LMFDB - Newform orbit 3131.1.de.a

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3131,1,Mod(30,3131)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3131, base_ring=CyclotomicField(50))

chi = DirichletCharacter(H, H._module([25, 47]))

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

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

chi := DirichletCharacter("3131.30");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 3131 = 31 \cdot 101 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	3131.de (of order $50$, degree $20$, 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:	$1.56257255455$
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:	$\Z[a_1, \ldots, a_{4}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Projective image:	$D_{50}$
Projective field:	Galois closure of $\mathbb{Q}[x]/(x^{50} - \cdots)$

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3131\mathbb{Z}\right)^\times$.

$n$	$809$	$2729$
$\chi(n)$	$-1$	$\zeta_{50}^{9}$

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} $

30.1

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

1.65875

−

0.316423i

1.72154

−

0.681604i

0.0235315

−

0.123357i

0.211645

1.67534i

1.21413

−

0.770513i

0.637424

−

0.770513i

−

0.212063i

123.1

0.450043

0.211774i

−0.479734

−

0.579899i

−0.791759

−

1.68257i

0.340480

0.362574i

−0.216787

−

0.844328i

−0.535827

−

0.844328i

−

0.924904i

247.1

−0.666178

1.68257i

−1.65829

−

1.55724i

−1.84489

0.730444i

−0.450043

1.75280i

2.08747

−

0.982287i

0.187381

−

0.982287i

−

3.59077i

278.1

1.96070

−

0.123357i

2.83700

−

0.358397i

−0.110048

1.74915i

−1.65875

1.05267i

3.58852

−

0.684547i

−0.728969

−

0.684547i

3.44314i

526.1

−0.742395

0.614163i

−0.0134265

0.0703844i

0.929324

−

1.12336i

0.961606

0.0604991i

−0.497433

−

0.904827i

0.425779

−

0.904827i

1.40473i

619.1

−0.961606

1.74915i

−1.59903

−

2.51967i

1.11716

−

0.614163i

−1.96070

0.374023i

3.95281

−

0.248690i

−0.968583

−

0.248690i

2.54466i

929.1

−1.05491

−

1.12336i

−0.0863221

1.37205i

−1.41213

−

1.32608i

0.742395

1.35041i

0.444986

−

0.368125i

0.929776

−

0.368125i

2.98522i

991.1

−0.0922765

−

0.730444i

0.443550

−

0.113884i

1.06320

0.134314i

0.666178

0.313480i

−0.395147

−

0.998027i

−0.0627905

−

0.998027i

−

0.789004i

1053.1

−0.340480

−

1.32608i

−0.766259

0.421255i

0.824805

0.211774i

1.05491

−

0.872693i

−0.117696

−

0.125333i

0.992115

−

0.125333i

−

1.16586i

1115.1

1.96070

0.123357i

2.83700

0.358397i

−0.110048

−

1.74915i

−1.65875

−

1.05267i

3.58852

0.684547i

−0.728969

0.684547i

−

3.44314i

1611.1

−1.05491

1.12336i

−0.0863221

−

1.37205i

−1.41213

1.32608i

0.742395

−

1.35041i

0.444986

0.368125i

0.929776

0.368125i

−

2.98522i

1766.1

−0.211645

−

0.134314i

−0.399026

−

0.847973i

0.200808

0.316423i

0.0922765

−

0.233064i

−0.0608596

0.481754i

−0.876307

0.481754i

−

0.0939404i

1952.1

−0.211645

0.134314i

−0.399026

0.847973i

0.200808

−

0.316423i

0.0922765

0.233064i

−0.0608596

−

0.481754i

−0.876307

−

0.481754i

0.0939404i

1983.1

1.65875

0.316423i

1.72154

0.681604i

0.0235315

0.123357i

0.211645

−

1.67534i

1.21413

0.770513i

0.637424

0.770513i

0.212063i

2231.1

−0.666178

−

1.68257i

−1.65829

1.55724i

−1.84489

−

0.730444i

−0.450043

−

1.75280i

2.08747

0.982287i

0.187381

0.982287i

3.59077i

2572.1

−0.340480

1.32608i

−0.766259

−

0.421255i

0.824805

−

0.211774i

1.05491

0.872693i

−0.117696

0.125333i

0.992115

0.125333i

1.16586i

2696.1

−0.961606

−

1.74915i

−1.59903

2.51967i

1.11716

0.614163i

−1.96070

−

0.374023i

3.95281

0.248690i

−0.968583

0.248690i

−

2.54466i

2851.1

0.450043

−

0.211774i

−0.479734

0.579899i

−0.791759

1.68257i

0.340480

−

0.362574i

−0.216787

0.844328i

−0.535827

0.844328i

0.924904i

2913.1

−0.0922765

0.730444i

0.443550

0.113884i

1.06320

−

0.134314i

0.666178

−

0.313480i

−0.395147

0.998027i

−0.0627905

0.998027i

0.789004i

3006.1

−0.742395

−

0.614163i

−0.0134265

−

0.0703844i

0.929324

1.12336i

0.961606

−

0.0604991i

−0.497433

0.904827i

0.425779

0.904827i

−

1.40473i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
31.b	odd	2	1	CM by $\Q(\sqrt{-31}) $
101.h	even	50	1	inner
3131.de	odd	50	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	3131.1.de.a	✓	20
31.b	odd	2	1	CM	3131.1.de.a	✓	20
101.h	even	50	1	inner	3131.1.de.a	✓	20
3131.de	odd	50	1	inner	3131.1.de.a	✓	20

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
3131.1.de.a	✓	20	1.a	even	1	1	trivial
3131.1.de.a	✓	20	31.b	odd	2	1	CM
3131.1.de.a	✓	20	101.h	even	50	1	inner
3131.1.de.a	✓	20	3131.de	odd	50	1	inner

This newform subspace can be constructed as the kernel of the linear operator $ T_{2}^{20} - 20 T_{2}^{17} - 5 T_{2}^{16} + 150 T_{2}^{14} + 75 T_{2}^{13} + 25 T_{2}^{12} - 525 T_{2}^{11} + \cdots + 5 $ T2^20 - 20*T2^17 - 5*T2^16 + 150*T2^14 + 75*T2^13 + 25*T2^12 - 525*T2^11 - 375*T2^10 - 250*T2^9 + 750*T2^8 + 750*T2^7 + 650*T2^6 - 5*T2^5 - 125*T2^3 + 25*T2^2 + 25*T2 + 5 acting on $S_{1}^{\mathrm{new}}(3131, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{20} - 20 T^{17} + \cdots + 5 $ T^20 - 20T^17 - 5T^16 + 150T^14 + 75T^13 + 25T^12 - 525T^11 - 375T^10 - 250T^9 + 750T^8 + 750T^7 + 650T^6 - 5T^5 - 125T^3 + 25T^2 + 25*T + 5
$3$	$ T^{20} $ T^20
$5$	$ 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
$7$	$ T^{20} + 5 T^{17} + \cdots + 5 $ T^20 + 5T^17 - 5T^16 + 25T^14 - 50T^13 + 25T^12 + 125T^11 - 375T^10 + 375T^9 + 500T^8 - 2525T^7 + 4400T^6 - 4380T^5 + 2750T^4 - 1125T^3 + 300T^2 - 50T + 5
$11$	$ T^{20} $ T^20
$13$	$ T^{20} $ T^20
$17$	$ T^{20} $ T^20
$19$	$ T^{20} + 5 T^{19} + \cdots + 1 $ T^20 + 5T^19 + 15T^18 + 35T^17 + 70T^16 + 122T^15 + 195T^14 + 310T^13 + 365T^12 + 205T^11 - 246T^10 - 815T^9 - 770T^8 - 55T^7 + 915T^6 + 998T^5 + 435T^4 + 80T^3 + 70T^2 + 15*T + 1
$23$	$ T^{20} $ T^20
$29$	$ T^{20} $ T^20
$31$	$ T^{20} - T^{15} + \cdots + 1 $ T^20 - T^15 + T^10 - T^5 + 1
$37$	$ T^{20} $ T^20
$41$	$ T^{20} - 5 T^{18} + \cdots + 5 $ T^20 - 5T^18 + 20T^16 - 75T^14 - 25T^13 + 275T^12 + 125T^11 - 370T^10 - 500T^9 + 475T^8 + 500T^7 - 375T^6 + 375T^4 - 50T^3 + 25T + 5
$43$	$ T^{20} $ T^20
$47$	$ T^{20} + 7 T^{15} + \cdots + 1 $ T^20 + 7T^15 + 124T^10 + 18*T^5 + 1
$53$	$ T^{20} $ T^20
$59$	$ T^{20} - 5 T^{19} + \cdots + 5 $ T^20 - 5T^19 + 15T^18 - 35T^17 + 70T^16 - 120T^15 + 200T^14 - 325T^13 + 475T^12 - 525T^11 + 135T^10 + 525T^9 - 925T^8 + 650T^7 + 75T^6 - 240T^5 + 400T^4 - 375T^3 + 200T^2 - 50*T + 5
$61$	$ T^{20} $ T^20
$67$	$ T^{20} - 5 T^{19} + \cdots + 5 $ T^20 - 5T^19 + 15T^18 - 35T^17 + 70T^16 - 120T^15 + 175T^14 - 200T^13 + 100T^12 + 135T^10 - 675T^9 + 1525T^8 - 1675T^7 + 1025T^6 - 240T^5 + 75T^4 - 50T^3 + 25*T + 5
$71$	$ T^{20} + 5 T^{19} + \cdots + 1 $ T^20 + 5T^19 + 15T^18 + 35T^17 + 70T^16 + 122T^15 + 195T^14 + 310T^13 + 540T^12 + 730T^11 + 629T^10 + 85T^9 - 645T^8 - 905T^7 - 735T^6 - 252T^5 + 435T^4 + 380T^3 + 120T^2 + 15*T + 1
$73$	$ T^{20} $ T^20
$79$	$ T^{20} $ T^20
$83$	$ T^{20} $ T^20
$89$	$ T^{20} $ T^20
$97$	$ T^{20} - 7 T^{15} + \cdots + 1 $ T^20 - 7T^15 + 124T^10 - 18*T^5 + 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:	\( N \)	\(=\)	\( 3131 = 31 \cdot 101 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	3131.de (of order \(50\), degree \(20\), minimal)

\(f(q)\)	\(=\)	\( q + (\zeta_{50}^{18} - \zeta_{50}^{16}) q^{2} + ( - \zeta_{50}^{11} + \cdots - \zeta_{50}^{7}) q^{4}+ \cdots + \zeta_{50}^{21} q^{9}+O(q^{10}) \) q + (z^18 - z^16) * q^2 + (-z^11 + z^9 - z^7) * q^4 + (-z^22 + z^19) * q^5 + (-z^4 + z^2) * q^7 + (z^23 + z^4 + z^2 - 1) * q^8 + z^21 * q^9	\( q + (\zeta_{50}^{18} - \zeta_{50}^{16}) q^{2} + ( - \zeta_{50}^{11} + \cdots - \zeta_{50}^{7}) q^{4}+ \cdots + ( - 2 \zeta_{50}^{24} + \cdots - \zeta_{50}) q^{98}+O(q^{100}) \) q + (z^18 - z^16) * q^2 + (-z^11 + z^9 - z^7) * q^4 + (-z^22 + z^19) * q^5 + (-z^4 + z^2) * q^7 + (z^23 + z^4 + z^2 - 1) * q^8 + z^21 * q^9 + (z^15 - z^13 - z^12 + z^10) * q^10 + (-z^22 + 2z^20 - z^18) q^14 + (z^22 - z^20 + z^18 - z^16 + z^14) * q^16 + (-z^14 + z^12) * q^18 + (z^10 + z^4) * q^19 + (-z^8 + z^6 + z^5 - z^4 - z^3 + z) * q^20 + (-z^19 + z^16 - z^13) * q^25 + (z^15 - 2z^13 + 2z^11 - z^9) * q^28 + z^3 * q^31 + (-z^15 + z^13 - z^11 + z^9 + z^7 + z^5) * q^32 + (-z^24 - z^23 + z^21 - z) * q^35 + (z^7 - z^5 + z^3) * q^36 + (z^22 - z^20 - z^3 + z) * q^38 + (z^24 + z^23 - z^22 - z^21 + z^20 + z^19 - z^17 + z) * q^40 + (-z^21 + z^9) * q^41 + (z^18 - z^15) * q^45 + (-z^21 - z) * q^47 + (z^8 - z^6 + z^4) * q^49 + (z^12 - z^10 - z^9 + z^7 + z^6 - z^4) * q^50 + (-z^8 + 2z^6 - 2z^4 + z^2 - 1) * q^56 + (-z^20 + z^16) * q^59 + (z^21 - z^19) * q^62 + (z^23 + 1) * q^63 + (z^23 - z^21 + z^8 + z^6 - z^4 + z^2 - 1) * q^64 + (-z^20 - z^9) * q^67 + (-z^19 + 2z^17 + z^16 - z^15 - 2z^14 + z^12) * q^70 + (z^16 - z^5) * q^71 + (-z^23 + z^21 - z^19 - 1) * q^72 + (-z^21 + z^19 - z^17 - z^15 + z^13 - z^11) * q^76 + (z^19 - z^17 - z^16 + z^15 + z^14 - z^13 - z^12 + z^11 + z^10 - z^8) * q^80 - z^17 * q^81 + (z^14 - z^12 - z^2 + 1) * q^82 + (-z^11 + z^9 + z^8 - z^6) * q^90 + (-z^19 + z^17 + z^14 - z^12) * q^94 + (z^23 + z^7 - z^4 + z) * q^95 + (-z^24 + z^19) * q^97 + (-2z^24 + 2z^22 - z^20 - z) * q^98
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q + 20 q^{8}+O(q^{10}) \) 20 * q + 20 * q^8	\( 20 q + 20 q^{8} - 10 q^{14} + 5 q^{16} - 5 q^{19} + 5 q^{20} + 5 q^{28} - 5 q^{36} + 5 q^{38} - 5 q^{40} - 5 q^{45} + 5 q^{50} - 20 q^{56} + 5 q^{59} + 20 q^{63} + 20 q^{64} + 5 q^{67} - 5 q^{70} - 5 q^{71} - 20 q^{72} - 5 q^{76} + 20 q^{82} + 5 q^{98}+O(q^{100}) \) 20 * q + 20 * q^8 - 10 * q^14 + 5 * q^16 - 5 * q^19 + 5 * q^20 + 5 * q^28 - 5 * q^36 + 5 * q^38 - 5 * q^40 - 5 * q^45 + 5 * q^50 - 20 * q^56 + 5 * q^59 + 20 * q^63 + 20 * q^64 + 5 * q^67 - 5 * q^70 - 5 * q^71 - 20 * q^72 - 5 * q^76 + 20 * q^82 + 5 * q^98

Introduction

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Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

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	3131.1.de.a

Level	$3131$
Weight	$1$
Character orbit	3131.de
Analytic conductor	$1.563$
Analytic rank	$0$
Dimension	$20$
Projective image	$D_{50}$
CM discriminant	-31
Inner twists	$4$

Self dual:	no
Analytic conductor:	\(1.56257255455\)
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:	\(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	yes
Projective image:	\(D_{50}\)
Projective field:	Galois closure of \(\mathbb{Q}[x]/(x^{50} - \cdots)\)

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 30.1
Significant digits:
Format:

$p$	$F_p(T)$
$2$	\( T^{20} - 20 T^{17} + \cdots + 5 \) T^20 - 20T^17 - 5T^16 + 150T^14 + 75T^13 + 25T^12 - 525T^11 - 375T^10 - 250T^9 + 750T^8 + 750T^7 + 650T^6 - 5T^5 - 125T^3 + 25T^2 + 25*T + 5
$3$	\( T^{20} \) T^20
$5$	\( T^{20} - 5 T^{17} + \cdots + 1 \) 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
$7$	\( T^{20} + 5 T^{17} + \cdots + 5 \) T^20 + 5T^17 - 5T^16 + 25T^14 - 50T^13 + 25T^12 + 125T^11 - 375T^10 + 375T^9 + 500T^8 - 2525T^7 + 4400T^6 - 4380T^5 + 2750T^4 - 1125T^3 + 300T^2 - 50T + 5
$11$	\( T^{20} \) T^20
$13$	\( T^{20} \) T^20
$17$	\( T^{20} \) T^20
$19$	\( T^{20} + 5 T^{19} + \cdots + 1 \) T^20 + 5T^19 + 15T^18 + 35T^17 + 70T^16 + 122T^15 + 195T^14 + 310T^13 + 365T^12 + 205T^11 - 246T^10 - 815T^9 - 770T^8 - 55T^7 + 915T^6 + 998T^5 + 435T^4 + 80T^3 + 70T^2 + 15*T + 1
$23$	\( T^{20} \) T^20
$29$	\( T^{20} \) T^20
$31$	\( T^{20} - T^{15} + \cdots + 1 \) T^20 - T^15 + T^10 - T^5 + 1
$37$	\( T^{20} \) T^20
$41$	\( T^{20} - 5 T^{18} + \cdots + 5 \) T^20 - 5T^18 + 20T^16 - 75T^14 - 25T^13 + 275T^12 + 125T^11 - 370T^10 - 500T^9 + 475T^8 + 500T^7 - 375T^6 + 375T^4 - 50T^3 + 25T + 5
$43$	\( T^{20} \) T^20
$47$	\( T^{20} + 7 T^{15} + \cdots + 1 \) T^20 + 7T^15 + 124T^10 + 18*T^5 + 1
$53$	\( T^{20} \) T^20
$59$	\( T^{20} - 5 T^{19} + \cdots + 5 \) T^20 - 5T^19 + 15T^18 - 35T^17 + 70T^16 - 120T^15 + 200T^14 - 325T^13 + 475T^12 - 525T^11 + 135T^10 + 525T^9 - 925T^8 + 650T^7 + 75T^6 - 240T^5 + 400T^4 - 375T^3 + 200T^2 - 50*T + 5
$61$	\( T^{20} \) T^20
$67$	\( T^{20} - 5 T^{19} + \cdots + 5 \) T^20 - 5T^19 + 15T^18 - 35T^17 + 70T^16 - 120T^15 + 175T^14 - 200T^13 + 100T^12 + 135T^10 - 675T^9 + 1525T^8 - 1675T^7 + 1025T^6 - 240T^5 + 75T^4 - 50T^3 + 25*T + 5
$71$	\( T^{20} + 5 T^{19} + \cdots + 1 \) T^20 + 5T^19 + 15T^18 + 35T^17 + 70T^16 + 122T^15 + 195T^14 + 310T^13 + 540T^12 + 730T^11 + 629T^10 + 85T^9 - 645T^8 - 905T^7 - 735T^6 - 252T^5 + 435T^4 + 380T^3 + 120T^2 + 15*T + 1
$73$	\( T^{20} \) T^20
$79$	\( T^{20} \) T^20
$83$	\( T^{20} \) T^20
$89$	\( T^{20} \) T^20
$97$	\( T^{20} - 7 T^{15} + \cdots + 1 \) T^20 - 7T^15 + 124T^10 - 18*T^5 + 1
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\(n\)	\(809\)	\(2729\)
\(\chi(n)\)	\(-1\)	\(\zeta_{50}^{9}\)