LMFDB - Newform orbit 1441.1.u.a

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1441,1,Mod(130,1441)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1441, base_ring=CyclotomicField(10))

chi = DirichletCharacter(H, H._module([6, 5]))

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

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

chi := DirichletCharacter("1441.130");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 1441 = 11 \cdot 131 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	1441.u (of order $10$, degree $4$, 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:	$0.719152683204$
Analytic rank:	$0$
Dimension:	$20$
Relative dimension:	$5$ over $\Q(\zeta_{10})$
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:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
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}^{21} - \zeta_{50}^{9}) q^{3} + \zeta_{50}^{10} q^{4} + (\zeta_{50}^{12} - \zeta_{50}^{3}) q^{5} + ( - \zeta_{50}^{23} + \zeta_{50}^{22}) q^{7} + (\zeta_{50}^{18} + \cdots - \zeta_{50}^{5}) q^{9}+O(q^{10}) $ q + (-z^21 - z^9) * q^3 + z^10 * q^4 + (z^12 - z^3) * q^5 + (-z^23 + z^22) * q^7 + (z^18 - z^17 - z^5) * q^9	$ q + ( - \zeta_{50}^{21} - \zeta_{50}^{9}) q^{3} + \zeta_{50}^{10} q^{4} + (\zeta_{50}^{12} - \zeta_{50}^{3}) q^{5} + ( - \zeta_{50}^{23} + \zeta_{50}^{22}) q^{7} + (\zeta_{50}^{18} + \cdots - \zeta_{50}^{5}) q^{9}+ \cdots + (\zeta_{50}^{22} + \cdots - \zeta_{50}^{9}) q^{99}+O(q^{100}) $ q + (-z^21 - z^9) * q^3 + z^10 * q^4 + (z^12 - z^3) * q^5 + (-z^23 + z^22) * q^7 + (z^18 - z^17 - z^5) * q^9 + z^4 * q^11 + (-z^19 + z^6) * q^12 + (-z^21 + z^14) * q^13 + (z^24 - z^21 + z^12 + z^8) * q^15 + z^20 * q^16 + (z^22 - z^13) * q^20 + (-z^19 + z^18 - z^7 + z^6) * q^21 + (z^24 - z^15 + z^6) * q^25 + (z^14 - z^13 + z^2 + z) * q^27 + (z^8 - z^7) * q^28 + (-z^13 + 1) * q^33 + (z^10 - z^9 - z + 1) * q^35 + (-z^15 - z^3 + z^2) * q^36 + (-z^23 - z^17 + z^10 - z^5) * q^39 + (z^4 - z) * q^41 + (-z^23 + z^2) * q^43 + z^14 * q^44 + (-z^21 + z^20 - z^17 + z^8 - z^5 + z^4) * q^45 + (z^16 + z^4) * q^48 + (-z^21 + z^20 - z^19) * q^49 + (z^24 + z^6) * q^52 + (z^20 - z^15) * q^53 + (z^16 - z^7) * q^55 + (-z^13 - z^7) * q^59 + (z^22 + z^18 - z^9 + z^6) * q^60 + (z^12 - z^3) * q^61 + (z^16 - 2z^15 + z^14 - z^3 + z^2) q^63 - z^5 * q^64 + (z^24 - z^17 + z^8 - z) * q^65 + (z^24 + z^20 - z^15 - z^11 + z^8 + z^2) * q^75 + (z^2 - z) * q^77 + (-z^23 - z^7) * q^80 + (-z^23 + z^22 - z^11 + z^10 - z^9) * q^81 + (-z^17 + z^16 + z^4 - z^3) * q^84 + (z^20 - z^5) * q^89 + (-z^19 + z^18 + z^12 - z^11) * q^91 + (z^22 - z^21 - z^9) * q^99
$\operatorname{Tr}(f)(q)$	$=$	$ 20 q - 5 q^{4} - 5 q^{9}+O(q^{10}) $ 20 * q - 5 * q^4 - 5 * q^9	$ 20 q - 5 q^{4} - 5 q^{9} - 5 q^{16} - 5 q^{25} + 20 q^{33} + 15 q^{35} - 5 q^{36} - 10 q^{39} - 10 q^{45} - 5 q^{49} - 10 q^{53} - 10 q^{63} - 5 q^{64} - 10 q^{75} - 5 q^{81} - 10 q^{89}+O(q^{100}) $ 20 * q - 5 * q^4 - 5 * q^9 - 5 * q^16 - 5 * q^25 + 20 * q^33 + 15 * q^35 - 5 * q^36 - 10 * q^39 - 10 * q^45 - 5 * q^49 - 10 * q^53 - 10 * q^63 - 5 * q^64 - 10 * q^75 - 5 * q^81 - 10 * q^89

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1441\mathbb{Z}\right)^\times$.

$n$	$133$	$1311$
$\chi(n)$	$-1$	$\zeta_{50}^{10}$

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} $

130.1

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

−1.56720

1.13864i

−0.809017

−

0.587785i

−0.115808

−

0.356420i

−1.41789

−

1.03016i

0.850604

−

2.61789i

130.2

−0.866986

0.629902i

−0.809017

−

0.587785i

0.450527

1.38658i

0.688925

0.500534i

0.0458709

−

0.141176i

130.3

−0.101597

0.0738147i

−0.809017

−

0.587785i

−0.263146

−

0.809880i

1.60528

1.16630i

−0.304144

0.936058i

130.4

1.03137

−

0.749337i

−0.809017

−

0.587785i

−0.613161

−

1.88711i

0.303189

0.220280i

0.193209

−

0.594636i

130.5

1.50441

−

1.09302i

−0.809017

−

0.587785i

0.541587

1.66683i

−1.17950

−

0.856954i

0.759544

−

2.33764i

654.1

−1.56720

−

1.13864i

−0.809017

0.587785i

−0.115808

0.356420i

−1.41789

1.03016i

0.850604

2.61789i

654.2

−0.866986

−

0.629902i

−0.809017

0.587785i

0.450527

−

1.38658i

0.688925

−

0.500534i

0.0458709

0.141176i

654.3

−0.101597

−

0.0738147i

−0.809017

0.587785i

−0.263146

0.809880i

1.60528

−

1.16630i

−0.304144

−

0.936058i

654.4

1.03137

0.749337i

−0.809017

0.587785i

−0.613161

1.88711i

0.303189

−

0.220280i

0.193209

0.594636i

654.5

1.50441

1.09302i

−0.809017

0.587785i

0.541587

−

1.66683i

−1.17950

0.856954i

0.759544

2.33764i

785.1

−0.613161

1.88711i

0.309017

0.951057i

1.03137

−

0.749337i

0.598617

1.84235i

−2.37622

−

1.72642i

785.2

−0.263146

0.809880i

0.309017

0.951057i

−0.101597

0.0738147i

−0.393950

−

1.21245i

0.222357

0.161552i

785.3

−0.115808

0.356420i

0.309017

0.951057i

−1.56720

1.13864i

−0.574633

−

1.76854i

0.695393

0.505233i

785.4

0.450527

−

1.38658i

0.309017

0.951057i

−0.866986

0.629902i

0.0388067

0.119435i

−0.910614

−

0.661600i

785.5

0.541587

−

1.66683i

0.309017

0.951057i

1.50441

−

1.09302i

0.331159

1.01920i

−1.67600

−

1.21769i

916.1

−0.613161

−

1.88711i

0.309017

−

0.951057i

1.03137

0.749337i

0.598617

−

1.84235i

−2.37622

1.72642i

916.2

−0.263146

−

0.809880i

0.309017

−

0.951057i

−0.101597

−

0.0738147i

−0.393950

1.21245i

0.222357

−

0.161552i

916.3

−0.115808

−

0.356420i

0.309017

−

0.951057i

−1.56720

−

1.13864i

−0.574633

1.76854i

0.695393

−

0.505233i

916.4

0.450527

1.38658i

0.309017

−

0.951057i

−0.866986

−

0.629902i

0.0388067

−

0.119435i

−0.910614

0.661600i

916.5

0.541587

1.66683i

0.309017

−

0.951057i

1.50441

1.09302i

0.331159

−

1.01920i

−1.67600

1.21769i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
131.b	odd	2	1	CM by $\Q(\sqrt{-131}) $
11.c	even	5	1	inner
1441.u	odd	10	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	1441.1.u.a	✓	20
11.c	even	5	1	inner	1441.1.u.a	✓	20
131.b	odd	2	1	CM	1441.1.u.a	✓	20
1441.u	odd	10	1	inner	1441.1.u.a	✓	20

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
1441.1.u.a	✓	20	1.a	even	1	1	trivial
1441.1.u.a	✓	20	11.c	even	5	1	inner
1441.1.u.a	✓	20	131.b	odd	2	1	CM
1441.1.u.a	✓	20	1441.u	odd	10	1	inner

Hecke kernels

This newform subspace is the entire newspace $S_{1}^{\mathrm{new}}(1441, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{20} $ T^20
$3$	$ T^{20} + 5 T^{18} + \cdots + 1 $ T^20 + 5T^18 + 20T^16 + 2T^15 + 75T^14 - 5T^13 + 275T^12 + 5T^11 + 374T^10 + 200T^9 + 485T^8 + 150T^7 + 715T^6 + 498T^5 + 675T^4 + 260T^3 + 100T^2 + 15*T + 1
$5$	$ T^{20} + 5 T^{18} + \cdots + 1 $ T^20 + 5T^18 + 20T^16 + 2T^15 + 75T^14 - 5T^13 + 275T^12 + 5T^11 + 374T^10 + 200T^9 + 485T^8 + 150T^7 + 715T^6 + 498T^5 + 675T^4 + 260T^3 + 100T^2 + 15*T + 1
$7$	$ T^{20} + 5 T^{18} + \cdots + 1 $ T^20 + 5T^18 + 20T^16 + 2T^15 + 75T^14 + 20T^13 + 275T^12 + 30T^11 + 374T^10 - 175T^9 + 510T^8 - 475T^7 + 740T^6 - 752T^5 + 800T^4 - 390T^3 + 100T^2 - 10*T + 1
$11$	$ T^{20} + T^{15} + \cdots + 1 $ T^20 + T^15 + T^10 + T^5 + 1
$13$	$ T^{20} + 5 T^{18} + \cdots + 1 $ T^20 + 5T^18 + 20T^16 + 2T^15 + 75T^14 + 20T^13 + 275T^12 + 30T^11 + 374T^10 - 175T^9 + 510T^8 - 475T^7 + 740T^6 - 752T^5 + 800T^4 - 390T^3 + 100T^2 - 10*T + 1
$17$	$ T^{20} $ T^20
$19$	$ T^{20} $ T^20
$23$	$ T^{20} $ T^20
$29$	$ T^{20} $ T^20
$31$	$ T^{20} $ T^20
$37$	$ T^{20} $ T^20
$41$	$ T^{20} + 5 T^{18} + \cdots + 1 $ T^20 + 5T^18 + 20T^16 + 2T^15 + 75T^14 - 5T^13 + 275T^12 + 5T^11 + 374T^10 + 200T^9 + 485T^8 + 150T^7 + 715T^6 + 498T^5 + 675T^4 + 260T^3 + 100T^2 + 15*T + 1
$43$	$ (T^{10} - 10 T^{8} + \cdots - 1)^{2} $ (T^10 - 10T^8 + 35T^6 + T^5 - 50T^4 - 5T^3 + 25T^2 + 5T - 1)^2
$47$	$ T^{20} $ T^20
$53$	$ (T^{4} + 2 T^{3} + 4 T^{2} + \cdots + 1)^{5} $ (T^4 + 2T^3 + 4T^2 + 3*T + 1)^5
$59$	$ T^{20} + 5 T^{18} + \cdots + 1 $ T^20 + 5T^18 + 20T^16 + 2T^15 + 75T^14 + 20T^13 + 275T^12 + 30T^11 + 374T^10 - 175T^9 + 510T^8 - 475T^7 + 740T^6 - 752T^5 + 800T^4 - 390T^3 + 100T^2 - 10*T + 1
$61$	$ T^{20} + 5 T^{18} + \cdots + 1 $ T^20 + 5T^18 + 20T^16 + 2T^15 + 75T^14 - 5T^13 + 275T^12 + 5T^11 + 374T^10 + 200T^9 + 485T^8 + 150T^7 + 715T^6 + 498T^5 + 675T^4 + 260T^3 + 100T^2 + 15*T + 1
$67$	$ T^{20} $ T^20
$71$	$ T^{20} $ T^20
$73$	$ T^{20} $ T^20
$79$	$ T^{20} $ T^20
$83$	$ T^{20} $ T^20
$89$	$ (T^{2} + T - 1)^{10} $ (T^2 + T - 1)^10
$97$	$ T^{20} $ T^20
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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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 1441 = 11 \cdot 131 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	1441.u (of order \(10\), degree \(4\), minimal)

\(f(q)\)	\(=\)	\( q + ( - \zeta_{50}^{21} - \zeta_{50}^{9}) q^{3} + \zeta_{50}^{10} q^{4} + (\zeta_{50}^{12} - \zeta_{50}^{3}) q^{5} + ( - \zeta_{50}^{23} + \zeta_{50}^{22}) q^{7} + (\zeta_{50}^{18} + \cdots - \zeta_{50}^{5}) q^{9}+O(q^{10}) \) q + (-z^21 - z^9) * q^3 + z^10 * q^4 + (z^12 - z^3) * q^5 + (-z^23 + z^22) * q^7 + (z^18 - z^17 - z^5) * q^9	\( q + ( - \zeta_{50}^{21} - \zeta_{50}^{9}) q^{3} + \zeta_{50}^{10} q^{4} + (\zeta_{50}^{12} - \zeta_{50}^{3}) q^{5} + ( - \zeta_{50}^{23} + \zeta_{50}^{22}) q^{7} + (\zeta_{50}^{18} + \cdots - \zeta_{50}^{5}) q^{9}+ \cdots + (\zeta_{50}^{22} + \cdots - \zeta_{50}^{9}) q^{99}+O(q^{100}) \) q + (-z^21 - z^9) * q^3 + z^10 * q^4 + (z^12 - z^3) * q^5 + (-z^23 + z^22) * q^7 + (z^18 - z^17 - z^5) * q^9 + z^4 * q^11 + (-z^19 + z^6) * q^12 + (-z^21 + z^14) * q^13 + (z^24 - z^21 + z^12 + z^8) * q^15 + z^20 * q^16 + (z^22 - z^13) * q^20 + (-z^19 + z^18 - z^7 + z^6) * q^21 + (z^24 - z^15 + z^6) * q^25 + (z^14 - z^13 + z^2 + z) * q^27 + (z^8 - z^7) * q^28 + (-z^13 + 1) * q^33 + (z^10 - z^9 - z + 1) * q^35 + (-z^15 - z^3 + z^2) * q^36 + (-z^23 - z^17 + z^10 - z^5) * q^39 + (z^4 - z) * q^41 + (-z^23 + z^2) * q^43 + z^14 * q^44 + (-z^21 + z^20 - z^17 + z^8 - z^5 + z^4) * q^45 + (z^16 + z^4) * q^48 + (-z^21 + z^20 - z^19) * q^49 + (z^24 + z^6) * q^52 + (z^20 - z^15) * q^53 + (z^16 - z^7) * q^55 + (-z^13 - z^7) * q^59 + (z^22 + z^18 - z^9 + z^6) * q^60 + (z^12 - z^3) * q^61 + (z^16 - 2z^15 + z^14 - z^3 + z^2) q^63 - z^5 * q^64 + (z^24 - z^17 + z^8 - z) * q^65 + (z^24 + z^20 - z^15 - z^11 + z^8 + z^2) * q^75 + (z^2 - z) * q^77 + (-z^23 - z^7) * q^80 + (-z^23 + z^22 - z^11 + z^10 - z^9) * q^81 + (-z^17 + z^16 + z^4 - z^3) * q^84 + (z^20 - z^5) * q^89 + (-z^19 + z^18 + z^12 - z^11) * q^91 + (z^22 - z^21 - z^9) * q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q - 5 q^{4} - 5 q^{9}+O(q^{10}) \) 20 * q - 5 * q^4 - 5 * q^9	\( 20 q - 5 q^{4} - 5 q^{9} - 5 q^{16} - 5 q^{25} + 20 q^{33} + 15 q^{35} - 5 q^{36} - 10 q^{39} - 10 q^{45} - 5 q^{49} - 10 q^{53} - 10 q^{63} - 5 q^{64} - 10 q^{75} - 5 q^{81} - 10 q^{89}+O(q^{100}) \) 20 * q - 5 * q^4 - 5 * q^9 - 5 * q^16 - 5 * q^25 + 20 * q^33 + 15 * q^35 - 5 * q^36 - 10 * q^39 - 10 * q^45 - 5 * q^49 - 10 * q^53 - 10 * q^63 - 5 * q^64 - 10 * q^75 - 5 * q^81 - 10 * q^89

Introduction

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Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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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	1441.1.u.a

Level	$1441$
Weight	$1$
Character orbit	1441.u
Analytic conductor	$0.719$
Analytic rank	$0$
Dimension	$20$
Projective image	$D_{25}$
CM discriminant	-131
Inner twists	$4$

Self dual:	no
Analytic conductor:	\(0.719152683204\)
Analytic rank:	\(0\)
Dimension:	\(20\)
Relative dimension:	\(5\) over \(\Q(\zeta_{10})\)
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:	\(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	yes
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 130.5
Significant digits:
Format:

$p$	$F_p(T)$
$2$	\( T^{20} \) T^20
$3$	\( T^{20} + 5 T^{18} + \cdots + 1 \) T^20 + 5T^18 + 20T^16 + 2T^15 + 75T^14 - 5T^13 + 275T^12 + 5T^11 + 374T^10 + 200T^9 + 485T^8 + 150T^7 + 715T^6 + 498T^5 + 675T^4 + 260T^3 + 100T^2 + 15*T + 1
$5$	\( T^{20} + 5 T^{18} + \cdots + 1 \) T^20 + 5T^18 + 20T^16 + 2T^15 + 75T^14 - 5T^13 + 275T^12 + 5T^11 + 374T^10 + 200T^9 + 485T^8 + 150T^7 + 715T^6 + 498T^5 + 675T^4 + 260T^3 + 100T^2 + 15*T + 1
$7$	\( T^{20} + 5 T^{18} + \cdots + 1 \) T^20 + 5T^18 + 20T^16 + 2T^15 + 75T^14 + 20T^13 + 275T^12 + 30T^11 + 374T^10 - 175T^9 + 510T^8 - 475T^7 + 740T^6 - 752T^5 + 800T^4 - 390T^3 + 100T^2 - 10*T + 1
$11$	\( T^{20} + T^{15} + \cdots + 1 \) T^20 + T^15 + T^10 + T^5 + 1
$13$	\( T^{20} + 5 T^{18} + \cdots + 1 \) T^20 + 5T^18 + 20T^16 + 2T^15 + 75T^14 + 20T^13 + 275T^12 + 30T^11 + 374T^10 - 175T^9 + 510T^8 - 475T^7 + 740T^6 - 752T^5 + 800T^4 - 390T^3 + 100T^2 - 10*T + 1
$17$	\( T^{20} \) T^20
$19$	\( T^{20} \) T^20
$23$	\( T^{20} \) T^20
$29$	\( T^{20} \) T^20
$31$	\( T^{20} \) T^20
$37$	\( T^{20} \) T^20
$41$	\( T^{20} + 5 T^{18} + \cdots + 1 \) T^20 + 5T^18 + 20T^16 + 2T^15 + 75T^14 - 5T^13 + 275T^12 + 5T^11 + 374T^10 + 200T^9 + 485T^8 + 150T^7 + 715T^6 + 498T^5 + 675T^4 + 260T^3 + 100T^2 + 15*T + 1
$43$	\( (T^{10} - 10 T^{8} + \cdots - 1)^{2} \) (T^10 - 10T^8 + 35T^6 + T^5 - 50T^4 - 5T^3 + 25T^2 + 5T - 1)^2
$47$	\( T^{20} \) T^20
$53$	\( (T^{4} + 2 T^{3} + 4 T^{2} + \cdots + 1)^{5} \) (T^4 + 2T^3 + 4T^2 + 3*T + 1)^5
$59$	\( T^{20} + 5 T^{18} + \cdots + 1 \) T^20 + 5T^18 + 20T^16 + 2T^15 + 75T^14 + 20T^13 + 275T^12 + 30T^11 + 374T^10 - 175T^9 + 510T^8 - 475T^7 + 740T^6 - 752T^5 + 800T^4 - 390T^3 + 100T^2 - 10*T + 1
$61$	\( T^{20} + 5 T^{18} + \cdots + 1 \) T^20 + 5T^18 + 20T^16 + 2T^15 + 75T^14 - 5T^13 + 275T^12 + 5T^11 + 374T^10 + 200T^9 + 485T^8 + 150T^7 + 715T^6 + 498T^5 + 675T^4 + 260T^3 + 100T^2 + 15*T + 1
$67$	\( T^{20} \) T^20
$71$	\( T^{20} \) T^20
$73$	\( T^{20} \) T^20
$79$	\( T^{20} \) T^20
$83$	\( T^{20} \) T^20
$89$	\( (T^{2} + T - 1)^{10} \) (T^2 + T - 1)^10
$97$	\( T^{20} \) T^20
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\(n\)	\(133\)	\(1311\)
\(\chi(n)\)	\(-1\)	\(\zeta_{50}^{10}\)