59) (x + 10, 37)  g  f
(x + 34, 59)  (x + 2, 5)  f
(x + 34, 59) (x + 31, 67)  f  g

f = 11A, g = 89A

P Q 979A 979B 979C 979D 979E 979F
3 2  f  g_{?}  f_{?}g_{?}
3 5  f  g  f_{?}
3 7  f  f_{?}g
5 2  g_{?}  f  fg_{?}
5 7  f  fg
7 2  g_{?}  fg_{?}
7 5  g  f

f = 11A, g = 89B

P Q 979A 979B 979C 979D 979E 979F
3 2 fg  g_{?}  g_{?}  f_{?}g_{?}
5 2  g  g_{?}  f  g_{?}  fg_{?}
7 2  g  g_{?}  g_{?}  fg_{?}

f = 11A, g = 89C (char 2 primes for g omitted)

P  Q  979A 979B 979C 979D 979E 979F
3  (x + 8, 11)  f  f_{?}
3 (x + 35, 113)  f  g  f_{?}
3 (x + 161, 179)  f  g  f_{?}
3 (x + 177, 607)  f  g  f_{?}
5  (x + 8, 11)  f  f
5 (x + 35, 113)  f  g  f
5 (x + 161, 179)  f  g  f
5 (x + 177, 607)  fg  f
7  (x + 8, 11)  f
7 (x + 35, 113)  g  f
7 (x + 161, 179)  g  f
7 (x + 177, 607)  g  f

f = 23A, g = 43A

P  Q 989A 989B 989C 989D 989E 989F
(x + 4, 11)  23  f  g  f
(x + 4, 11)  5  fg  f
(x + 8, 11)  23  f  g
(x + 8, 11)  5  fg
(x^{2} + x + 1, 2) 23  f_{?}  f_{?}  f_{?}  f_{?}g
(x^{2} + x + 1, 2) 5  f_{?}  f_{?}  f_{?}g  f_{?}

f = 23A, g = 43B

P  Q  989A 989B 989C 989D 989E 989F
(x + 4, 11)  (x + 142, 593)  g  f  f
(x + 4, 11)  (x + 4, 7)  f  f
(x + 4, 11)  (x + 59, 71)  f  fg
(x + 8, 11)  (x + 142, 593)  g  f
(x + 8, 11)  (x + 4, 7)  f
(x + 8, 11)  (x + 59, 71)  f  g
(x^{2} + x + 1, 2) (x + 142, 593)  f_{?}  f_{?}g  f_{?}  f_{?}
(x^{2} + x + 1, 2)  (x + 4, 7)  f_{?}  f_{?}  f_{?}  f_{?}
(x^{2} + x + 1, 2) (x + 59, 71)  f_{?}  f_{?}  f_{?}  f_{?}  g

f = 17A, g = 59A (char 2 primes for g omitted)

P  Q  1003A 1003B 1003C 1003D 1003E 1003F 1003G 1003H 1003I 1003J
2  (x + 3, 19)  f  f_{?}  g  f_{?}
2  (x + 26, 29)  f  f_{?}  f_{?}
2  (x + 22, 97)  f  f_{?}  g  f_{?}
2 (x + 61609, 97001)  f  f_{?}g  f_{?}
3  (x + 3, 19)  f  f_{?}g  f
3  (x + 26, 29)  f  f_{?}  f
3  (x + 22, 97)  f  f_{?}g  f
3 (x + 61609, 97001)  f  g  f_{?}  f

f = 19A, g = 53A

P Q 1007A 1007B 1007C 1007D 1007E
11 3  g  f
11 5  g  fg_{?}
2 3  fg  f_{?}
2 5  f  g  f_{?}  g_{?}
3 5  g  fg_{?}
7 3  g  f
7 5  f  g  g_{?}

f = 19A, g = 53B

P  Q  1007A 1007B 1007C 1007D 1007E
11 (x + 10, 13)  f
11 (x + 24, 13177)  g  f
11 (x + 315, 331)  g  f
2  (x + 10, 13)  f  f_{?}
2 (x + 24, 13177)  f  f_{?}g
2 (x + 315, 331)  f  g  f_{?}
3  (x + 10, 13)  f
3 (x + 24, 13177)  g  f
3 (x + 315, 331)  g  f
7  (x + 10, 13)  f
7 (x + 24, 13177)  f  g
7 (x + 315, 331)  fg

f = 17A, g = 61A

P Q 1037A 1037B 1037C 1037D 1037E
13 11  g  f
13 2  g_{?}  fg_{?}
13 7  fg
2 11  g  f_{?}  f_{?}  f
2 7  f_{?}  f_{?}g  f
3 11  g  f
3 2  g_{?}  f  g_{?}
3 7  f  g

f = 17A, g = 61B

P  Q  1037A 1037B 1037C 1037D 1037E
13 (x + 141, 1259)  f  g
13  (x + 1, 2)  g_{?}  fg_{?}
13 (x + 177, 317)  g  f
13  (x + 2, 5)  f
2 (x + 141, 1259)  f_{?}  f_{?}  fg
2 (x + 177, 317)  f_{?}  g  f_{?}  f
2  (x + 2, 5)  f_{?}  f_{?}  f
3 (x + 141, 1259)  f  g
3  (x + 1, 2)  g_{?}  f  g_{?}
3 (x + 177, 317)  fg
3  (x + 2, 5)  f

f = 11A, g = 97A

P  Q  1067A 1067B 1067C 1067D 1067E 1067F
13 (x + 28, 41)  fg
13 (x + 58, 71)  g  f
13 (x + 11, 881)  f  g
3 (x + 28, 41)  f  g
3 (x + 58, 71)  g  f
3 (x + 11, 881)  f  g
5 (x + 28, 41)  g
5 (x + 58, 71)  g
5 (x + 11, 881)  g
7 (x + 28, 41)  f  f  g
7 (x + 58, 71)  f  g  f
7 (x + 11, 881)  f  f  g

f = 11A, g = 97B

P  Q  1067A 1067B 1067C 1067D 1067E 1067F
13  (x + 4, 11)  g  f
13  (x + 1, 2)  fg_{?}  g  g_{?}
13 (x + 1040, 3361)  f  g
13  (x + 31, 61)  g  f
3  (x + 4, 11)  fg
3  (x + 1, 2)  f  g_{?}  g  g_{?}
3 (x + 1040, 3361)  f  g
3  (x + 31, 61)  fg
5  (x + 4, 11)  g
5  (x + 1, 2)  g_{?}  g  g_{?}
5 (x + 1040, 3361)  g
5  (x + 31, 61)  g
7  (x + 4, 11)  f  fg
7  (x + 1, 2)  f  f  g_{?}  g  g_{?}
7 (x + 1040, 3361)  f  f  g
7  (x + 31, 61)  f  fg

4 Table of Eigenforms

The characteristic polynomial of T_{2} on S_{2}(Γ_{0}(N)) has distinct roots for all prime levels ≤ 97. Thus a conjugacy class of each such eigenform is determined by giving a factor of the characteristic polynomial. The same is not the case for the levels NM considered, but at least some T_{p} determines each newform.

Form Factor of T_{2}
11A x + 2
17A x + 1
19A x
23A x^{2} + x − 1
29A x^{2} + 2x − 1
31A x^{2} − x − 1
37A x + 2
37B x
41A x^{3} + x^{2} − 5x − 1
43A x + 2
43B x^{2} − 2
47A x^{4} − x^{3} − 5x^{2} + 5x − 1
53A x + 1
53B x^{3} + x^{2} − 3x − 1
59A x^{5} − 9x^{3} + 2x^{2} + 16x − 8
61A x + 1
61B x^{3} − x^{2} − 3x + 1
67A x − 2
67B x^{2} + x − 1
67C x^{2} + 3x + 1
71A x^{3} − 5x + 3
71B x^{3} + x^{2} − 4x − 3
73A x − 1
73B x^{2} − x − 3
73C x^{2} + 3x + 1
79A x + 1
79B x^{5} − 6x^{3} + 8x − 1
83A x + 1
83B x^{6} − x^{5} − 9x^{4} + 7x^{3} + 20x^{2} − 12x − 8
89A x + 1
89B x − 1
89C x^{5} + x^{4} − 10x^{3} − 10x^{2} + 21x + 17
97A x^{3} + 4x^{2} + 3x − 1
97B x^{4} − 3x^{3} − x^{2} + 6x − 1

Form Factor of T_{2}
143A x^{4} − 3x^{3} − x^{2} + 5x + 1
143B x^{6} − 10x^{4} + 2x^{3} + 24x^{2} − 7x − 12
143C x

187

Form Factor of T_{3}
187A x − 1
187B x^{2} + x − 4
187C x^{2} − 3
187D x^{3} + 3x^{2} − x − 5
187E x^{4} − x^{3} − 11x^{2} + 9x + 20
187F x

209

Form Factor of T_{2}
209A x^{2} − 2
209B x^{5} − 2x^{4} − 6x^{3} + 10x^{2} + 5x − 4
209C x^{7} + x^{6} − 14x^{5} − 10x^{4} + 59x^{3} + 27x^{2} − 66x − 30
209D x

221

Form Factor of T_{2}
221A x − 1
221B x + 1
221C x^{2} − 5
221D x^{2} + x − 1
221E x^{3} − 4x + 1
221F x^{6} − x^{5} − 9x^{4} + 6x^{3} + 19x^{2} − 5x − 3
221G x^{2} + x − 5

247

Form Factor of T_{2}
247A x^{2} − x − 1
247B x^{3} + 3x^{2} − 3
247C x^{5} − 4x^{4} + 12x^{2} − 5x − 5
247D x^{5} − 9x^{3} − x^{2} + 19x + 4
247E x^{4} + 3x^{3} − 2x^{2} − 9x − 4

253

Form Factor of T_{2}
253A x^{3} − 3x^{2} + 3
253B x^{3} + x^{2} − 4x + 1
253C x^{5} + 4x^{4} − 14x^{2} − 13x − 1
253D x^{6} − 3x^{5} − 4x^{4} + 16x^{3} − 3x^{2} − 10x + 1

299

Form Factor of T_{5}
299A x^{2} + 3x + 1
299B x^{2} − 3x − 3
299C x^{2} − 2x − 4
299D x^{2} + x − 1
299E x^{10} − 3x^{9} − 37x^{8} + 112x^{7} + 443x^{6} − 1401x^{5} − 1817x^{4} + 6424x^{3} + 1108x^{2} − 6140x − 2372
299F x^{2} − x − 4
299G x^{3} − x^{2} − 7x − 3

Form Factor of T_{2}
319A x − 2
319B x^{3} − 3x − 1
319C x^{4} + 2x^{3} − 3x^{2} − 3x + 2
319D x^{7} − 3x^{6} − 4x^{5} + 15x^{4} + x^{3} − 14x^{2} + 1
319E x^{8} − 13x^{6} − x^{5} + 50x^{4} + 7x^{3} − 54x^{2} − 5x + 1

323

Form Factor of T_{2}
323A x^{2} + x − 4
323B x^{4} − 6x^{2} − x + 7
323C x^{5} + 3x^{4} − 2x^{3} − 7x^{2} + 2x + 1
323D x^{6} − 2x^{5} − 9x^{4} + 15x^{3} + 23x^{2} − 23x − 21
323E x^{7} − x^{6} − 10x^{5} + 9x^{4} + 26x^{3} − 19x^{2} − 12x + 8
323F x

341

Form Factor of T_{5}
341A x^{2} + 3x + 1
341B x^{4} + x^{3} − 8x^{2} − 11x + 1
341C x^{8} + 5x^{7} − 12x^{6} − 77x^{5} − 11x^{4} + 176x^{3} + 35x^{2} − 77x + 9
341D x^{11} − 3x^{10} − 35x^{9} + 106x^{8} + 423x^{7} − 1261x^{6} − 2318x^{5}

+6533x^{4} + 5956x^{3} − 14599x^{2} − 6045x + 10618

377

Form Factor of T_{2}
377A x − 1
377B x^{2} − 3
377C x^{5} + x^{4} − 5x^{3} − 3x^{2} + 6x + 1
377D x^{7} − 3x^{6} − 8x^{5} + 26x^{4} + 9x^{3} − 36x^{2} − 14x + 3
377E x^{5} + 3x^{4} − 3x^{3} − 13x^{2} − 8x − 1
377F x^{9} − x^{8} − 13x^{7} + 13x^{6} + 51x^{5} − 50x^{4} − 59x^{3} + 45x^{2} + 20x − 3

391

Form Factor of T_{2}
391A x^{3} + x^{2} − 4x − 3
391B x^{3} + x^{2} − 4x + 1
391C x^{9} − 2x^{8} − 12x^{7} + 23x^{6} + 43x^{5} − 79x^{4} − 43x^{3} + 78x^{2} + 11x − 21
391D x^{2} + x − 1
391E x^{12} − 4x^{11} − 12x^{10} + 62x^{9} + 27x^{8} − 321x^{7} + 108x^{6} + 625x^{5} − 362x^{4} − 372x^{3} + 116x^{2} + 97x + 13

403

Form Factor of T_{2}
403A x^{2} − 3x + 1
403B x^{7} − 2x^{6} − 9x^{5} + 17x^{4} + 20x^{3} − 37x^{2} + x + 4
403C x^{8} + x^{7} − 11x^{6} − 10x^{5} + 37x^{4} + 33x^{3} − 36x^{2} − 33x − 4
403D x^{6} + 2x^{5} − 7x^{4} − 13x^{3} + 6x^{2} + 7x − 3
403E x^{8} + 5x^{7} − 30x^{5} − 24x^{4} + 54x^{3} + 54x^{2} − 28x − 29

407

Form Factor of T_{2}
407A x^{4} + x^{3} − 4x^{2} + 1
407B x^{4} − x^{3} − 4x^{2} + 2x + 3
407C x^{12} − x^{11} − 18x^{10} + 18x^{9} + 111x^{8} − 104x^{7} − 274x^{6} + 212x^{5} + 255x^{4} − 129x^{3} − 78x^{2} + 4x + 1
407D x^{11} − 2x^{10} − 16x^{9} + 32x^{8} + 89x^{7} − 179x^{6} − 201x^{5} + 407x^{4} + 168x^{3} − 333x^{2} − 51x + 75

Form Factor of T_{5}
437A x − 1
437B x + 1
437C x^{2} + 2x − 4
437D x^{2} + 2x − 1
437E x^{2} − 2x − 4
437F x^{5} + x^{4} − 7x^{3} − 5x^{2} + 10x + 4
437G x^{8} − 2x^{7} − 19x^{6} + 18x^{5} + 116x^{4} − 12x^{3} − 240x^{2} − 128x + 16
437H x^{12} + x^{11} − 43x^{10} − 29x^{9} + 690x^{8} + 304x^{7} − 5116x^{6} − 1600x^{5}

+17904x^{4} + 4400x^{3} − 26240x^{2} − 4736x + 10368

451

Form Factor of T_{2}
451A x^{5} + 2x^{4} − 5x^{3} − 10x^{2} + 4x + 9
451B x^{5} + 2x^{4} − 3x^{3} − 4x^{2} + 2x + 1
451C x^{10} − 4x^{9} − 6x^{8} + 38x^{7} − 7x^{6} − 105x^{5} + 74x^{4} + 77x^{3} − 74x^{2} + 8
451D x^{12} − 3x^{11} − 16x^{10} + 48x^{9} + 93x^{8} − 270x^{7} − 251x^{6} + 633x^{5} + 359x^{4}

−582x^{3} − 248x^{2} + 136x + 32

451E x

473

Form Factor of T_{7}
473A x^{2} + 4x − 1
473B x^{2} − 5
473C x^{5} + 9x^{4} + 23x^{3} + 9x^{2} − 30x − 25
473D x^{5} + 15x^{4} + 71x^{3} + 53x^{2} − 420x − 799
473E x^{11} − 17x^{10} + 96x^{9} − 92x^{8} − 1043x^{7} + 3448x^{6} − 142x^{5} − 12971x^{4}

+13700x^{3} + 4760x^{2} − 6848x + 1024

473F x^{9} − 19x^{8} + 140x^{7} − 482x^{6} + 627x^{5} + 570x^{4} − 2324x^{3} + 1223x^{2} + 1580x − 1368
473G x

481

Form Factor of T_{2}
481A x − 1
481B x^{7} + x^{6} − 8x^{5} − 7x^{4} + 17x^{3} + 12x^{2} − 9x − 6
481C x^{11} − 3x^{10} − 14x^{9} + 45x^{8} + 64x^{7} − 237x^{6} − 99x^{5} + 529x^{4} − 7x^{3} − 460x^{2} + 67x + 110
481D x^{11} − 3x^{10} − 12x^{9} + 39x^{8} + 38x^{7} − 149x^{6} − 23x^{5} + 175x^{4} − 5x^{3} − 48x^{2} + 5x + 2
481E x^{7} + 5x^{6} + 2x^{5} − 21x^{4} − 25x^{3} + 8x^{2} + 13x − 2

493

Form Factor of T_{3}
493A x + 3
493B x^{2} + x − 4
493C x^{4} − 3x^{3} − 7x^{2} + 27x − 16
493D x^{5} + 2x^{4} − 5x^{3} − 10x^{2} + 1
493E x^{6} − 2x^{5} − 11x^{4} + 14x^{3} + 38x^{2} − 15x − 26
493F x^{10} + 9x^{9} + 22x^{8} − 16x^{7} − 115x^{6} − 54x^{5} + 155x^{4} + 95x^{3} − 65x^{2} − 24x + 2
493G x^{8} − 6x^{7} + x^{6} + 44x^{5} − 42x^{4} − 85x^{3} + 80x^{2} + 42x − 8
493H x

Form Factor of T_{17}
517A x − 4
517B x + 6
517C x^{2} − 4x − 28
517D x^{2} − 8x + 14
517E x^{2} − 4x + 2
517F x^{2} − 12
517G x^{3} + 6x^{2} − 18x − 50
517H x^{3} − 4x^{2} − 20x + 66
517I x^{10} + 6x^{9} − 69x^{8} − 354x^{7} + 1881x^{6} + 7074x^{5} − 25042x^{4}

−54922x^{3} + 148036x^{2} + 135080x − 264560

517J x^{10} + 16x^{9} + 49x^{8} − 288x^{7} − 1157x^{6} + 2224x^{5} + 5640x^{4} − 9922x^{3} − 6252x^{2} + 17096x − 7408
517K x

527

Form Factor of T_{2}
527A x^{5} + 2x^{4} − 5x^{3} − 7x^{2} + 7x + 1
527B x^{2} − x − 1
527C x^{7} + x^{6} − 8x^{5} − 8x^{4} + 15x^{3} + 13x^{2} − 8x − 5
527D x^{11} + 2x^{10} − 17x^{9} − 34x^{8} + 101x^{7} + 202x^{6} − 238x^{5} − 469x^{4} + 182x^{3} + 295x^{2} − 83x − 3
527E x^{16} − 3x^{15} − 22x^{14} + 70x^{13} + 179x^{12} − 631x^{11} − 642x^{10} + 2789x^{9} + 792x^{8} − 6335x^{7}

+903x^{6} + 6928x^{5} − 3096x^{4} − 2631x^{3} + 2063x^{2} − 344x − 11

533

Form Factor of T_{2}
533A x^{2} − 2x − 1
533B x^{13} − 21x^{11} + 166x^{9} + x^{8} − 613x^{7} − 16x^{6} + 1074x^{5}

+72x^{4} − 822x^{3} − 76x^{2} + 215x + 27

533C x^{11} + 2x^{10} − 14x^{9} − 26x^{8} + 64x^{7} + 103x^{6}

−117x^{5} − 149x^{4} + 65x^{3} + 71x^{2} + 5x − 1

533D x^{7} + 2x^{6} − 6x^{5} − 12x^{4} + 9x^{3} + 19x^{2} − x − 5
533E x^{8} − x^{7} − 10x^{6} + 8x^{5} + 31x^{4} − 22x^{3} − 28x^{2} + 22x − 3

551

Form Factor of T_{17}
551A x − 2
551B x − 3
551C x + 1
551D x^{2} + 2x − 4
551E x^{3} + 5x^{2} − 7x − 1
551F x^{16} − 187x^{14} + 28x^{13} + 13484x^{12} − 3248x^{11} − 473664x^{10} + 123616x^{9} + 8434432x^{8}

−1894272x^{7} − 70540288x^{6} + 12555264x^{5} + 211791872x^{4} − 65138688x^{3}

−89800704x^{2} + 24330240x + 6635520

551G x^{18} − 7x^{17} − 179x^{16} + 1237x^{15} + 13154x^{14} − 90768x^{13} − 504248x^{12} + 3575872x^{11} + 10366592x^{10}

−81432448x^{9} − 95461888x^{8} + 1065127936x^{7} − 129328640x^{6} − 7221715968x^{5} + 8790904832x^{4}

+16087502848x^{3} − 43953332224x^{2} + 34624585728x − 8982429696

551H x

559

Form Factor of T_{2}
559A x^{3} + 3x^{2} − 3
559B x^{15} − 2x^{14} − 22x^{13} + 43x^{12} + 187x^{11} − 354x^{10} − 769x^{9} + 1395x^{8}

+1553x^{7} − 2684x^{6} − 1328x^{5} + 2265x^{4} + 241x^{3} − 606x^{2} + 33x + 13

559C x^{7} + 2x^{6} − 6x^{5} − 9x^{4} + 11x^{3} + 10x^{2} − 5x − 1
559D x^{14} − 7x^{13} + 3x^{12} + 78x^{11} − 145x^{10} − 243x^{9} + 758x^{8}

+83x^{7} − 1422x^{6} + 532x^{5} + 1004x^{4} − 525x^{3} − 224x^{2} + 82x + 23

559E x^{4} + x^{3} − 5x^{2} − 3x + 1

Form Factor of T_{3}
583A x + 1
583B x − 1
583C x − 3
583D x^{2} + 2x − 1
583E x^{2} − 3
583F x^{6} − 2x^{5} − 8x^{4} + 16x^{3} + 10x^{2} − 20x − 1
583G x^{8} + 8x^{7} + 17x^{6} − 10x^{5} − 58x^{4} − 16x^{3} + 45x^{2} + 10x − 1
583H x^{12} − 6x^{11} − 9x^{10} + 110x^{9} − 66x^{8} − 672x^{7} + 937x^{6} + 1492x^{5} − 3291x^{4}

−160x^{3} + 3592x^{2} − 2256x + 352

583I x^{10} + x^{9} − 22x^{8} − 16x^{7} + 170x^{6} + 90x^{5} − 525x^{4} − 227x^{3} + 496x^{2} + 264x − 16

589

Form Factor of T_{2}
589A x^{9} − 14x^{7} + 64x^{5} + 2x^{4} − 106x^{3} − 5x^{2} + 49x − 7
589B x^{10} − 13x^{8} + 53x^{6} − 4x^{5} − 77x^{4} + 13x^{3} + 38x^{2} − 10x − 2
589C x^{3} − x^{2} − 4x + 2
589D x^{14} − 23x^{12} + 204x^{10} − 880x^{8} + x^{7} + 1913x^{6} − 12x^{5} − 1943x^{4} + 37x^{3} + 730x^{2} − 26x − 18
589E x^{9} + 4x^{8} − 5x^{7} − 34x^{6} − 7x^{5} + 90x^{4} + 61x^{3} − 69x^{2} − 64x − 6

611

Form Factor of T_{2}
611A x − 2
611B x^{14} − 23x^{12} − x^{11} + 206x^{10} + 23x^{9} − 907x^{8} − 181x^{7}

+2027x^{6} + 615x^{5} − 2104x^{4} − 884x^{3} + 704x^{2} + 400x + 32

611C x^{5} − 5x^{3} − x^{2} + 5x + 1
611D x^{18} + 2x^{17} − 29x^{16} − 57x^{15} + 338x^{14} + 648x^{13} − 2033x^{12} − 3742x^{11} + 6794x^{10} + 11595x^{9}

−12875x^{8} − 18843x^{7} + 13831x^{6} + 14703x^{5} − 7711x^{4} − 4190x^{3} + 1528x^{2} + 144x − 48

611E x^{9} + 3x^{8} − 6x^{7} − 21x^{6} + 8x^{5} + 42x^{4} + 2x^{3} − 25x^{2} − 6x + 1

629

Form Factor of T_{2}
629A x − 1
629B x + 1
629C x − 2
629D x^{5} − 7x^{3} + 2x^{2} + 12x − 7
629E x^{17} + x^{16} − 30x^{15} − 26x^{14} + 367x^{13} + 266x^{12} − 2349x^{11} − 1339x^{10} + 8394x^{9}

+3341x^{8} − 16544x^{7} − 3504x^{6}

+16591x^{5} + 760x^{4} − 6981x^{3} − 59x^{2} + 1031x + 48

629F x^{15} − 2x^{14} − 24x^{13} + 46x^{12} + 225x^{11} − 407x^{10} − 1050x^{9} + 1751x^{8} + 2573x^{7}

−3822x^{6} − 3134x^{5} + 3962x^{4} + 1409x^{3} − 1497x^{2} + 144x + 17

629G x^{8} − 10x^{6} + 29x^{4} + 3x^{3} − 24x^{2} − 7x + 2
629H x

649

Form Factor of T_{5}
649A x + 1
649B x^{2} + x − 9
649C x^{4} + 5x^{3} + 6x^{2} − 1
649D x^{11} + x^{10} − 31x^{9} − 14x^{8} + 351x^{7} − 24x^{6} − 1716x^{5} + 836x^{4}

+3140x^{3} − 2293x^{2} − 1242x + 1013

649E x^{17} − 7x^{16} − 29x^{15} + 275x^{14} + 233x^{13} − 4337x^{12} + 722x^{11} + 35220x^{10} − 20733x^{9}

−156048x^{8} + 114636x^{7} + 366067x^{6} − 243583x^{5} − 410049x^{4} + 137150x^{3}

+199305x^{2} + 8095x − 14958

649F x^{12} + 5x^{11} − 24x^{10} − 152x^{9} + 28x^{8} + 1128x^{7} + 1428x^{6} − 680x^{5}

−1705x^{4} − 504x^{3} + 67x^{2} + 31x + 1

Form Factor of T_{2}
667A x^{10} + 3x^{9} − 10x^{8} − 32x^{7} + 32x^{6} + 118x^{5} − 29x^{4} − 182x^{3} − 28x^{2} + 101x + 43
667B x^{12} + 3x^{11} − 13x^{10} − 41x^{9} + 54x^{8} + 188x^{7} − 77x^{6} − 342x^{5} + 13x^{4} + 215x^{3}

+9x^{2} − 37x − 5

667C x^{16} − 3x^{15} − 22x^{14} + 68x^{13} + 187x^{12} − 597x^{11} − 795x^{10} + 2592x^{9} + 1860x^{8}

−5877x^{7} − 2496x^{6} + 6612x^{5} + 1842x^{4} − 3011x^{3} − 505x^{2} + 336x + 64

667D x^{13} − 4x^{12} − 11x^{11} + 58x^{10} + 24x^{9} − 298x^{8} + 97x^{7} + 641x^{6} − 402x^{5}

−547x^{4} + 352x^{3} + 219x^{2} − 88x − 40

671

Form Factor of T_{2}
671A x^{5} + 2x^{4} − 3x^{3} − 4x^{2} + 2x + 1
671B x^{6} − 7x^{4} + 2x^{3} + 12x^{2} − 5x − 2
671C x^{21} − 37x^{19} + 2x^{18} + 582x^{17} − 65x^{16} − 5074x^{15} + 876x^{14} + 26808x^{13} − 6340x^{12}

−88228x^{11} + 26710x^{10} + 179236x^{9} − 66498x^{8} − 215174x^{7} + 94807x^{6} + 138664x^{5}

−70301x^{4} − 39180x^{3} + 21784x^{2} + 3447x − 2082

671D x^{19} − 5x^{18} − 18x^{17} + 122x^{16} + 78x^{15} − 1177x^{14} + 387x^{13} + 5755x^{12} − 4673x^{11}

−15053x^{10} + 16875x^{9} + 20141x^{8} − 28019x^{7} − 11589x^{6} + 21077x^{5} + 1674x^{4}

−6404x^{3} + 241x^{2} + 643x − 43

689

Form Factor of T_{3}
689A x + 2
689B x^{3} + x^{2} − 5x + 2
689C x^{18} − x^{17} − 36x^{16} + 37x^{15} + 509x^{14} − 518x^{13} − 3591x^{12} + 3481x^{11} + 13297x^{10}

−11814x^{9} − 24999x^{8} + 19921x^{7} + 21688x^{6} − 16429x^{5} − 7147x^{4} + 5528x^{3} + 440x^{2} − 464x + 16

689D x^{2} − x − 1
689E x^{2} + x − 1
689F x^{18} − 5x^{17} − 28x^{16} + 165x^{15} + 291x^{14} − 2260x^{13} − 1207x^{12} + 16757x^{11} − 713x^{10}

−73440x^{9} + 26157x^{8} + 194393x^{7} − 104038x^{6} − 302555x^{5} + 191313x^{4} + 251884x^{3}

−168800x^{2} − 85056x + 55616

689G x^{9} + 3x^{8} − 10x^{7} − 33x^{6} + 24x^{5} + 105x^{4} − 7x^{3} − 116x^{2} − 16x + 32

697

Form Factor of T_{3}
697A x^{3} + 2x^{2} − 4x − 6
697B x^{15} + 8x^{14} + 2x^{13} − 134x^{12} − 281x^{11} + 613x^{10} + 2238x^{9} − 281x^{8} − 6500x^{7} − 3592x^{6} + 7246x^{5}

+6704x^{4} − 1852x^{3} − 2832x^{2} − 164x + 220

697C x^{3} − 2x^{2} − 2x + 2
697D x^{7} + 2x^{6} − 12x^{5} − 18x^{4} + 35x^{3} + 21x^{2} − 14x + 1
697E x^{15} − 4x^{14} − 24x^{13} + 106x^{12} + 213x^{11} − 1119x^{10} − 786x^{9} + 5993x^{8} + 390x^{7} − 17050x^{6}

+5244x^{5} + 24068x^{4} − 13256x^{3} − 12640x^{2} + 9184x − 324

697F x^{10} − 2x^{9} − 20x^{8} + 34x^{7} + 133x^{6} − 167x^{5} − 324x^{4} + 209x^{3} + 190x^{2} − 100x − 2

703

Form Factor of T_{2}
703A x + 2
703B x^{2} − 2
703C x^{4} − 6x^{2} + 4x + 2
703D x^{5} + 3x^{4} − 3x^{3} − 14x^{2} − 7x + 3
703E x^{11} − 5x^{10} − 6x^{9} + 60x^{8} − 25x^{7} − 228x^{6} + 193x^{5} + 312x^{4} − 257x^{3} − 194x^{2} + 94x + 54
703F x^{18} − 3x^{17} − 25x^{16} + 78x^{15} + 251x^{14} − 827x^{13} − 1294x^{12} + 4619x^{11} + 3615x^{10} − 14679x^{9} − 5160x^{8}

+26821x^{7} + 2497x^{6} − 27186x^{5} + 2136x^{4} + 13772x^{3} − 2914x^{2} − 2636x + 836

703G x^{4} − 3x^{3} − 2x^{2} + 9x − 4
703H x^{9} + 3x^{8} − 9x^{7} − 28x^{6} + 21x^{5} + 69x^{4} − 12x^{3} − 44x^{2} − 4x + 4
703I x

Form Factor of T_{2}
713A x − 1
713B x^{2} − x − 1
713C x^{5} + 4x^{4} + x^{3} − 9x^{2} − 5x + 1
713D x^{15} − x^{14} − 24x^{13} + 23x^{12} + 225x^{11} − 207x^{10} − 1031x^{9} + 912x^{8} + 2343x^{7} − 1992x^{6}

−2324x^{5} + 1872x^{4} + 675x^{3} − 428x^{2} − 69x + 9

713E x^{18} − 5x^{17} − 18x^{16} + 120x^{15} + 86x^{14} − 1141x^{13} + 216x^{12} + 5545x^{11} − 3248x^{10} − 14881x^{9} + 11254x^{8}

+22409x^{7} − 17505x^{6} − 18664x^{5} + 12837x^{4} + 7970x^{3} − 3926x^{2} − 1353x + 271

713F x^{5} − 7x^{3} + 11x − 1
713G x^{9} + 5x^{8} − x^{7} − 38x^{6} − 43x^{5} + 52x^{4} + 102x^{3} + 31x^{2} − 11x − 3

731

Form Factor of T_{3}
731A x − 1
731B x^{2} − x − 4
731C x^{6} + 3x^{5} − 3x^{4} − 10x^{3} + x^{2} + 4x − 1
731D x^{8} + 3x^{7} − 10x^{6} − 29x^{5} + 28x^{4} + 74x^{3} − 30x^{2} − 50x + 5
731E x^{21} + x^{20} − 48x^{19} − 41x^{18} + 964x^{17} + 674x^{16} − 10525x^{15} − 5705x^{14} + 67848x^{13}

+26590x^{12} − 262265x^{11} − 68190x^{10} + 590444x^{9} + 93377x^{8} − 719141x^{7} − 67956x^{6} + 414111x^{5}

+34290x^{4} − 99217x^{3} − 11164x^{2} + 8076x + 1384

731F x^{19} − 5x^{18} − 30x^{17} + 179x^{16} + 318x^{15} − 2624x^{14} − 1061x^{13} + 20311x^{12} − 4812x^{11}

−89702x^{10} + 50281x^{9} + 230680x^{8} − 159654x^{7} − 344023x^{6} + 234519x^{5} + 280066x^{4}

−154817x^{3} − 98750x^{2} + 33639x + 2568

737

Form Factor of T_{2}
737A x + 2
737B x^{2} − 2
737C x^{17} − x^{16} − 26x^{15} + 25x^{14} + 272x^{13} − 244x^{12} − 1472x^{11} + 1186x^{10} + 4406x^{9} − 3042x^{8} − 7180x^{7}

+4055x^{6} + 5665x^{5} − 2655x^{4} − 1418x^{3} + 810x^{2} − 76x − 2

737D x^{12} + 2x^{11} − 14x^{10} − 25x^{9} + 72x^{8} + 109x^{7} − 165x^{6} − 196x^{5} + 165x^{4} + 134x^{3} − 64x^{2} − 30x + 6
737E x^{15} − 2x^{14} − 22x^{13} + 43x^{12} + 185x^{11} − 353x^{10} − 745x^{9} + 1391x^{8} + 1463x^{7} − 2685x^{6} − 1277x^{5}

+2284x^{4} + 457x^{3} − 718x^{2} − 58x + 72

737F x^{8} − 2x^{7} − 8x^{6} + 15x^{5} + 12x^{4} − 19x^{3} − 7x^{2} + 6x + 1

767

Form Factor of T_{5}
767A x^{3} + x^{2} − 7x − 9
767B x^{4} + x^{3} − 3x^{2} − x + 1
767C x^{10} + 6x^{9} − 2x^{8} − 63x^{7} − 61x^{6} + 134x^{5} + 96x^{4} − 132x^{3} + 13x^{2} + 7x − 1
767D x^{2} + 2x − 1
767E x^{23} − 4x^{22} − 81x^{21} + 317x^{20} + 2801x^{19} − 10657x^{18} − 54111x^{17} + 198312x^{16} + 643019x^{15}

−2238143x^{14} − 4894174x^{13} + 15814887x^{12} + 24187057x^{11} − 69900276x^{10} − 77130544x^{9}

+187733632x^{8} + 153123344x^{7} − 285750528x^{6} − 173058880x^{5} + 212348800x^{4}

+92806144x^{3} − 54402560x^{2} − 20271104x + 657408

767F x^{17} − 4x^{16} − 51x^{15} + 224x^{14} + 962x^{13} − 4947x^{12} − 7716x^{11} + 54891x^{10} + 13542x^{9}

−321428x^{8} + 167128x^{7} + 929632x^{6} − 981312x^{5} − 980800x^{4} + 1623296x^{3}

−148480x^{2} − 376320x + 25600

779

Form Factor of T_{2}
779A x^{2} − 2
779B x^{8} + 2x^{7} − 9x^{6} − 18x^{5} + 20x^{4} + 37x^{3} − 17x^{2} − 21x + 4
779C x^{11} − 13x^{9} + 58x^{7} + x^{6} − 105x^{5} − 9x^{4} + 72x^{3} + 10x^{2} − 12x − 2
779D x^{19} − 31x^{17} + 403x^{15} + x^{14} − 2856x^{13} − 21x^{12} + 12018x^{11} + 167x^{10} − 30705x^{9} − 609x^{8}

+46868x^{7} + 898x^{6} − 40530x^{5} + 48x^{4} + 17736x^{3} − 1056x^{2} − 2912x + 576

779E x^{21} − x^{20} − 35x^{19} + 33x^{18} + 517x^{17} − 454x^{16} − 4203x^{15} + 3401x^{14} + 20547x^{13} − 15245x^{12}

−61842x^{11} + 42414x^{10} + 112133x^{9} − 73432x^{8} − 113232x^{7} + 76256x^{6} + 51782x^{5} − 42048x^{4}

−3816x^{3} + 8096x^{2} − 1696x + 64

Form Factor of T_{7}
781A x + 3
781B x − 3
781C x^{2} − x − 3
781D x^{6} + 6x^{5} + 6x^{4} − 20x^{3} − 41x^{2} − 22x − 3
781E x^{15} + 4x^{14} − 54x^{13} − 209x^{12} + 1091x^{11} + 3742x^{10} − 11591x^{9} − 28217x^{8} + 78036x^{7} + 82851x^{6}

−306450x^{5} + 40124x^{4} + 479896x^{3} − 500400x^{2} + 173536x − 12352

781F x^{12} − 7x^{11} − 5x^{10} + 100x^{9} − 43x^{8} − 503x^{7} + 280x^{6} + 1099x^{5} − 371x^{4} − 964x^{3} − 36x^{2} + 128x − 16
781G x^{20} + 2x^{19} − 94x^{18} − 145x^{17} + 3679x^{16} + 3686x^{15} − 77889x^{14} − 31965x^{13} + 960848x^{12} − 170343x^{11}

−6812398x^{10} + 4828640x^{9} + 25056784x^{8} − 28885904x^{7} − 37008736x^{6} + 56238784x^{5}

+17695232x^{4} − 37068288x^{3} − 4181504x^{2} + 7863296x + 1466368

793

Form Factor of T_{2}
793A x − 1
793B x^{16} − 3x^{15} − 20x^{14} + 63x^{13} + 148x^{12} − 507x^{11} − 487x^{10} + 1959x^{9} + 625x^{8} − 3706x^{7}

−35x^{6} + 3151x^{5} − 203x^{4} − 1134x^{3} + 22x^{2} + 123x + 13

793C x^{15} + 8x^{14} + 8x^{13} − 89x^{12} − 231x^{11} + 242x^{10} + 1295x^{9} + 332x^{8} − 2847x^{7} − 2275x^{6}

+2304x^{5} + 2831x^{4} − 194x^{3} − 792x^{2} − 54x + 27

793D x^{9} + 3x^{8} − 9x^{7} − 32x^{6} + 15x^{5} + 94x^{4} + 13x^{3} − 85x^{2} − 29x + 1
793E x^{4} + x^{3} − 5x^{2} − x + 1
793F x^{16} − 5x^{15} − 12x^{14} + 91x^{13} + 14x^{12} − 637x^{11} + 375x^{10} + 2141x^{9} − 2071x^{8} − 3516x^{7}

+4367x^{6} + 2507x^{5} − 4115x^{4} − 320x^{3} + 1504x^{2} − 249x − 77

799

Form Factor of T_{5}
799A x − 4
799B x^{2} + x − 1
799C x^{12} + 11x^{11} + 30x^{10} − 82x^{9} − 572x^{8} − 805x^{7} + 713x^{6} + 2431x^{5} + 656x^{4}

−1927x^{3} − 1021x^{2} + 422x + 250

799D x^{20} − 13x^{19} + 14x^{18} + 482x^{17} − 1818x^{16} − 5729x^{15} + 38323x^{14} + 9389x^{13}

−358596x^{12} + 319657x^{11} + 1663371x^{10} − 2674360x^{9} − 3552042x^{8} + 8622044x^{7} + 2202456x^{6}

−12131764x^{5} + 1528708x^{4} + 7557516x^{3} − 1964948x^{2} − 1720968x + 468504

799E x^{8} + 10x^{7} + 23x^{6} − 53x^{5} − 226x^{4} − 22x^{3} + 463x^{2} + 286x + 7
799F x^{17} − 11x^{16} + 18x^{15} + 192x^{14} − 672x^{13} − 999x^{12} + 6135x^{11} + 503x^{10} − 24124x^{9} + 7077x^{8}

+44295x^{7} − 6560x^{6} − 39544x^{5} − 9768x^{4} + 8072x^{3} + 3900x^{2} + 420x − 8

799G x

803

Form Factor of T_{3}
803A x^{2} − x − 5
803B x^{5} − 2x^{4} − 9x^{3} + 8x^{2} + 25x + 7
803C x^{19} − 3x^{18} − 35x^{17} + 111x^{16} + 482x^{15} − 1658x^{14} − 3304x^{13} + 12959x^{12} + 11430x^{11}

−57301x^{10} − 15408x^{9} + 144599x^{8} − 12824x^{7} − 200394x^{6} + 57748x^{5} + 139435x^{4} − 49971x^{3}

−42564x^{2} + 12386x + 4307

803D x^{16} − 4x^{15} − 29x^{14} + 130x^{13} + 287x^{12} − 1615x^{11} − 856x^{10} + 9461x^{9} − 3102x^{8} − 25236x^{7}

+22922x^{6} + 20598x^{5} − 34448x^{4} + 10606x^{3} + 2476x^{2} − 1167x − 20

803E x^{9} + 5x^{8} − 27x^{6} − 19x^{5} + 41x^{4} + 30x^{3} − 21x^{2} − 9x + 1
803F x^{10} + 3x^{9} − 13x^{8} − 35x^{7} + 67x^{6} + 136x^{5} − 156x^{4} − 188x^{3} + 106x^{2} + 89x + 7

Form Factor of T_{7}
817A x − 4
817B x + 4
817C x^{13} + 15x^{12} + 71x^{11} − 3x^{10} − 1037x^{9} − 2573x^{8} + 2149x^{7} + 15828x^{6} + 12639x^{5} − 20749x^{4}

−26976x^{3} + 7432x^{2} + 10728x − 2384

817D x^{15} + 13x^{14} + 13x^{13} − 453x^{12} − 1403x^{11} + 5879x^{10} + 24651x^{9} − 39302x^{8} − 191153x^{7} + 177467x^{6}

+718900x^{5} − 651322x^{4} − 1046764x^{3} + 1330320x^{2} − 276448x − 51552

817E x^{15} − 15x^{14} + 66x^{13} + 46x^{12} − 1072x^{11} + 2066x^{10} + 3118x^{9} − 14751x^{8} + 11848x^{7} + 17325x^{6}

−41887x^{5} + 35611x^{4} − 15584x^{3} + 3624x^{2} − 408x + 16

817F x^{18} − 13x^{17} + 8x^{16} + 554x^{15} − 1928x^{14} − 7628x^{13} + 46850x^{12} + 16595x^{11} − 477198x^{10}

+501983x^{9} + 2190865x^{8} − 4639631x^{7} − 3079348x^{6} + 15233622x^{5} − 6887244x^{4} − 16339616x^{3}

+18852320x^{2} − 3841248x − 1607936

851

Form Factor of T_{5}
851A x^{2} − x − 10
851B x^{3} − 9x + 9
851C x^{3} + 4x^{2} + 3x − 1
851D x^{5} + 3x^{4} − 5x^{3} − 5x^{2} + 3x + 2
851E x^{20} − 13x^{19} + 19x^{18} + 443x^{17} − 2001x^{16} − 3445x^{15} + 36558x^{14} − 25663x^{13}

−276167x^{12} + 502325x^{11} + 940829x^{10} − 2798094x^{9} − 1158810x^{8} + 7343270x^{7} − 539281x^{6}

−9850887x^{5} + 1677633x^{4} + 6808852x^{3} − 224506x^{2} − 1951231x − 421276

851F x^{12} + 5x^{11} − 25x^{10} − 153x^{9} + 124x^{8} + 1495x^{7} + 688x^{6} − 5232x^{5}

−5286x^{4} + 4299x^{3} + 5237x^{2} − 10x − 484

851G x^{21} − 4x^{20} − 59x^{19} + 240x^{18} + 1389x^{17} − 5878x^{16} − 16622x^{15} + 76555x^{14}

+103986x^{13} − 577015x^{12} − 276364x^{11} + 2553842x^{10} − 280214x^{9} − 6368404x^{8} + 3463991x^{7}

+7822938x^{6} − 7350113x^{5} − 2824907x^{4} + 5264627x^{3} − 1375068x^{2} − 354840x + 142728

851H x

869

Form Factor of T_{7}
869A x − 1
869B x − 4
869C x + 2
869D x + 5
869E x^{11} + 3x^{10} − 28x^{9} − 72x^{8} + 256x^{7} + 484x^{6} − 1056x^{5} − 1088x^{4} + 1855x^{3} + 436x^{2} − 665x − 125
869F x^{11} + 7x^{10} − 14x^{9} − 174x^{8} − 64x^{7} + 1264x^{6} + 690x^{5} − 4200x^{4} − 573x^{3} + 5980x^{2} − 2631x − 277
869G x^{18} − 10x^{17} − 23x^{16} + 516x^{15} − 652x^{14} − 9304x^{13} + 25946x^{12} + 64888x^{11} − 294601x^{10}

−83585x^{9} + 1368375x^{8} − 574658x^{7} − 2942339x^{6} + 1653448x^{5} + 3292400x^{4}

−1009632x^{3} − 1835808x^{2} − 376512x − 4736

869H x^{21} − 10x^{20} − 51x^{19} + 802x^{18} + 34x^{17} − 24720x^{16} + 43996x^{15} + 362766x^{14} − 1083285x^{13}

−2442801x^{12} + 11200143x^{11} + 4060240x^{10} − 54269061x^{9} + 28349170x^{8} + 107743960x^{7}

−109829248x^{6} − 44052576x^{5} + 59249280x^{4}

+15322368x^{3} − 10312960x^{2} − 3903488x − 305152

871

Form Factor of T_{3}
871A x − 2
871B x^{3} − 6x + 2
871C x^{8} + 4x^{7} − x^{6} − 17x^{5} − 6x^{4} + 21x^{3} + 8x^{2} − 6x − 1
871D x^{20} − 49x^{18} − 3x^{17} + 1008x^{16} + 115x^{15} − 11336x^{14} − 1758x^{13} + 76077x^{12} + 13646x^{11}

−312262x^{10} − 56448x^{9} + 772280x^{8} + 118560x^{7} − 1088640x^{6} − 107392x^{5} + 772224x^{4}

+36608x^{3} − 193024x^{2} − 21504x + 2048

871E x^{23} − 4x^{22} − 51x^{21} + 217x^{20} + 1062x^{19} − 4933x^{18} − 11512x^{17} + 61028x^{16} + 67947x^{15}

−447186x^{14} − 201188x^{13} + 1985836x^{12} + 186230x^{11} − 5291936x^{10} + 244520x^{9} + 8232128x^{8}

−51264x^{7} − 7197696x^{6} − 1144704x^{5} + 3128064x^{4} + 1121792x^{3} − 437248x^{2} − 288768x − 40960

871F x^{12} − 2x^{11} − 13x^{10} + 23x^{9} + 60x^{8} − 91x^{7} − 120x^{6} + 154x^{5} + 107x^{4}

−114x^{3} − 36x^{2} + 30x + 2

Form Factor of T_{2}
893A x^{18} − x^{17} − 28x^{16} + 25x^{15} + 322x^{14} − 247x^{13} − 1971x^{12} + 1231x^{11} + 6953x^{10} − 3283x^{9}

−14235x^{8} + 4562x^{7} + 15962x^{6} − 2882x^{5} − 8159x^{4} + 606x^{3} + 890x^{2} − 179x + 9

893B x^{23} − x^{22} − 38x^{21} + 37x^{20} + 622x^{19} − 586x^{18} − 5746x^{17} + 5195x^{16} + 32986x^{15}

−28295x^{14} − 122210x^{13} + 97857x^{12} + 294219x^{11} − 214815x^{10} − 452516x^{9} + 290939x^{8}

+425340x^{7} − 227487x^{6} − 224027x^{5} + 88871x^{4} + 55896x^{3} − 11568x^{2} − 4727x − 315

893C x^{16} + 4x^{15} − 13x^{14} − 65x^{13} + 47x^{12} + 390x^{11} + 4x^{10} − 1115x^{9} − 320x^{8}

+1639x^{7} + 618x^{6} − 1250x^{5} − 487x^{4} + 456x^{3} + 179x^{2} − 62x − 25

893D x^{12} + x^{11} − 15x^{10} − 14x^{9} + 84x^{8} + 76x^{7} − 213x^{6} − 196x^{5} + 225x^{4}

+229x^{3} − 49x^{2} − 83x − 17

899

Form Factor of T_{2}
899A x − 2
899B x − 1
899C x^{11} + 4x^{10} − 5x^{9} − 35x^{8} − 9x^{7} + 95x^{6} + 68x^{5} − 87x^{4} − 86x^{3} + 9x^{2} + 21x + 3
899D x^{23} − 2x^{22} − 37x^{21} + 72x^{20} + 589x^{19} − 1114x^{18} − 5272x^{17} + 9694x^{16} + 29077x^{15} − 52143x^{14}

−101680x^{13} + 179479x^{12} + 222656x^{11} − 395841x^{10} − 286995x^{9} + 544647x^{8} + 181121x^{7}

−439160x^{6} − 11748x^{5} + 181376x^{4} − 38720x^{3} − 26048x^{2} + 11328x − 1152

899E x^{25} − 43x^{23} − x^{22} + 803x^{21} + 41x^{20} − 8552x^{19} − 705x^{18} + 57371x^{17} + 6615x^{16} − 252748x^{15}

−36904x^{14} + 739489x^{13} + 124981x^{12} − 1419906x^{11} − 250727x^{10} + 1726030x^{9} + 273427x^{8}

−1241767x^{7} − 126943x^{6} + 471802x^{5} + 1736x^{4} − 77984x^{3} + 7008x^{2} + 3968x − 576

899F x^{10} − 12x^{8} + 48x^{6} − 75x^{4} + 38x^{2} + 3x − 1

901

Form Factor of T_{13}
901A x − 2
901B x − 5
901C x − 6
901D x + 7
901E x − 3
901F x − 1
901G x^{13} − 8x^{12} − 64x^{11} + 690x^{10} + 423x^{9} − 18583x^{8} + 34111x^{7} + 150966x^{6} − 527232x^{5}

+49798x^{4} + 1393785x^{3} − 1262290x^{2} − 94868x + 275192

901H x^{12} + 16x^{11} + 49x^{10} − 340x^{9} − 1609x^{8} + 2819x^{7} + 14122x^{6} − 17107x^{5} − 41391x^{4}

+49378x^{3} + 36149x^{2} − 48120x + 4637

901I x^{20} − 121x^{18} − 108x^{17} + 5622x^{16} + 9275x^{15} − 124711x^{14} − 284725x^{13} + 1336016x^{12}

+3867263x^{11} − 5865527x^{10} − 22349231x^{9} + 4694844x^{8} + 41540018x^{7} − 3767109x^{6}

−29164520x^{5} + 5810385x^{4} + 6416556x^{3} − 1536920x^{2} − 433536x + 111424

901J x^{18} + 12x^{17} − 68x^{16} − 1388x^{15} − 934x^{14} + 59487x^{13} + 196711x^{12} − 1059678x^{11}

−6454793x^{10} + 2816281x^{9} + 85456840x^{8} + 139334462x^{7} − 357393148x^{6} − 1366160472x^{5}

−956290865x^{4} + 1655736784x^{3} + 2911246368x^{2} + 1179800512x − 88282240

913

Form Factor of T_{2}
913A x + 1
913B x − 2
913C x^{12} − 2x^{11} − 20x^{10} + 38x^{9} + 145x^{8} − 260x^{7} − 444x^{6} + 749x^{5} + 470x^{4}

−778x^{3} − 7x^{2} + 169x − 29

913D x^{16} + 5x^{15} − 10x^{14} − 80x^{13} + 3x^{12} + 492x^{11} + 296x^{10} − 1442x^{9} − 1323x^{8}

+1982x^{7} + 2237x^{6} − 988x^{5} − 1401x^{4} + x^{3} + 179x^{2} − 5x − 2

913E x^{17} − 9x^{16} + 15x^{15} + 91x^{14} − 345x^{13} − 66x^{12} + 1859x^{11} − 1804x^{10} − 3615x^{9}

+6684x^{8} + 1122x^{7} − 8208x^{6} + 3278x^{5} + 2853x^{4} − 2196x^{3} + 358x^{2} − 5x − 1

913F x^{5} − x^{4} − 5x^{3} + 3x^{2} + 4x − 1
913G x^{15} + 7x^{14} + 2x^{13} − 82x^{12} − 129x^{11} + 328x^{10} + 746x^{9} − 524x^{8} − 1707x^{7}

+286x^{6} + 1801x^{5} + 86x^{4} − 831x^{3} − 151x^{2} + 133x + 37

Form Factor of T_{2}
923A x^{16} + 4x^{15} − 13x^{14} − 64x^{13} + 52x^{12} + 391x^{11} − 28x^{10} − 1164x^{9} − 289x^{8}

+1761x^{7} + 715x^{6} − 1247x^{5} − 608x^{4} + 309x^{3} + 152x^{2} − 14x − 5

923B x^{10} + x^{9} − 11x^{8} − 11x^{7} + 38x^{6} + 35x^{5} − 49x^{4} − 39x^{3} + 18x^{2} + 11x − 1
923C x^{18} − 3x^{17} − 25x^{16} + 77x^{15} + 249x^{14} − 798x^{13} − 1256x^{12} + 4310x^{11} + 3296x^{10}

−13052x^{9} − 3694x^{8} + 22151x^{7} − 1086x^{6} − 19675x^{5} + 5971x^{4} + 7226x^{3}

−3768x^{2} − 160x + 240

923D x^{26} − 3x^{25} − 39x^{24} + 119x^{23} + 659x^{22} − 2050x^{21} − 6346x^{20} + 20159x^{19} + 38611x^{18}

−125213x^{17} − 155933x^{16} + 513936x^{15} + 428618x^{14} − 1418685x^{13} − 809799x^{12} + 2630864x^{11}

+1043228x^{10} − 3216012x^{9} − 878901x^{8} + 2486222x^{7} + 428867x^{6} − 1126309x^{5} − 77770x^{4}

+259800x^{3} − 15120x^{2} − 22928x + 3840

923E x

943

Form Factor of T_{2}
943A x − 1
943B x^{16} + 7x^{15} + x^{14} − 91x^{13} − 144x^{12} + 405x^{11} + 995x^{10} − 661x^{9} − 2772x^{8} + 23x^{7} + 3671x^{6}

+938x^{5} − 2288x^{4} − 777x^{3} + 593x^{2} + 176x − 41

943C x^{19} − 6x^{18} − 11x^{17} + 122x^{16} − 23x^{15} − 1007x^{14} + 851x^{13} + 4351x^{12} − 4949x^{11}

−10657x^{10} + 13195x^{9} + 15208x^{8} − 17642x^{7} − 12700x^{6} + 11260x^{5} + 5628x^{4}

−2855x^{3} − 898x^{2} + 237x + 39

943D x^{24} − 9x^{23} + x^{22} + 213x^{21} − 462x^{20} − 1885x^{19} + 6877x^{18} + 6549x^{17} − 47095x^{16}

+6084x^{15} + 179298x^{14} − 127891x^{13} − 393376x^{12} + 452990x^{11} + 471970x^{10} − 789081x^{9}

−237645x^{8} + 733769x^{7} − 52371x^{6} − 341822x^{5} + 101462x^{4} + 59331x^{3} − 28895x^{2} + 1832x + 171

943E x^{13} + 6x^{12} − 60x^{10} − 78x^{9} + 183x^{8} + 370x^{7} − 154x^{6} − 601x^{5} − 120x^{4}

+300x^{3} + 157x^{2} + 23x + 1

949

Form Factor of T_{2}
949A x^{2} − 3x + 1
949B x^{2} + x − 1
949C x^{16} + x^{15} − 24x^{14} − 20x^{13} + 234x^{12} + 157x^{11} − 1191x^{10} − 620x^{9} + 3367x^{8} + 1308x^{7} − 5160x^{6}

−1437x^{5} + 3799x^{4} + 732x^{3} − 937x^{2} − 133x − 1

949D x^{16} + 3x^{15} − 17x^{14} − 53x^{13} + 108x^{12} + 355x^{11} − 320x^{10} − 1132x^{9} + 440x^{8} + 1756x^{7} − 208x^{6}

−1219x^{5} − 68x^{4} + 363x^{3} + 62x^{2} − 37x − 9

949E x^{19} − 4x^{18} − 21x^{17} + 96x^{16} + 165x^{15} − 937x^{14} − 556x^{13} + 4791x^{12} + 393x^{11} − 13804x^{10}

+2384x^{9} + 22463x^{8} − 6559x^{7} − 19460x^{6} + 6378x^{5} + 7664x^{4} − 2215x^{3} − 898x^{2} + 64x − 1

949F x^{18} + 3x^{17} − 22x^{16} − 68x^{15} + 193x^{14} + 622x^{13} − 872x^{12} − 2985x^{11} + 2168x^{10} + 8148x^{9}

−2902x^{8} − 12815x^{7} + 1818x^{6} + 11232x^{5} − 216x^{4} − 4962x^{3} − 201x^{2} + 835x + 37

979

Form Factor of T_{2}
979A x − 1
979B x^{17} + 4x^{16} − 16x^{15} − 74x^{14} + 95x^{13} + 547x^{12} − 241x^{11} − 2054x^{10} + 141x^{9} + 4103x^{8} + 489x^{7}

−4118x^{6} − 966x^{5} + 1663x^{4} + 596x^{3} − 94x^{2} − 43x − 1

979C x^{11} + 2x^{10} − 11x^{9} − 22x^{8} + 37x^{7} + 79x^{6} − 38x^{5} − 105x^{4} − 5x^{3} + 35x^{2} + 5x − 1
979D x^{18} − 4x^{17} − 19x^{16} + 90x^{15} + 124x^{14} − 811x^{13} − 225x^{12} + 3743x^{11} − 952x^{10}

−9378x^{9} + 5261x^{8} + 12404x^{7} − 9477x^{6} − 7697x^{5} + 7275x^{4} + 1621x^{3}

−2164x^{2} + 32x + 160

979E x^{25} − 4x^{24} − 34x^{23} + 150x^{22} + 472x^{21} − 2405x^{20} − 3367x^{19} + 21590x^{18} + 12144x^{17}

−119518x^{16} − 11478x^{15} + 424322x^{14} − 75545x^{13} − 975398x^{12} + 328509x^{11} + 1432194x^{10}

−602131x^{9} − 1290914x^{8} + 586606x^{7} + 662736x^{6} − 303037x^{5} − 172209x^{4}

+73684x^{3} + 18592x^{2} − 6016x − 672

979F x

Form Factor of T_{3}
989A x − 3
989B x^{4} + 2x^{3} − 5x^{2} − 6x + 4
989C x^{24} − 8x^{23} − 20x^{22} + 305x^{21} − 125x^{20} − 4784x^{19} + 7159x^{18} + 39563x^{17} − 87958x^{16}

−181345x^{15} + 547998x^{14} + 421434x^{13} − 1945873x^{12} − 223618x^{11} + 3950948x^{10} − 1046125x^{9}

−4283043x^{8} + 2176204x^{7} + 2045523x^{6} − 1457455x^{5} − 159844x^{4}

+252973x^{3} − 52506x^{2} + 3196x + 16

989D x^{22} − 11x^{21} + 17x^{20} + 222x^{19} − 851x^{18} − 1163x^{17} + 9774x^{16} − 3541x^{15} − 51223x^{14} + 57070x^{13}

+140536x^{12} − 235130x^{11} − 202509x^{10} + 485883x^{9} + 128009x^{8} − 554328x^{7} + 9227x^{6} + 345207x^{5}

−50118x^{4} − 106767x^{3} + 19237x^{2} + 12406x − 1964

989E x^{16} + 11x^{15} + 31x^{14} − 68x^{13} − 468x^{12} − 324x^{11} + 1924x^{10} + 3278x^{9} − 2190x^{8}

−7920x^{7} − 1828x^{6} + 6736x^{5} + 4031x^{4} − 1527x^{3} − 1509x^{2} − 138x + 52

989F x^{10} + 5x^{9} − 5x^{8} − 52x^{7} − 23x^{6} + 135x^{5} + 53x^{4} − 136x^{3} + 11x^{2} + 15x + 1

1003

Form Factor of T_{3}
1003A x − 3
1003B x − 1
1003C x − 2
1003D x^{3} − 2x^{2} − 5x + 2
1003E x^{10} + 7x^{9} + 8x^{8} − 34x^{7} − 56x^{6} + 67x^{5} + 94x^{4} − 66x^{3} − 39x^{2} + 18x − 1
1003F x^{22} − 7x^{21} − 24x^{20} + 260x^{19} + 65x^{18} − 3924x^{17} + 3122x^{16} + 31442x^{15}

−41051x^{14} − 147389x^{13} + 232957x^{12} + 425192x^{11} − 712621x^{10} − 787756x^{9} + 1209743x^{8}

+971082x^{7} − 1052280x^{6} − 768268x^{5} + 324016x^{4} + 292480x^{3} + 49984x^{2}

+1472x − 128

1003G x^{16} + 7x^{15} − 8x^{14} − 154x^{13} − 157x^{12} + 1228x^{11} + 2482x^{10} − 3944x^{9}

−13179x^{8} + 1539x^{7} + 30581x^{6} + 17216x^{5} − 25563x^{4} − 27478x^{3} − 1953x^{2}

+4266x + 540

1003H x^{4} + 2x^{3} − 5x^{2} − 6x + 4
1003I x^{18} − x^{17} − 35x^{16} + 29x^{15} + 504x^{14} − 331x^{13} − 3862x^{12} + 1901x^{11}

+16995x^{10} − 5784x^{9} − 43240x^{8} + 8592x^{7} + 60913x^{6} − 3090x^{5} − 42652x^{4}

−5572x^{3} + 11672x^{2} + 3760x + 128

1003J x

1007

Form Factor of T_{2}
1007A x − 2
1007B x^{12} + 3x^{11} − 12x^{10} − 38x^{9} + 47x^{8} + 164x^{7} − 68x^{6} − 286x^{5}

+37x^{4} + 196x^{3} − 9x^{2} − 37x + 7

1007C x^{12} + 5x^{11} − 2x^{10} − 42x^{9} − 31x^{8} + 116x^{7} + 124x^{6} − 130x^{5}

−151x^{4} + 54x^{3} + 55x^{2} − 3x − 1

1007D x^{28} − 3x^{27} − 42x^{26} + 128x^{25} + 773x^{24} − 2398x^{23} − 8210x^{22} + 25984x^{21}

+55795x^{20} − 180640x^{19} − 254355x^{18} + 845427x^{17} + 792781x^{16} − 2721726x^{15}

−1686216x^{14} + 6057517x^{13} + 2381593x^{12} − 9243830x^{11} − 2066032x^{10} + 9460131x^{9}

+842962x^{8} − 6228602x^{7} + 139290x^{6} + 2443827x^{5} − 281692x^{4} − 487131x^{3}

+89497x^{2} + 31575x − 5878

1007E x^{26} − 6x^{25} − 24x^{24} + 204x^{23} + 143x^{22} − 2957x^{21} + 1313x^{20} + 23773x^{19}

−25728x^{18} − 115036x^{17} + 181081x^{16} + 336456x^{15} − 713409x^{14} − 543569x^{13}

+1698815x^{12} + 280790x^{11} − 2424843x^{10} + 511527x^{9} + 1923915x^{8} − 940458x^{7}

−679950x^{6} + 547980x^{5} + 10922x^{4} − 95345x^{3} + 27015x^{2} − 2450x + 33

Form Factor of T_{2}
1027A x^{18} + 6x^{17} − 8x^{16} − 106x^{15} − 57x^{14} + 715x^{13} + 859x^{12} − 2323x^{11}

−3741x^{10} + 3768x^{9} + 7603x^{8} − 2785x^{7} − 7585x^{6} + 510x^{5} + 3292x^{4} + 241x^{3}

−361x^{2} − 32x + 8

1027B x^{17} + 10x^{16} + 22x^{15} − 90x^{14} − 431x^{13} − 29x^{12} + 2401x^{11} + 2695x^{10}

−5361x^{9} − 10562x^{8} + 3217x^{7} + 16039x^{6} + 4113x^{5} − 9336x^{4} − 4872x^{3}

+1239x^{2} + 959x + 64

1027C x^{22} − 8x^{21} − 3x^{20} + 176x^{19} − 281x^{18} − 1465x^{17} + 3952x^{16} + 5282x^{15}

−23404x^{14} − 3893x^{13} + 72142x^{12} − 27611x^{11} − 118500x^{10} + 81451x^{9}

+98907x^{8} − 85881x^{7} − 39195x^{6} + 38199x^{5} + 6767x^{4} − 6456x^{3} − 880x^{2} + 384x + 64

1027D x^{21} − 9x^{20} + 8x^{19} + 152x^{18} − 425x^{17} − 792x^{16} + 4160x^{15} + 64x^{14}

−18654x^{13} + 13873x^{12} + 43575x^{11} − 55324x^{10} − 51008x^{9} + 99433x^{8} + 19464x^{7}

−91617x^{6} + 14906x^{5} + 40363x^{4} − 15992x^{3} − 5632x^{2} + 4032x − 576

1027E x

1037

Form Factor of T_{3}
1037A x^{2} − 3x + 1
1037B x^{18} + 15x^{17} + 74x^{16} + 32x^{15} − 926x^{14} − 2681x^{13} + 1400x^{12}

+17484x^{11} + 19234x^{10} − 33596x^{9} − 85376x^{8} − 19066x^{7} + 105512x^{6} + 102553x^{5}

−8083x^{4} − 61166x^{3} − 35970x^{2} − 8416x − 704

1037C x^{16} + 12x^{15} + 42x^{14} − 35x^{13} − 502x^{12} − 615x^{11} + 1771x^{10} + 3959x^{9}

−2292x^{8} − 9624x^{7} + 120x^{6} + 11416x^{5} + 1974x^{4} − 6425x^{3} − 1318x^{2}

+1236x + 279

1037D x^{23} − 10x^{22} + x^{21} + 291x^{20} − 681x^{19} − 3174x^{18} + 12221x^{17} + 14285x^{16}

−99365x^{15} − 1412x^{14} + 443287x^{13} − 234259x^{12} − 1144878x^{11} + 904603x^{10}

+1756306x^{9} − 1468367x^{8} − 1677167x^{7} + 1055365x^{6} + 1036867x^{5} − 217298x^{4} − 323651x^{3}

−67998x^{2} − 672x + 416

1037E x^{22} − 14x^{21} + 53x^{20} + 131x^{19} − 1375x^{18} + 1742x^{17} + 9953x^{16}

−29511x^{15} − 17727x^{14} + 156246x^{13} − 89995x^{12} − 367795x^{11} + 465214x^{10} + 342047x^{9}

−794956x^{8} + 43663x^{7} + 542531x^{6} − 226103x^{5} − 91437x^{4} + 64966x^{3}

−6527x^{2} − 1202x + 146

1067

Form Factor of T_{13}
1067A x^{2} + 2x − 17
1067B x^{2} + 6x + 7
1067C x^{14} + 25x^{13} + 211x^{12} + 319x^{11} − 5205x^{10} − 26815x^{9} + 6316x^{8}

+322481x^{7} + 509743x^{6} − 1144277x^{5} − 3041934x^{4} + 773584x^{3} + 4970173x^{2}

+1136520x − 1051945

1067D x^{23} − 31x^{22} + 323x^{21} − 373x^{20} − 17818x^{19} + 122314x^{18} + 23968x^{17}

−3280979x^{16} + 10485339x^{15} + 21218891x^{14} − 176189305x^{13} + 155668558x^{12}

+988194140x^{11} − 2112849646x^{10} − 2047373230x^{9} + 8552932523x^{8} − 102049391x^{7}

−16065070244x^{6} + 5765616786x^{5} + 13791034932x^{4} − 6488157369x^{3} − 3539208616x^{2}

+2153862224x − 205611392

1067E x^{15} + 23x^{14} + 179x^{13} + 213x^{12} − 4773x^{11} − 28337x^{10} − 44114x^{9}

+124591x^{8} + 578605x^{7} + 653875x^{6} − 372448x^{5} − 1226400x^{4} − 550665x^{3}

+154818x^{2} + 53831x + 3416

1067F x^{25} − 39x^{24} + 547x^{23} − 2007x^{22} − 28941x^{21} + 340125x^{20}

−618675x^{19} − 9773072x^{18} + 59314293x^{17} + 20147003x^{16}

−1190555831x^{15} + 2833660753x^{14} + 8271323665x^{13} − 44669586321x^{12} + 10591618031x^{11}

+268605910227x^{10} − 398322683771x^{9} − 599389230806x^{8} + 1720189666248x^{7}

−42102336229x^{6} − 2561015519636x^{5} + 1187441668674x^{4} + 1349674851756x^{3}

−789142890382x^{2} − 121020476167x + 5645501986