LMFDB - Newform orbit 784.3.d.a

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [784,3,Mod(687,784)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(784, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("784.687");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 784 = 2^{4} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	784.d (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:	$21.3624527258$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
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)

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/784\mathbb{Z}\right)^\times$.

$n$	$197$	$687$	$689$
$\chi(n)$	$1$	$-1$	$1$

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} $

687.1

−8.00000

9.00000

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	784.3.d.a	✓	1
4.b	odd	2	1	CM	784.3.d.a	✓	1
7.b	odd	2	1		784.3.d.c	yes	1
7.c	even	3	2		784.3.r.f		2
7.d	odd	6	2		784.3.r.c		2
28.d	even	2	1		784.3.d.c	yes	1
28.f	even	6	2		784.3.r.c		2
28.g	odd	6	2		784.3.r.f		2

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
784.3.d.a	✓	1	1.a	even	1	1	trivial
784.3.d.a	✓	1	4.b	odd	2	1	CM
784.3.d.c	yes	1	7.b	odd	2	1
784.3.d.c	yes	1	28.d	even	2	1
784.3.r.c		2	7.d	odd	6	2
784.3.r.c		2	28.f	even	6	2
784.3.r.f		2	7.c	even	3	2
784.3.r.f		2	28.g	odd	6	2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{3}^{\mathrm{new}}(784, [\chi])$:

$ T_{3} $

$ T_{5} + 8 $

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T $ T
$3$	$ T $ T
$5$	$ T + 8 $ T + 8
$7$	$ T $ T
$11$	$ T $ T
$13$	$ T + 24 $ T + 24
$17$	$ T - 16 $ T - 16
$19$	$ T $ T
$23$	$ T $ T
$29$	$ T + 42 $ T + 42
$31$	$ T $ T
$37$	$ T - 70 $ T - 70
$41$	$ T - 80 $ T - 80
$43$	$ T $ T
$47$	$ T $ T
$53$	$ T - 90 $ T - 90
$59$	$ T $ T
$61$	$ T - 120 $ T - 120
$67$	$ T $ T
$71$	$ T $ T
$73$	$ T - 96 $ T - 96
$79$	$ T $ T
$83$	$ T $ T
$89$	$ T + 160 $ T + 160
$97$	$ T + 144 $ T + 144
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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:	\( N \)	\(=\)	\( 784 = 2^{4} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	784.d (of order \(2\), degree \(1\), minimal)

\(n\)	\(197\)	\(687\)	\(689\)
\(\chi(n)\)	\(1\)	\(-1\)	\(1\)

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