LMFDB - Newform orbit 2500.1.p.a

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2500,1,Mod(51,2500)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2500, base_ring=CyclotomicField(50))

chi = DirichletCharacter(H, H._module([25, 28]))

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

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

chi := DirichletCharacter("2500.51");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 2500 = 2^{2} \cdot 5^{4} $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	2500.p (of order $50$, degree $20$, not 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:	$1.24766253158$
Analytic rank:	$0$
Dimension:	$20$
Coefficient field:	$\Q(\zeta_{50})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{20} - x^{15} + x^{10} - x^{5} + 1 $ x^20 - x^15 + x^10 - x^5 + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 500)
Projective image:	$D_{25}$
Projective field:	Galois closure of $\mathbb{Q}[x]/(x^{25} - \cdots)$

$q$-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_{50}^{3} q^{2} + \zeta_{50}^{6} q^{4} + \zeta_{50}^{9} q^{8} - \zeta_{50}^{17} q^{9} +O(q^{10}) $ q + z^3 * q^2 + z^6 * q^4 + z^9 * q^8 - z^17 * q^9	\( q + \zeta_{50}^{3} q^{2} + \zeta_{50}^{6} q^{4} + \zeta_{50}^{9} q^{8} - \zeta_{50}^{17} q^{9} + ( - \zeta_{50}^{18} - \zeta_{50}^{16}) q^{13} + \zeta_{50}^{12} q^{16} + (\zeta_{50}^{11} - \zeta_{50}^{2}) q^{17} - \zeta_{50}^{20} q^{18} + ( - \zeta_{50}^{21} - \zeta_{50}^{19}) q^{26} + (\zeta_{50}^{24} - \zeta_{50}^{23}) q^{29} + \zeta_{50}^{15} q^{32} + (\zeta_{50}^{14} - \zeta_{50}^{5}) q^{34} - \zeta_{50}^{23} q^{36} + (\zeta_{50}^{19} + \zeta_{50}^{5}) q^{37} + ( - \zeta_{50}^{13} - \zeta_{50}) q^{41} + \zeta_{50}^{10} q^{49} + ( - \zeta_{50}^{24} - \zeta_{50}^{22}) q^{52} + (\zeta_{50}^{15} - \zeta_{50}^{12}) q^{53} + ( - \zeta_{50}^{2} + \zeta_{50}) q^{58} + (\zeta_{50}^{22} + \zeta_{50}^{14}) q^{61} + \zeta_{50}^{18} q^{64} + (\zeta_{50}^{17} - \zeta_{50}^{8}) q^{68} + \zeta_{50} q^{72} + ( - \zeta_{50}^{14} + \zeta_{50}^{7}) q^{73} + (\zeta_{50}^{22} + \zeta_{50}^{8}) q^{74} - \zeta_{50}^{9} q^{81} + ( - \zeta_{50}^{16} - \zeta_{50}^{4}) q^{82} + (\zeta_{50}^{20} - \zeta_{50}) q^{89} + (\zeta_{50}^{13} + \zeta_{50}^{9}) q^{97} + \zeta_{50}^{13} q^{98} +O(q^{100}) \) q + z^3 * q^2 + z^6 * q^4 + z^9 * q^8 - z^17 * q^9 + (-z^18 - z^16) * q^13 + z^12 * q^16 + (z^11 - z^2) * q^17 - z^20 * q^18 + (-z^21 - z^19) * q^26 + (z^24 - z^23) * q^29 + z^15 * q^32 + (z^14 - z^5) * q^34 - z^23 * q^36 + (z^19 + z^5) * q^37 + (-z^13 - z) * q^41 + z^10 * q^49 + (-z^24 - z^22) * q^52 + (z^15 - z^12) * q^53 + (-z^2 + z) * q^58 + (z^22 + z^14) * q^61 + z^18 * q^64 + (z^17 - z^8) * q^68 + z * q^72 + (-z^14 + z^7) * q^73 + (z^22 + z^8) * q^74 - z^9 * q^81 + (-z^16 - z^4) * q^82 + (z^20 - z) * q^89 + (z^13 + z^9) * q^97 + z^13 * q^98
$\operatorname{Tr}(f)(q)$	$=$	$ 20 q+O(q^{10}) $ 20 * q	$ 20 q + 5 q^{18} + 5 q^{32} - 5 q^{34} + 5 q^{37} - 5 q^{49} + 5 q^{53} - 5 q^{89}+O(q^{100}) $ 20 * q + 5 * q^18 + 5 * q^32 - 5 * q^34 + 5 * q^37 - 5 * q^49 + 5 * q^53 - 5 * q^89

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2500\mathbb{Z}\right)^\times$.

$n$	$1251$	$1877$
$\chi(n)$	$-1$	$\zeta_{50}^{6}$

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} $

51.1

−0.0627905	+	0.998027i
0.187381	−	0.982287i
0.425779	−	0.904827i
−0.535827	+	0.844328i
0.637424	−	0.770513i
−0.728969	+	0.684547i
−0.876307	+	0.481754i
0.929776	−	0.368125i
−0.968583	+	0.248690i
0.992115	−	0.125333i
0.992115	+	0.125333i
−0.968583	−	0.248690i
0.929776	+	0.368125i
−0.876307	−	0.481754i
−0.728969	−	0.684547i
0.637424	+	0.770513i
−0.535827	−	0.844328i
0.425779	+	0.904827i
0.187381	+	0.982287i
−0.0627905	−	0.998027i

0.187381

−

0.982287i

−0.929776

−

0.368125i

−0.535827

0.844328i

0.876307

−

0.481754i

151.1

−0.535827

0.844328i

−0.425779

−

0.904827i

0.992115

0.125333i

0.0627905

−

0.998027i

351.1

−0.968583

0.248690i

0.876307

−

0.481754i

−0.728969

0.684547i

−0.929776

0.368125i

451.1

0.992115

0.125333i

0.968583

0.248690i

0.929776

0.368125i

−0.187381

0.982287i

551.1

−0.876307

−

0.481754i

0.535827

0.844328i

−0.0627905

−

0.998027i

0.728969

0.684547i

651.1

0.637424

0.770513i

−0.187381

0.982287i

−0.876307

0.481754i

0.968583

−

0.248690i

851.1

−0.0627905

0.998027i

−0.992115

−

0.125333i

0.187381

−

0.982287i

−0.637424

−

0.770513i

951.1

0.425779

−

0.904827i

−0.637424

−

0.770513i

−0.968583

0.248690i

−0.992115

0.125333i

1051.1

−0.728969

0.684547i

0.0627905

−

0.998027i

0.637424

0.770513i

−0.425779

0.904827i

1151.1

0.929776

−

0.368125i

0.728969

−

0.684547i

0.425779

−

0.904827i

0.535827

0.844328i

1351.1

0.929776

0.368125i

0.728969

0.684547i

0.425779

0.904827i

0.535827

−

0.844328i

1451.1

−0.728969

−

0.684547i

0.0627905

0.998027i

0.637424

−

0.770513i

−0.425779

−

0.904827i

1551.1

0.425779

0.904827i

−0.637424

0.770513i

−0.968583

−

0.248690i

−0.992115

−

0.125333i

1651.1

−0.0627905

−

0.998027i

−0.992115

0.125333i

0.187381

0.982287i

−0.637424

0.770513i

1851.1

0.637424

−

0.770513i

−0.187381

−

0.982287i

−0.876307

−

0.481754i

0.968583

0.248690i

1951.1

−0.876307

0.481754i

0.535827

−

0.844328i

−0.0627905

0.998027i

0.728969

−

0.684547i

2051.1

0.992115

−

0.125333i

0.968583

−

0.248690i

0.929776

−

0.368125i

−0.187381

−

0.982287i

2151.1

−0.968583

−

0.248690i

0.876307

0.481754i

−0.728969

−

0.684547i

−0.929776

−

0.368125i

2351.1

−0.535827

−

0.844328i

−0.425779

0.904827i

0.992115

−

0.125333i

0.0627905

0.998027i

2451.1

0.187381

0.982287i

−0.929776

0.368125i

−0.535827

−

0.844328i

0.876307

0.481754i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
4.b	odd	2	1	CM by $\Q(\sqrt{-1}) $
125.g	even	25	1	inner
500.p	odd	50	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	2500.1.p.a		20
4.b	odd	2	1	CM	2500.1.p.a		20
5.b	even	2	1		500.1.p.a	✓	20
5.c	odd	4	2		2500.1.n.a		40
20.d	odd	2	1		500.1.p.a	✓	20
20.e	even	4	2		2500.1.n.a		40
125.g	even	25	1	inner	2500.1.p.a		20
125.h	even	50	1		500.1.p.a	✓	20
125.i	odd	100	2		2500.1.n.a		40
500.n	odd	50	1		500.1.p.a	✓	20
500.p	odd	50	1	inner	2500.1.p.a		20
500.r	even	100	2		2500.1.n.a		40

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
500.1.p.a	✓	20	5.b	even	2	1
500.1.p.a	✓	20	20.d	odd	2	1
500.1.p.a	✓	20	125.h	even	50	1
500.1.p.a	✓	20	500.n	odd	50	1
2500.1.n.a		40	5.c	odd	4	2
2500.1.n.a		40	20.e	even	4	2
2500.1.n.a		40	125.i	odd	100	2
2500.1.n.a		40	500.r	even	100	2
2500.1.p.a		20	1.a	even	1	1	trivial
2500.1.p.a		20	4.b	odd	2	1	CM
2500.1.p.a		20	125.g	even	25	1	inner
2500.1.p.a		20	500.p	odd	50	1	inner

Hecke kernels

This newform subspace is the entire newspace $S_{1}^{\mathrm{new}}(2500, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{20} - T^{15} + \cdots + 1 $ T^20 - T^15 + T^10 - T^5 + 1
$3$	$ T^{20} $ T^20
$5$	$ T^{20} $ T^20
$7$	$ T^{20} $ T^20
$11$	$ T^{20} $ T^20
$13$	$ T^{20} + 20 T^{17} + \cdots + 1 $ T^20 + 20T^17 + 5T^16 - 2T^15 + 150T^14 + 75T^13 - 5T^12 + 505T^11 + 379T^10 + 100T^9 + 800T^8 + 640T^7 + 360T^6 + 747T^5 + 600T^4 + 175T^3 + 85T^2 + 15*T + 1
$17$	$ T^{20} - 5 T^{17} + \cdots + 1 $ T^20 - 5T^17 + 5T^16 - 2T^15 + 25T^14 + 75T^13 + 20T^12 - 145T^11 - 246T^10 + 225T^9 + 675T^8 + 615T^7 + 385T^6 + 122T^5 - 25T^4 - 200T^3 + 85T^2 - 10*T + 1
$19$	$ T^{20} $ T^20
$23$	$ T^{20} $ T^20
$29$	$ T^{20} + 5 T^{17} + \cdots + 1 $ T^20 + 5T^17 + 5T^16 + 2T^15 + 25T^14 + 50T^13 + 45T^12 + 145T^11 + 379T^10 + 525T^9 + 1050T^8 + 2685T^7 + 4160T^6 + 3628T^5 + 1850T^4 + 575T^3 + 110T^2 + 10*T + 1
$31$	$ T^{20} $ T^20
$37$	$ T^{20} - 5 T^{19} + \cdots + 1 $ T^20 - 5T^19 + 15T^18 - 35T^17 + 70T^16 - 122T^15 + 170T^14 - 185T^13 + 165T^12 - 205T^11 + 629T^10 - 1235T^9 + 1655T^8 - 1520T^7 + 665T^6 + 252T^5 - 340T^4 + 45T^3 + 70T^2 + 10*T + 1
$41$	$ T^{20} + 5 T^{17} + \cdots + 1 $ T^20 + 5T^17 + 5T^16 + 2T^15 + 25T^14 + 50T^13 + 45T^12 + 145T^11 + 379T^10 + 525T^9 + 1050T^8 + 2685T^7 + 4160T^6 + 3628T^5 + 1850T^4 + 575T^3 + 110T^2 + 10*T + 1
$43$	$ T^{20} $ T^20
$47$	$ T^{20} $ T^20
$53$	$ T^{20} - 5 T^{19} + \cdots + 1 $ T^20 - 5T^19 + 15T^18 - 35T^17 + 70T^16 - 122T^15 + 170T^14 - 185T^13 + 165T^12 - 205T^11 + 629T^10 - 1235T^9 + 1655T^8 - 1520T^7 + 665T^6 + 252T^5 - 340T^4 + 45T^3 + 70T^2 + 10*T + 1
$59$	$ T^{20} $ T^20
$61$	$ T^{20} - 20 T^{17} + \cdots + 1 $ T^20 - 20T^17 + 5T^16 + 2T^15 + 150T^14 - 75T^13 - 5T^12 - 505T^11 + 379T^10 - 100T^9 + 800T^8 - 640T^7 + 360T^6 - 747T^5 + 600T^4 - 175T^3 + 85T^2 - 15*T + 1
$67$	$ T^{20} $ T^20
$71$	$ T^{20} $ T^20
$73$	$ T^{20} - 5 T^{17} + \cdots + 1 $ T^20 - 5T^17 + 5T^16 - 2T^15 + 25T^14 - 50T^13 + 45T^12 - 145T^11 + 379T^10 - 525T^9 + 1050T^8 - 2685T^7 + 4160T^6 - 3628T^5 + 1850T^4 - 575T^3 + 110T^2 - 10*T + 1
$79$	$ T^{20} $ T^20
$83$	$ T^{20} $ T^20
$89$	$ T^{20} + 5 T^{19} + \cdots + 1 $ T^20 + 5T^19 + 15T^18 + 35T^17 + 70T^16 + 122T^15 + 170T^14 + 185T^13 + 165T^12 + 205T^11 + 629T^10 + 1235T^9 + 1655T^8 + 1520T^7 + 665T^6 - 252T^5 - 340T^4 - 45T^3 + 70T^2 - 10*T + 1
$97$	$ T^{20} - 5 T^{17} + \cdots + 1 $ T^20 - 5T^17 + 5T^16 - 2T^15 + 25T^14 + 75T^13 + 20T^12 - 145T^11 - 246T^10 + 225T^9 + 675T^8 + 615T^7 + 385T^6 + 122T^5 - 25T^4 - 200T^3 + 85T^2 - 10*T + 1
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Overview	Random
Universe	Knowledge

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:	\( N \)	\(=\)	\( 2500 = 2^{2} \cdot 5^{4} \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	2500.p (of order \(50\), degree \(20\), not minimal)

\(f(q)\)	\(=\)	\( q + \zeta_{50}^{3} q^{2} + \zeta_{50}^{6} q^{4} + \zeta_{50}^{9} q^{8} - \zeta_{50}^{17} q^{9} +O(q^{10}) \) q + z^3 * q^2 + z^6 * q^4 + z^9 * q^8 - z^17 * q^9	\( q + \zeta_{50}^{3} q^{2} + \zeta_{50}^{6} q^{4} + \zeta_{50}^{9} q^{8} - \zeta_{50}^{17} q^{9} + ( - \zeta_{50}^{18} - \zeta_{50}^{16}) q^{13} + \zeta_{50}^{12} q^{16} + (\zeta_{50}^{11} - \zeta_{50}^{2}) q^{17} - \zeta_{50}^{20} q^{18} + ( - \zeta_{50}^{21} - \zeta_{50}^{19}) q^{26} + (\zeta_{50}^{24} - \zeta_{50}^{23}) q^{29} + \zeta_{50}^{15} q^{32} + (\zeta_{50}^{14} - \zeta_{50}^{5}) q^{34} - \zeta_{50}^{23} q^{36} + (\zeta_{50}^{19} + \zeta_{50}^{5}) q^{37} + ( - \zeta_{50}^{13} - \zeta_{50}) q^{41} + \zeta_{50}^{10} q^{49} + ( - \zeta_{50}^{24} - \zeta_{50}^{22}) q^{52} + (\zeta_{50}^{15} - \zeta_{50}^{12}) q^{53} + ( - \zeta_{50}^{2} + \zeta_{50}) q^{58} + (\zeta_{50}^{22} + \zeta_{50}^{14}) q^{61} + \zeta_{50}^{18} q^{64} + (\zeta_{50}^{17} - \zeta_{50}^{8}) q^{68} + \zeta_{50} q^{72} + ( - \zeta_{50}^{14} + \zeta_{50}^{7}) q^{73} + (\zeta_{50}^{22} + \zeta_{50}^{8}) q^{74} - \zeta_{50}^{9} q^{81} + ( - \zeta_{50}^{16} - \zeta_{50}^{4}) q^{82} + (\zeta_{50}^{20} - \zeta_{50}) q^{89} + (\zeta_{50}^{13} + \zeta_{50}^{9}) q^{97} + \zeta_{50}^{13} q^{98} +O(q^{100}) \) q + z^3 * q^2 + z^6 * q^4 + z^9 * q^8 - z^17 * q^9 + (-z^18 - z^16) * q^13 + z^12 * q^16 + (z^11 - z^2) * q^17 - z^20 * q^18 + (-z^21 - z^19) * q^26 + (z^24 - z^23) * q^29 + z^15 * q^32 + (z^14 - z^5) * q^34 - z^23 * q^36 + (z^19 + z^5) * q^37 + (-z^13 - z) * q^41 + z^10 * q^49 + (-z^24 - z^22) * q^52 + (z^15 - z^12) * q^53 + (-z^2 + z) * q^58 + (z^22 + z^14) * q^61 + z^18 * q^64 + (z^17 - z^8) * q^68 + z * q^72 + (-z^14 + z^7) * q^73 + (z^22 + z^8) * q^74 - z^9 * q^81 + (-z^16 - z^4) * q^82 + (z^20 - z) * q^89 + (z^13 + z^9) * q^97 + z^13 * q^98
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q+O(q^{10}) \) 20 * q	\( 20 q + 5 q^{18} + 5 q^{32} - 5 q^{34} + 5 q^{37} - 5 q^{49} + 5 q^{53} - 5 q^{89}+O(q^{100}) \) 20 * q + 5 * q^18 + 5 * q^32 - 5 * q^34 + 5 * q^37 - 5 * q^49 + 5 * q^53 - 5 * q^89

Introduction

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Modular forms

Varieties

Fields

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Database

Properties

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Learn more

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	2500.1.p.a

Level	$2500$
Weight	$1$
Character orbit	2500.p
Analytic conductor	$1.248$
Analytic rank	$0$
Dimension	$20$
Projective image	$D_{25}$
CM discriminant	-4
Inner twists	$4$

Self dual:	no
Analytic conductor:	\(1.24766253158\)
Analytic rank:	\(0\)
Dimension:	\(20\)
Coefficient field:	\(\Q(\zeta_{50})\)
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	\( x^{20} - x^{15} + x^{10} - x^{5} + 1 \) x^20 - x^15 + x^10 - x^5 + 1
Coefficient ring:	\(\Z[a_1, a_2]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 500)
Projective image:	\(D_{25}\)
Projective field:	Galois closure of \(\mathbb{Q}[x]/(x^{25} - \cdots)\)

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 51.1
Significant digits:
Format:

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
4.b	odd	2	1	CM by \(\Q(\sqrt{-1}) \)
125.g	even	25	1	inner
500.p	odd	50	1	inner

$p$	$F_p(T)$
$2$	\( T^{20} - T^{15} + \cdots + 1 \) T^20 - T^15 + T^10 - T^5 + 1
$3$	\( T^{20} \) T^20
$5$	\( T^{20} \) T^20
$7$	\( T^{20} \) T^20
$11$	\( T^{20} \) T^20
$13$	\( T^{20} + 20 T^{17} + \cdots + 1 \) T^20 + 20T^17 + 5T^16 - 2T^15 + 150T^14 + 75T^13 - 5T^12 + 505T^11 + 379T^10 + 100T^9 + 800T^8 + 640T^7 + 360T^6 + 747T^5 + 600T^4 + 175T^3 + 85T^2 + 15*T + 1
$17$	\( T^{20} - 5 T^{17} + \cdots + 1 \) T^20 - 5T^17 + 5T^16 - 2T^15 + 25T^14 + 75T^13 + 20T^12 - 145T^11 - 246T^10 + 225T^9 + 675T^8 + 615T^7 + 385T^6 + 122T^5 - 25T^4 - 200T^3 + 85T^2 - 10*T + 1
$19$	\( T^{20} \) T^20
$23$	\( T^{20} \) T^20
$29$	\( T^{20} + 5 T^{17} + \cdots + 1 \) T^20 + 5T^17 + 5T^16 + 2T^15 + 25T^14 + 50T^13 + 45T^12 + 145T^11 + 379T^10 + 525T^9 + 1050T^8 + 2685T^7 + 4160T^6 + 3628T^5 + 1850T^4 + 575T^3 + 110T^2 + 10*T + 1
$31$	\( T^{20} \) T^20
$37$	\( T^{20} - 5 T^{19} + \cdots + 1 \) T^20 - 5T^19 + 15T^18 - 35T^17 + 70T^16 - 122T^15 + 170T^14 - 185T^13 + 165T^12 - 205T^11 + 629T^10 - 1235T^9 + 1655T^8 - 1520T^7 + 665T^6 + 252T^5 - 340T^4 + 45T^3 + 70T^2 + 10*T + 1
$41$	\( T^{20} + 5 T^{17} + \cdots + 1 \) T^20 + 5T^17 + 5T^16 + 2T^15 + 25T^14 + 50T^13 + 45T^12 + 145T^11 + 379T^10 + 525T^9 + 1050T^8 + 2685T^7 + 4160T^6 + 3628T^5 + 1850T^4 + 575T^3 + 110T^2 + 10*T + 1
$43$	\( T^{20} \) T^20
$47$	\( T^{20} \) T^20
$53$	\( T^{20} - 5 T^{19} + \cdots + 1 \) T^20 - 5T^19 + 15T^18 - 35T^17 + 70T^16 - 122T^15 + 170T^14 - 185T^13 + 165T^12 - 205T^11 + 629T^10 - 1235T^9 + 1655T^8 - 1520T^7 + 665T^6 + 252T^5 - 340T^4 + 45T^3 + 70T^2 + 10*T + 1
$59$	\( T^{20} \) T^20
$61$	\( T^{20} - 20 T^{17} + \cdots + 1 \) T^20 - 20T^17 + 5T^16 + 2T^15 + 150T^14 - 75T^13 - 5T^12 - 505T^11 + 379T^10 - 100T^9 + 800T^8 - 640T^7 + 360T^6 - 747T^5 + 600T^4 - 175T^3 + 85T^2 - 15*T + 1
$67$	\( T^{20} \) T^20
$71$	\( T^{20} \) T^20
$73$	\( T^{20} - 5 T^{17} + \cdots + 1 \) T^20 - 5T^17 + 5T^16 - 2T^15 + 25T^14 - 50T^13 + 45T^12 - 145T^11 + 379T^10 - 525T^9 + 1050T^8 - 2685T^7 + 4160T^6 - 3628T^5 + 1850T^4 - 575T^3 + 110T^2 - 10*T + 1
$79$	\( T^{20} \) T^20
$83$	\( T^{20} \) T^20
$89$	\( T^{20} + 5 T^{19} + \cdots + 1 \) T^20 + 5T^19 + 15T^18 + 35T^17 + 70T^16 + 122T^15 + 170T^14 + 185T^13 + 165T^12 + 205T^11 + 629T^10 + 1235T^9 + 1655T^8 + 1520T^7 + 665T^6 - 252T^5 - 340T^4 - 45T^3 + 70T^2 - 10*T + 1
$97$	\( T^{20} - 5 T^{17} + \cdots + 1 \) T^20 - 5T^17 + 5T^16 - 2T^15 + 25T^14 + 75T^13 + 20T^12 - 145T^11 - 246T^10 + 225T^9 + 675T^8 + 615T^7 + 385T^6 + 122T^5 - 25T^4 - 200T^3 + 85T^2 - 10*T + 1
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\(n\)	\(1251\)	\(1877\)
\(\chi(n)\)	\(-1\)	\(\zeta_{50}^{6}\)