LMFDB - Newform orbit 912.6.a.d

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [912,6,Mod(1,912)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(912, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("912.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$N$	$=$	$912 = 2^{4} \cdot 3 \cdot 19$
Weight:	$k$	$=$	$6$
Character orbit:	$[\chi]$	$=$	912.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:	$146.270043669$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$1$
Twist minimal:	no (minimal twist has level 57)
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} + 6 q^{5} + 176 q^{7} + 81 q^{9} + 496 q^{11} - 178 q^{13} - 54 q^{15} + 202 q^{17} + 361 q^{19} - 1584 q^{21} - 4396 q^{23} - 3089 q^{25} - 729 q^{27} - 5902 q^{29} - 5760 q^{31} - 4464 q^{33}+ \cdots + 40176 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

6.00000

176.000

81.0000

Atkin-Lehner signs

$p$	Sign
$2$	$-1$
$3$	$+1$
$19$	$-1$

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	912.6.a.d		1
4.b	odd	2	1		57.6.a.b	✓	1
12.b	even	2	1		171.6.a.a		1
76.d	even	2	1		1083.6.a.a		1

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
57.6.a.b	✓	1	4.b	odd	2	1
171.6.a.a		1	12.b	even	2	1
912.6.a.d		1	1.a	even	1	1	trivial
1083.6.a.a		1	76.d	even	2	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $T_{5} - 6$ T5 - 6 acting on $S_{6}^{\mathrm{new}}(\Gamma_0(912))$ .

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T$ T
$3$	$T + 9$ T + 9
$5$	$T - 6$ T - 6
$7$	$T - 176$ T - 176
$11$	$T - 496$ T - 496
$13$	$T + 178$ T + 178
$17$	$T - 202$ T - 202
$19$	$T - 361$ T - 361
$23$	$T + 4396$ T + 4396
$29$	$T + 5902$ T + 5902
$31$	$T + 5760$ T + 5760
$37$	$T + 3906$ T + 3906
$41$	$T - 15774$ T - 15774
$43$	$T - 7492$ T - 7492
$47$	$T - 7452$ T - 7452
$53$	$T + 29014$ T + 29014
$59$	$T + 13604$ T + 13604
$61$	$T + 12466$ T + 12466
$67$	$T + 43436$ T + 43436
$71$	$T + 28800$ T + 28800
$73$	$T - 80746$ T - 80746
$79$	$T + 76456$ T + 76456
$83$	$T - 56880$ T - 56880
$89$	$T + 103266$ T + 103266
$97$	$T - 82490$ T - 82490
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