Tables of modular Galois representations

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Tables of modular Galois representations

Nicolas Mascot

To cite this version:

Nicolas Mascot. Tables of modular Galois representations. 2014. �hal-01110252�

Tables of modular Galois representations

Nicolas Mascot

University of Warwick, supported by EPSRC Programme Grant “LMF: L-Functions and Modular Forms”.

Formerly IMB, Universit´e Bordeaux 1, UMR 5251, F-33400 Talence, France. CNRS, IMB, UMR 5251, F-33400 Talence, France. INRIA, project LFANT, F-33400 Talence, France.

December 17, 2014

Abstract

We give tables of modular Galois representations obtained using the algo- rithm which we described in [Mas13]. We computed Galois representations modulo primes up to 31 for the first time. In particular, we computed the representations attached to a newform with non-rational (but of course alge- braic) coefficients, which had never been done before. These computations take place in the jacobian of modular curves of genus up to 26. We also show how these computation results can be partially proved.

Acknowledgements

I heartily thank my advisor J.-M. Couveignes for offering me to work on this beautiful subject. The computations presented here would not have been amenable without Bill Allombert, who suggested to me the idea of step-by-step polynomial reduction, and Karim Belabas and Denis Simon, who provided me their [BS14] script. I am therefore extremely grateful to them.

The computations presented in this paper were carried out using the PlaFRIM experimental testbed, being developed under the Inria PlaFRIM development action with support from LABRI and IMB and other entities:

Conseil R´ egional d’Aquitaine, FeDER, Universit´ e de Bordeaux and CNRS (see https://plafrim.bordeaux.inria.fr/). The softwares used were [SAGE] and [Pari/GP].

This research was supported by the French ANR-12-BS01-0010-01 through

the project PEACE, and by the DGA maˆıtrise de l’information.

We begin with a short summary about Galois representations attached to mod- ular forms and how we used these in [Mas13] to compute Fourier coefficients of modular forms in section 1. This computation becomes much easier if the polyno- mial in Q [X] defining the representation is reduced, and we show new ideas to do so efficiently in section 2. We then present tables of results of our computations in the last section 3. Finally, since these results rely on the identification of rational numbers given in approximate form, we present in section 4 a method to formally prove that the number field cut out by the Galois representation has been correctly computed.

1 Background summary

Let f = q + P

n=2

a

q

∈ S

Γ

(N ), ε

be a classical newform of weight k ∈ N

, level N ∈ N

and nebentypus ε. Jean-Pierre Serre conjectured and Pierre Deligne proved in [Del71] that for every finite prime l of the number field K

= Q (a

, n > 2) spanned by the coefficients a

of the q-expansion of f at infinity, there exists a Galois representation

Gal( Q / Q ) −→ GL

( Z

)

which is unramified outside `N and such that the image of any Frobenius element at p - `N has characteristic polynomial x

− a

x + ε(p)p

∈ Z

[x], where Z

denotes the l-adic completion of the ring of integers Z

of K

, and ` is the rational prime lying below l.

Let F

be the residue field of l. By reducing the above l-adic Galois representation modulo l, we get a modulo l Galois representation

ρ

: Gal( Q / Q ) −→ GL

( F

),

which is unramified outside `N and such that the image of any Frobenius element at p - `N has characteristic polynomial x

− a

x + ε(p)p

∈ F

[x]. In particular, the trace of this image is a

mod l.

In [Mas13], we described an algorithm based on ideas from the book [CE11]

edited by Jean-Marc Couveignes and Bas Edixhoven to compute such modulo l Galois representations, provided that the image of the Galois representation contains SL

( F

), that k < `, and that l has inertia degree 1, so that F

= F

. This gives a way to quickly compute the coefficients a

modulo l for huge primes p.

The condition that the image of the Galois representation contain SL

( F

) is a very weak one. Indeed, by [Rib85, theorem 2.1] and [Swi72, lemma 2], for any non- CM newform f (and in particular for any newform f of level 1), the image of the representation ρ

contains SL

( F

) for almost every l. The finitely many l for which SL

( F

) 6⊂ Im ρ

are called exceptional primes for f , and we exclude them. They were explicitly determined by Sir Peter Swinnerton-Dyer in [Swi72] for the known

newforms f of level 1 whose coefficients a

are rational. In our case, this means we exclude l = 23 for f = ∆ and l = 31 for f = E

∆ (cf. the notations of section 3 below).

1According to Maeda’s conjecture (cf [FW02]), there are only 6 such forms: ∆, E₄∆, E₆∆, E₈∆,E₁₀∆ andE₁₄∆ in the notation of section 3 below, of respective weights 12, 16, 18, 20, 22

This algorithm relies on the fact that if k < `, then the Galois representation ρ

is afforded with multiplicity 1 by a subspace V

of the `-torsion of the jacobian J

(`N ) of the modular curve X

(`N ) under the natural Gal( Q / Q )-action, cf [Gro90]

and [Mas13, section 1].

The algorithm first computes the number field L = Q

cut out by the Galois representation, by evaluating a well-chosen function α ∈ Q J

(`N )

in the nonzero points of V

and forming the polynomial

F (X) = Y

x∈V_f,l x6=0

X − α(x)

∈ Q [X]

of degree `

− 1 whose decomposition field is L. The algorithm then uses a method

from T. and V. Dokchitser (cf [Dok10]) to compute the image of the Frobenius

element at p given a rational prime p - `N . Since such a Frobenius element is

defined only up to conjugation, the output is a similarity class in GL

( F

).

2 Reducing the polynomials

Unfortunately, the coefficients of the polynomial F (X) tend to have larger and larger height as ` grows. More precisely, the following table, which shows the genus g =

of the modular curves X

(`) and the rough number h of decimal digits in the common denominator of the polynomials F (X) associated to newforms of level N = 1 (cf the Tables section below) which we computed using the algorithm described in [Mas13], seems to indicate the heuristic h ≈ g

:

` g h

11 1 0

13 2 5

17 5 50

19 7 150

23 12 500

29 22 1800

31 26 2500

While this is rather harmless for l 6 17, it makes the Dokchitser’s method intractable as soon as ` > 29. It is thus necessary to reduce this polynomial, that is to say to compute another polynomial whose splitting field is isomorphic to the splitting field of F (X) but whose coefficients are much nicer. An algorithm to perform this task based on LLL lattice reduction is described in [Coh93, section 4.4.2]

and implemented in [Pari/GP] under the name polred. However, the polynomial F (X) has degree `

− 1 and tends to have ugly coefficients, and this is too much for polred to be practical, even for small values of `. Indeed, the fact that polred is based on LLL reduction means that its execution time is especially sensitive to the degree of the polynomial.

On the other hand, it would be possible to apply the polred algorithm to the polynomial

F

(X) = Y

W∈PVf,l





 X − X

x∈W x6=0

α(x)





 ∈ Q [X]

whose splitting field is the number field L

cut out by the projective Galois rep- resentation

ρ

: Gal( Q / Q )

GL

( F

)

PGL

( F

)

since the degree of this polynomial is only ` + 1, but this representation does not contain enough information to recover the values of a

mod l.

However, we noted in [Mas13, section 3.7.2] that if S ⊂ F

denotes the largest subgroup of F

not containing −1, then the knowledge of the quotient representation

ρ

: Gal( Q / Q )

GL

( F

)

GL

( F

)/S ,

combined with the fact that the image in GL

( F

) of a Frobenius element at p

has determinant p

ε(p) mod l, is enough to recover the values of a

mod l. It is

therefore enough for our purpose to compute this quotient representation, by first forming the polynomial

F

(X) = Y

Sx∈V_f,l/S x6=0

X − X

s∈S

α(sx)

!

∈ Q [X],

whose splitting field is the number field L

cut out by ρ

, and then to apply the Dokchitsers’ method on it in order to compute the images of the Frobenius elements by ρ

, cf [Mas13, section 3.7.2].

This is practical provided that we manage to apply the polred algorithm to F

(X), that is to say if the degree of F

(X) is not too large. Let ` − 1 = 2

s with s ∈ N odd. Since we have |S| = s, the degree of F

is 2

(` + 1), so we can polred F

in the cases ` = 19 or 23, but the cases ` = 29 or 31 remain impractical.

For these remaining cases, Bill Allombert suggested to the author that one can still reduce F

(X) in several steps, as we now explain. Since F

is cyclic, we have a filtration

F

= S

⊃

2

S

⊃

2

· · · ⊃

2

S

= S with [S

: S

] = 2 for all i, namely

S

= Im

F

−→ F

x 7−→ x

.

For each i 6 r, let us define F

(X) = Y

Six∈V_f,l/Si

x6=0

X − X

s∈S_i

α(sx)

!

∈ Q [X],

K

= Q [X]/F

(X),

and L

= normal closure of K

= the number field cut out by the quotient represen- tation

ρ

: Gal( Q / Q )

GL

( F

)

GL

( F

)/S

.

In particular, we have ρ

= ρ

, L

= L

, and we are looking for a nice model of K

.

The fields K

fit in an extension tower K

2

2^r

.. .

2

K

2

K

and we are going to polred the polynomials F

(X) along this tower recursively from bottom up, as we now explain.

First, we apply directly the polred algorithm to F

(X) = F

(X). Since the degree of this polynomial is only ` + 1, this is amenable, as mentioned above.

Then, assuming we have managed to reduce F

(X), we have a nice model for K

= Q [X]/F

(X), so we can factor F

(X) in K

[X]. Since the extension K

= Q [X]/F

(X) is quadratic over K

, there must be at least one factor of degree 2.

Let G

(X) be one of those, and let ∆

∈ K

be its discriminant, so that we have K

' K

[X]/G

(X) ' K

p

∆

.

In order to complete the recursion, all we have to do is to strip ∆

from the largest square factor we can find, say ∆

= A

δ

with A

, δ

∈ K

and δ

as small as possible.

Indeed we then have K

= K

√ δ

, and actually even K

= Q

√ δ

unless we are very unlucky

, so that if we denote by χ

(X) ∈ Q [X] the characteristic polynomial of δ

, then we have

K

' Q [X]/χ

(X

),

so that χ

(X

) is a reduced version of F

. If its degree and coefficients are not too big, we can even apply the polred algorithm to this polynomial in order to further reduce it, which is what we do in practice.

In order to write ∆

= A

δ

, we would like to factor ∆

in K

, but even if K

is principal, this is not amenable whatsoever. We can however consider the ideal

generated by ∆

in K

, and remove its `N -part. The fractional ideal B

we obtain

must then be a perfect square, since K

is unramified outside `N (since L is), and

the very efficient idealsqrt script from [BS14] can explicitly factor it into B

= A

.

If A

denotes an element in A

close to being a generator of A

(an actual generator,

if amenable, would be even better), then δ

:= ∆

/A

must be small.

3 The tables

Notation

Following the tradition, we define E

= 1 + 240

+∞

X

n=1

σ

(n)q

, E

= 1 − 504

+∞

X

n=1

σ

(n)q

, and ∆ = E

− E

1728 , where σ

(n) = P

0<d|n

d

.

We computed the Galois representations modulo the primes ` ranging from 11 to 31 and attached to the newforms f ∈ S

(1) of level N = 1 and of weight k < `.

According to Maeda’s conjecture, for each weight k, there is only one newform in S

(1) up to Gal( Q / Q )-conjugation. This conjecture has been verified in [FW02] for k up to 2000, and since we work with newforms of level 1 and weight k up to only 30 (because of the condition k < `), we may denote without ambiguity one of the newforms in S

(1) by f

, and the coefficients of its q-expansion at infinity by τ

(n).

Then, for each k, the newform f

and the sequence τ

(n)

n>2

are well-defined up to Gal( Q / Q )-action, and the newforms in S

(1) are the Gal( Q / Q )-conjugates of

f

= q +

+∞

X

n=2

τ

(n)q

.

Thus for instance we have f

= ∆, τ

= τ is Ramanujan’s τ-function, f

= E

∆ = q + P

n=2

τ

(n)q

is the only newform of level 1 and weight 16, and so on.

For each Galois representation ρ

, we denote by L the number field it cuts off, and we give the image of the Frobenius element

L/Q p

at p for the 40 first primes p above 10

. Since these p are unramified, these Frobenius elements are well- defined up to conjugacy, so their images are well-defined up to similarity. Instead of representing a similarity class in GL

( F

) by a matrix as we did in [Mas13], we deemed it more elegant to give its minimal polynomial in factored form over F

. As we are dealing with matrices of size 2, this is a faithful representation. Indeed, recall that GL

( F

) splits into similarity classes as follows:

Type of class Representative Minimal polynomial no of classes no of elements in class Scalar

λ 0 0 λ

x − λ ` − 1 1

Split semisimple

λ 0 0 µ

(x − λ)(x − µ)

`(` + 1)

Non-split semisimple

0 −n 1 t

x

− tx + n irreducible over F

`(`−1)

2

`(` − 1)

Non-semisimple

λ 1 0 λ

(x − λ)

` − 1 (` + 1)(` − 1)

For each p, we also give the trace of the image of

L/Q

, which is none other

` = 11

f

= ∆ =

+∞

X

n=1

τ(n)q

= q − 24q

+ 252q

+ O(q

)

p Similarity class of

Q p

τ(p) mod 11

10

+ 453 (x − 9)(x − 4) 2

10

+ 1357 (x − 8)(x − 2) 10

10

+ 2713 x

+ x + 8 10

10

+ 4351 (x − 6)(x − 3) 9

10

+ 5733 x

+ 3x + 3 8

10

+ 7383 x

+ 3x + 3 8

10

+ 10401 (x − 8)(x − 5) 2

10

+ 11979 x

+ 1 0

10

+ 17557 (x − 10)(x − 9) 8

10

+ 21567 x

+ 10x + 8 1

10

+ 22273 (x − 9)(x − 6) 4 10

+ 24493 (x − 8)(x − 1) 9 10

+ 25947 (x − 9)(x − 6) 4

10

+ 27057 x

+ 4x + 9 7

10

+ 29737 (x − 9)(x − 3) 1

10

+ 41599 x

+ 9 0

10

+ 43789 x

+ 6x + 10 5

10

+ 46227 (x − 7)(x − 4) 0 10

+ 46339 (x − 8)(x − 1) 9

10

+ 52423 (x − 3)

6

10

+ 55831 x

+ 10x + 7 1

10

+ 57867 (x − 8)(x − 1) 9 10

+ 59743 (x − 3)(x − 1) 4

10

+ 61053 (x − 9)

7

10

+ 61353 x

+ x + 7 10

10

+ 63729 x

+ x + 7 10

10

+ 64047 (x − 3)(x − 2) 5 10

+ 64749 (x − 10)(x − 7) 6

10

+ 68139 x

+ 2x + 6 9

10

+ 68367 (x − 3)(x − 1) 4 10

+ 70897 (x − 10)(x − 8) 7 10

+ 72237 (x − 4)(x − 3) 7 10

+ 77611 (x − 8)(x − 5) 2 10

+ 78199 (x − 6)(x − 2) 8 10

+ 79237 (x − 5)(x − 1) 6

10

+ 79767 x

+ 4x + 7 7

10

+ 82767 x

+ 2x + 4 9

10

+ 93559 (x − 4)

8

10

+ 95107 (x − 10)(x − 9) 8

10

+ 100003 (x − 9)(x − 4) 2

` = 13

f

= ∆ =

+∞

X

n=1

τ(n)q

= q − 24q

+ 252q

+ O(q

)

p Similarity class of

Q p

τ(p) mod 13

10

+ 453 x

+ 3x + 1 10

10

+ 1357 (x − 10)(x − 7) 4 10

+ 2713 x

+ 12x + 12 1

10

+ 4351 x

+ x + 12 12

10

+ 5733 (x − 12)(x − 4) 3

10

+ 7383 x

+ 6x + 7 7

10

+ 10401 (x − 5)(x − 2) 7

10

+ 11979 (x − 9)

5

10

+ 17557 x

+ 6x + 4 7

10

+ 21567 x

+ 5x + 9 8

10

+ 22273 (x − 10)(x − 8) 5

10

+ 24493 x

+ 8x + 10 5

10

+ 25947 x

+ 10x + 7 3

10

+ 27057 (x − 7)(x − 4) 11

10

+ 29737 x

+ x + 3 12

10

+ 41599 (x − 11)(x − 3) 1 10

+ 43789 (x − 10)(x − 7) 4

10

+ 46227 x

+ 10x + 7 3

10

+ 46339 (x − 8)(x − 7) 2 10

+ 52423 (x − 10)(x − 3) 0 10

+ 55831 (x − 4)(x − 3) 7 10

+ 57867 (x − 2)(x − 1) 3

10

+ 59743 x

+ 6 0

10

+ 61053 x

+ x + 5 12

10

+ 61353 (x − 11)(x − 5) 3 10

+ 63729 (x − 11)(x − 1) 12 10

+ 64047 (x − 10)(x − 9) 6 10

+ 64749 (x − 4)(x − 3) 7

10

+ 68139 x

+ 6x + 3 7

10

+ 68367 (x − 7)(x − 5) 12 10

+ 70897 (x − 12)(x − 7) 6 10

+ 72237 x

+ 11x + 12 2

10

+ 77611 x

+ 5x + 10 8

10

+ 78199 (x − 7)(x − 4) 11 10

+ 79237 (x − 4)(x − 2) 6

10

+ 79767 x

+ 7x + 7 6

10

+ 82767 x

+ 9x + 12 4

10

+ 93559 x

+ 8x + 1 5

10

+ 95107 x

+ 10x + 7 3

10

+ 100003 x

+ 6x + 4 7

` = 17

f

= ∆ =

+∞

X

n=1

τ(n)q

= q − 24q

+ 252q

+ O(q

)

p Similarity class of

Q p

τ(p) mod 17

10

+ 453 x

+ 3 0

10

+ 1357 (x − 15)

13

10

+ 2713 (x − 14)(x − 12) 9 10

+ 4351 (x − 10)(x − 6) 16 10

+ 5733 (x − 6)(x − 4) 10 10

+ 7383 (x − 15)(x − 2) 0 10

+ 10401 (x − 7)(x − 3) 10

10

+ 11979 x

+ 6x + 3 11

10

+ 17557 x

+ 11x + 6 6

10

+ 21567 x

+ 16x + 3 1

10

+ 22273 x

+ 16x + 8 1

10

+ 24493 x

+ 8x + 6 9

10

+ 25947 x

+ 2x + 13 15 10

+ 27057 (x − 16)(x − 2) 1

10

+ 29737 x

+ 5x + 7 12

10

+ 41599 x

+ 6x + 16 11 10

+ 43789 (x − 11)(x − 5) 16

10

+ 46227 x

+ 4x + 7 13

10

+ 46339 (x − 14)(x − 10) 7 10

+ 52423 (x − 16)(x − 5) 4

10

+ 55831 x

+ 14x + 8 3

10

+ 57867 x

+ 8x + 9 9

10

+ 59743 (x − 14)(x − 7) 4 10

+ 61053 x

+ 15x + 11 2 10

+ 61353 x

+ 6x + 16 11

10

+ 63729 x

+ 6 0

10

+ 64047 x

+ 7x + 14 10 10

+ 64749 (x − 6)(x − 1) 7 10

+ 68139 (x − 11)(x − 10) 4 10

+ 68367 (x − 16)(x − 2) 1

10

+ 70897 x

+ 5x + 5 12

10

+ 72237 x

+ 7 0

10

+ 77611 x

+ 15x + 11 2 10

+ 78199 (x − 16)(x − 8) 7 10

+ 79237 (x − 10)(x − 5) 15 10

+ 79767 (x − 8)(x − 1) 9

10

+ 82767 x

+ 16x + 3 1

10

+ 93559 x

+ 5x + 14 12

10

+ 95107 (x − 11)

5

10

+ 100003 (x − 14)(x − 5) 2

f

= E

∆ =

+∞

X

n=1

τ

(n)q

= q + 216q

− 3348q

+ O(q

)

p Similarity class of

L/Q p

τ

(p) mod 17

10

+ 453 x

+ 5x + 12 12

10

+ 1357 x

+ 3x + 4 14

10

+ 2713 x

+ 8x + 2 9

10

+ 4351 x

+ 14x + 8 3

10

+ 5733 x

+ 11x + 6 6

10

+ 7383 (x − 8)

16

10

+ 10401 (x − 16)(x − 13) 12 10

+ 11979 (x − 9)(x − 7) 16 10

+ 17557 (x − 5)(x − 2) 7 10

+ 21567 x

+ 12x + 12 5

10

+ 22273 x

+ 13x + 9 4

10

+ 24493 x

+ 10 0

10

+ 25947 (x − 16)(x − 4) 3 10

+ 27057 (x − 10)(x − 7) 0

10

+ 29737 x

+ 9x + 6 8

10

+ 41599 x

+ 4x + 16 13 10

+ 43789 (x − 4)(x − 1) 5 10

+ 46227 (x − 12)(x − 9) 4

10

+ 46339 x

+ 15x + 4 2

10

+ 52423 (x − 11)(x − 9) 3

10

+ 55831 x

+ 9x + 9 8

10

+ 57867 x

+ 12x + 8 5

10

+ 59743 (x − 8)

16

10

+ 61053 (x − 15)(x − 5) 3 10

+ 61353 x

+ 16x + 16 1 10

+ 63729 x

+ 14x + 10 3

10

+ 64047 x

+ 12x + 5 5

10

+ 64749 x

+ 10 0

10

+ 68139 (x − 10)(x − 6) 16

10

+ 68367 x

+ 8x + 2 9

10

+ 70897 (x − 16)(x − 14) 13 10

+ 72237 (x − 13)(x − 7) 3 10

+ 77611 (x − 6)(x − 4) 10 10

+ 78199 (x − 8)(x − 1) 9 10

+ 79237 x

+ 13x + 16 4

10

+ 79767 x

+ 4x + 9 13

10

+ 82767 x

+ 5x + 12 12

10

+ 93559 x

+ 5 0

10

+ 95107 (x − 7)

14

10

+ 100003 (x − 10)

3

` = 19

f

= ∆ =

+∞

X

n=1

τ(n)q

= q − 24q

+ 252q

+ O(q

)

p Similarity class of

Q p

τ(p) mod 19 10

+ 453 (x − 15)(x − 10) 6

10

+ 1357 (x − 17)

15

10

+ 2713 (x − 11)(x − 4) 15 10

+ 4351 (x − 6)(x − 4) 10 10

+ 5733 (x − 16)(x − 1) 17

10

+ 7383 (x − 1)

2

10

+ 10401 x

+ 11x + 4 8

10

+ 11979 (x − 16)(x − 13) 10 10

+ 17557 x

+ 8x + 14 11

10

+ 21567 (x − 11)

3

10

+ 22273 (x − 13)(x − 1) 14 10

+ 24493 (x − 14)(x − 10) 5 10

+ 25947 x

+ 14x + 15 5 10

+ 27057 (x − 10)(x − 9) 0

10

+ 29737 x

+ 12x + 7 7

10

+ 41599 (x − 18)(x − 15) 14 10

+ 43789 (x − 13)(x − 11) 5

10

+ 46227 x

+ 5 0

10

+ 46339 x

+ x + 11 18

10

+ 52423 x

+ 7x + 7 12

10

+ 55831 (x − 16)(x − 13) 10 10

+ 57867 (x − 17)(x − 2) 0

10

+ 59743 x

+ 5x + 9 14

10

+ 61053 x

+ 9x + 3 10

10

+ 61353 (x − 14)(x − 10) 5

10

+ 63729 x

+ 15x + 8 4

10

+ 64047 (x − 6)(x − 5) 11

10

+ 64749 (x − 13)

7

10

+ 68139 x

+ 15x + 13 4 10

+ 68367 (x − 14)(x − 5) 0 10

+ 70897 (x − 18)(x − 15) 14 10

+ 72237 (x − 10)(x − 5) 15

10

+ 77611 x

+ 13x + 6 6

10

+ 78199 (x − 15)

11

10

+ 79237 x

+ 12x + 9 7

10

+ 79767 x

+ 13x + 13 6

10

+ 82767 x

+ 3x + 8 16

10

+ 93559 x

+ 4x + 8 15

10

+ 95107 x

+ 13x + 15 6

10

+ 100003 x

+ 5x + 3 14

f

= E

∆ =

+∞

X

n=1

τ

(n)q

= q + 216q

− 3348q

+ O(q

)

p Similarity class of

L/Q p

τ

(p) mod 19 10

+ 453 (x − 15)(x − 2) 17 10

+ 1357 (x − 18)(x − 12) 11

10

+ 2713 x

+ 6x + 7 13

10

+ 4351 x

+ 9x + 11 10

10

+ 5733 (x − 17)(x − 4) 2

10

+ 7383 x

+ 5x + 1 14

10

+ 10401 x

+ 13x + 7 6

10

+ 11979 (x − 16)(x − 13) 10 10

+ 17557 (x − 9)(x − 3) 12

10

+ 21567 x

+ 5x + 1 14

10

+ 22273 (x − 17)(x − 13) 11 10

+ 24493 (x − 17)(x − 9) 7 10

+ 25947 (x − 18)(x − 7) 6

10

+ 27057 x

+ 5x + 8 14

10

+ 29737 (x − 13)(x − 3) 16

10

+ 41599 x

+ 7x + 7 12

10

+ 43789 x

+ 9x + 12 10 10

+ 46227 x

+ 16x + 11 3 10

+ 46339 (x − 17)(x − 9) 7 10

+ 52423 (x − 15)(x − 14) 10 10

+ 55831 (x − 14)(x − 4) 18 10

+ 57867 x

+ 18x + 12 1

10

+ 59743 x

+ 7 0

10

+ 61053 (x − 17)(x − 15) 13 10

+ 61353 (x − 10)(x − 2) 12 10

+ 63729 x

+ 16x + 18 3 10

+ 64047 (x − 10)(x − 2) 12 10

+ 64749 x

+ 10x + 11 9 10

+ 68139 (x − 10)(x − 5) 15 10

+ 68367 (x − 18)(x − 7) 6

10

+ 70897 x

+ 6x + 7 13

10

+ 72237 x

+ 6x + 18 13

10

+ 77611 x

+ 13x + 7 6

10

+ 78199 (x − 7)

14

10

+ 79237 (x − 14)(x − 10) 5

10

+ 79767 (x − 12)(x − 1) 13

10

+ 82767 (x − 16)(x − 13) 10

10

+ 93559 x

+ 2x + 18 17

10

+ 95107 x

+ 18x + 12 1

10

+ 100003 (x − 14)(x − 6) 1

f

= E

∆ =

+∞

X

n=1

τ

(n)q

= q − 528q

− 4284q

+ O(q

)

p Similarity class of

L/Q p

τ

(p) mod 19

10

+ 453 (x − 7)(x − 5) 12

10

+ 1357 (x − 14)(x − 2) 16 10

+ 2713 (x − 13)(x − 12) 6 10

+ 4351 (x − 15)(x − 10) 6 10

+ 5733 (x − 12)(x − 2) 14

10

+ 7383 x

+ 8x + 1 11

10

+ 10401 (x − 17)(x − 5) 3 10

+ 11979 (x − 15)(x − 5) 1

10

+ 17557 x

+ x + 2 18

10

+ 21567 x

+ 9x + 7 10

10

+ 22273 x

+ 9x + 15 10 10

+ 24493 (x − 13)(x − 2) 15 10

+ 25947 (x − 18)(x − 9) 8

10

+ 27057 x

+ x + 2 18

10

+ 29737 x

+ 13x + 7 6

10

+ 41599 x

+ 9 0

10

+ 43789 (x − 16)(x − 2) 18 10

+ 46227 x

+ 17x + 17 2

10

+ 46339 x

+ x + 11 18

10

+ 52423 (x − 7)(x − 1) 8 10

+ 55831 (x − 18)(x − 1) 0 10

+ 57867 (x − 16)(x − 3) 0 10

+ 59743 (x − 6)(x − 1) 7 10

+ 61053 x

+ 18x + 14 1

10

+ 61353 x

+ 17x + 7 2

10

+ 63729 x

+ 15x + 8 4

10

+ 64047 (x − 7)

14

10

+ 64749 (x − 7)(x − 5) 12 10

+ 68139 (x − 5)(x − 3) 8 10

+ 68367 (x − 9)(x − 8) 17

10

+ 70897 (x − 16)

13

10

+ 72237 x

+ 14x + 12 5

10

+ 77611 x

+ 9x + 4 10

10

+ 78199 (x − 18)(x − 14) 13 10

+ 79237 (x − 15)(x − 8) 4 10

+ 79767 (x − 18)(x − 4) 3 10

+ 82767 (x − 16)(x − 10) 7

10

+ 93559 x

+ 4x + 8 15

10

+ 95107 x

+ 10x + 10 9

10

+ 100003 (x − 15)(x − 6) 2

` = 23

f

= E

∆ =

+∞

X

n=1

τ

(n)q

= q + 216q

− 3348q

+ O(q

)

p Similarity class of

Q p

τ

(p) mod 23 10

+ 453 (x − 15)(x − 5) 20 10

+ 1357 (x − 19)(x − 15) 11 10

+ 2713 x

+ 11x + 21 12

10

+ 4351 x

+ 7x + 11 16

10

+ 5733 (x − 18)(x − 14) 9 10

+ 7383 (x − 13)(x − 6) 19

10

+ 10401 x

+ 4x + 7 19

10

+ 11979 (x − 15)(x − 7) 22

10

+ 17557 x

+ 8x + 1 15

10

+ 21567 x

+ 8x + 6 15

10

+ 22273 (x − 17)(x − 5) 22 10

+ 24493 (x − 8)(x − 5) 13 10

+ 25947 (x − 21)(x − 13) 11 10

+ 27057 (x − 8)(x − 2) 10 10

+ 29737 x

+ 12x + 17 11 10

+ 41599 (x − 20)(x − 7) 4

10

+ 43789 (x − 15)

7

10

+ 46227 (x − 9)(x − 2) 11 10

+ 46339 (x − 22)(x − 18) 17 10

+ 52423 (x − 19)(x − 6) 2 10

+ 55831 x

+ 4x + 12 19 10

+ 57867 x

+ 16x + 21 7 10

+ 59743 (x − 7)(x − 6) 13

10

+ 61053 x

+ 21x + 3 2

10

+ 61353 (x − 11)(x − 8) 19 10

+ 63729 x

+ 5x + 13 18 10

+ 64047 (x − 22)(x − 21) 20 10

+ 64749 x

+ 16x + 11 7 10

+ 68139 (x − 18)(x − 3) 21

10

+ 68367 x

+ 2x + 3 21

10

+ 70897 x

+ 21x + 3 2

10

+ 72237 x

+ 14x + 5 9

10

+ 77611 x

+ 14x + 16 9 10

+ 78199 x

+ 6x + 21 17

10

+ 79237 x

+ 9x + 4 14

10

+ 79767 x

+ 15x + 20 8 10

+ 82767 (x − 8)(x − 1) 9

10

+ 93559 x

+ x + 10 22

10

+ 95107 (x − 15)(x − 11) 3

10

+ 100003 (x − 14)(x − 13) 4

f

= E

∆ =

+∞

X

n=1

τ

(n)q

= q − 528q

− 4284q

+ O(q

)

p Similarity class of

L/Q p

τ

(p) mod 23 10

+ 453 (x − 18)(x − 4) 22

10

+ 1357 x

+ 10x + 4 13

10

+ 2713 x

+ 13x + 10 10 10

+ 4351 (x − 6)(x − 5) 11

10

+ 5733 x

+ x + 22 22

10

+ 7383 (x − 20)(x − 14) 11 10

+ 10401 (x − 7)(x − 4) 11 10

+ 11979 (x − 11)(x − 5) 16

10

+ 17557 x

+ 7x + 1 16

10

+ 21567 (x − 15)(x − 14) 6 10

+ 22273 x

+ 22x + 18 1 10

+ 24493 (x − 15)(x − 9) 1 10

+ 25947 (x − 7)(x − 3) 10 10

+ 27057 x

+ 5x + 18 18 10

+ 29737 x

+ 5x + 20 18 10

+ 41599 (x − 13)(x − 1) 14

10

+ 43789 x

+ 8x + 6 15

10

+ 46227 x

+ 4x + 6 19

10

+ 46339 (x − 18)(x − 15) 10 10

+ 52423 x

+ 15x + 22 8 10

+ 55831 (x − 17)(x − 5) 22 10

+ 57867 x

+ 22x + 10 1 10

+ 59743 x

+ 13x + 15 10 10

+ 61053 (x − 20)(x − 7) 4 10

+ 61353 (x − 15)(x − 1) 16

10

+ 63729 x

+ 9 0

10

+ 64047 (x − 17)

11

10

+ 64749 x

+ 22x + 7 1

10

+ 68139 (x − 9)

18

10

+ 68367 x

+ 8x + 2 15

10

+ 70897 (x − 2)(x − 1) 3 10

+ 72237 (x − 17)(x − 1) 18 10

+ 77611 (x − 19)(x − 7) 3 10

+ 78199 (x − 17)(x − 6) 0

10

+ 79237 x

+ 4x + 8 19

10

+ 79767 x

+ 11x + 21 12

10

+ 82767 x

+ x + 12 22

10

+ 93559 (x − 10)(x − 6) 16

10

+ 95107 x

+ 13x + 8 10

10

+ 100003 (x − 14)(x − 4) 18

f

= E

∆ =

+∞

X

n=1

τ

(n)q

= q + 456q

+ 50652q

+ O(q

)

p Similarity class of

L/Q p

τ

(p) mod 23

10

+ 453 x

+ 5x + 13 18

10

+ 1357 x

+ 22x + 12 1

10

+ 2713 x

+ 11x + 19 12 10

+ 4351 (x − 22)(x − 6) 5

10

+ 5733 x

+ 22x + 22 1

10

+ 7383 (x − 15)(x − 10) 2 10

+ 10401 x

+ 13x + 20 10 10

+ 11979 x

+ 10x + 8 13 10

+ 17557 (x − 16)(x − 13) 6 10

+ 21567 (x − 18)(x − 2) 20 10

+ 22273 (x − 22)(x − 20) 19 10

+ 24493 (x − 11)(x − 3) 14 10

+ 25947 x

+ 18x + 14 5

10

+ 27057 x

+ 16x + 3 7

10

+ 29737 x

+ 22x + 10 1 10

+ 41599 (x − 22)(x − 19) 18

10

+ 43789 x

+ 2 0

10

+ 46227 (x − 14)(x − 10) 1 10

+ 46339 (x − 18)(x − 5) 0 10

+ 52423 (x − 21)(x − 12) 10 10

+ 55831 (x − 21)(x − 20) 18 10

+ 57867 (x − 22)(x − 4) 3 10

+ 59743 (x − 11)(x − 9) 20

10

+ 61053 x

+ 9x + 9 14

10

+ 61353 (x − 22)(x − 16) 15

10

+ 63729 x

+ 19x + 8 4

10

+ 64047 (x − 18)(x − 13) 8 10

+ 64749 (x − 21)(x − 3) 1 10

+ 68139 (x − 18)(x − 1) 19

10

+ 68367 x

+ x + 9 22

10

+ 70897 x

+ 21x + 9 2

10

+ 72237 (x − 11)(x − 4) 15 10

+ 77611 (x − 15)(x − 14) 6 10

+ 78199 (x − 17)(x − 16) 10 10

+ 79237 x

+ 5x + 16 18 10

+ 79767 x

+ 18x + 14 5 10

+ 82767 (x − 11)(x − 10) 21 10

+ 93559 x

+ 6x + 15 17

10

+ 95107 (x − 19)

15

10

+ 100003 x

+ 15x + 19 8

f

= E

∆ =

+∞

X

n=1

τ

(n)q

= q − 288q

− 128844q

+ O(q

)

p Similarity class of

L/Q p

τ

(p) mod 23 10

+ 453 (x − 19)(x − 7) 3

10

+ 1357 x

+ 13 0

10

+ 2713 x

+ 8x + 20 15

10

+ 4351 (x − 16)(x − 11) 4

10

+ 5733 x

+ 19x + 22 4

10

+ 7383 (x − 19)(x − 14) 10 10

+ 10401 (x − 16)(x − 5) 21 10

+ 11979 (x − 17)(x − 15) 9 10

+ 17557 (x − 19)(x − 17) 13 10

+ 21567 (x − 19)(x − 7) 3 10

+ 22273 x

+ 14x + 12 9 10

+ 24493 (x − 7)(x − 4) 11 10

+ 25947 x

+ 4x + 17 19 10

+ 27057 x

+ 3x + 12 20

10

+ 29737 x

+ 5x + 5 18

10

+ 41599 (x − 7)

14

10

+ 43789 x

+ 18x + 16 5 10

+ 46227 x

+ 19x + 16 4

10

+ 46339 x

+ 22x + 7 1

10

+ 52423 (x − 22)(x − 1) 0 10

+ 55831 x

+ 12x + 8 11 10

+ 57867 (x − 17)(x − 12) 6 10

+ 59743 (x − 21)(x − 16) 14

10

+ 61053 x

+ 4x + 6 19

10

+ 61353 (x − 19)(x − 8) 4

10

+ 63729 (x − 5)

10

10

+ 64047 (x − 12)(x − 6) 18 10

+ 64749 (x − 13)(x − 10) 0

10

+ 68139 (x − 21)

19

10

+ 68367 (x − 19)(x − 10) 6

10

+ 70897 x

+ 14x + 6 9

10

+ 72237 (x − 20)(x − 13) 10 10

+ 77611 (x − 4)(x − 3) 7 10

+ 78199 (x − 14)(x − 8) 22

10

+ 79237 x

+ 20x + 9 3

10

+ 79767 x

+ 8x + 17 15

10

+ 82767 x

+ 16x + 4 7

10

+ 93559 (x − 14)(x − 13) 4

10

+ 95107 (x − 3)

6

10

+ 100003 (x − 19)(x − 18) 14