LMFDB - Newform orbit 784.4.a.m

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 196)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$ q + 4 q^{3} - 20 q^{5} - 11 q^{9}+O(q^{10}) $

$ q + 4 q^{3} - 20 q^{5} - 11 q^{9} - 44 q^{11} - 44 q^{13} - 80 q^{15} + 72 q^{17} - 100 q^{19} + 120 q^{23} + 275 q^{25} - 152 q^{27} + 218 q^{29} + 280 q^{31} - 176 q^{33} - 30 q^{37} - 176 q^{39} + 120 q^{41} - 220 q^{43} + 220 q^{45} - 88 q^{47} + 288 q^{51} + 110 q^{53} + 880 q^{55} - 400 q^{57} - 580 q^{59} + 380 q^{61} + 880 q^{65} + 980 q^{67} + 480 q^{69} + 112 q^{71} - 640 q^{73} + 1100 q^{75} + 488 q^{79} - 311 q^{81} - 660 q^{83} - 1440 q^{85} + 872 q^{87} + 320 q^{89} + 1120 q^{93} + 2000 q^{95} + 248 q^{97} + 484 q^{99}+O(q^{100}) $

Display 100 coefficients 10 coefficients

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

4.00000

−20.0000

−11.0000

Atkin-Lehner signs

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

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	784.4.a.m		1
4.b	odd	2	1		196.4.a.a	✓	1
7.b	odd	2	1		784.4.a.f		1
12.b	even	2	1		1764.4.a.m		1
28.d	even	2	1		196.4.a.c	yes	1
28.f	even	6	2		196.4.e.b		2
28.g	odd	6	2		196.4.e.e		2
84.h	odd	2	1		1764.4.a.a		1
84.j	odd	6	2		1764.4.k.p		2
84.n	even	6	2		1764.4.k.a		2

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
196.4.a.a	✓	1	4.b	odd	2	1
196.4.a.c	yes	1	28.d	even	2	1
196.4.e.b		2	28.f	even	6	2
196.4.e.e		2	28.g	odd	6	2
784.4.a.f		1	7.b	odd	2	1
784.4.a.m		1	1.a	even	1	1	trivial
1764.4.a.a		1	84.h	odd	2	1
1764.4.a.m		1	12.b	even	2	1
1764.4.k.a		2	84.n	even	6	2
1764.4.k.p		2	84.j	odd	6	2

Hecke kernels

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

$ T_{3} - 4 $

$ T_{5} + 20 $

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T $ T
$3$	$ T - 4 $ T - 4
$5$	$ T + 20 $ T + 20
$7$	$ T $ T
$11$	$ T + 44 $ T + 44
$13$	$ T + 44 $ T + 44
$17$	$ T - 72 $ T - 72
$19$	$ T + 100 $ T + 100
$23$	$ T - 120 $ T - 120
$29$	$ T - 218 $ T - 218
$31$	$ T - 280 $ T - 280
$37$	$ T + 30 $ T + 30
$41$	$ T - 120 $ T - 120
$43$	$ T + 220 $ T + 220
$47$	$ T + 88 $ T + 88
$53$	$ T - 110 $ T - 110
$59$	$ T + 580 $ T + 580
$61$	$ T - 380 $ T - 380
$67$	$ T - 980 $ T - 980
$71$	$ T - 112 $ T - 112
$73$	$ T + 640 $ T + 640
$79$	$ T - 488 $ T - 488
$83$	$ T + 660 $ T + 660
$89$	$ T - 320 $ T - 320
$97$	$ T - 248 $ T - 248
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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:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	784.a (trivial)

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