LMFDB - Newform orbit 3332.1.bp.b

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3332,1,Mod(263,3332)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3332, base_ring=CyclotomicField(24))

chi = DirichletCharacter(H, H._module([12, 16, 15]))

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

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

chi := DirichletCharacter("3332.263");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$N$	$=$	$3332 = 2^{2} \cdot 7^{2} \cdot 17$
Weight:	$k$	$=$	$1$
Character orbit:	$[\chi]$	$=$	3332.bp (of order $24$ , 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:	$1.66288462209$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\Q(\zeta_{24})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$x^{8} - x^{4} + 1$ x^8 - x^4 + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$1$
Twist minimal:	yes
Projective image:	$D_{8}$
Projective field:	Galois closure of 8.2.3089659810545728.4

$q$ -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.

$f(q)$	$=$	$q - \zeta_{24}^{7} q^{2} - \zeta_{24}^{2} q^{4} + ( - \zeta_{24}^{10} - \zeta_{24}^{7}) q^{5} + \zeta_{24}^{9} q^{8} - \zeta_{24} q^{9} + ( - \zeta_{24}^{5} - \zeta_{24}^{2}) q^{10} + \zeta_{24}^{4} q^{16} + \cdots + (\zeta_{24}^{6} + \zeta_{24}^{3}) q^{97} +O(q^{100})$ q - z^7 * q^2 - z^2 * q^4 + (-z^10 - z^7) * q^5 + z^9 * q^8 - z * q^9 + (-z^5 - z^2) * q^10 + z^4 * q^16 + z^11 * q^17 + z^8 * q^18 + (z^9 - 1) * q^20 + (-z^8 - z^5 - z^2) * q^25 + (-z^6 - z^3) * q^29 - z^11 * q^32 + z^6 * q^34 + z^3 * q^36 + (-z^4 - z) * q^37 + (z^7 + z^4) * q^40 + (z^3 - 1) * q^41 + (z^11 + z^8) * q^45 + (z^9 - z^3 - 1) * q^50 + (-z^8 + z^2) * q^53 + (z^10 - z) * q^58 + (-z^7 + z^4) * q^61 - z^6 * q^64 + z * q^68 - z^10 * q^72 + (-z^11 + z^2) * q^73 + (z^11 + z^8) * q^74 + (-z^11 + z^2) * q^80 + z^2 * q^81 + (-z^10 + z^7) * q^82 + (z^9 + z^6) * q^85 + 2z^10 q^89 + (z^6 + z^3) * q^90 + (z^6 + z^3) * q^97
$\operatorname{Tr}(f)(q)$	$=$	$8 q + 4 q^{16} - 4 q^{18} - 8 q^{20} + 4 q^{25} - 4 q^{37} + 4 q^{40} - 8 q^{41} - 4 q^{45} - 8 q^{50} + 4 q^{53} + 4 q^{61} - 4 q^{74}+O(q^{100})$ 8 * q + 4 * q^16 - 4 * q^18 - 8 * q^20 + 4 * q^25 - 4 * q^37 + 4 * q^40 - 8 * q^41 - 4 * q^45 - 8 * q^50 + 4 * q^53 + 4 * q^61 - 4 * q^74

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3332\mathbb{Z}\right)^\times$ .

$n$	$785$	$885$	$1667$
$\chi(n)$	$-\zeta_{24}^{9}$	$-\zeta_{24}^{4}$	$-1$

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}

263.1

0.258819	−	0.965926i
−0.258819	+	0.965926i
0.965926	+	0.258819i
0.258819	+	0.965926i
−0.965926	+	0.258819i
0.965926	−	0.258819i
−0.965926	−	0.258819i
−0.258819	−	0.965926i

0.965926

0.258819i

0.866025

0.500000i

0.0999004

0.758819i

0.707107

0.707107i

−0.258819

0.965926i

−0.0999004

0.758819i

655.1

−0.965926

−

0.258819i

0.866025

0.500000i

−1.83195

0.241181i

−0.707107

−

0.707107i

0.258819

−

0.965926i

1.83195

0.241181i

1243.1

0.258819

−

0.965926i

−0.866025

−

0.500000i

1.12484

−

1.46593i

−0.707107

0.707107i

−0.965926

−

0.258819i

−1.12484

−

1.46593i

1647.1

0.965926

−

0.258819i

0.866025

−

0.500000i

0.0999004

−

0.758819i

0.707107

−

0.707107i

−0.258819

−

0.965926i

−0.0999004

−

0.758819i

2235.1

−0.258819

−

0.965926i

−0.866025

0.500000i

0.607206

−

0.465926i

0.707107

0.707107i

0.965926

−

0.258819i

−0.607206

−

0.465926i

2627.1

0.258819

0.965926i

−0.866025

0.500000i

1.12484

1.46593i

−0.707107

−

0.707107i

−0.965926

0.258819i

−1.12484

1.46593i

3007.1

−0.258819

0.965926i

−0.866025

−

0.500000i

0.607206

0.465926i

0.707107

−

0.707107i

0.965926

0.258819i

−0.607206

0.465926i

3215.1

−0.965926

0.258819i

0.866025

−

0.500000i

−1.83195

−

0.241181i

−0.707107

0.707107i

0.258819

0.965926i

1.83195

−

0.241181i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
4.b	odd	2	1	CM by $\Q(\sqrt{-1})$
7.c	even	3	1	inner
17.d	even	8	1	inner
28.g	odd	6	1	inner
68.g	odd	8	1	inner
119.q	even	24	1	inner
476.bg	odd	24	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	3332.1.bp.b		8
4.b	odd	2	1	CM	3332.1.bp.b		8
7.b	odd	2	1		3332.1.bp.c		8
7.c	even	3	1		3332.1.w.b	✓	4
7.c	even	3	1	inner	3332.1.bp.b		8
7.d	odd	6	1		3332.1.w.c	yes	4
7.d	odd	6	1		3332.1.bp.c		8
17.d	even	8	1	inner	3332.1.bp.b		8
28.d	even	2	1		3332.1.bp.c		8
28.f	even	6	1		3332.1.w.c	yes	4
28.f	even	6	1		3332.1.bp.c		8
28.g	odd	6	1		3332.1.w.b	✓	4
28.g	odd	6	1	inner	3332.1.bp.b		8
68.g	odd	8	1	inner	3332.1.bp.b		8
119.l	odd	8	1		3332.1.bp.c		8
119.q	even	24	1		3332.1.w.b	✓	4
119.q	even	24	1	inner	3332.1.bp.b		8
119.r	odd	24	1		3332.1.w.c	yes	4
119.r	odd	24	1		3332.1.bp.c		8
476.w	even	8	1		3332.1.bp.c		8
476.bg	odd	24	1		3332.1.w.b	✓	4
476.bg	odd	24	1	inner	3332.1.bp.b		8
476.bj	even	24	1		3332.1.w.c	yes	4
476.bj	even	24	1		3332.1.bp.c		8

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
3332.1.w.b	✓	4	7.c	even	3	1
3332.1.w.b	✓	4	28.g	odd	6	1
3332.1.w.b	✓	4	119.q	even	24	1
3332.1.w.b	✓	4	476.bg	odd	24	1
3332.1.w.c	yes	4	7.d	odd	6	1
3332.1.w.c	yes	4	28.f	even	6	1
3332.1.w.c	yes	4	119.r	odd	24	1
3332.1.w.c	yes	4	476.bj	even	24	1
3332.1.bp.b		8	1.a	even	1	1	trivial
3332.1.bp.b		8	4.b	odd	2	1	CM
3332.1.bp.b		8	7.c	even	3	1	inner
3332.1.bp.b		8	17.d	even	8	1	inner
3332.1.bp.b		8	28.g	odd	6	1	inner
3332.1.bp.b		8	68.g	odd	8	1	inner
3332.1.bp.b		8	119.q	even	24	1	inner
3332.1.bp.b		8	476.bg	odd	24	1	inner
3332.1.bp.c		8	7.b	odd	2	1
3332.1.bp.c		8	7.d	odd	6	1
3332.1.bp.c		8	28.d	even	2	1
3332.1.bp.c		8	28.f	even	6	1
3332.1.bp.c		8	119.l	odd	8	1
3332.1.bp.c		8	119.r	odd	24	1
3332.1.bp.c		8	476.w	even	8	1
3332.1.bp.c		8	476.bj	even	24	1

This newform subspace can be constructed as the kernel of the linear operator $T_{5}^{8} - 2T_{5}^{6} + 8T_{5}^{5} + 2T_{5}^{4} - 8T_{5}^{3} + 12T_{5}^{2} - 8T_{5} + 4$ T5^8 - 2*T5^6 + 8*T5^5 + 2*T5^4 - 8*T5^3 + 12*T5^2 - 8*T5 + 4 acting on $S_{1}^{\mathrm{new}}(3332, [\chi])$ .

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T^{8} - T^{4} + 1$ T^8 - T^4 + 1
$3$	$T^{8}$ T^8
$5$	$T^{8} - 2 T^{6} + \cdots + 4$ T^8 - 2T^6 + 8T^5 + 2T^4 - 8T^3 + 12T^2 - 8T + 4
$7$	$T^{8}$ T^8
$11$	$T^{8}$ T^8
$13$	$T^{8}$ T^8
$17$	$T^{8} - T^{4} + 1$ T^8 - T^4 + 1
$19$	$T^{8}$ T^8
$23$	$T^{8}$ T^8
$29$	$(T^{4} + 2 T^{2} - 4 T + 2)^{2}$ (T^4 + 2T^2 - 4T + 2)^2
$31$	$T^{8}$ T^8
$37$	$T^{8} + 4 T^{7} + \cdots + 4$ T^8 + 4T^7 + 10T^6 + 16T^5 + 18T^4 + 8T^3 + 4T^2 + 8*T + 4
$41$	$(T^{4} + 4 T^{3} + 6 T^{2} + \cdots + 2)^{2}$ (T^4 + 4T^3 + 6T^2 + 4*T + 2)^2
$43$	$T^{8}$ T^8
$47$	$T^{8}$ T^8
$53$	$(T^{4} - 2 T^{3} + 2 T^{2} + \cdots + 4)^{2}$ (T^4 - 2T^3 + 2T^2 - 4*T + 4)^2
$59$	$T^{8}$ T^8
$61$	$T^{8} - 4 T^{7} + \cdots + 4$ T^8 - 4T^7 + 10T^6 - 16T^5 + 18T^4 - 8T^3 + 4T^2 - 8*T + 4
$67$	$T^{8}$ T^8
$71$	$T^{8}$ T^8
$73$	$T^{8} - 2 T^{6} + \cdots + 4$ T^8 - 2T^6 - 8T^5 + 2T^4 + 8T^3 + 12T^2 + 8T + 4
$79$	$T^{8}$ T^8
$83$	$T^{8}$ T^8
$89$	$(T^{4} - 4 T^{2} + 16)^{2}$ (T^4 - 4*T^2 + 16)^2
$97$	$(T^{4} + 2 T^{2} + 4 T + 2)^{2}$ (T^4 + 2T^2 + 4T + 2)^2
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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 $\F_{q}$

$n$ :		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 263.1
Significant digits:
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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	3332.1.bp.b

Level	$3332$
Weight	$1$
Character orbit	3332.bp
Analytic conductor	$1.663$
Analytic rank	$0$
Dimension	$8$
Projective image	$D_{8}$
CM discriminant	-4
Inner twists	$8$