LMFDB - Newform orbit 983.1.b.a

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [983,1,Mod(982,983)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(983, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1]))

N = Newforms(chi, 1, names="a")

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

chi := DirichletCharacter("983.982");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 983 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	983.b (of order $2$, degree $1$, 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:	$0.490580907418$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Projective image:	$D_{3}$
Projective field:	Galois closure of 3.1.983.1
Artin image:	$S_3$
Artin field:	Galois closure of 3.1.983.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$ q - q^{2} - q^{3} + q^{6} - q^{7} + q^{8}+O(q^{10}) $

$ q - q^{2} - q^{3} + q^{6} - q^{7} + q^{8} + q^{14} - q^{16} - q^{19} + q^{21} - q^{23} - q^{24} + q^{25} + q^{27} + 2 q^{31} + 2 q^{37} + q^{38} + 2 q^{41} - q^{42} - q^{43} + q^{46} - q^{47} + q^{48} - q^{50} - q^{53} - q^{54} - q^{56} + q^{57} + 2 q^{59} - 2 q^{62} + q^{64} + 2 q^{67} + q^{69} - 2 q^{74} - q^{75} + 2 q^{79} - q^{81} - 2 q^{82} + q^{86} - q^{89} - 2 q^{93} + q^{94}+O(q^{100}) $

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/983\mathbb{Z}\right)^\times$.

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

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} $

982.1

−1.00000

1.00000

−1.00000

1.00000

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
983.b	odd	2	1	CM by $\Q(\sqrt{-983}) $

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	983.1.b.a	✓	1
983.b	odd	2	1	CM	983.1.b.a	✓	1

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
983.1.b.a	✓	1	1.a	even	1	1	trivial
983.1.b.a	✓	1	983.b	odd	2	1	CM

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $ T_{2} + 1 $ T2 + 1 acting on $S_{1}^{\mathrm{new}}(983, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T + 1 $ T + 1
$3$	$ T + 1 $ T + 1
$5$	$ T $ T
$7$	$ T + 1 $ T + 1
$11$	$ T $ T
$13$	$ T $ T
$17$	$ T $ T
$19$	$ T + 1 $ T + 1
$23$	$ T + 1 $ T + 1
$29$	$ T $ T
$31$	$ T - 2 $ T - 2
$37$	$ T - 2 $ T - 2
$41$	$ T - 2 $ T - 2
$43$	$ T + 1 $ T + 1
$47$	$ T + 1 $ T + 1
$53$	$ T + 1 $ T + 1
$59$	$ T - 2 $ T - 2
$61$	$ T $ T
$67$	$ T - 2 $ T - 2
$71$	$ T $ T
$73$	$ T $ T
$79$	$ T - 2 $ T - 2
$83$	$ T $ T
$89$	$ T + 1 $ T + 1
$97$	$ T $ T
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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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 983 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	983.b (of order \(2\), degree \(1\), minimal)

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

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