LMFDB - Newform orbit 784.4.a.k

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [784,4,Mod(1,784)]

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([0, 0, 0]))

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("784.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 784 = 2^{4} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	784.a (trivial)

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:	$46.2574974445$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 49)
Fricke sign:	$1$
Sato-Tate group:	$N(\mathrm{U}(1))$

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

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} $

1.1

−27.0000

Atkin-Lehner signs

$ p $	Sign
$2$	$-1$
$7$	$-1$

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
7.b	odd	2	1	CM by $\Q(\sqrt{-7}) $

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	784.4.a.k		1
4.b	odd	2	1		49.4.a.a	✓	1
7.b	odd	2	1	CM	784.4.a.k		1
12.b	even	2	1		441.4.a.m		1
20.d	odd	2	1		1225.4.a.l		1
28.d	even	2	1		49.4.a.a	✓	1
28.f	even	6	2		49.4.c.d		2
28.g	odd	6	2		49.4.c.d		2
84.h	odd	2	1		441.4.a.m		1
84.j	odd	6	2		441.4.e.a		2
84.n	even	6	2		441.4.e.a		2
140.c	even	2	1		1225.4.a.l		1

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
49.4.a.a	✓	1	4.b	odd	2	1
49.4.a.a	✓	1	28.d	even	2	1
49.4.c.d		2	28.f	even	6	2
49.4.c.d		2	28.g	odd	6	2
441.4.a.m		1	12.b	even	2	1
441.4.a.m		1	84.h	odd	2	1
441.4.e.a		2	84.j	odd	6	2
441.4.e.a		2	84.n	even	6	2
784.4.a.k		1	1.a	even	1	1	trivial
784.4.a.k		1	7.b	odd	2	1	CM
1225.4.a.l		1	20.d	odd	2	1
1225.4.a.l		1	140.c	even	2	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{4}^{\mathrm{new}}(\Gamma_0(784))$:

$ T_{3} $

$ T_{5} $

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T $ T
$3$	$ T $ T
$5$	$ T $ T
$7$	$ T $ T
$11$	$ T - 68 $ T - 68
$13$	$ T $ T
$17$	$ T $ T
$19$	$ T $ T
$23$	$ T - 40 $ T - 40
$29$	$ T + 166 $ T + 166
$31$	$ T $ T
$37$	$ T - 450 $ T - 450
$41$	$ T $ T
$43$	$ T - 180 $ T - 180
$47$	$ T $ T
$53$	$ T - 590 $ T - 590
$59$	$ T $ T
$61$	$ T $ T
$67$	$ T - 740 $ T - 740
$71$	$ T + 688 $ T + 688
$73$	$ T $ T
$79$	$ T - 1384 $ T - 1384
$83$	$ T $ T
$89$	$ T $ T
$97$	$ T $ T
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Overview	Random
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Classical	Maass
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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:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	784.a (trivial)

\(n\):		e.g. 2-40 or 990-1000
Significant digits:
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