LMFDB - Newform orbit 32.9.d.a

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [32,9,Mod(15,32)]

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([1, 1]))

N = Newforms(chi, 9, names="a")

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

chi := DirichletCharacter("32.15");

S:= CuspForms(chi, 9);

N := Newforms(S);

Level:	$ N $	$=$	$ 32 = 2^{5} $
Weight:	$ k $	$=$	$ 9 $
Character orbit:	$[\chi]$	$=$	32.d (of order $2$, degree $1$, not 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:	yes
Analytic conductor:	$13.0361155220$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 8)
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$ q - 34 q^{3} - 5405 q^{9}+O(q^{10}) $

$ q - 34 q^{3} - 5405 q^{9} + 27166 q^{11} + 162434 q^{17} + 72286 q^{19} + 390625 q^{25} + 406844 q^{27} - 923644 q^{33} - 4099006 q^{41} - 5426402 q^{43} + 5764801 q^{49} - 5522756 q^{51} - 2457724 q^{57} + 24178078 q^{59} + 13944286 q^{67} + 33567554 q^{73} - 13281250 q^{75} + 21629509 q^{81} - 30209954 q^{83} - 95519806 q^{89} - 77418238 q^{97} - 146832230 q^{99}+O(q^{100}) $

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/32\mathbb{Z}\right)^\times$.

$n$	$5$	$31$
$\chi(n)$	$-1$	$-1$

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} $

15.1

−34.0000

−5405.00

Inner twists

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

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	32.9.d.a		1
3.b	odd	2	1		288.9.b.a		1
4.b	odd	2	1		8.9.d.a	✓	1
8.b	even	2	1		8.9.d.a	✓	1
8.d	odd	2	1	CM	32.9.d.a		1
12.b	even	2	1		72.9.b.a		1
16.e	even	4	2		256.9.c.f		2
16.f	odd	4	2		256.9.c.f		2
24.f	even	2	1		288.9.b.a		1
24.h	odd	2	1		72.9.b.a		1

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
8.9.d.a	✓	1	4.b	odd	2	1
8.9.d.a	✓	1	8.b	even	2	1
32.9.d.a		1	1.a	even	1	1	trivial
32.9.d.a		1	8.d	odd	2	1	CM
72.9.b.a		1	12.b	even	2	1
72.9.b.a		1	24.h	odd	2	1
256.9.c.f		2	16.e	even	4	2
256.9.c.f		2	16.f	odd	4	2
288.9.b.a		1	3.b	odd	2	1
288.9.b.a		1	24.f	even	2	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $ T_{3} + 34 $ T3 + 34 acting on $S_{9}^{\mathrm{new}}(32, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T $ T
$3$	$ T + 34 $ T + 34
$5$	$ T $ T
$7$	$ T $ T
$11$	$ T - 27166 $ T - 27166
$13$	$ T $ T
$17$	$ T - 162434 $ T - 162434
$19$	$ T - 72286 $ T - 72286
$23$	$ T $ T
$29$	$ T $ T
$31$	$ T $ T
$37$	$ T $ T
$41$	$ T + 4099006 $ T + 4099006
$43$	$ T + 5426402 $ T + 5426402
$47$	$ T $ T
$53$	$ T $ T
$59$	$ T - 24178078 $ T - 24178078
$61$	$ T $ T
$67$	$ T - 13944286 $ T - 13944286
$71$	$ T $ T
$73$	$ T - 33567554 $ T - 33567554
$79$	$ T $ T
$83$	$ T + 30209954 $ T + 30209954
$89$	$ T + 95519806 $ T + 95519806
$97$	$ T + 77418238 $ T + 77418238
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Elliptic curves over $\Q$
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Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 32 = 2^{5} \)
Weight:	\( k \)	\(=\)	\( 9 \)
Character orbit:	\([\chi]\)	\(=\)	32.d (of order \(2\), degree \(1\), not minimal)

\(n\)	\(5\)	\(31\)
\(\chi(n)\)	\(-1\)	\(-1\)

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