LMFDB - Newform orbit 504.8.a.b

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [504,8,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, 8, names="a")

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

chi := DirichletCharacter("504.1");

S:= CuspForms(chi, 8);

N := Newforms(S);

Level:	$N$	$=$	$504 = 2^{3} \cdot 3^{2} \cdot 7$
Weight:	$k$	$=$	$8$
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:	$157.442052844$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$1$
Twist minimal:	no (minimal twist has level 56)
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 + 160 q^{5} - 343 q^{7} + 6840 q^{11} - 2900 q^{13} - 16566 q^{17} - 6718 q^{19} + 976 q^{23} - 52525 q^{25} + 61662 q^{29} - 69236 q^{31} - 54880 q^{35} - 533062 q^{37} - 183158 q^{41} + 966864 q^{43}+ \cdots - 1824794 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

160.000

−343.000

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	504.8.a.b		1
3.b	odd	2	1		56.8.a.b	✓	1
12.b	even	2	1		112.8.a.a		1
21.c	even	2	1		392.8.a.a		1
24.f	even	2	1		448.8.a.h		1
24.h	odd	2	1		448.8.a.c		1

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
56.8.a.b	✓	1	3.b	odd	2	1
112.8.a.a		1	12.b	even	2	1
392.8.a.a		1	21.c	even	2	1
448.8.a.c		1	24.h	odd	2	1
448.8.a.h		1	24.f	even	2	1
504.8.a.b		1	1.a	even	1	1	trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $T_{5} - 160$ T5 - 160 acting on $S_{8}^{\mathrm{new}}(\Gamma_0(504))$ .

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T$ T
$3$	$T$ T
$5$	$T - 160$ T - 160
$7$	$T + 343$ T + 343
$11$	$T - 6840$ T - 6840
$13$	$T + 2900$ T + 2900
$17$	$T + 16566$ T + 16566
$19$	$T + 6718$ T + 6718
$23$	$T - 976$ T - 976
$29$	$T - 61662$ T - 61662
$31$	$T + 69236$ T + 69236
$37$	$T + 533062$ T + 533062
$41$	$T + 183158$ T + 183158
$43$	$T - 966864$ T - 966864
$47$	$T - 190268$ T - 190268
$53$	$T - 785010$ T - 785010
$59$	$T + 2893594$ T + 2893594
$61$	$T + 95896$ T + 95896
$67$	$T + 991644$ T + 991644
$71$	$T + 1068160$ T + 1068160
$73$	$T - 2523458$ T - 2523458
$79$	$T - 285848$ T - 285848
$83$	$T + 7094938$ T + 7094938
$89$	$T - 252390$ T - 252390
$97$	$T + 1824794$ T + 1824794
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