LMFDB - Newform orbit 400.6.a.j

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [400,6,Mod(1,400)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(400, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 6, names="a")

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

chi := DirichletCharacter("400.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$N$	$=$	$400 = 2^{4} \cdot 5^{2}$
Weight:	$k$	$=$	$6$
Character orbit:	$[\chi]$	$=$	400.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:	$64.1535279252$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$1$
Twist minimal:	no (minimal twist has level 50)
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)

f(q)

=

q + 11 q^{3} + 142 q^{7} - 122 q^{9} - 777 q^{11} + 884 q^{13} - 27 q^{17} - 1145 q^{19} + 1562 q^{21} - 1854 q^{23} - 4015 q^{27} - 4920 q^{29} - 1802 q^{31} - 8547 q^{33} + 13178 q^{37} + 9724 q^{39}+ \cdots + 94794 q^{99}+O(q^{100})

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

11.0000

142.000

−122.000

Atkin-Lehner signs

$p$	Sign
$2$	$-1$
$5$	$-1$

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	400.6.a.j		1
4.b	odd	2	1		50.6.a.a	✓	1
5.b	even	2	1		400.6.a.e		1
5.c	odd	4	2		400.6.c.g		2
12.b	even	2	1		450.6.a.n		1
20.d	odd	2	1		50.6.a.f	yes	1
20.e	even	4	2		50.6.b.c		2
60.h	even	2	1		450.6.a.j		1
60.l	odd	4	2		450.6.c.a		2

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
50.6.a.a	✓	1	4.b	odd	2	1
50.6.a.f	yes	1	20.d	odd	2	1
50.6.b.c		2	20.e	even	4	2
400.6.a.e		1	5.b	even	2	1
400.6.a.j		1	1.a	even	1	1	trivial
400.6.c.g		2	5.c	odd	4	2
450.6.a.j		1	60.h	even	2	1
450.6.a.n		1	12.b	even	2	1
450.6.c.a		2	60.l	odd	4	2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $T_{3} - 11$ T3 - 11 acting on $S_{6}^{\mathrm{new}}(\Gamma_0(400))$ .

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T$ T
$3$	$T - 11$ T - 11
$5$	$T$ T
$7$	$T - 142$ T - 142
$11$	$T + 777$ T + 777
$13$	$T - 884$ T - 884
$17$	$T + 27$ T + 27
$19$	$T + 1145$ T + 1145
$23$	$T + 1854$ T + 1854
$29$	$T + 4920$ T + 4920
$31$	$T + 1802$ T + 1802
$37$	$T - 13178$ T - 13178
$41$	$T + 15123$ T + 15123
$43$	$T + 7844$ T + 7844
$47$	$T - 6732$ T - 6732
$53$	$T - 3414$ T - 3414
$59$	$T + 33960$ T + 33960
$61$	$T - 47402$ T - 47402
$67$	$T - 13177$ T - 13177
$71$	$T - 7548$ T - 7548
$73$	$T + 59821$ T + 59821
$79$	$T + 75830$ T + 75830
$83$	$T + 46299$ T + 46299
$89$	$T + 30585$ T + 30585
$97$	$T - 104018$ T - 104018
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