LMFDB - Newform orbit 384.4.a.d

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [384,4,Mod(1,384)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(384, 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("384.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$N$	$=$	$384 = 2^{7} \cdot 3$
Weight:	$k$	$=$	$4$
Character orbit:	$[\chi]$	$=$	384.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:	$22.6567334422$
Analytic rank:	$0$
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 - 3 q^{3} + 8 q^{5} + 10 q^{7} + 9 q^{9} + 68 q^{11} - 46 q^{13} - 24 q^{15} - 74 q^{17} + 16 q^{19} - 30 q^{21} + 20 q^{23} - 61 q^{25} - 27 q^{27} + 228 q^{29} + 162 q^{31} - 204 q^{33} + 80 q^{35}+ \cdots + 612 q^{99}+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

−3.00000

8.00000

10.0000

9.00000

Atkin-Lehner signs

$p$	Sign
$2$	$+1$
$3$	$+1$

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	384.4.a.d	yes	1
3.b	odd	2	1		1152.4.a.b		1
4.b	odd	2	1		384.4.a.h	yes	1
8.b	even	2	1		384.4.a.e	yes	1
8.d	odd	2	1		384.4.a.a	✓	1
12.b	even	2	1		1152.4.a.a		1
16.e	even	4	2		768.4.d.e		2
16.f	odd	4	2		768.4.d.l		2
24.f	even	2	1		1152.4.a.k		1
24.h	odd	2	1		1152.4.a.l		1

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
384.4.a.a	✓	1	8.d	odd	2	1
384.4.a.d	yes	1	1.a	even	1	1	trivial
384.4.a.e	yes	1	8.b	even	2	1
384.4.a.h	yes	1	4.b	odd	2	1
768.4.d.e		2	16.e	even	4	2
768.4.d.l		2	16.f	odd	4	2
1152.4.a.a		1	12.b	even	2	1
1152.4.a.b		1	3.b	odd	2	1
1152.4.a.k		1	24.f	even	2	1
1152.4.a.l		1	24.h	odd	2	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{4}^{\mathrm{new}}(\Gamma_0(384))$ :

T_{5} - 8

T_{7} - 10

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T$ T
$3$	$T + 3$ T + 3
$5$	$T - 8$ T - 8
$7$	$T - 10$ T - 10
$11$	$T - 68$ T - 68
$13$	$T + 46$ T + 46
$17$	$T + 74$ T + 74
$19$	$T - 16$ T - 16
$23$	$T - 20$ T - 20
$29$	$T - 228$ T - 228
$31$	$T - 162$ T - 162
$37$	$T - 262$ T - 262
$41$	$T - 30$ T - 30
$43$	$T - 264$ T - 264
$47$	$T + 124$ T + 124
$53$	$T + 204$ T + 204
$59$	$T - 340$ T - 340
$61$	$T - 950$ T - 950
$67$	$T + 436$ T + 436
$71$	$T - 780$ T - 780
$73$	$T - 518$ T - 518
$79$	$T - 1010$ T - 1010
$83$	$T - 852$ T - 852
$89$	$T + 686$ T + 686
$97$	$T + 806$ T + 806
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