LMFDB - Newform orbit 4160.2.a.bi

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4160,2,Mod(1,4160)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4160, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("4160.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$N$	$=$	$4160 = 2^{6} \cdot 5 \cdot 13$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	4160.a (trivial)

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:	yes
Analytic conductor:	$33.2177672409$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{2})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$x^{2} - 2$ x^2 - 2
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$1$
Twist minimal:	no (minimal twist has level 2080)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

$q$ -expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

Coefficients of the $q$ -expansion are expressed in terms of $\beta = \sqrt{2}$ . We also show the integral $q$ -expansion of the trace form.

$f(q)$	$=$	$q + \beta q^{3} + q^{5} - q^{9} - 3 \beta q^{11} - q^{13} + \beta q^{15} - 2 q^{17} + 3 \beta q^{19} - \beta q^{23} + q^{25} - 4 \beta q^{27} + \beta q^{31} - 6 q^{33} - 4 q^{37} - \beta q^{39} - 10 q^{41} + \cdots + 3 \beta q^{99} +O(q^{100})$ q + b * q^3 + q^5 - q^9 - 3b q^11 - q^13 + b * q^15 - 2 * q^17 + 3b q^19 - b * q^23 + q^25 - 4b q^27 + b * q^31 - 6 * q^33 - 4 * q^37 - b * q^39 - 10 * q^41 - 7b q^43 - q^45 + 8b q^47 - 7 * q^49 - 2b q^51 + 10 * q^53 - 3b q^55 + 6 * q^57 + b * q^59 + 4 * q^61 - q^65 + 2b q^67 - 2 * q^69 - 5b q^71 - 4 * q^73 + b * q^75 - 6b q^79 - 5 * q^81 - 4b q^83 - 2 * q^85 - 18 * q^89 + 2 * q^93 + 3b q^95 - 18 * q^97 + 3b q^99
$\operatorname{Tr}(f)(q)$	$=$	$2 q + 2 q^{5} - 2 q^{9} - 2 q^{13} - 4 q^{17} + 2 q^{25} - 12 q^{33} - 8 q^{37} - 20 q^{41} - 2 q^{45} - 14 q^{49} + 20 q^{53} + 12 q^{57} + 8 q^{61} - 2 q^{65} - 4 q^{69} - 8 q^{73} - 10 q^{81} - 4 q^{85}+ \cdots - 36 q^{97}+O(q^{100})$ 2 * q + 2 * q^5 - 2 * q^9 - 2 * q^13 - 4 * q^17 + 2 * q^25 - 12 * q^33 - 8 * q^37 - 20 * q^41 - 2 * q^45 - 14 * q^49 + 20 * q^53 + 12 * q^57 + 8 * q^61 - 2 * q^65 - 4 * q^69 - 8 * q^73 - 10 * q^81 - 4 * q^85 - 36 * q^89 + 4 * q^93 - 36 * q^97

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}

1.1

−1.41421
1.41421

−1.41421

1.00000

−1.00000

1.2

1.41421

1.00000

−1.00000

Atkin-Lehner signs

$p$	Sign
$2$	$-1$
$5$	$-1$
$13$	$+1$

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
4.b	odd	2	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	4160.2.a.bi		2
4.b	odd	2	1	inner	4160.2.a.bi		2
8.b	even	2	1		2080.2.a.g	✓	2
8.d	odd	2	1		2080.2.a.g	✓	2

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
2080.2.a.g	✓	2	8.b	even	2	1
2080.2.a.g	✓	2	8.d	odd	2	1
4160.2.a.bi		2	1.a	even	1	1	trivial
4160.2.a.bi		2	4.b	odd	2	1	inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{2}^{\mathrm{new}}(\Gamma_0(4160))$ :

T_{3}^{2} - 2

T_{7}

T_{11}^{2} - 18

T_{17} + 2

T_{19}^{2} - 18

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T^{2}$ T^2
$3$	$T^{2} - 2$ T^2 - 2
$5$	$(T - 1)^{2}$ (T - 1)^2
$7$	$T^{2}$ T^2
$11$	$T^{2} - 18$ T^2 - 18
$13$	$(T + 1)^{2}$ (T + 1)^2
$17$	$(T + 2)^{2}$ (T + 2)^2
$19$	$T^{2} - 18$ T^2 - 18
$23$	$T^{2} - 2$ T^2 - 2
$29$	$T^{2}$ T^2
$31$	$T^{2} - 2$ T^2 - 2
$37$	$(T + 4)^{2}$ (T + 4)^2
$41$	$(T + 10)^{2}$ (T + 10)^2
$43$	$T^{2} - 98$ T^2 - 98
$47$	$T^{2} - 128$ T^2 - 128
$53$	$(T - 10)^{2}$ (T - 10)^2
$59$	$T^{2} - 2$ T^2 - 2
$61$	$(T - 4)^{2}$ (T - 4)^2
$67$	$T^{2} - 8$ T^2 - 8
$71$	$T^{2} - 50$ T^2 - 50
$73$	$(T + 4)^{2}$ (T + 4)^2
$79$	$T^{2} - 72$ T^2 - 72
$83$	$T^{2} - 32$ T^2 - 32
$89$	$(T + 18)^{2}$ (T + 18)^2
$97$	$(T + 18)^{2}$ (T + 18)^2
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