LMFDB - Newform orbit 2116.1.c.b

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2116,1,Mod(1059,2116)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2116.1059"); S:= CuspForms(chi, 1); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2116, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 0])) B = ModularForms(chi, 1).cuspidal_submodule().basis() N = [B[i] for i in range(len(B))]

Level:	$ N $	$=$	$ 2116 = 2^{2} \cdot 23^{2} $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	2116.c (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [1,1,0,1,-1] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)

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

Self dual:	yes
Analytic conductor:	$1.05602156673$
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 3.1.2116.1
Artin image:	$S_3$
Artin field:	Galois closure of 3.1.2116.1
Stark unit:	Root of $x^{3} - 361103x^{2} + 1199x - 1$

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2116\mathbb{Z}\right)^\times$.

$n$	$5$	$1059$
$\chi(n)$	$1$	$-1$

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} $

1059.1

1.00000

−1.00000

1.00000

−1.00000

Inner twists

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

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	2116.1.c.b	✓	1
4.b	odd	2	1	CM	2116.1.c.b	✓	1
23.b	odd	2	1		2116.1.c.c	yes	1
23.c	even	11	10		2116.1.g.b		10
23.d	odd	22	10		2116.1.g.a		10
92.b	even	2	1		2116.1.c.c	yes	1
92.g	odd	22	10		2116.1.g.b		10
92.h	even	22	10		2116.1.g.a		10

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
2116.1.c.b	✓	1	1.a	even	1	1	trivial
2116.1.c.b	✓	1	4.b	odd	2	1	CM
2116.1.c.c	yes	1	23.b	odd	2	1
2116.1.c.c	yes	1	92.b	even	2	1
2116.1.g.a		10	23.d	odd	22	10
2116.1.g.a		10	92.h	even	22	10
2116.1.g.b		10	23.c	even	11	10
2116.1.g.b		10	92.g	odd	22	10

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{1}^{\mathrm{new}}(2116, [\chi])$:

$ T_{3} $

T3

$ T_{5} + 1 $

T5 + 1

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T - 1 $ T - 1
$3$	$ T $ T
$5$	$ T + 1 $ T + 1
$7$	$ T $ T
$11$	$ T $ T
$13$	$ T + 1 $ T + 1
$17$	$ T - 2 $ T - 2
$19$	$ T $ T
$23$	$ T $ T
$29$	$ T + 1 $ T + 1
$31$	$ T $ T
$37$	$ T - 2 $ T - 2
$41$	$ T + 1 $ T + 1
$43$	$ T $ T
$47$	$ T $ T
$53$	$ T + 1 $ T + 1
$59$	$ T $ T
$61$	$ T + 1 $ T + 1
$67$	$ T $ T
$71$	$ T $ T
$73$	$ T + 1 $ T + 1
$79$	$ T $ T
$83$	$ T $ T
$89$	$ T + 1 $ T + 1
$97$	$ T + 1 $ T + 1
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Level:	\( N \)	\(=\)	\( 2116 = 2^{2} \cdot 23^{2} \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	2116.c (of order \(2\), degree \(1\), minimal)

\(n\)	\(5\)	\(1059\)
\(\chi(n)\)	\(1\)	\(-1\)

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