LMFDB - Newform orbit 378.4.a.g

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [378,4,Mod(1,378)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(378, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("378.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$N$	$=$	$378 = 2 \cdot 3^{3} \cdot 7$
Weight:	$k$	$=$	$4$
Character orbit:	$[\chi]$	$=$	378.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:	$22.3027219822$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$1$
Twist minimal:	yes
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 + 2 q^{2} + 4 q^{4} - 15 q^{5} + 7 q^{7} + 8 q^{8} - 30 q^{10} + 42 q^{11} - 88 q^{13} + 14 q^{14} + 16 q^{16} + 45 q^{17} - 106 q^{19} - 60 q^{20} + 84 q^{22} - 114 q^{23} + 100 q^{25} - 176 q^{26}+ \cdots + 98 q^{98}+O(q^{100})

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

2.00000

4.00000

−15.0000

7.00000

8.00000

−30.0000

Atkin-Lehner signs

$p$	Sign
$2$	$-1$
$3$	$-1$
$7$	$-1$

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	378.4.a.g	yes	1
3.b	odd	2	1		378.4.a.f	✓	1

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
378.4.a.f	✓	1	3.b	odd	2	1
378.4.a.g	yes	1	1.a	even	1	1	trivial

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{4}^{\mathrm{new}}(\Gamma_0(378))$ :

T_{5} + 15

T_{11} - 42

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T - 2$ T - 2
$3$	$T$ T
$5$	$T + 15$ T + 15
$7$	$T - 7$ T - 7
$11$	$T - 42$ T - 42
$13$	$T + 88$ T + 88
$17$	$T - 45$ T - 45
$19$	$T + 106$ T + 106
$23$	$T + 114$ T + 114
$29$	$T + 66$ T + 66
$31$	$T + 304$ T + 304
$37$	$T + 187$ T + 187
$41$	$T + 69$ T + 69
$43$	$T - 29$ T - 29
$47$	$T + 471$ T + 471
$53$	$T - 414$ T - 414
$59$	$T - 597$ T - 597
$61$	$T - 218$ T - 218
$67$	$T + 628$ T + 628
$71$	$T + 288$ T + 288
$73$	$T - 1190$ T - 1190
$79$	$T + 295$ T + 295
$83$	$T - 1311$ T - 1311
$89$	$T + 1206$ T + 1206
$97$	$T + 1186$ T + 1186
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