LMFDB - Newform orbit 2312.1.p.e

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2312,1,Mod(155,2312)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2312, base_ring=CyclotomicField(8))

chi = DirichletCharacter(H, H._module([4, 4, 7]))

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

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

chi := DirichletCharacter("2312.155");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$N$	$=$	$2312 = 2^{3} \cdot 17^{2}$
Weight:	$k$	$=$	$1$
Character orbit:	$[\chi]$	$=$	2312.p (of order $8$ , degree $4$ , 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:	$1.15383830921$
Analytic rank:	$0$
Dimension:	$8$
Relative dimension:	$2$ over $\Q(\zeta_{8})$
Coefficient field:	$\Q(\zeta_{16})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$x^{8} + 1$ x^8 + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$2^{2}$
Twist minimal:	no (minimal twist has level 136)
Projective image:	$D_{4}$
Projective field:	Galois closure of 4.0.314432.1

$q$ -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.

$f(q)$	$=$	$q + \zeta_{16}^{6} q^{2} + ( - \zeta_{16}^{7} - \zeta_{16}^{3}) q^{3} - \zeta_{16}^{4} q^{4} + (\zeta_{16}^{5} + \zeta_{16}) q^{6} + \zeta_{16}^{2} q^{8} - \zeta_{16}^{2} q^{9} + (\zeta_{16}^{5} + \zeta_{16}) q^{11} + \cdots + ( - \zeta_{16}^{7} - \zeta_{16}^{3}) q^{99} +O(q^{100})$ q + z^6 * q^2 + (-z^7 - z^3) * q^3 - z^4 * q^4 + (z^5 + z) * q^6 + z^2 * q^8 - z^2 * q^9 + (z^5 + z) * q^11 + (z^7 - z^3) * q^12 - q^16 + q^18 - 2z^6 q^19 + (z^7 - z^3) * q^22 + (-z^5 + z) * q^24 + z^2 * q^25 - z^6 * q^32 + 2 * q^33 + z^6 * q^36 + 2z^4 q^38 + (z^5 - z) * q^41 + (-z^5 + z) * q^44 + (z^7 + z^3) * q^48 - z^6 * q^49 - q^50 + (-2z^5 - 2z) * q^57 + z^4 * q^64 + 2z^6 q^66 - z^4 * q^72 + (z^7 - z^3) * q^73 + (-z^5 + z) * q^75 - 2z^2 q^76 - z^4 * q^81 + (-z^7 - z^3) * q^82 + (z^7 + z^3) * q^88 + (-z^5 - z) * q^96 + (z^7 - z^3) * q^97 + z^4 * q^98 + (-z^7 - z^3) * q^99
$\operatorname{Tr}(f)(q)$	$=$	$8 q - 8 q^{16} + 8 q^{18} + 16 q^{33} - 8 q^{50}+O(q^{100})$ 8 * q - 8 * q^16 + 8 * q^18 + 16 * q^33 - 8 * q^50

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2312\mathbb{Z}\right)^\times$ .

$n$	$1157$	$1735$	$1737$
$\chi(n)$	$-1$	$-1$	$-\zeta_{16}^{2}$

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}

155.1

−0.382683	−	0.923880i
0.382683	+	0.923880i
−0.382683	+	0.923880i
0.382683	−	0.923880i
−0.923880	−	0.382683i
0.923880	+	0.382683i
−0.923880	+	0.382683i
0.923880	−	0.382683i

0.707107

0.707107i

−1.30656

0.541196i

1.00000i

−1.30656

−

0.541196i

−0.707107

0.707107i

0.707107

−

0.707107i

155.2

0.707107

0.707107i

1.30656

−

0.541196i

1.00000i

1.30656

0.541196i

−0.707107

0.707107i

0.707107

−

0.707107i

179.1

0.707107

−

0.707107i

−1.30656

−

0.541196i

−

1.00000i

−1.30656

0.541196i

−0.707107

−

0.707107i

0.707107

0.707107i

179.2

0.707107

−

0.707107i

1.30656

0.541196i

−

1.00000i

1.30656

−

0.541196i

−0.707107

−

0.707107i

0.707107

0.707107i

1555.1

−0.707107

0.707107i

−0.541196

1.30656i

−

1.00000i

−0.541196

−

1.30656i

0.707107

0.707107i

−0.707107

−

0.707107i

1555.2

−0.707107

0.707107i

0.541196

−

1.30656i

−

1.00000i

0.541196

1.30656i

0.707107

0.707107i

−0.707107

−

0.707107i

1579.1

−0.707107

−

0.707107i

−0.541196

−

1.30656i

1.00000i

−0.541196

1.30656i

0.707107

−

0.707107i

−0.707107

0.707107i

1579.2

−0.707107

−

0.707107i

0.541196

1.30656i

1.00000i

0.541196

−

1.30656i

0.707107

−

0.707107i

−0.707107

0.707107i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
8.d	odd	2	1	CM by $\Q(\sqrt{-2})$
17.b	even	2	1	inner
17.c	even	4	2	inner
17.d	even	8	4	inner
136.e	odd	2	1	inner
136.j	odd	4	2	inner
136.p	odd	8	4	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	2312.1.p.e		8
8.d	odd	2	1	CM	2312.1.p.e		8
17.b	even	2	1	inner	2312.1.p.e		8
17.c	even	4	2	inner	2312.1.p.e		8
17.d	even	8	4	inner	2312.1.p.e		8
17.e	odd	16	2		136.1.j.a	✓	2
17.e	odd	16	2		2312.1.e.a		2
17.e	odd	16	2		2312.1.f.b		2
17.e	odd	16	2		2312.1.j.b		2
51.i	even	16	2		1224.1.s.a		2
68.i	even	16	2		544.1.n.a		2
85.o	even	16	1		3400.1.bc.a		2
85.o	even	16	1		3400.1.bc.b		2
85.p	odd	16	2		3400.1.y.a		2
85.r	even	16	1		3400.1.bc.a		2
85.r	even	16	1		3400.1.bc.b		2
136.e	odd	2	1	inner	2312.1.p.e		8
136.j	odd	4	2	inner	2312.1.p.e		8
136.p	odd	8	4	inner	2312.1.p.e		8
136.q	odd	16	2		544.1.n.a		2
136.s	even	16	2		136.1.j.a	✓	2
136.s	even	16	2		2312.1.e.a		2
136.s	even	16	2		2312.1.f.b		2
136.s	even	16	2		2312.1.j.b		2
408.bg	odd	16	2		1224.1.s.a		2
680.ch	odd	16	1		3400.1.bc.a		2
680.ch	odd	16	1		3400.1.bc.b		2
680.co	even	16	2		3400.1.y.a		2
680.cr	odd	16	1		3400.1.bc.a		2
680.cr	odd	16	1		3400.1.bc.b		2

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
136.1.j.a	✓	2	17.e	odd	16	2
136.1.j.a	✓	2	136.s	even	16	2
544.1.n.a		2	68.i	even	16	2
544.1.n.a		2	136.q	odd	16	2
1224.1.s.a		2	51.i	even	16	2
1224.1.s.a		2	408.bg	odd	16	2
2312.1.e.a		2	17.e	odd	16	2
2312.1.e.a		2	136.s	even	16	2
2312.1.f.b		2	17.e	odd	16	2
2312.1.f.b		2	136.s	even	16	2
2312.1.j.b		2	17.e	odd	16	2
2312.1.j.b		2	136.s	even	16	2
2312.1.p.e		8	1.a	even	1	1	trivial
2312.1.p.e		8	8.d	odd	2	1	CM
2312.1.p.e		8	17.b	even	2	1	inner
2312.1.p.e		8	17.c	even	4	2	inner
2312.1.p.e		8	17.d	even	8	4	inner
2312.1.p.e		8	136.e	odd	2	1	inner
2312.1.p.e		8	136.j	odd	4	2	inner
2312.1.p.e		8	136.p	odd	8	4	inner
3400.1.y.a		2	85.p	odd	16	2
3400.1.y.a		2	680.co	even	16	2
3400.1.bc.a		2	85.o	even	16	1
3400.1.bc.a		2	85.r	even	16	1
3400.1.bc.a		2	680.ch	odd	16	1
3400.1.bc.a		2	680.cr	odd	16	1
3400.1.bc.b		2	85.o	even	16	1
3400.1.bc.b		2	85.r	even	16	1
3400.1.bc.b		2	680.ch	odd	16	1
3400.1.bc.b		2	680.cr	odd	16	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $T_{3}^{8} + 16$ T3^8 + 16 acting on $S_{1}^{\mathrm{new}}(2312, [\chi])$ .

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$(T^{4} + 1)^{2}$ (T^4 + 1)^2
$3$	$T^{8} + 16$ T^8 + 16
$5$	$T^{8}$ T^8
$7$	$T^{8}$ T^8
$11$	$T^{8} + 16$ T^8 + 16
$13$	$T^{8}$ T^8
$17$	$T^{8}$ T^8
$19$	$(T^{4} + 16)^{2}$ (T^4 + 16)^2
$23$	$T^{8}$ T^8
$29$	$T^{8}$ T^8
$31$	$T^{8}$ T^8
$37$	$T^{8}$ T^8
$41$	$T^{8} + 16$ T^8 + 16
$43$	$T^{8}$ T^8
$47$	$T^{8}$ T^8
$53$	$T^{8}$ T^8
$59$	$T^{8}$ T^8
$61$	$T^{8}$ T^8
$67$	$T^{8}$ T^8
$71$	$T^{8}$ T^8
$73$	$T^{8} + 16$ T^8 + 16
$79$	$T^{8}$ T^8
$83$	$T^{8}$ T^8
$89$	$T^{8}$ T^8
$97$	$T^{8} + 16$ T^8 + 16
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Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

$n$ :		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 155.2
Significant digits:
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Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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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	2312.1.p.e

Level	$2312$
Weight	$1$
Character orbit	2312.p
Analytic conductor	$1.154$
Analytic rank	$0$
Dimension	$8$
Projective image	$D_{4}$
CM discriminant	-8
Inner twists	$16$