LMFDB - Newform orbit 1712.1.g.a

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1712,1,Mod(641,1712)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1712, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 1]))

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

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

chi := DirichletCharacter("1712.641");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 1712 = 2^{4} \cdot 107 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	1712.g (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:	$0.854399301628$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 107)
Projective image:	$D_{3}$
Projective field:	Galois closure of 3.1.107.1
Artin image:	$D_6$
Artin field:	Galois closure of 6.0.732736.2

$q$-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 $\left(\mathbb{Z}/1712\mathbb{Z}\right)^\times$.

$n$	$1071$	$1285$	$1393$
$\chi(n)$	$1$	$1$	$-1$

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} $

641.1

1.00000

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
107.b	odd	2	1	CM by $\Q(\sqrt{-107}) $

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	1712.1.g.a		1
4.b	odd	2	1		107.1.b.a	✓	1
12.b	even	2	1		963.1.b.a		1
20.d	odd	2	1		2675.1.c.a		1
20.e	even	4	2		2675.1.d.a		2
107.b	odd	2	1	CM	1712.1.g.a		1
428.b	even	2	1		107.1.b.a	✓	1
1284.h	odd	2	1		963.1.b.a		1
2140.f	even	2	1		2675.1.c.a		1
2140.i	odd	4	2		2675.1.d.a		2

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
107.1.b.a	✓	1	4.b	odd	2	1
107.1.b.a	✓	1	428.b	even	2	1
963.1.b.a		1	12.b	even	2	1
963.1.b.a		1	1284.h	odd	2	1
1712.1.g.a		1	1.a	even	1	1	trivial
1712.1.g.a		1	107.b	odd	2	1	CM
2675.1.c.a		1	20.d	odd	2	1
2675.1.c.a		1	2140.f	even	2	1
2675.1.d.a		2	20.e	even	4	2
2675.1.d.a		2	2140.i	odd	4	2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $ T_{3} - 1 $ T3 - 1 acting on $S_{1}^{\mathrm{new}}(1712, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T $ T
$3$	$ T - 1 $ T - 1
$5$	$ T $ T
$7$	$ T $ T
$11$	$ T - 1 $ T - 1
$13$	$ T + 1 $ T + 1
$17$	$ T $ T
$19$	$ T - 1 $ T - 1
$23$	$ T - 1 $ T - 1
$29$	$ T - 2 $ T - 2
$31$	$ T $ T
$37$	$ T + 1 $ T + 1
$41$	$ T + 1 $ T + 1
$43$	$ T $ T
$47$	$ T + 2 $ T + 2
$53$	$ T + 1 $ T + 1
$59$	$ T $ T
$61$	$ T + 1 $ T + 1
$67$	$ T $ T
$71$	$ T $ T
$73$	$ T $ T
$79$	$ T - 1 $ T - 1
$83$	$ T + 2 $ T + 2
$89$	$ T + 1 $ T + 1
$97$	$ T $ T
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Level:	\( N \)	\(=\)	\( 1712 = 2^{4} \cdot 107 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	1712.g (of order \(2\), degree \(1\), not minimal)

\(n\)	\(1071\)	\(1285\)	\(1393\)
\(\chi(n)\)	\(1\)	\(1\)	\(-1\)

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