LMFDB - Newform orbit 3871.1.ch.a

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [3871,1,Mod(31,3871)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("3871.31"); S:= CuspForms(chi, 1); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3871, base_ring=CyclotomicField(78)) chi = DirichletCharacter(H, H._module([13, 56])) B = ModularForms(chi, 1).cuspidal_submodule().basis() N = [B[i] for i in range(len(B))]

Level:	$ N $	$=$	$ 3871 = 7^{2} \cdot 79 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	3871.ch (of order $78$, degree $24$, not minimal)

Newform invariants

comment:select newform

sage:traces = [] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(0)] == traces)

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

Self dual:	no
Analytic conductor:	$1.93188066390$
Analytic rank:	$0$
Dimension:	$24$
Coefficient field:	$\Q(\zeta_{39})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{24} - x^{23} + x^{21} - x^{20} + x^{18} - x^{17} + x^{15} - x^{14} + x^{12} - x^{10} + x^{9} + \cdots + 1 $ x^24 - x^23 + x^21 - x^20 + x^18 - x^17 + x^15 - x^14 + x^12 - x^10 + x^9 - x^7 + x^6 - x^4 + x^3 - x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{4}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 553)
Projective image:	$D_{39}$
Projective field:	Galois closure of $\mathbb{Q}[x]/(x^{39} - \cdots)$

$q$-expansion

comment:q-expansion

sage:f.q_expansion() # note that sage often uses an isomorphic number field

gp:mfcoefs(f, 20)

The $q$-expansion and trace form are shown below.

$f(q)$	$=$	$ q + ( - \zeta_{78}^{33} - \zeta_{78}^{29}) q^{2} + ( - \zeta_{78}^{27} + \cdots - \zeta_{78}^{19}) q^{4} + ( - \zeta_{78}^{21} + \cdots - \zeta_{78}^{9}) q^{8} - \zeta_{78}^{9} q^{9} + ( - \zeta_{78} + 1) q^{11} + \cdots + (\zeta_{78}^{10} - \zeta_{78}^{9}) q^{99} +O(q^{100}) $ q + (-z^33 - z^29) * q^2 + (-z^27 - z^23 - z^19) * q^4 + (-z^21 - z^17 - z^13 - z^9) * q^8 - z^9 * q^9 + (-z + 1) * q^11 + (z^38 - z^15 - z^11 - z^7 - z^3) * q^16 + (z^38 - z^3) * q^18 + (z^34 - z^33 + z^30 - z^29) * q^22 + (z^34 - z^5) * q^23 + z^12 * q^25 + (-z^31 + z^16) * q^29 + (z^36 + z^32 + z^28 - z^9 - z^5 - z) * q^32 + (z^36 + z^32 + z^28) * q^36 + (-z^37 + z^4) * q^37 + (-z^23 + z^2) * q^43 + (z^28 - z^27 + z^24 - z^23 + z^20 - z^19) * q^44 + (z^38 + z^34 + z^28 + z^24) * q^46 + (z^6 + z^2) * q^50 + (z^26 - z^17) * q^53 + (-z^25 - z^21 + z^10 + z^6) * q^58 + (z^38 + z^34 + z^30 + z^26 + z^22 + z^18 - z^3) * q^64 + (-z^23 + z^6) * q^67 + (z^36 - z^33) * q^71 + (z^30 + z^26 + z^22 + z^18) * q^72 + (-z^37 - z^33 - z^31 - z^27) * q^74 - z^17 * q^79 + z^18 * q^81 + (-z^35 - z^31 - z^17 - z^13) * q^86 + (z^22 - z^21 + z^18 - z^17 + z^14 - z^13 + z^10 - z^9) * q^88 + (z^32 + z^28 + z^24 + z^22 + z^18 + z^14) * q^92 + (z^10 - z^9) * q^99
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q - q^{2} - 15 q^{8} - 2 q^{9} + 25 q^{11} - q^{16} - q^{18} - 2 q^{22} + 2 q^{23} - 2 q^{25} + 2 q^{29} + 2 q^{37} + 2 q^{43} + q^{46} - q^{50} - 11 q^{53} - 2 q^{58} - 15 q^{64} - q^{67} - 4 q^{71}+ \cdots - q^{99}+O(q^{100}) $ 24 * q - q^2 - 15 * q^8 - 2 * q^9 + 25 * q^11 - q^16 - q^18 - 2 * q^22 + 2 * q^23 - 2 * q^25 + 2 * q^29 + 2 * q^37 + 2 * q^43 + q^46 - q^50 - 11 * q^53 - 2 * q^58 - 15 * q^64 - q^67 - 4 * q^71 - 15 * q^72 - 2 * q^74 + q^79 - 2 * q^81 - 9 * q^86 - 14 * q^88 - q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3871\mathbb{Z}\right)^\times$.

$n$	$1030$	$2845$
$\chi(n)$	$\zeta_{78}^{22}$	$-\zeta_{78}^{26}$

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} $

31.1

−0.996757	+	0.0804666i
−0.0402659	+	0.999189i
0.278217	+	0.960518i
−0.845190	+	0.534466i
−0.200026	+	0.979791i
0.987050	−	0.160411i
0.428693	−	0.903450i
−0.996757	−	0.0804666i
0.948536	−	0.316668i
−0.919979	−	0.391967i
0.692724	+	0.721202i
−0.200026	−	0.979791i
0.799443	−	0.600742i
0.987050	+	0.160411i
−0.632445	+	0.774605i
0.799443	+	0.600742i
0.428693	+	0.903450i
0.948536	+	0.316668i
−0.919979	+	0.391967i
−0.845190	−	0.534466i

1.57818

1.18593i

0.806016

2.78269i

−1.32800

3.50165i

−0.748511

0.663123i

129.1

−1.89092

0.631282i

2.37762

−

1.78667i

−2.23556

3.23877i

−0.354605

0.935016i

313.1

1.06907

−

1.30938i

−0.371525

−

1.81985i

−1.28330

−

0.673526i

0.568065

−

0.822984i

325.1

−0.171499

−

0.840058i

0.243694

−

0.103828i

−0.616065

−

0.892525i

−0.354605

−

0.935016i

411.1

0.0740877

1.83847i

−2.37771

0.191949i

−0.307268

−

2.53058i

−0.970942

−

0.239316i

803.1

0.527799

1.82217i

−2.19655

1.38902i

−2.27038

−

2.01139i

0.120537

−

0.992709i

950.1

1.16367

−

0.495795i

0.415599

−

0.432684i

−0.179438

0.473138i

−0.748511

0.663123i

999.1

1.57818

−

1.18593i

0.806016

−

2.78269i

−1.32800

−

3.50165i

−0.748511

−

0.663123i

1354.1

−1.35136

0.854550i

0.667232

−

1.40616i

0.107239

0.883189i

−0.970942

−

0.239316i

1599.1

−1.38096

0.111482i

0.907561

−

0.147493i

0.108331

−

0.0267011i

0.885456

0.464723i

1783.1

−0.0794890

0.0129182i

−0.942385

0.314614i

0.142152

−

0.0746073i

0.568065

0.822984i

2138.1

0.0740877

−

1.83847i

−2.37771

−

0.191949i

−0.307268

2.53058i

−0.970942

0.239316i

2175.1

0.238540

−

0.502711i

0.436628

0.534772i

0.913255

−

0.225097i

0.885456

0.464723i

2285.1

0.527799

−

1.82217i

−2.19655

−

1.38902i

−2.27038

2.01139i

0.120537

0.992709i

2469.1

−0.277125

−

0.288518i

0.0338217

−

0.839278i

−0.550962

0.488110i

0.120537

0.992709i

2481.1

0.238540

0.502711i

0.436628

−

0.534772i

0.913255

0.225097i

0.885456

−

0.464723i

2726.1

1.16367

0.495795i

0.415599

0.432684i

−0.179438

−

0.473138i

−0.748511

−

0.663123i

2959.1

−1.35136

−

0.854550i

0.667232

1.40616i

0.107239

−

0.883189i

−0.970942

0.239316i

3106.1

−1.38096

−

0.111482i

0.907561

0.147493i

0.108331

0.0267011i

0.885456

−

0.464723i

3204.1

−0.171499

0.840058i

0.243694

0.103828i

−0.616065

0.892525i

−0.354605

0.935016i

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}) $
553.z	even	39	1	inner
553.bn	odd	78	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	3871.1.ch.a		24
7.b	odd	2	1	CM	3871.1.ch.a		24
7.c	even	3	1		553.1.bg.a	✓	24
7.c	even	3	1		3871.1.bw.a		24
7.d	odd	6	1		553.1.bg.a	✓	24
7.d	odd	6	1		3871.1.bw.a		24
79.g	even	39	1		3871.1.bw.a		24
553.y	even	39	1		553.1.bg.a	✓	24
553.z	even	39	1	inner	3871.1.ch.a		24
553.bc	odd	78	1		553.1.bg.a	✓	24
553.bg	odd	78	1		3871.1.bw.a		24
553.bn	odd	78	1	inner	3871.1.ch.a		24

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
553.1.bg.a	✓	24	7.c	even	3	1
553.1.bg.a	✓	24	7.d	odd	6	1
553.1.bg.a	✓	24	553.y	even	39	1
553.1.bg.a	✓	24	553.bc	odd	78	1
3871.1.bw.a		24	7.c	even	3	1
3871.1.bw.a		24	7.d	odd	6	1
3871.1.bw.a		24	79.g	even	39	1
3871.1.bw.a		24	553.bg	odd	78	1
3871.1.ch.a		24	1.a	even	1	1	trivial
3871.1.ch.a		24	7.b	odd	2	1	CM
3871.1.ch.a		24	553.z	even	39	1	inner
3871.1.ch.a		24	553.bn	odd	78	1	inner

Hecke kernels

This newform subspace is the entire newspace $S_{1}^{\mathrm{new}}(3871, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{24} + T^{23} + \cdots + 1 $ T^24 + T^23 + 12T^21 + 12T^20 + 105T^18 + 105T^17 + 13T^16 + 454T^15 + 441T^14 - 234T^13 + 1171T^12 + 1404T^11 - 1457T^10 + 181T^9 + 1781T^8 + 2107T^7 + 3875T^6 + 1378T^5 + 1052T^4 + 532T^3 + 221T^2 + 27T + 1
$3$	$ T^{24} $ T^24
$5$	$ T^{24} $ T^24
$7$	$ T^{24} $ T^24
$11$	$ T^{24} - 25 T^{23} + \cdots + 1 $ T^24 - 25T^23 + 299T^22 - 2276T^21 + 12375T^20 - 51129T^19 + 166726T^18 - 439945T^17 + 955604T^16 - 1729001T^15 + 2627130T^14 - 3370029T^13 + 3660541T^12 - 3370029T^11 + 2627117T^10 - 1728936T^9 + 955396T^8 - 439503T^7 + 166089T^6 - 50492T^5 + 11933T^4 - 2068T^3 + 234T^2 - 12*T + 1
$13$	$ T^{24} $ T^24
$17$	$ T^{24} $ T^24
$19$	$ T^{24} $ T^24
$23$	$ (T^{12} - T^{11} - 12 T^{10} + \cdots + 1)^{2} $ (T^12 - T^11 - 12T^10 + 12T^9 + 53T^8 - 53T^7 - 103T^6 + 103T^5 + 79T^4 - 79T^3 - 12T^2 + 12T + 1)^2
$29$	$ T^{24} - 2 T^{23} + \cdots + 1 $ T^24 - 2T^23 - 18T^21 + 36T^20 + 129T^18 - 258T^17 + 13T^16 - 476T^15 + 900T^14 - 39T^13 + 898T^12 - 1638T^11 + 22T^10 - 590T^9 + 1235T^8 + 397T^7 - 157T^6 - 494T^5 + 95T^4 + 70T^3 + 26T^2 + 6*T + 1
$31$	$ T^{24} $ T^24
$37$	$ T^{24} - 2 T^{23} + \cdots + 1 $ T^24 - 2T^23 - 18T^21 + 36T^20 + 129T^18 - 258T^17 + 13T^16 - 476T^15 + 900T^14 - 39T^13 + 898T^12 - 1638T^11 + 22T^10 - 590T^9 + 1235T^8 + 397T^7 - 157T^6 - 494T^5 + 95T^4 + 70T^3 + 26T^2 + 6*T + 1
$41$	$ T^{24} $ T^24
$43$	$ T^{24} - 2 T^{23} + \cdots + 1 $ T^24 - 2T^23 + 8T^21 - 16T^20 + 26T^19 - 14T^18 - 76T^17 + 208T^16 - 138T^15 + 55T^14 + 156T^13 - 337T^12 + 52T^11 + 906T^10 + 8T^9 - 91T^8 - 188T^7 + 519T^6 + 533T^5 + 407T^4 + 96T^3 + 13T^2 + 6T + 1
$47$	$ T^{24} $ T^24
$53$	$ T^{24} + 11 T^{23} + \cdots + 1 $ T^24 + 11T^23 + 65T^22 + 268T^21 + 855T^20 + 2223T^19 + 4861T^18 + 9128T^17 + 14924T^16 + 21442T^15 + 27264T^14 + 30810T^13 + 30967T^12 + 27768T^11 + 22265T^10 + 15582T^9 + 9334T^8 + 5376T^7 + 3249T^6 + 1456T^5 + 251T^4 - 34T^3 + 6T + 1
$59$	$ T^{24} $ T^24
$61$	$ T^{24} $ T^24
$67$	$ T^{24} + T^{23} + \cdots + 1 $ T^24 + T^23 + 12T^21 + 12T^20 + 105T^18 + 105T^17 + 13T^16 + 454T^15 + 441T^14 - 234T^13 + 1171T^12 + 1404T^11 - 1457T^10 + 181T^9 + 1781T^8 + 2107T^7 + 3875T^6 + 1378T^5 + 1052T^4 + 532T^3 + 221T^2 + 27T + 1
$71$	$ (T^{12} + 2 T^{11} + \cdots + 1)^{2} $ (T^12 + 2T^11 + 4T^10 + 8T^9 + 16T^8 + 32T^7 + 64T^6 + 115T^5 + 139T^4 + 96T^3 + 36T^2 + 7*T + 1)^2
$73$	$ T^{24} $ T^24
$79$	$ T^{24} - T^{23} + \cdots + 1 $ T^24 - T^23 + T^21 - T^20 + T^18 - T^17 + T^15 - T^14 + T^12 - T^10 + T^9 - T^7 + T^6 - T^4 + T^3 - T + 1
$83$	$ T^{24} $ T^24
$89$	$ T^{24} $ T^24
$97$	$ T^{24} $ T^24
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Overview	Random
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Classical	Maass
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Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 3871 = 7^{2} \cdot 79 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	3871.ch (of order \(78\), degree \(24\), not minimal)

\(f(q)\)	\(=\)	\( q + ( - \zeta_{78}^{33} - \zeta_{78}^{29}) q^{2} + ( - \zeta_{78}^{27} + \cdots - \zeta_{78}^{19}) q^{4} + ( - \zeta_{78}^{21} + \cdots - \zeta_{78}^{9}) q^{8} - \zeta_{78}^{9} q^{9} + ( - \zeta_{78} + 1) q^{11} + \cdots + (\zeta_{78}^{10} - \zeta_{78}^{9}) q^{99} +O(q^{100}) \) q + (-z^33 - z^29) * q^2 + (-z^27 - z^23 - z^19) * q^4 + (-z^21 - z^17 - z^13 - z^9) * q^8 - z^9 * q^9 + (-z + 1) * q^11 + (z^38 - z^15 - z^11 - z^7 - z^3) * q^16 + (z^38 - z^3) * q^18 + (z^34 - z^33 + z^30 - z^29) * q^22 + (z^34 - z^5) * q^23 + z^12 * q^25 + (-z^31 + z^16) * q^29 + (z^36 + z^32 + z^28 - z^9 - z^5 - z) * q^32 + (z^36 + z^32 + z^28) * q^36 + (-z^37 + z^4) * q^37 + (-z^23 + z^2) * q^43 + (z^28 - z^27 + z^24 - z^23 + z^20 - z^19) * q^44 + (z^38 + z^34 + z^28 + z^24) * q^46 + (z^6 + z^2) * q^50 + (z^26 - z^17) * q^53 + (-z^25 - z^21 + z^10 + z^6) * q^58 + (z^38 + z^34 + z^30 + z^26 + z^22 + z^18 - z^3) * q^64 + (-z^23 + z^6) * q^67 + (z^36 - z^33) * q^71 + (z^30 + z^26 + z^22 + z^18) * q^72 + (-z^37 - z^33 - z^31 - z^27) * q^74 - z^17 * q^79 + z^18 * q^81 + (-z^35 - z^31 - z^17 - z^13) * q^86 + (z^22 - z^21 + z^18 - z^17 + z^14 - z^13 + z^10 - z^9) * q^88 + (z^32 + z^28 + z^24 + z^22 + z^18 + z^14) * q^92 + (z^10 - z^9) * q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q - q^{2} - 15 q^{8} - 2 q^{9} + 25 q^{11} - q^{16} - q^{18} - 2 q^{22} + 2 q^{23} - 2 q^{25} + 2 q^{29} + 2 q^{37} + 2 q^{43} + q^{46} - q^{50} - 11 q^{53} - 2 q^{58} - 15 q^{64} - q^{67} - 4 q^{71}+ \cdots - q^{99}+O(q^{100}) \) 24 * q - q^2 - 15 * q^8 - 2 * q^9 + 25 * q^11 - q^16 - q^18 - 2 * q^22 + 2 * q^23 - 2 * q^25 + 2 * q^29 + 2 * q^37 + 2 * q^43 + q^46 - q^50 - 11 * q^53 - 2 * q^58 - 15 * q^64 - q^67 - 4 * q^71 - 15 * q^72 - 2 * q^74 + q^79 - 2 * q^81 - 9 * q^86 - 14 * q^88 - 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	3871.1.ch.a

Level	$3871$
Weight	$1$
Character orbit	3871.ch
Analytic conductor	$1.932$
Analytic rank	$0$
Dimension	$24$
Projective image	$D_{39}$
CM discriminant	-7
Inner twists	$4$

Self dual:	no
Analytic conductor:	\(1.93188066390\)
Analytic rank:	\(0\)
Dimension:	\(24\)
Coefficient field:	\(\Q(\zeta_{39})\)
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	\( x^{24} - x^{23} + x^{21} - x^{20} + x^{18} - x^{17} + x^{15} - x^{14} + x^{12} - x^{10} + x^{9} + \cdots + 1 \) x^24 - x^23 + x^21 - x^20 + x^18 - x^17 + x^15 - x^14 + x^12 - x^10 + x^9 - x^7 + x^6 - x^4 + x^3 - x + 1
Coefficient ring:	\(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 553)
Projective image:	\(D_{39}\)
Projective field:	Galois closure of \(\mathbb{Q}[x]/(x^{39} - \cdots)\)

\(n\)	\(1030\)	\(2845\)
\(\chi(n)\)	\(\zeta_{78}^{22}\)	\(-\zeta_{78}^{26}\)

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 31.1
Significant digits:
Format:

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
7.b	odd	2	1	CM by \(\Q(\sqrt{-7}) \)
553.z	even	39	1	inner
553.bn	odd	78	1	inner

$p$	$F_p(T)$
$2$	\( T^{24} + T^{23} + \cdots + 1 \) T^24 + T^23 + 12T^21 + 12T^20 + 105T^18 + 105T^17 + 13T^16 + 454T^15 + 441T^14 - 234T^13 + 1171T^12 + 1404T^11 - 1457T^10 + 181T^9 + 1781T^8 + 2107T^7 + 3875T^6 + 1378T^5 + 1052T^4 + 532T^3 + 221T^2 + 27T + 1
$3$	\( T^{24} \) T^24
$5$	\( T^{24} \) T^24
$7$	\( T^{24} \) T^24
$11$	\( T^{24} - 25 T^{23} + \cdots + 1 \) T^24 - 25T^23 + 299T^22 - 2276T^21 + 12375T^20 - 51129T^19 + 166726T^18 - 439945T^17 + 955604T^16 - 1729001T^15 + 2627130T^14 - 3370029T^13 + 3660541T^12 - 3370029T^11 + 2627117T^10 - 1728936T^9 + 955396T^8 - 439503T^7 + 166089T^6 - 50492T^5 + 11933T^4 - 2068T^3 + 234T^2 - 12*T + 1
$13$	\( T^{24} \) T^24
$17$	\( T^{24} \) T^24
$19$	\( T^{24} \) T^24
$23$	\( (T^{12} - T^{11} - 12 T^{10} + \cdots + 1)^{2} \) (T^12 - T^11 - 12T^10 + 12T^9 + 53T^8 - 53T^7 - 103T^6 + 103T^5 + 79T^4 - 79T^3 - 12T^2 + 12T + 1)^2
$29$	\( T^{24} - 2 T^{23} + \cdots + 1 \) T^24 - 2T^23 - 18T^21 + 36T^20 + 129T^18 - 258T^17 + 13T^16 - 476T^15 + 900T^14 - 39T^13 + 898T^12 - 1638T^11 + 22T^10 - 590T^9 + 1235T^8 + 397T^7 - 157T^6 - 494T^5 + 95T^4 + 70T^3 + 26T^2 + 6*T + 1
$31$	\( T^{24} \) T^24
$37$	\( T^{24} - 2 T^{23} + \cdots + 1 \) T^24 - 2T^23 - 18T^21 + 36T^20 + 129T^18 - 258T^17 + 13T^16 - 476T^15 + 900T^14 - 39T^13 + 898T^12 - 1638T^11 + 22T^10 - 590T^9 + 1235T^8 + 397T^7 - 157T^6 - 494T^5 + 95T^4 + 70T^3 + 26T^2 + 6*T + 1
$41$	\( T^{24} \) T^24
$43$	\( T^{24} - 2 T^{23} + \cdots + 1 \) T^24 - 2T^23 + 8T^21 - 16T^20 + 26T^19 - 14T^18 - 76T^17 + 208T^16 - 138T^15 + 55T^14 + 156T^13 - 337T^12 + 52T^11 + 906T^10 + 8T^9 - 91T^8 - 188T^7 + 519T^6 + 533T^5 + 407T^4 + 96T^3 + 13T^2 + 6T + 1
$47$	\( T^{24} \) T^24
$53$	\( T^{24} + 11 T^{23} + \cdots + 1 \) T^24 + 11T^23 + 65T^22 + 268T^21 + 855T^20 + 2223T^19 + 4861T^18 + 9128T^17 + 14924T^16 + 21442T^15 + 27264T^14 + 30810T^13 + 30967T^12 + 27768T^11 + 22265T^10 + 15582T^9 + 9334T^8 + 5376T^7 + 3249T^6 + 1456T^5 + 251T^4 - 34T^3 + 6T + 1
$59$	\( T^{24} \) T^24
$61$	\( T^{24} \) T^24
$67$	\( T^{24} + T^{23} + \cdots + 1 \) T^24 + T^23 + 12T^21 + 12T^20 + 105T^18 + 105T^17 + 13T^16 + 454T^15 + 441T^14 - 234T^13 + 1171T^12 + 1404T^11 - 1457T^10 + 181T^9 + 1781T^8 + 2107T^7 + 3875T^6 + 1378T^5 + 1052T^4 + 532T^3 + 221T^2 + 27T + 1
$71$	\( (T^{12} + 2 T^{11} + \cdots + 1)^{2} \) (T^12 + 2T^11 + 4T^10 + 8T^9 + 16T^8 + 32T^7 + 64T^6 + 115T^5 + 139T^4 + 96T^3 + 36T^2 + 7*T + 1)^2
$73$	\( T^{24} \) T^24
$79$	\( T^{24} - T^{23} + \cdots + 1 \) T^24 - T^23 + T^21 - T^20 + T^18 - T^17 + T^15 - T^14 + T^12 - T^10 + T^9 - T^7 + T^6 - T^4 + T^3 - T + 1
$83$	\( T^{24} \) T^24
$89$	\( T^{24} \) T^24
$97$	\( T^{24} \) T^24
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