LMFDB - Newform orbit 3875.1.dk.a

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3875,1,Mod(154,3875)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3875, base_ring=CyclotomicField(50))

chi = DirichletCharacter(H, H._module([31, 25]))

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

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

chi := DirichletCharacter("3875.154");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 3875 = 5^{3} \cdot 31 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	3875.dk (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:	$1.93387692395$
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 $\left(\mathbb{Z}/3875\mathbb{Z}\right)^\times$.

$n$	$251$	$3752$
$\chi(n)$	$-1$	$\zeta_{50}^{7}$

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} $

154.1

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

0.567290

−

1.43281i

−1.00216

−

0.941090i

−0.968583

−

0.248690i

−0.566335

0.779494i

−0.522555

0.245896i

−0.535827

0.844328i

−0.905793

1.24672i

309.1

−0.120759

−

0.219661i

0.502159

−

0.791276i

0.187381

−

0.982287i

−0.700215

0.227513i

−0.484624

−

0.0304900i

−0.728969

0.684547i

−0.238398

0.0774602i

464.1

−0.120759

0.219661i

0.502159

0.791276i

0.187381

0.982287i

−0.700215

−

0.227513i

−0.484624

0.0304900i

−0.728969

−

0.684547i

−0.238398

−

0.0774602i

619.1

−0.496398

1.93334i

−2.61510

−

1.43766i

0.637424

−

0.770513i

−1.15475

−

1.58937i

2.71124

−

2.88717i

0.929776

0.368125i

1.17325

1.61484i

929.1

−1.30113

−

1.07639i

0.346947

1.81876i

−0.0627905

−

0.998027i

−0.147338

0.202793i

0.692757

−

1.26012i

−0.968583

−

0.248690i

−0.992567

1.36615i

1084.1

1.77760

−

0.339095i

2.11510

−

0.837427i

0.992115

−

0.125333i

−0.473036

0.153699i

1.94789

−

1.23617i

−0.876307

−

0.481754i

1.72108

−

0.559214i

1239.1

1.34484

1.43211i

−0.179553

2.85391i

−0.876307

0.481754i

1.60601

0.521823i

−2.81486

2.32866i

0.425779

−

0.904827i

−1.86842

−

0.607087i

1394.1

0.419952

0.266509i

−0.320447

−

0.680985i

0.929776

0.368125i

0.804733

1.10762i

0.109255

−

0.864841i

−0.0627905

0.998027i

0.292352

0.402389i

1704.1

0.419952

−

0.266509i

−0.320447

0.680985i

0.929776

−

0.368125i

0.804733

−

1.10762i

0.109255

0.864841i

−0.0627905

−

0.998027i

0.292352

−

0.402389i

1859.1

0.171593

−

1.35830i

−0.846947

−

0.217459i

0.425779

0.904827i

1.46560

−

0.476203i

0.0632925

−

0.159859i

0.187381

−

0.982287i

1.30209

−

0.423073i

2014.1

0.871808

−

0.410241i

−0.0456737

0.0552100i

−0.728969

−

0.684547i

−1.89836

−

0.616814i

−0.256784

1.00011i

0.992115

0.125333i

−0.916350

−

0.297740i

2169.1

−1.30113

1.07639i

0.346947

−

1.81876i

−0.0627905

0.998027i

−0.147338

−

0.202793i

0.692757

1.26012i

−0.968583

0.248690i

−0.992567

−

1.36615i

2479.1

−0.496398

−

1.93334i

−2.61510

1.43766i

0.637424

0.770513i

−1.15475

1.58937i

2.71124

2.88717i

0.929776

−

0.368125i

1.17325

−

1.61484i

2634.1

0.871808

0.410241i

−0.0456737

−

0.0552100i

−0.728969

0.684547i

−1.89836

0.616814i

−0.256784

−

1.00011i

0.992115

−

0.125333i

−0.916350

0.297740i

2789.1

0.171593

1.35830i

−0.846947

0.217459i

0.425779

−

0.904827i

1.46560

0.476203i

0.0632925

0.159859i

0.187381

0.982287i

1.30209

0.423073i

2944.1

0.567290

1.43281i

−1.00216

0.941090i

−0.968583

0.248690i

−0.566335

−

0.779494i

−0.522555

−

0.245896i

−0.535827

−

0.844328i

−0.905793

−

1.24672i

3254.1

−0.734796

−

0.0462295i

−0.454326

−

0.0573948i

−0.535827

0.844328i

1.06369

−

1.46404i

1.05439

0.201136i

0.637424

0.770513i

0.432756

−

0.595638i

3409.1

1.34484

−

1.43211i

−0.179553

−

2.85391i

−0.876307

−

0.481754i

1.60601

−

0.521823i

−2.81486

−

2.32866i

0.425779

0.904827i

−1.86842

0.607087i

3564.1

1.77760

0.339095i

2.11510

0.837427i

0.992115

0.125333i

−0.473036

−

0.153699i

1.94789

1.23617i

−0.876307

0.481754i

1.72108

0.559214i

3719.1

−0.734796

0.0462295i

−0.454326

0.0573948i

−0.535827

−

0.844328i

1.06369

1.46404i

1.05439

−

0.201136i

0.637424

−

0.770513i

0.432756

0.595638i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
31.b	odd	2	1	CM by $\Q(\sqrt{-31}) $
125.h	even	50	1	inner
3875.dk	odd	50	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	3875.1.dk.a	✓	20
31.b	odd	2	1	CM	3875.1.dk.a	✓	20
125.h	even	50	1	inner	3875.1.dk.a	✓	20
3875.dk	odd	50	1	inner	3875.1.dk.a	✓	20

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
3875.1.dk.a	✓	20	1.a	even	1	1	trivial
3875.1.dk.a	✓	20	31.b	odd	2	1	CM
3875.1.dk.a	✓	20	125.h	even	50	1	inner
3875.1.dk.a	✓	20	3875.dk	odd	50	1	inner

This newform subspace can be constructed as the kernel of the linear operator $ T_{2}^{20} - 5 T_{2}^{19} + 15 T_{2}^{18} - 35 T_{2}^{17} + 70 T_{2}^{16} - 120 T_{2}^{15} + 175 T_{2}^{14} + \cdots + 5 $ T2^20 - 5*T2^19 + 15*T2^18 - 35*T2^17 + 70*T2^16 - 120*T2^15 + 175*T2^14 - 275*T2^13 + 475*T2^12 - 775*T2^11 + 1010*T2^10 - 675*T2^9 - 75*T2^8 + 900*T2^7 - 1025*T2^6 + 260*T2^5 + 350*T2^4 - 200*T2^3 + 25*T2^2 + 5 acting on $S_{1}^{\mathrm{new}}(3875, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{20} - 5 T^{19} + \cdots + 5 $ T^20 - 5T^19 + 15T^18 - 35T^17 + 70T^16 - 120T^15 + 175T^14 - 275T^13 + 475T^12 - 775T^11 + 1010T^10 - 675T^9 - 75T^8 + 900T^7 - 1025T^6 + 260T^5 + 350T^4 - 200T^3 + 25T^2 + 5
$3$	$ T^{20} $ T^20
$5$	$ T^{20} - T^{15} + \cdots + 1 $ T^20 - T^15 + T^10 - T^5 + 1
$7$	$ T^{20} - 5 T^{18} + \cdots + 5 $ T^20 - 5T^18 + 20T^16 - 75T^14 + 275T^12 - 100T^11 - 370T^10 + 125T^9 + 500T^8 - 125T^7 - 650T^6 + 750T^4 + 650T^3 + 250T^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 - 235T^13 + 165T^12 + 45T^11 - 246T^10 - 85T^9 + 805T^8 - 1270T^7 + 1315T^6 - 998T^5 + 660T^4 - 130T^3 - 5T^2 + 10*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^{17} + \cdots + 1 $ T^20 + 5T^17 + 5T^16 + 2T^15 + 25T^14 + 50T^13 + 45T^12 + 120T^11 + 379T^10 - 225T^9 + 925T^8 - 715T^7 + 1010T^6 - 747T^5 + 600T^4 - 175T^3 - 15T^2 + 10*T + 1
$43$	$ T^{20} $ T^20
$47$	$ T^{20} + 5 T^{17} + \cdots + 5 $ T^20 + 5T^17 - 5T^16 + 25T^14 + 75T^13 + 50T^12 + 125T^11 + 250T^10 - 125T^9 + 375T^8 + 475T^7 + 575T^6 + 620T^5 + 625T^4 + 500T^3 + 225T^2 + 50T + 5
$53$	$ T^{20} $ T^20
$59$	$ T^{20} + 5 T^{17} + \cdots + 1 $ T^20 + 5T^17 - 20T^16 + 2T^15 + 25T^14 - 75T^13 + 145T^12 + 95T^11 - 246T^10 + 275T^9 + 175T^8 - 465T^7 + 335T^6 - 122T^5 + 475T^4 - 300T^3 + 110T^2 - 15*T + 1
$61$	$ T^{20} $ T^20
$67$	$ T^{20} + 20 T^{19} + \cdots + 5 $ T^20 + 20T^19 + 190T^18 + 1140T^17 + 4845T^16 + 15505T^15 + 38775T^14 + 77625T^13 + 126425T^12 + 169325T^11 + 187760T^10 + 172975T^9 + 132450T^8 + 84075T^7 + 43975T^6 + 18760T^5 + 6425T^4 + 1725T^3 + 350T^2 + 50*T + 5
$71$	$ T^{20} - 32 T^{15} + \cdots + 1048576 $ T^20 - 32T^15 + 1024T^10 - 32768*T^5 + 1048576
$73$	$ T^{20} $ T^20
$79$	$ T^{20} $ T^20
$83$	$ T^{20} $ T^20
$89$	$ T^{20} $ T^20
$97$	$ T^{20} - 5 T^{17} + \cdots + 5 $ T^20 - 5T^17 - 5T^16 + 25T^14 - 75T^13 + 50T^12 - 125T^11 + 250T^10 + 125T^9 + 375T^8 - 475T^7 + 575T^6 - 620T^5 + 625T^4 - 500T^3 + 225T^2 - 50T + 5
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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:	\( N \)	\(=\)	\( 3875 = 5^{3} \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	3875.dk (of order \(50\), degree \(20\), minimal)

\(f(q)\)	\(=\)	\( q + ( - \zeta_{50}^{20} + \zeta_{50}^{12}) q^{2} + (\zeta_{50}^{24} + \cdots + \zeta_{50}^{7}) q^{4}+ \cdots - \zeta_{50}^{24} q^{9} +O(q^{10}) \) q + (-z^20 + z^12) * q^2 + (z^24 - z^15 + z^7) * q^4 + z^19 * q^5 + (z^23 + z^22) * q^7 + (z^19 - z^11 - z^10 - z^2) * q^8 - z^24 * q^9	\( q + ( - \zeta_{50}^{20} + \zeta_{50}^{12}) q^{2} + (\zeta_{50}^{24} + \cdots + \zeta_{50}^{7}) q^{4}+ \cdots + ( - \zeta_{50}^{16} + \cdots + \zeta_{50}^{6}) q^{98}+O(q^{100}) \) q + (-z^20 + z^12) * q^2 + (z^24 - z^15 + z^7) * q^4 + z^19 * q^5 + (z^23 + z^22) * q^7 + (z^19 - z^11 - z^10 - z^2) * q^8 - z^24 * q^9 + (z^14 - z^6) * q^10 + (z^18 + z^17 - z^10 - z^9) * q^14 + (-z^23 - z^22 + z^14 + z^6 - z^5) * q^16 + (-z^19 + z^11) * q^18 + (-z^16 - z^10) * q^19 + (-z^18 + z^9 - z) * q^20 - z^13 * q^25 + (-z^22 - z^21 + z^13 + z^12 - z^5 - z^4) * q^28 - z^18 * q^31 + (-z^18 - z^17 + z^10 + z^9 + z - 1) * q^32 + (-z^17 - z^16) * q^35 + (z^23 - z^14 + z^6) * q^36 + (-z^22 - z^11 - z^5 + z^3) * q^38 + (z^21 - z^13 + z^5 + z^4) * q^40 + (z^6 + z^2) * q^41 + z^18 * q^45 + (-z^16 - z^13) * q^47 + (-z^21 - z^20 - z^19) * q^49 + (-z^8 + 1) * q^50 + (z^24 - z^17 - z^16 + z^9 + 2z^8 + z^7 - 1) q^56 + (-z^7 + z^6) * q^59 + (-z^13 + z^5) * q^62 + (z^22 + z^21) * q^63 + (z^22 + z^21 + z^20 - z^13 - z^12 - z^5 - z^4) * q^64 + (z^16 - 1) * q^67 + (-z^12 - z^11 + z^4 + z^3) * q^70 - z^2 * q^71 + (z^18 - z^10 - z^9 + z) * q^72 + (-z^23 - z^17 + z^15 + z^9 - z^6 - 1) * q^76 + (-z^24 + z^17 + z^16 - z^8 + 1) * q^80 - z^23 * q^81 + (-z^22 + z^18 + z^14 + z) * q^82 + (z^13 - z^5) * q^90 + (-z^11 - z^8 + z^3 + 1) * q^94 + (z^10 + z^4) * q^95 + (z^23 - z^11) * q^97 + (-z^16 - z^15 - z^14 + z^8 + z^7 + z^6) * q^98
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q + 5 q^{2} - 5 q^{4} + 5 q^{8}+O(q^{10}) \) 20 * q + 5 * q^2 - 5 * q^4 + 5 * q^8	\( 20 q + 5 q^{2} - 5 q^{4} + 5 q^{8} + 5 q^{14} - 5 q^{16} + 5 q^{19} - 5 q^{28} - 25 q^{32} - 5 q^{38} + 5 q^{40} + 5 q^{49} + 20 q^{50} - 20 q^{56} + 5 q^{62} - 20 q^{67} + 5 q^{72} - 15 q^{76} + 20 q^{80} - 5 q^{90} + 20 q^{94} - 5 q^{95} - 5 q^{98}+O(q^{100}) \) 20 * q + 5 * q^2 - 5 * q^4 + 5 * q^8 + 5 * q^14 - 5 * q^16 + 5 * q^19 - 5 * q^28 - 25 * q^32 - 5 * q^38 + 5 * q^40 + 5 * q^49 + 20 * q^50 - 20 * q^56 + 5 * q^62 - 20 * q^67 + 5 * q^72 - 15 * q^76 + 20 * q^80 - 5 * q^90 + 20 * q^94 - 5 * q^95 - 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	3875.1.dk.a

Level	$3875$
Weight	$1$
Character orbit	3875.dk
Analytic conductor	$1.934$
Analytic rank	$0$
Dimension	$20$
Projective image	$D_{50}$
CM discriminant	-31
Inner twists	$4$

Self dual:	no
Analytic conductor:	\(1.93387692395\)
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 154.1
Significant digits:
Format:

$p$	$F_p(T)$
$2$	\( T^{20} - 5 T^{19} + \cdots + 5 \) T^20 - 5T^19 + 15T^18 - 35T^17 + 70T^16 - 120T^15 + 175T^14 - 275T^13 + 475T^12 - 775T^11 + 1010T^10 - 675T^9 - 75T^8 + 900T^7 - 1025T^6 + 260T^5 + 350T^4 - 200T^3 + 25T^2 + 5
$3$	\( T^{20} \) T^20
$5$	\( T^{20} - T^{15} + \cdots + 1 \) T^20 - T^15 + T^10 - T^5 + 1
$7$	\( T^{20} - 5 T^{18} + \cdots + 5 \) T^20 - 5T^18 + 20T^16 - 75T^14 + 275T^12 - 100T^11 - 370T^10 + 125T^9 + 500T^8 - 125T^7 - 650T^6 + 750T^4 + 650T^3 + 250T^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 - 235T^13 + 165T^12 + 45T^11 - 246T^10 - 85T^9 + 805T^8 - 1270T^7 + 1315T^6 - 998T^5 + 660T^4 - 130T^3 - 5T^2 + 10*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
$41$	\( T^{20} + 5 T^{17} + \cdots + 1 \) T^20 + 5T^17 + 5T^16 + 2T^15 + 25T^14 + 50T^13 + 45T^12 + 120T^11 + 379T^10 - 225T^9 + 925T^8 - 715T^7 + 1010T^6 - 747T^5 + 600T^4 - 175T^3 - 15T^2 + 10*T + 1
$43$	\( T^{20} \) T^20
$47$	\( T^{20} + 5 T^{17} + \cdots + 5 \) T^20 + 5T^17 - 5T^16 + 25T^14 + 75T^13 + 50T^12 + 125T^11 + 250T^10 - 125T^9 + 375T^8 + 475T^7 + 575T^6 + 620T^5 + 625T^4 + 500T^3 + 225T^2 + 50T + 5
$53$	\( T^{20} \) T^20
$59$	\( T^{20} + 5 T^{17} + \cdots + 1 \) T^20 + 5T^17 - 20T^16 + 2T^15 + 25T^14 - 75T^13 + 145T^12 + 95T^11 - 246T^10 + 275T^9 + 175T^8 - 465T^7 + 335T^6 - 122T^5 + 475T^4 - 300T^3 + 110T^2 - 15*T + 1
$61$	\( T^{20} \) T^20
$67$	\( T^{20} + 20 T^{19} + \cdots + 5 \) T^20 + 20T^19 + 190T^18 + 1140T^17 + 4845T^16 + 15505T^15 + 38775T^14 + 77625T^13 + 126425T^12 + 169325T^11 + 187760T^10 + 172975T^9 + 132450T^8 + 84075T^7 + 43975T^6 + 18760T^5 + 6425T^4 + 1725T^3 + 350T^2 + 50*T + 5
$71$	\( T^{20} - 32 T^{15} + \cdots + 1048576 \) T^20 - 32T^15 + 1024T^10 - 32768*T^5 + 1048576
$73$	\( T^{20} \) T^20
$79$	\( T^{20} \) T^20
$83$	\( T^{20} \) T^20
$89$	\( T^{20} \) T^20
$97$	\( T^{20} - 5 T^{17} + \cdots + 5 \) T^20 - 5T^17 - 5T^16 + 25T^14 - 75T^13 + 50T^12 - 125T^11 + 250T^10 + 125T^9 + 375T^8 - 475T^7 + 575T^6 - 620T^5 + 625T^4 - 500T^3 + 225T^2 - 50T + 5
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\(n\)	\(251\)	\(3752\)
\(\chi(n)\)	\(-1\)	\(\zeta_{50}^{7}\)