LMFDB - Newform orbit 2156.1.bk.b

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2156,1,Mod(471,2156)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2156, base_ring=CyclotomicField(30))

chi = DirichletCharacter(H, H._module([15, 10, 18]))

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

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

chi := DirichletCharacter("2156.471");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 2156 = 2^{2} \cdot 7^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	2156.bk (of order $30$, degree $8$, 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:	$1.07598416724$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\Q(\zeta_{15})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - x^{7} + x^{5} - x^{4} + x^{3} - x + 1 $ x^8 - x^7 + x^5 - x^4 + x^3 - x + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Projective image:	$D_{10}$
Projective field:	Galois closure of 10.0.527027889439744.1

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2156\mathbb{Z}\right)^\times$.

$n$	$981$	$1079$	$1277$
$\chi(n)$	$-\zeta_{30}^{3}$	$-1$	$\zeta_{30}^{10}$

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} $

471.1

−0.978148	−	0.207912i
0.913545	+	0.406737i
0.669131	+	0.743145i
0.913545	−	0.406737i
0.669131	−	0.743145i
−0.978148	+	0.207912i
−0.104528	+	0.994522i
−0.104528	−	0.994522i

0.104528

0.994522i

−0.978148

0.207912i

−0.309017

−

0.951057i

0.669131

0.743145i

851.1

0.978148

−

0.207912i

0.913545

−

0.406737i

0.809017

−

0.587785i

−0.104528

0.994522i

863.1

−0.913545

0.406737i

0.669131

−

0.743145i

−0.309017

0.951057i

−0.978148

−

0.207912i

1059.1

0.978148

0.207912i

0.913545

0.406737i

0.809017

0.587785i

−0.104528

−

0.994522i

1439.1

−0.913545

−

0.406737i

0.669131

0.743145i

−0.309017

−

0.951057i

−0.978148

0.207912i

1831.1

0.104528

−

0.994522i

−0.978148

−

0.207912i

−0.309017

0.951057i

0.669131

−

0.743145i

2027.1

−0.669131

0.743145i

−0.104528

−

0.994522i

0.809017

0.587785i

0.913545

0.406737i

2039.1

−0.669131

−

0.743145i

−0.104528

0.994522i

0.809017

−

0.587785i

0.913545

−

0.406737i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
7.b	odd	2	1	CM by $\Q(\sqrt{-7}) $
7.c	even	3	1	inner
7.d	odd	6	1	inner
44.h	odd	10	1	inner
308.t	even	10	1	inner
308.bb	odd	30	1	inner
308.be	even	30	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	2156.1.bk.b		8
4.b	odd	2	1		2156.1.bk.a		8
7.b	odd	2	1	CM	2156.1.bk.b		8
7.c	even	3	1		2156.1.v.a	✓	4
7.c	even	3	1	inner	2156.1.bk.b		8
7.d	odd	6	1		2156.1.v.a	✓	4
7.d	odd	6	1	inner	2156.1.bk.b		8
11.c	even	5	1		2156.1.bk.a		8
28.d	even	2	1		2156.1.bk.a		8
28.f	even	6	1		2156.1.v.b	yes	4
28.f	even	6	1		2156.1.bk.a		8
28.g	odd	6	1		2156.1.v.b	yes	4
28.g	odd	6	1		2156.1.bk.a		8
44.h	odd	10	1	inner	2156.1.bk.b		8
77.j	odd	10	1		2156.1.bk.a		8
77.m	even	15	1		2156.1.v.b	yes	4
77.m	even	15	1		2156.1.bk.a		8
77.p	odd	30	1		2156.1.v.b	yes	4
77.p	odd	30	1		2156.1.bk.a		8
308.t	even	10	1	inner	2156.1.bk.b		8
308.bb	odd	30	1		2156.1.v.a	✓	4
308.bb	odd	30	1	inner	2156.1.bk.b		8
308.be	even	30	1		2156.1.v.a	✓	4
308.be	even	30	1	inner	2156.1.bk.b		8

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
2156.1.v.a	✓	4	7.c	even	3	1
2156.1.v.a	✓	4	7.d	odd	6	1
2156.1.v.a	✓	4	308.bb	odd	30	1
2156.1.v.a	✓	4	308.be	even	30	1
2156.1.v.b	yes	4	28.f	even	6	1
2156.1.v.b	yes	4	28.g	odd	6	1
2156.1.v.b	yes	4	77.m	even	15	1
2156.1.v.b	yes	4	77.p	odd	30	1
2156.1.bk.a		8	4.b	odd	2	1
2156.1.bk.a		8	11.c	even	5	1
2156.1.bk.a		8	28.d	even	2	1
2156.1.bk.a		8	28.f	even	6	1
2156.1.bk.a		8	28.g	odd	6	1
2156.1.bk.a		8	77.j	odd	10	1
2156.1.bk.a		8	77.m	even	15	1
2156.1.bk.a		8	77.p	odd	30	1
2156.1.bk.b		8	1.a	even	1	1	trivial
2156.1.bk.b		8	7.b	odd	2	1	CM
2156.1.bk.b		8	7.c	even	3	1	inner
2156.1.bk.b		8	7.d	odd	6	1	inner
2156.1.bk.b		8	44.h	odd	10	1	inner
2156.1.bk.b		8	308.t	even	10	1	inner
2156.1.bk.b		8	308.bb	odd	30	1	inner
2156.1.bk.b		8	308.be	even	30	1	inner

This newform subspace can be constructed as the kernel of the linear operator $ T_{71}^{4} + 5T_{71}^{3} + 10T_{71}^{2} + 10T_{71} + 5 $ T71^4 + 5*T71^3 + 10*T71^2 + 10*T71 + 5 acting on $S_{1}^{\mathrm{new}}(2156, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{8} + T^{7} - T^{5} + \cdots + 1 $ T^8 + T^7 - T^5 - T^4 - T^3 + T + 1
$3$	$ T^{8} $ T^8
$5$	$ T^{8} $ T^8
$7$	$ T^{8} $ T^8
$11$	$ T^{8} - T^{7} + T^{5} + \cdots + 1 $ T^8 - T^7 + T^5 - T^4 + T^3 - T + 1
$13$	$ T^{8} $ T^8
$17$	$ T^{8} $ T^8
$19$	$ T^{8} $ T^8
$23$	$ T^{8} - 5 T^{6} + \cdots + 25 $ T^8 - 5T^6 + 20T^4 - 25*T^2 + 25
$29$	$ (T^{4} - 2 T^{3} + 4 T^{2} + \cdots + 1)^{2} $ (T^4 - 2T^3 + 4T^2 - 3*T + 1)^2
$31$	$ T^{8} $ T^8
$37$	$ T^{8} - 3 T^{7} + \cdots + 1 $ T^8 - 3T^7 + 5T^6 - 8T^5 + 9T^4 - 2T^3 - 2T + 1
$41$	$ T^{8} $ T^8
$43$	$ (T^{4} + 5 T^{2} + 5)^{2} $ (T^4 + 5*T^2 + 5)^2
$47$	$ T^{8} $ T^8
$53$	$ T^{8} + 3 T^{7} + \cdots + 1 $ T^8 + 3T^7 + 5T^6 + 8T^5 + 9T^4 + 2T^3 + 2T + 1
$59$	$ T^{8} $ T^8
$61$	$ T^{8} $ T^8
$67$	$ T^{8} - 5 T^{6} + \cdots + 25 $ T^8 - 5T^6 + 20T^4 - 25*T^2 + 25
$71$	$ (T^{4} + 5 T^{3} + 10 T^{2} + \cdots + 5)^{2} $ (T^4 + 5T^3 + 10T^2 + 10*T + 5)^2
$73$	$ T^{8} $ T^8
$79$	$ T^{8} + 5 T^{7} + \cdots + 25 $ T^8 + 5T^7 + 15T^6 + 30T^5 + 45T^4 + 50T^3 + 50T^2 + 50*T + 25
$83$	$ T^{8} $ T^8
$89$	$ T^{8} $ T^8
$97$	$ T^{8} $ T^8
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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:	\( N \)	\(=\)	\( 2156 = 2^{2} \cdot 7^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	2156.bk (of order \(30\), degree \(8\), not minimal)

\(f(q)\)	\(=\)	\( q + \zeta_{30}^{7} q^{2} + \zeta_{30}^{14} q^{4} - \zeta_{30}^{6} q^{8} + \zeta_{30}^{4} q^{9} +O(q^{10}) \) q + z^7 * q^2 + z^14 * q^4 - z^6 * q^8 + z^4 * q^9	\( q + \zeta_{30}^{7} q^{2} + \zeta_{30}^{14} q^{4} - \zeta_{30}^{6} q^{8} + \zeta_{30}^{4} q^{9} - \zeta_{30}^{11} q^{11} - \zeta_{30}^{13} q^{16} + \zeta_{30}^{11} q^{18} + \zeta_{30}^{3} q^{22} + ( - \zeta_{30}^{13} + \zeta_{30}^{7}) q^{23} - \zeta_{30}^{2} q^{25} + ( - \zeta_{30}^{12} + \zeta_{30}^{9}) q^{29} + \zeta_{30}^{5} q^{32} - \zeta_{30}^{3} q^{36} + ( - \zeta_{30}^{10} + \zeta_{30}) q^{37} + (\zeta_{30}^{9} + \zeta_{30}^{6}) q^{43} + \zeta_{30}^{10} q^{44} + (\zeta_{30}^{14} + \zeta_{30}^{5}) q^{46} - \zeta_{30}^{9} q^{50} + (\zeta_{30}^{8} - \zeta_{30}^{5}) q^{53} + (\zeta_{30}^{4} - \zeta_{30}) q^{58} + \zeta_{30}^{12} q^{64} + ( - \zeta_{30}^{14} - \zeta_{30}^{11}) q^{67} + (\zeta_{30}^{12} - 1) q^{71} - \zeta_{30}^{10} q^{72} + (\zeta_{30}^{8} + \zeta_{30}^{2}) q^{74} + (\zeta_{30}^{13} + \zeta_{30}^{10}) q^{79} + \zeta_{30}^{8} q^{81} + (\zeta_{30}^{13} - \zeta_{30}) q^{86} - \zeta_{30}^{2} q^{88} + (\zeta_{30}^{12} - \zeta_{30}^{6}) q^{92} + q^{99} +O(q^{100}) \) q + z^7 * q^2 + z^14 * q^4 - z^6 * q^8 + z^4 * q^9 - z^11 * q^11 - z^13 * q^16 + z^11 * q^18 + z^3 * q^22 + (-z^13 + z^7) * q^23 - z^2 * q^25 + (-z^12 + z^9) * q^29 + z^5 * q^32 - z^3 * q^36 + (-z^10 + z) * q^37 + (z^9 + z^6) * q^43 + z^10 * q^44 + (z^14 + z^5) * q^46 - z^9 * q^50 + (z^8 - z^5) * q^53 + (z^4 - z) * q^58 + z^12 * q^64 + (-z^14 - z^11) * q^67 + (z^12 - 1) * q^71 - z^10 * q^72 + (z^8 + z^2) * q^74 + (z^13 + z^10) * q^79 + z^8 * q^81 + (z^13 - z) * q^86 - z^2 * q^88 + (z^12 - z^6) * q^92 + q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - q^{2} + q^{4} + 2 q^{8} + q^{9}+O(q^{10}) \) 8 * q - q^2 + q^4 + 2 * q^8 + q^9	\( 8 q - q^{2} + q^{4} + 2 q^{8} + q^{9} + q^{11} + q^{16} - q^{18} + 2 q^{22} - q^{25} + 4 q^{29} + 4 q^{32} - 2 q^{36} + 3 q^{37} - 4 q^{44} + 5 q^{46} - 2 q^{50} - 3 q^{53} + 2 q^{58} - 2 q^{64} - 10 q^{71} + 4 q^{72} + 2 q^{74} - 5 q^{79} + q^{81} - q^{88} + 8 q^{99}+O(q^{100}) \) 8 * q - q^2 + q^4 + 2 * q^8 + q^9 + q^11 + q^16 - q^18 + 2 * q^22 - q^25 + 4 * q^29 + 4 * q^32 - 2 * q^36 + 3 * q^37 - 4 * q^44 + 5 * q^46 - 2 * q^50 - 3 * q^53 + 2 * q^58 - 2 * q^64 - 10 * q^71 + 4 * q^72 + 2 * q^74 - 5 * q^79 + q^81 - q^88 + 8 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

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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	2156.1.bk.b

Level	$2156$
Weight	$1$
Character orbit	2156.bk
Analytic conductor	$1.076$
Analytic rank	$0$
Dimension	$8$
Projective image	$D_{10}$
CM discriminant	-7
Inner twists	$8$

Self dual:	no
Analytic conductor:	\(1.07598416724\)
Analytic rank:	\(0\)
Dimension:	\(8\)
Coefficient field:	\(\Q(\zeta_{15})\)
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	\( x^{8} - x^{7} + x^{5} - x^{4} + x^{3} - x + 1 \) x^8 - x^7 + x^5 - x^4 + x^3 - x + 1
Coefficient ring:	\(\Z[a_1, a_2]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	yes
Projective image:	\(D_{10}\)
Projective field:	Galois closure of 10.0.527027889439744.1

\(n\)	\(981\)	\(1079\)	\(1277\)
\(\chi(n)\)	\(-\zeta_{30}^{3}\)	\(-1\)	\(\zeta_{30}^{10}\)

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 471.1
Significant digits:
Format:

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
7.b	odd	2	1	CM by \(\Q(\sqrt{-7}) \)
7.c	even	3	1	inner
7.d	odd	6	1	inner
44.h	odd	10	1	inner
308.t	even	10	1	inner
308.bb	odd	30	1	inner
308.be	even	30	1	inner

$p$	$F_p(T)$
$2$	\( T^{8} + T^{7} - T^{5} + \cdots + 1 \) T^8 + T^7 - T^5 - T^4 - T^3 + T + 1
$3$	\( T^{8} \) T^8
$5$	\( T^{8} \) T^8
$7$	\( T^{8} \) T^8
$11$	\( T^{8} - T^{7} + T^{5} + \cdots + 1 \) T^8 - T^7 + T^5 - T^4 + T^3 - T + 1
$13$	\( T^{8} \) T^8
$17$	\( T^{8} \) T^8
$19$	\( T^{8} \) T^8
$23$	\( T^{8} - 5 T^{6} + \cdots + 25 \) T^8 - 5T^6 + 20T^4 - 25*T^2 + 25
$29$	\( (T^{4} - 2 T^{3} + 4 T^{2} + \cdots + 1)^{2} \) (T^4 - 2T^3 + 4T^2 - 3*T + 1)^2
$31$	\( T^{8} \) T^8
$37$	\( T^{8} - 3 T^{7} + \cdots + 1 \) T^8 - 3T^7 + 5T^6 - 8T^5 + 9T^4 - 2T^3 - 2T + 1
$41$	\( T^{8} \) T^8
$43$	\( (T^{4} + 5 T^{2} + 5)^{2} \) (T^4 + 5*T^2 + 5)^2
$47$	\( T^{8} \) T^8
$53$	\( T^{8} + 3 T^{7} + \cdots + 1 \) T^8 + 3T^7 + 5T^6 + 8T^5 + 9T^4 + 2T^3 + 2T + 1
$59$	\( T^{8} \) T^8
$61$	\( T^{8} \) T^8
$67$	\( T^{8} - 5 T^{6} + \cdots + 25 \) T^8 - 5T^6 + 20T^4 - 25*T^2 + 25
$71$	\( (T^{4} + 5 T^{3} + 10 T^{2} + \cdots + 5)^{2} \) (T^4 + 5T^3 + 10T^2 + 10*T + 5)^2
$73$	\( T^{8} \) T^8
$79$	\( T^{8} + 5 T^{7} + \cdots + 25 \) T^8 + 5T^7 + 15T^6 + 30T^5 + 45T^4 + 50T^3 + 50T^2 + 50*T + 25
$83$	\( T^{8} \) T^8
$89$	\( T^{8} \) T^8
$97$	\( T^{8} \) T^8
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