LMFDB - Newform orbit 9.4.a.a

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [9,4,Mod(1,9)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(9, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0])) N = Newforms(chi, 4, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("9.1"); S:= CuspForms(chi, 4); N := Newforms(S);

Level:	$N$	$=$	$9 = 3^{2}$
Weight:	$k$	$=$	$4$
Character orbit:	$[\chi]$	$=$	9.a (trivial)

Newform invariants

comment:select newform

sage:traces = [] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(0)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$0.531017190052$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$1$
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$N(\mathrm{U}(1))$

$q$ -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^{4} + 20 q^{7} - 70 q^{13} + 64 q^{16} + 56 q^{19} - 125 q^{25} - 160 q^{28} + 308 q^{31} + 110 q^{37} - 520 q^{43} + 57 q^{49} + 560 q^{52} + 182 q^{61} - 512 q^{64} - 880 q^{67} + 1190 q^{73}+ \cdots - 1330 q^{97}+O(q^{100})

q - 8 * q^4 + 20 * q^7 - 70 * q^13 + 64 * q^16 + 56 * q^19 - 125 * q^25 - 160 * q^28 + 308 * q^31 + 110 * q^37 - 520 * q^43 + 57 * q^49 + 560 * q^52 + 182 * q^61 - 512 * q^64 - 880 * q^67 + 1190 * q^73 - 448 * q^76 + 884 * q^79 - 1400 * q^91 - 1330 * q^97

Expression as an eta quotient

$f(z) = \eta(3z)^{8}=q\prod_{n=1}^\infty(1 - q^{3n})^{8}$

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

20.0000

Atkin-Lehner signs

$p$	Sign
$3$	$+1$

Inner twists

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

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	9.4.a.a	✓	1
3.b	odd	2	1	CM	9.4.a.a	✓	1
4.b	odd	2	1		144.4.a.d		1
5.b	even	2	1		225.4.a.d		1
5.c	odd	4	2		225.4.b.g		2
7.b	odd	2	1		441.4.a.f		1
7.c	even	3	2		441.4.e.i		2
7.d	odd	6	2		441.4.e.j		2
8.b	even	2	1		576.4.a.m		1
8.d	odd	2	1		576.4.a.l		1
9.c	even	3	2		81.4.c.b		2
9.d	odd	6	2		81.4.c.b		2
11.b	odd	2	1		1089.4.a.g		1
12.b	even	2	1		144.4.a.d		1
13.b	even	2	1		1521.4.a.g		1
15.d	odd	2	1		225.4.a.d		1
15.e	even	4	2		225.4.b.g		2
21.c	even	2	1		441.4.a.f		1
21.g	even	6	2		441.4.e.j		2
21.h	odd	6	2		441.4.e.i		2
24.f	even	2	1		576.4.a.l		1
24.h	odd	2	1		576.4.a.m		1
33.d	even	2	1		1089.4.a.g		1
39.d	odd	2	1		1521.4.a.g		1

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
9.4.a.a	✓	1	1.a	even	1	1	trivial
9.4.a.a	✓	1	3.b	odd	2	1	CM
81.4.c.b		2	9.c	even	3	2
81.4.c.b		2	9.d	odd	6	2
144.4.a.d		1	4.b	odd	2	1
144.4.a.d		1	12.b	even	2	1
225.4.a.d		1	5.b	even	2	1
225.4.a.d		1	15.d	odd	2	1
225.4.b.g		2	5.c	odd	4	2
225.4.b.g		2	15.e	even	4	2
441.4.a.f		1	7.b	odd	2	1
441.4.a.f		1	21.c	even	2	1
441.4.e.i		2	7.c	even	3	2
441.4.e.i		2	21.h	odd	6	2
441.4.e.j		2	7.d	odd	6	2
441.4.e.j		2	21.g	even	6	2
576.4.a.l		1	8.d	odd	2	1
576.4.a.l		1	24.f	even	2	1
576.4.a.m		1	8.b	even	2	1
576.4.a.m		1	24.h	odd	2	1
1089.4.a.g		1	11.b	odd	2	1
1089.4.a.g		1	33.d	even	2	1
1521.4.a.g		1	13.b	even	2	1
1521.4.a.g		1	39.d	odd	2	1

Hecke kernels

This newform subspace is the entire newspace $S_{4}^{\mathrm{new}}(\Gamma_0(9))$ .

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T$ T
$3$	$T$ T
$5$	$T$ T
$7$	$T - 20$ T - 20
$11$	$T$ T
$13$	$T + 70$ T + 70
$17$	$T$ T
$19$	$T - 56$ T - 56
$23$	$T$ T
$29$	$T$ T
$31$	$T - 308$ T - 308
$37$	$T - 110$ T - 110
$41$	$T$ T
$43$	$T + 520$ T + 520
$47$	$T$ T
$53$	$T$ T
$59$	$T$ T
$61$	$T - 182$ T - 182
$67$	$T + 880$ T + 880
$71$	$T$ T
$73$	$T - 1190$ T - 1190
$79$	$T - 884$ T - 884
$83$	$T$ T
$89$	$T$ T
$97$	$T + 1330$ T + 1330
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 $\F_{q}$
Belyi maps

$n$ :		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