LMFDB - Newform orbit 2736.1.bk.b

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2736,1,Mod(847,2736)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2736, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([3, 0, 0, 4]))

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

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

chi := DirichletCharacter("2736.847");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$N$	$=$	$2736 = 2^{4} \cdot 3^{2} \cdot 19$
Weight:	$k$	$=$	$1$
Character orbit:	$[\chi]$	$=$	2736.bk (of order $6$ , degree $2$ , 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:	$1.36544187456$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{6})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$x^{2} - x + 1$ x^2 - x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$1$
Twist minimal:	yes
Projective image:	$D_{6}$
Projective field:	Galois closure of 6.2.225194688.1

$q$ -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.

$f(q)$	$=$	$q + (\zeta_{6}^{2} + \zeta_{6}) q^{7} - \zeta_{6} q^{13} + q^{19} + \zeta_{6} q^{25} + (\zeta_{6}^{2} + \zeta_{6}) q^{31} - q^{37} + (\zeta_{6} + 1) q^{43} + (\zeta_{6}^{2} - \zeta_{6} - 1) q^{49} + \zeta_{6} q^{61} + \cdots - 2 \zeta_{6}^{2} q^{97} +O(q^{100})$ q + (z^2 + z) * q^7 - z * q^13 + q^19 + z * q^25 + (z^2 + z) * q^31 - q^37 + (z + 1) * q^43 + (z^2 - z - 1) * q^49 + z * q^61 + (z^2 - 1) * q^67 + z^2 * q^73 + (-z - 1) * q^79 + (-z^2 + 1) * q^91 - 2z^2 q^97
$\operatorname{Tr}(f)(q)$	$=$	$2 q - q^{13} + 2 q^{19} + q^{25} - 2 q^{37} + 3 q^{43} - 4 q^{49} + q^{61} - 3 q^{67} - q^{73} - 3 q^{79} + 3 q^{91} + 2 q^{97}+O(q^{100})$ 2 * q - q^13 + 2 * q^19 + q^25 - 2 * q^37 + 3 * q^43 - 4 * q^49 + q^61 - 3 * q^67 - q^73 - 3 * q^79 + 3 * q^91 + 2 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2736\mathbb{Z}\right)^\times$ .

$n$	$1009$	$1217$	$1711$	$2053$
$\chi(n)$	$-\zeta_{6}$	$1$	$-1$	$1$

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}

847.1

0.500000	+	0.866025i
0.500000	−	0.866025i

1.73205i

2287.1

−

1.73205i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
3.b	odd	2	1	CM by $\Q(\sqrt{-3})$
76.g	odd	6	1	inner
228.m	even	6	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	2736.1.bk.b	yes	2
3.b	odd	2	1	CM	2736.1.bk.b	yes	2
4.b	odd	2	1		2736.1.bk.a	✓	2
12.b	even	2	1		2736.1.bk.a	✓	2
19.c	even	3	1		2736.1.bk.a	✓	2
57.h	odd	6	1		2736.1.bk.a	✓	2
76.g	odd	6	1	inner	2736.1.bk.b	yes	2
228.m	even	6	1	inner	2736.1.bk.b	yes	2

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
2736.1.bk.a	✓	2	4.b	odd	2	1
2736.1.bk.a	✓	2	12.b	even	2	1
2736.1.bk.a	✓	2	19.c	even	3	1
2736.1.bk.a	✓	2	57.h	odd	6	1
2736.1.bk.b	yes	2	1.a	even	1	1	trivial
2736.1.bk.b	yes	2	3.b	odd	2	1	CM
2736.1.bk.b	yes	2	76.g	odd	6	1	inner
2736.1.bk.b	yes	2	228.m	even	6	1	inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $T_{43}^{2} - 3T_{43} + 3$ T43^2 - 3*T43 + 3 acting on $S_{1}^{\mathrm{new}}(2736, [\chi])$ .

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T^{2}$ T^2
$3$	$T^{2}$ T^2
$5$	$T^{2}$ T^2
$7$	$T^{2} + 3$ T^2 + 3
$11$	$T^{2}$ T^2
$13$	$T^{2} + T + 1$ T^2 + T + 1
$17$	$T^{2}$ T^2
$19$	$(T - 1)^{2}$ (T - 1)^2
$23$	$T^{2}$ T^2
$29$	$T^{2}$ T^2
$31$	$T^{2} + 3$ T^2 + 3
$37$	$(T + 1)^{2}$ (T + 1)^2
$41$	$T^{2}$ T^2
$43$	$T^{2} - 3T + 3$ T^2 - 3*T + 3
$47$	$T^{2}$ T^2
$53$	$T^{2}$ T^2
$59$	$T^{2}$ T^2
$61$	$T^{2} - T + 1$ T^2 - T + 1
$67$	$T^{2} + 3T + 3$ T^2 + 3*T + 3
$71$	$T^{2}$ T^2
$73$	$T^{2} + T + 1$ T^2 + T + 1
$79$	$T^{2} + 3T + 3$ T^2 + 3*T + 3
$83$	$T^{2}$ T^2
$89$	$T^{2}$ T^2
$97$	$T^{2} - 2T + 4$ T^2 - 2*T + 4
show more
show less

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
Significant digits:
Format:

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more