LMFDB - Newform orbit 504.6.a.h

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [504,6,Mod(1,504)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(504, 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("504.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$N$	$=$	$504 = 2^{3} \cdot 3^{2} \cdot 7$
Weight:	$k$	$=$	$6$
Character orbit:	$[\chi]$	$=$	504.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:	$80.8334451857$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$1$
Twist minimal:	no (minimal twist has level 168)
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 + 64 q^{5} + 49 q^{7} + 54 q^{11} + 738 q^{13} + 848 q^{17} - 1604 q^{19} + 3670 q^{23} + 971 q^{25} + 4330 q^{29} - 4760 q^{31} + 3136 q^{35} - 2094 q^{37} + 6116 q^{41} + 7916 q^{43} - 6572 q^{47}+ \cdots + 159670 q^{97}+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

64.0000

49.0000

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	504.6.a.h		1
3.b	odd	2	1		168.6.a.d	✓	1
4.b	odd	2	1		1008.6.a.z		1
12.b	even	2	1		336.6.a.c		1

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
168.6.a.d	✓	1	3.b	odd	2	1
336.6.a.c		1	12.b	even	2	1
504.6.a.h		1	1.a	even	1	1	trivial
1008.6.a.z		1	4.b	odd	2	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{6}^{\mathrm{new}}(\Gamma_0(504))$ :

T_{5} - 64

T_{11} - 54

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T$ T
$3$	$T$ T
$5$	$T - 64$ T - 64
$7$	$T - 49$ T - 49
$11$	$T - 54$ T - 54
$13$	$T - 738$ T - 738
$17$	$T - 848$ T - 848
$19$	$T + 1604$ T + 1604
$23$	$T - 3670$ T - 3670
$29$	$T - 4330$ T - 4330
$31$	$T + 4760$ T + 4760
$37$	$T + 2094$ T + 2094
$41$	$T - 6116$ T - 6116
$43$	$T - 7916$ T - 7916
$47$	$T + 6572$ T + 6572
$53$	$T - 7894$ T - 7894
$59$	$T - 41664$ T - 41664
$61$	$T + 26570$ T + 26570
$67$	$T + 41736$ T + 41736
$71$	$T + 83574$ T + 83574
$73$	$T + 42314$ T + 42314
$79$	$T - 508$ T - 508
$83$	$T - 8364$ T - 8364
$89$	$T - 49220$ T - 49220
$97$	$T - 159670$ T - 159670
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