LMFDB - Newform orbit 784.4.a.c

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 14)
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 - 5 q^{3} + 9 q^{5} - 2 q^{9}+O(q^{10}) $

$ q - 5 q^{3} + 9 q^{5} - 2 q^{9} + 57 q^{11} + 70 q^{13} - 45 q^{15} - 51 q^{17} + 5 q^{19} - 69 q^{23} - 44 q^{25} + 145 q^{27} + 114 q^{29} + 23 q^{31} - 285 q^{33} - 253 q^{37} - 350 q^{39} + 42 q^{41} + 124 q^{43} - 18 q^{45} + 201 q^{47} + 255 q^{51} - 393 q^{53} + 513 q^{55} - 25 q^{57} + 219 q^{59} + 709 q^{61} + 630 q^{65} - 419 q^{67} + 345 q^{69} + 96 q^{71} + 313 q^{73} + 220 q^{75} - 461 q^{79} - 671 q^{81} - 588 q^{83} - 459 q^{85} - 570 q^{87} + 1017 q^{89} - 115 q^{93} + 45 q^{95} + 1834 q^{97} - 114 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

−5.00000

9.00000

−2.00000

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.c		1
4.b	odd	2	1		98.4.a.f		1
7.b	odd	2	1		784.4.a.p		1
7.d	odd	6	2		112.4.i.a		2
12.b	even	2	1		882.4.a.c		1
20.d	odd	2	1		2450.4.a.d		1
28.d	even	2	1		98.4.a.d		1
28.f	even	6	2		14.4.c.a	✓	2
28.g	odd	6	2		98.4.c.a		2
56.j	odd	6	2		448.4.i.e		2
56.m	even	6	2		448.4.i.b		2
84.h	odd	2	1		882.4.a.f		1
84.j	odd	6	2		126.4.g.d		2
84.n	even	6	2		882.4.g.u		2
140.c	even	2	1		2450.4.a.q		1
140.s	even	6	2		350.4.e.e		2
140.x	odd	12	4		350.4.j.b		4

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
14.4.c.a	✓	2	28.f	even	6	2
98.4.a.d		1	28.d	even	2	1
98.4.a.f		1	4.b	odd	2	1
98.4.c.a		2	28.g	odd	6	2
112.4.i.a		2	7.d	odd	6	2
126.4.g.d		2	84.j	odd	6	2
350.4.e.e		2	140.s	even	6	2
350.4.j.b		4	140.x	odd	12	4
448.4.i.b		2	56.m	even	6	2
448.4.i.e		2	56.j	odd	6	2
784.4.a.c		1	1.a	even	1	1	trivial
784.4.a.p		1	7.b	odd	2	1
882.4.a.c		1	12.b	even	2	1
882.4.a.f		1	84.h	odd	2	1
882.4.g.u		2	84.n	even	6	2
2450.4.a.d		1	20.d	odd	2	1
2450.4.a.q		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} + 5 $

$ T_{5} - 9 $

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T $ T
$3$	$ T + 5 $ T + 5
$5$	$ T - 9 $ T - 9
$7$	$ T $ T
$11$	$ T - 57 $ T - 57
$13$	$ T - 70 $ T - 70
$17$	$ T + 51 $ T + 51
$19$	$ T - 5 $ T - 5
$23$	$ T + 69 $ T + 69
$29$	$ T - 114 $ T - 114
$31$	$ T - 23 $ T - 23
$37$	$ T + 253 $ T + 253
$41$	$ T - 42 $ T - 42
$43$	$ T - 124 $ T - 124
$47$	$ T - 201 $ T - 201
$53$	$ T + 393 $ T + 393
$59$	$ T - 219 $ T - 219
$61$	$ T - 709 $ T - 709
$67$	$ T + 419 $ T + 419
$71$	$ T - 96 $ T - 96
$73$	$ T - 313 $ T - 313
$79$	$ T + 461 $ T + 461
$83$	$ T + 588 $ T + 588
$89$	$ T - 1017 $ T - 1017
$97$	$ T - 1834 $ T - 1834
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Overview	Random
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Classical	Maass
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Elliptic curves over $\Q$
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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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