LMFDB - Newform orbit 32.4.a.a

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [32,4,Mod(1,32)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(32, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("32.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$N$	$=$	$32 = 2^{5}$
Weight:	$k$	$=$	$4$
Character orbit:	$[\chi]$	$=$	32.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:	$1.88806112018$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$1$
Twist minimal:	yes
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 - 8 q^{3} - 10 q^{5} - 16 q^{7} + 37 q^{9} + 40 q^{11} - 50 q^{13} + 80 q^{15} - 30 q^{17} - 40 q^{19} + 128 q^{21} - 48 q^{23} - 25 q^{25} - 80 q^{27} - 34 q^{29} - 320 q^{31} - 320 q^{33} + 160 q^{35}+ \cdots + 1480 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

−8.00000

−10.0000

−16.0000

37.0000

Atkin-Lehner signs

$p$	Sign
$2$	$-1$

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	32.4.a.a	✓	1
3.b	odd	2	1		288.4.a.h		1
4.b	odd	2	1		32.4.a.c	yes	1
5.b	even	2	1		800.4.a.k		1
5.c	odd	4	2		800.4.c.b		2
7.b	odd	2	1		1568.4.a.o		1
8.b	even	2	1		64.4.a.e		1
8.d	odd	2	1		64.4.a.a		1
12.b	even	2	1		288.4.a.i		1
16.e	even	4	2		256.4.b.e		2
16.f	odd	4	2		256.4.b.c		2
20.d	odd	2	1		800.4.a.a		1
20.e	even	4	2		800.4.c.a		2
24.f	even	2	1		576.4.a.h		1
24.h	odd	2	1		576.4.a.g		1
28.d	even	2	1		1568.4.a.c		1
40.e	odd	2	1		1600.4.a.bw		1
40.f	even	2	1		1600.4.a.e		1

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
32.4.a.a	✓	1	1.a	even	1	1	trivial
32.4.a.c	yes	1	4.b	odd	2	1
64.4.a.a		1	8.d	odd	2	1
64.4.a.e		1	8.b	even	2	1
256.4.b.c		2	16.f	odd	4	2
256.4.b.e		2	16.e	even	4	2
288.4.a.h		1	3.b	odd	2	1
288.4.a.i		1	12.b	even	2	1
576.4.a.g		1	24.h	odd	2	1
576.4.a.h		1	24.f	even	2	1
800.4.a.a		1	20.d	odd	2	1
800.4.a.k		1	5.b	even	2	1
800.4.c.a		2	20.e	even	4	2
800.4.c.b		2	5.c	odd	4	2
1568.4.a.c		1	28.d	even	2	1
1568.4.a.o		1	7.b	odd	2	1
1600.4.a.e		1	40.f	even	2	1
1600.4.a.bw		1	40.e	odd	2	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $T_{3} + 8$ T3 + 8 acting on $S_{4}^{\mathrm{new}}(\Gamma_0(32))$ .

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T$ T
$3$	$T + 8$ T + 8
$5$	$T + 10$ T + 10
$7$	$T + 16$ T + 16
$11$	$T - 40$ T - 40
$13$	$T + 50$ T + 50
$17$	$T + 30$ T + 30
$19$	$T + 40$ T + 40
$23$	$T + 48$ T + 48
$29$	$T + 34$ T + 34
$31$	$T + 320$ T + 320
$37$	$T - 310$ T - 310
$41$	$T - 410$ T - 410
$43$	$T + 152$ T + 152
$47$	$T - 416$ T - 416
$53$	$T + 410$ T + 410
$59$	$T - 200$ T - 200
$61$	$T - 30$ T - 30
$67$	$T + 776$ T + 776
$71$	$T + 400$ T + 400
$73$	$T + 630$ T + 630
$79$	$T - 1120$ T - 1120
$83$	$T + 552$ T + 552
$89$	$T + 326$ T + 326
$97$	$T + 110$ T + 110
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