LMFDB - Newform orbit 192.6.a.m

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [192,6,Mod(1,192)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(192, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 6, names="a")

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

chi := DirichletCharacter("192.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$N$	$=$	$192 = 2^{6} \cdot 3$
Weight:	$k$	$=$	$6$
Character orbit:	$[\chi]$	$=$	192.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:	$30.7936934041$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$1$
Twist minimal:	no (minimal twist has level 96)
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 + 9 q^{3} + 14 q^{5} + 100 q^{7} + 81 q^{9} - 220 q^{11} + 818 q^{13} + 126 q^{15} - 774 q^{17} + 1436 q^{19} + 900 q^{21} + 192 q^{23} - 2929 q^{25} + 729 q^{27} + 7022 q^{29} + 1436 q^{31} - 1980 q^{33}+ \cdots - 17820 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

9.00000

14.0000

100.000

81.0000

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	192.6.a.m		1
3.b	odd	2	1		576.6.a.p		1
4.b	odd	2	1		192.6.a.e		1
8.b	even	2	1		96.6.a.b	✓	1
8.d	odd	2	1		96.6.a.e	yes	1
12.b	even	2	1		576.6.a.o		1
16.e	even	4	2		768.6.d.g		2
16.f	odd	4	2		768.6.d.l		2
24.f	even	2	1		288.6.a.f		1
24.h	odd	2	1		288.6.a.g		1

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
96.6.a.b	✓	1	8.b	even	2	1
96.6.a.e	yes	1	8.d	odd	2	1
192.6.a.e		1	4.b	odd	2	1
192.6.a.m		1	1.a	even	1	1	trivial
288.6.a.f		1	24.f	even	2	1
288.6.a.g		1	24.h	odd	2	1
576.6.a.o		1	12.b	even	2	1
576.6.a.p		1	3.b	odd	2	1
768.6.d.g		2	16.e	even	4	2
768.6.d.l		2	16.f	odd	4	2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{6}^{\mathrm{new}}(\Gamma_0(192))$ :

T_{5} - 14

T_{7} - 100

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T$ T
$3$	$T - 9$ T - 9
$5$	$T - 14$ T - 14
$7$	$T - 100$ T - 100
$11$	$T + 220$ T + 220
$13$	$T - 818$ T - 818
$17$	$T + 774$ T + 774
$19$	$T - 1436$ T - 1436
$23$	$T - 192$ T - 192
$29$	$T - 7022$ T - 7022
$31$	$T - 1436$ T - 1436
$37$	$T - 3410$ T - 3410
$41$	$T + 7838$ T + 7838
$43$	$T - 16036$ T - 16036
$47$	$T - 22712$ T - 22712
$53$	$T + 27578$ T + 27578
$59$	$T - 28828$ T - 28828
$61$	$T + 12438$ T + 12438
$67$	$T - 70948$ T - 70948
$71$	$T - 58832$ T - 58832
$73$	$T - 79386$ T - 79386
$79$	$T + 46948$ T + 46948
$83$	$T + 67284$ T + 67284
$89$	$T - 16106$ T - 16106
$97$	$T + 4238$ T + 4238
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