LMFDB - Newform orbit 3311.1.ex.b

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3311,1,Mod(118,3311)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3311, base_ring=CyclotomicField(70))

chi = DirichletCharacter(H, H._module([35, 21, 15]))

N = Newforms(chi, 1, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("3311.118");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 3311 = 7 \cdot 11 \cdot 43 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	3311.ex (of order $70$, degree $24$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$1.65240425683$
Analytic rank:	$0$
Dimension:	$24$
Coefficient field:	$\Q(\zeta_{35})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{24} - x^{23} + x^{19} - x^{18} + x^{17} - x^{16} + x^{14} - x^{13} + x^{12} - x^{11} + x^{10} + \cdots + 1 $ x^24 - x^23 + x^19 - x^18 + x^17 - x^16 + x^14 - x^13 + x^12 - x^11 + x^10 - x^8 + x^7 - x^6 + x^5 - x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{4}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Projective image:	$D_{70}$
Projective field:	Galois closure of $\mathbb{Q}[x]/(x^{70} - \cdots)$

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3311\mathbb{Z}\right)^\times$.

$n$	$904$	$1893$	$2927$
$\chi(n)$	$-\zeta_{70}^{14}$	$-1$	$-\zeta_{70}^{30}$

Embeddings

For each embedding $\iota_m$ of the coefficient field, the values $\iota_m(a_n)$ are shown below.

For more information on an embedded modular form you can click on its label.

comment: embeddings in the coefficient field

gp: mfembed(f)

Label

$\iota_m(\nu)$

$ a_{2} $

$ a_{3} $

$ a_{4} $

$ a_{5} $

$ a_{6} $

$ a_{7} $

$ a_{8} $

$ a_{9} $

$ a_{10} $

118.1

0.936235	+	0.351375i
0.983930	+	0.178557i
−0.550897	+	0.834573i
0.983930	−	0.178557i
−0.995974	−	0.0896393i
−0.393025	−	0.919528i
0.473869	+	0.880596i
0.753071	−	0.657939i
0.134233	+	0.990950i
−0.0448648	+	0.998993i
0.858449	−	0.512899i
−0.963963	+	0.266037i
−0.963963	−	0.266037i
−0.691063	+	0.722795i
0.858449	+	0.512899i
−0.0448648	−	0.998993i
−0.691063	−	0.722795i
−0.550897	−	0.834573i
0.134233	−	0.990950i
0.753071	+	0.657939i

1.33232

0.367696i

0.781420

0.466877i

0.309017

−

0.951057i

−0.0857040

−

0.0896393i

−0.753071

−

0.657939i

237.1

−0.105514

0.778936i

0.368355

0.101659i

−0.809017

0.587785i

−0.426990

0.998993i

−0.936235

−

0.351375i

426.1

−0.0120447

0.268196i

0.924190

0.0831786i

0.309017

−

0.951057i

−0.0694769

0.512899i

0.393025

0.919528i

475.1

−0.105514

−

0.778936i

0.368355

−

0.101659i

−0.809017

−

0.587785i

−0.426990

−

0.998993i

−0.936235

0.351375i

524.1

−0.829730

0.724913i

0.0287192

−

0.212014i

0.309017

0.951057i

−0.477113

−

0.722795i

−0.983930

−

0.178557i

629.1

1.92016

0.172818i

2.67324

0.485121i

−0.809017

−

0.587785i

3.19077

0.880596i

0.691063

−

0.722795i

776.1

−1.04084

−

1.08863i

−0.0569095

1.26719i

−0.809017

0.587785i

0.304504

−

0.266037i

0.550897

−

0.834573i

930.1

−0.0770283

0.0460223i

−0.470053

0.873505i

−0.809017

0.587785i

−0.00801900

−

0.178557i

−0.134233

0.990950i

1140.1

−0.943922

−

1.75410i

−1.63498

2.47690i

0.309017

0.951057i

3.90409

0.351375i

0.963963

−

0.266037i

1322.1

0.887305

−

0.333011i

−0.0766581

0.0669742i

−0.809017

−

0.587785i

−0.494819

0.919528i

0.995974

0.0896393i

1427.1

−0.735927

−

1.72179i

−1.73190

1.81143i

0.309017

−

0.951057i

2.64038

0.990950i

−0.473869

0.880596i

1623.1

−1.08409

1.64232i

−1.12895

−

2.64132i

−0.809017

0.587785i

3.62554

0.657939i

−0.858449

0.512899i

1630.1

−1.08409

−

1.64232i

−1.12895

2.64132i

−0.809017

−

0.587785i

3.62554

−

0.657939i

−0.858449

−

0.512899i

1679.1

1.68931

0.306564i

1.82354

0.684386i

0.309017

0.951057i

1.39684

0.834573i

0.0448648

0.998993i

1833.1

−0.735927

1.72179i

−1.73190

−

1.81143i

0.309017

0.951057i

2.64038

−

0.990950i

−0.473869

−

0.880596i

1931.1

0.887305

0.333011i

−0.0766581

−

0.0669742i

−0.809017

0.587785i

−0.494819

−

0.919528i

0.995974

−

0.0896393i

2043.1

1.68931

−

0.306564i

1.82354

−

0.684386i

0.309017

−

0.951057i

1.39684

−

0.834573i

0.0448648

−

0.998993i

2526.1

−0.0120447

−

0.268196i

0.924190

−

0.0831786i

0.309017

0.951057i

−0.0694769

−

0.512899i

0.393025

−

0.919528i

2582.1

−0.943922

1.75410i

−1.63498

−

2.47690i

0.309017

−

0.951057i

3.90409

−

0.351375i

0.963963

0.266037i

2631.1

−0.0770283

−

0.0460223i

−0.470053

−

0.873505i

−0.809017

−

0.587785i

−0.00801900

0.178557i

−0.134233

−

0.990950i

See all 24 embeddings

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
7.b	odd	2	1	CM by $\Q(\sqrt{-7}) $
473.bb	even	70	1	inner
3311.ex	odd	70	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	3311.1.ex.b	yes	24
7.b	odd	2	1	CM	3311.1.ex.b	yes	24
11.d	odd	10	1		3311.1.ex.a	✓	24
43.f	odd	14	1		3311.1.ex.a	✓	24
77.l	even	10	1		3311.1.ex.a	✓	24
301.w	even	14	1		3311.1.ex.a	✓	24
473.bb	even	70	1	inner	3311.1.ex.b	yes	24
3311.ex	odd	70	1	inner	3311.1.ex.b	yes	24

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
3311.1.ex.a	✓	24	11.d	odd	10	1
3311.1.ex.a	✓	24	43.f	odd	14	1
3311.1.ex.a	✓	24	77.l	even	10	1
3311.1.ex.a	✓	24	301.w	even	14	1
3311.1.ex.b	yes	24	1.a	even	1	1	trivial
3311.1.ex.b	yes	24	7.b	odd	2	1	CM
3311.1.ex.b	yes	24	473.bb	even	70	1	inner
3311.1.ex.b	yes	24	3311.ex	odd	70	1	inner

This newform subspace can be constructed as the kernel of the linear operator $ T_{2}^{24} - 2 T_{2}^{23} - 23 T_{2}^{21} + 51 T_{2}^{20} - T_{2}^{19} + 189 T_{2}^{18} - 502 T_{2}^{17} + \cdots + 1 $ T2^24 - 2*T2^23 - 23*T2^21 + 51*T2^20 - T2^19 + 189*T2^18 - 502*T2^17 + 53*T2^16 - 624*T2^15 + 2360*T2^14 - 705*T2^13 + 593*T2^12 - 5172*T2^11 + 3272*T2^10 - 209*T2^9 + 4066*T2^8 - 3715*T2^7 + 1457*T2^6 - 2514*T2^5 + 1386*T2^4 + 114*T2^3 + 117*T2^2 + 18*T2 + 1 acting on $S_{1}^{\mathrm{new}}(3311, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{24} - 2 T^{23} + \cdots + 1 $ T^24 - 2T^23 - 23T^21 + 51T^20 - T^19 + 189T^18 - 502T^17 + 53T^16 - 624T^15 + 2360T^14 - 705T^13 + 593T^12 - 5172T^11 + 3272T^10 - 209T^9 + 4066T^8 - 3715T^7 + 1457T^6 - 2514T^5 + 1386T^4 + 114T^3 + 117T^2 + 18*T + 1
$3$	$ T^{24} $ T^24
$5$	$ T^{24} $ T^24
$7$	$ (T^{4} + T^{3} + T^{2} + \cdots + 1)^{6} $ (T^4 + T^3 + T^2 + T + 1)^6
$11$	$ T^{24} + T^{23} + \cdots + 1 $ T^24 + T^23 - T^19 - T^18 - T^17 - T^16 + T^14 + T^13 + T^12 + T^11 + T^10 - T^8 - T^7 - T^6 - T^5 + T + 1
$13$	$ T^{24} $ T^24
$17$	$ T^{24} $ T^24
$19$	$ T^{24} $ T^24
$23$	$ T^{24} + 2 T^{23} + \cdots + 1 $ T^24 + 2T^23 + 5T^22 + 10T^21 + 27T^20 + 38T^19 + 85T^18 + 142T^17 + 315T^16 + 368T^15 + 976T^14 + 1849T^13 + 4071T^12 + 5585T^11 + 7136T^10 + 6387T^9 + 6041T^8 + 4737T^7 + 3326T^6 + 1418T^5 + 1980T^4 + 165T^3 + 117T^2 + 19*T + 1
$29$	$ T^{24} + 2 T^{23} + \cdots + 1 $ T^24 + 2T^23 - 5T^21 + 2T^20 + 8T^19 - 23T^17 + 25T^16 - 62T^15 + T^14 + 124T^13 + 481T^12 - 1555T^11 + 486T^10 + 1357T^9 + 4346T^8 - 16753T^7 + 21491T^6 - 14202T^5 + 5320T^4 - 1220T^3 + 187T^2 - 11T + 1
$31$	$ T^{24} $ T^24
$37$	$ T^{24} - 5 T^{23} + \cdots + 1 $ T^24 - 5T^23 + 8T^22 - 55T^19 + 81T^18 + 150T^17 - 373T^16 - 205T^15 + 1184T^14 - 715T^13 - 528T^12 - 1025T^11 + 3756T^10 - 1880T^9 - 1928T^8 + 1885T^7 + 33T^6 - 625T^5 + 333T^4 - 15T^3 - 12T^2 + 5*T + 1
$41$	$ T^{24} $ T^24
$43$	$ T^{24} + T^{23} + \cdots + 1 $ T^24 + T^23 - T^19 - T^18 - T^17 - T^16 + T^14 + T^13 + T^12 + T^11 + T^10 - T^8 - T^7 - T^6 - T^5 + T + 1
$47$	$ T^{24} $ T^24
$53$	$ T^{24} + 10 T^{23} + \cdots + 1 $ T^24 + 10T^23 + 54T^22 + 208T^21 + 637T^20 + 1632T^19 + 3585T^18 + 6809T^17 + 11108T^16 + 15126T^15 + 15914T^14 + 10030T^13 - 3306T^12 - 18972T^11 - 26900T^10 - 20103T^9 - 2236T^8 + 13788T^7 + 17332T^6 + 10543T^5 + 3306T^4 + 331T^3 - 35T^2 - 12*T + 1
$59$	$ T^{24} $ T^24
$61$	$ T^{24} $ T^24
$67$	$ T^{24} - 2 T^{23} + \cdots + 1 $ T^24 - 2T^23 + 5T^22 - 3T^21 + 6T^20 + 4T^19 + 29T^18 - 37T^17 + 203T^16 - 144T^15 + 465T^14 + 300T^13 + 788T^12 - 832T^11 + 1557T^10 - 759T^9 + 1456T^8 - 3435T^7 + 6182T^6 - 6269T^5 + 3751T^4 - 1061T^3 + 187T^2 - 12*T + 1
$71$	$ T^{24} + 5 T^{23} + \cdots + 15625 $ T^24 + 5T^23 + 15T^22 + 35T^21 + 70T^20 + 125T^19 + 200T^18 + 375T^17 + 775T^16 + 1500T^15 + 2500T^14 + 3375T^13 + 3000T^12 - 1250T^11 - 7500T^10 - 11875T^9 - 11875T^8 - 9375T^7 - 9375T^6 - 12500T^5 - 3125T^4 + 15625T^3 + 31250T^2 + 31250*T + 15625
$73$	$ T^{24} $ T^24
$79$	$ T^{24} + 5 T^{23} + \cdots + 1 $ T^24 + 5T^23 + 8T^22 + 55T^19 + 81T^18 - 150T^17 - 373T^16 + 205T^15 + 1184T^14 + 715T^13 - 528T^12 + 1025T^11 + 3756T^10 + 1880T^9 - 1928T^8 - 1885T^7 + 33T^6 + 625T^5 + 333T^4 + 15T^3 - 12T^2 - 5*T + 1
$83$	$ T^{24} $ T^24
$89$	$ T^{24} $ T^24
$97$	$ T^{24} $ T^24
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Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 3311 = 7 \cdot 11 \cdot 43 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	3311.ex (of order \(70\), degree \(24\), minimal)

\(f(q)\)	\(=\)	\( q + (\zeta_{70}^{16} - \zeta_{70}^{3}) q^{2} + (\zeta_{70}^{32} + \cdots + \zeta_{70}^{6}) q^{4}+ \cdots - \zeta_{70}^{2} q^{9} +O(q^{10}) \) q + (z^16 - z^3) * q^2 + (z^32 - z^19 + z^6) * q^4 + z^14 * q^7 + (z^22 - z^13 - z^9 - 1) * q^8 - z^2 * q^9	\( q + (\zeta_{70}^{16} - \zeta_{70}^{3}) q^{2} + (\zeta_{70}^{32} + \cdots + \zeta_{70}^{6}) q^{4}+ \cdots - \zeta_{70}^{5} q^{99} +O(q^{100}) \) q + (z^16 - z^3) * q^2 + (z^32 - z^19 + z^6) * q^4 + z^14 * q^7 + (z^22 - z^13 - z^9 - 1) * q^8 - z^2 * q^9 + z^3 * q^11 + (z^30 - z^17) * q^14 + (-z^29 - z^25 - z^16 + z^12 + z^3) * q^16 + (-z^18 + z^5) * q^18 + (z^19 - z^6) * q^22 + (-z^34 - z^26) * q^23 + z * q^25 + (-z^33 + z^20 - z^11) * q^28 + (-z^22 + z^11) * q^29 + (z^32 + z^28 - z^19 - z^15 + z^10 - z^6) * q^32 + (-z^34 + z^21 - z^8) * q^36 + (z^18 - z^10) * q^37 + z^33 * q^43 + (-z^22 + z^9 - 1) * q^44 + (z^29 + z^15 + z^7 - z^2) * q^46 + z^28 * q^49 + (z^17 - z^4) * q^50 + (z^20 + z^14) * q^53 + (-z^27 - z^23 + z^14 - z) * q^56 + (z^27 + z^25 - z^14 + z^3) * q^58 - z^16 * q^63 + (-z^31 + z^26 + z^22 + z^18 + z^13 + z^9 - 1) * q^64 + (-z^27 + z^18) * q^67 + (-z^26 - z^5) * q^71 + (-z^24 + z^15 + z^11 - z^2) * q^72 + (z^34 - z^26 - z^21 + z^13) * q^74 + z^17 * q^77 + (z^9 - z^5) * q^79 + z^4 * q^81 + (-z^14 + z) * q^86 + (z^25 - z^16 - z^12 + z^3) * q^88 + (-z^32 + z^31 + z^23 - z^18 - z^10 + z^5) * q^92 + (-z^31 - z^9) * q^98 - z^5 * q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q + 2 q^{2} + 3 q^{4} - 6 q^{7} + 27 q^{8} - q^{9}+O(q^{10}) \) 24 * q + 2 * q^2 + 3 * q^4 - 6 * q^7 + 27 * q^8 - q^9	\( 24 q + 2 q^{2} + 3 q^{4} - 6 q^{7} + 27 q^{8} - q^{9} - q^{11} - 3 q^{14} + 3 q^{18} - 2 q^{22} - 2 q^{23} - q^{25} - 2 q^{28} - 2 q^{29} - 11 q^{32} + 4 q^{36} + 5 q^{37} - q^{43} - 26 q^{44} + 8 q^{46} - 6 q^{49} - 2 q^{50} - 10 q^{53} - 3 q^{56} + 8 q^{58} - q^{63} + 30 q^{64} + 2 q^{67} - 5 q^{71} + q^{72} - 7 q^{74} - q^{77} - 5 q^{79} + q^{81} + 5 q^{86} + q^{88} + 4 q^{92} + 2 q^{98} - 4 q^{99}+O(q^{100}) \) 24 * q + 2 * q^2 + 3 * q^4 - 6 * q^7 + 27 * q^8 - q^9 - q^11 - 3 * q^14 + 3 * q^18 - 2 * q^22 - 2 * q^23 - q^25 - 2 * q^28 - 2 * q^29 - 11 * q^32 + 4 * q^36 + 5 * q^37 - q^43 - 26 * q^44 + 8 * q^46 - 6 * q^49 - 2 * q^50 - 10 * q^53 - 3 * q^56 + 8 * q^58 - q^63 + 30 * q^64 + 2 * q^67 - 5 * q^71 + q^72 - 7 * q^74 - q^77 - 5 * q^79 + q^81 + 5 * q^86 + q^88 + 4 * q^92 + 2 * q^98 - 4 * q^99

Introduction

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Newspace parameters

Newform invariants

$q$-expansion

Character values

Embeddings

Inner twists

Twists

Hecke kernels

Hecke characteristic polynomials

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	3311.1.ex.b

Level	$3311$
Weight	$1$
Character orbit	3311.ex
Analytic conductor	$1.652$
Analytic rank	$0$
Dimension	$24$
Projective image	$D_{70}$
CM discriminant	-7
Inner twists	$4$

Self dual:	no
Analytic conductor:	\(1.65240425683\)
Analytic rank:	\(0\)
Dimension:	\(24\)
Coefficient field:	\(\Q(\zeta_{35})\)
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	\( x^{24} - x^{23} + x^{19} - x^{18} + x^{17} - x^{16} + x^{14} - x^{13} + x^{12} - x^{11} + x^{10} + \cdots + 1 \) x^24 - x^23 + x^19 - x^18 + x^17 - x^16 + x^14 - x^13 + x^12 - x^11 + x^10 - x^8 + x^7 - x^6 + x^5 - x + 1
Coefficient ring:	\(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	yes
Projective image:	\(D_{70}\)
Projective field:	Galois closure of \(\mathbb{Q}[x]/(x^{70} - \cdots)\)

\(n\)	\(904\)	\(1893\)	\(2927\)
\(\chi(n)\)	\(-\zeta_{70}^{14}\)	\(-1\)	\(-\zeta_{70}^{30}\)

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 118.1
Significant digits:
Format:

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
7.b	odd	2	1	CM by \(\Q(\sqrt{-7}) \)
473.bb	even	70	1	inner
3311.ex	odd	70	1	inner

$p$	$F_p(T)$
$2$	\( T^{24} - 2 T^{23} + \cdots + 1 \) T^24 - 2T^23 - 23T^21 + 51T^20 - T^19 + 189T^18 - 502T^17 + 53T^16 - 624T^15 + 2360T^14 - 705T^13 + 593T^12 - 5172T^11 + 3272T^10 - 209T^9 + 4066T^8 - 3715T^7 + 1457T^6 - 2514T^5 + 1386T^4 + 114T^3 + 117T^2 + 18*T + 1
$3$	\( T^{24} \) T^24
$5$	\( T^{24} \) T^24
$7$	\( (T^{4} + T^{3} + T^{2} + \cdots + 1)^{6} \) (T^4 + T^3 + T^2 + T + 1)^6
$11$	\( T^{24} + T^{23} + \cdots + 1 \) T^24 + T^23 - T^19 - T^18 - T^17 - T^16 + T^14 + T^13 + T^12 + T^11 + T^10 - T^8 - T^7 - T^6 - T^5 + T + 1
$13$	\( T^{24} \) T^24
$17$	\( T^{24} \) T^24
$19$	\( T^{24} \) T^24
$23$	\( T^{24} + 2 T^{23} + \cdots + 1 \) T^24 + 2T^23 + 5T^22 + 10T^21 + 27T^20 + 38T^19 + 85T^18 + 142T^17 + 315T^16 + 368T^15 + 976T^14 + 1849T^13 + 4071T^12 + 5585T^11 + 7136T^10 + 6387T^9 + 6041T^8 + 4737T^7 + 3326T^6 + 1418T^5 + 1980T^4 + 165T^3 + 117T^2 + 19*T + 1
$29$	\( T^{24} + 2 T^{23} + \cdots + 1 \) T^24 + 2T^23 - 5T^21 + 2T^20 + 8T^19 - 23T^17 + 25T^16 - 62T^15 + T^14 + 124T^13 + 481T^12 - 1555T^11 + 486T^10 + 1357T^9 + 4346T^8 - 16753T^7 + 21491T^6 - 14202T^5 + 5320T^4 - 1220T^3 + 187T^2 - 11T + 1
$31$	\( T^{24} \) T^24
$37$	\( T^{24} - 5 T^{23} + \cdots + 1 \) T^24 - 5T^23 + 8T^22 - 55T^19 + 81T^18 + 150T^17 - 373T^16 - 205T^15 + 1184T^14 - 715T^13 - 528T^12 - 1025T^11 + 3756T^10 - 1880T^9 - 1928T^8 + 1885T^7 + 33T^6 - 625T^5 + 333T^4 - 15T^3 - 12T^2 + 5*T + 1
$41$	\( T^{24} \) T^24
$43$	\( T^{24} + T^{23} + \cdots + 1 \) T^24 + T^23 - T^19 - T^18 - T^17 - T^16 + T^14 + T^13 + T^12 + T^11 + T^10 - T^8 - T^7 - T^6 - T^5 + T + 1
$47$	\( T^{24} \) T^24
$53$	\( T^{24} + 10 T^{23} + \cdots + 1 \) T^24 + 10T^23 + 54T^22 + 208T^21 + 637T^20 + 1632T^19 + 3585T^18 + 6809T^17 + 11108T^16 + 15126T^15 + 15914T^14 + 10030T^13 - 3306T^12 - 18972T^11 - 26900T^10 - 20103T^9 - 2236T^8 + 13788T^7 + 17332T^6 + 10543T^5 + 3306T^4 + 331T^3 - 35T^2 - 12*T + 1
$59$	\( T^{24} \) T^24
$61$	\( T^{24} \) T^24
$67$	\( T^{24} - 2 T^{23} + \cdots + 1 \) T^24 - 2T^23 + 5T^22 - 3T^21 + 6T^20 + 4T^19 + 29T^18 - 37T^17 + 203T^16 - 144T^15 + 465T^14 + 300T^13 + 788T^12 - 832T^11 + 1557T^10 - 759T^9 + 1456T^8 - 3435T^7 + 6182T^6 - 6269T^5 + 3751T^4 - 1061T^3 + 187T^2 - 12*T + 1
$71$	\( T^{24} + 5 T^{23} + \cdots + 15625 \) T^24 + 5T^23 + 15T^22 + 35T^21 + 70T^20 + 125T^19 + 200T^18 + 375T^17 + 775T^16 + 1500T^15 + 2500T^14 + 3375T^13 + 3000T^12 - 1250T^11 - 7500T^10 - 11875T^9 - 11875T^8 - 9375T^7 - 9375T^6 - 12500T^5 - 3125T^4 + 15625T^3 + 31250T^2 + 31250*T + 15625
$73$	\( T^{24} \) T^24
$79$	\( T^{24} + 5 T^{23} + \cdots + 1 \) T^24 + 5T^23 + 8T^22 + 55T^19 + 81T^18 - 150T^17 - 373T^16 + 205T^15 + 1184T^14 + 715T^13 - 528T^12 + 1025T^11 + 3756T^10 + 1880T^9 - 1928T^8 - 1885T^7 + 33T^6 + 625T^5 + 333T^4 + 15T^3 - 12T^2 - 5*T + 1
$83$	\( T^{24} \) T^24
$89$	\( T^{24} \) T^24
$97$	\( T^{24} \) T^24
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