LMFDB - Newform orbit 2352.4.a.k

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2352,4,Mod(1,2352)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2352, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("2352.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$N$	$=$	$2352 = 2^{4} \cdot 3 \cdot 7^{2}$
Weight:	$k$	$=$	$4$
Character orbit:	$[\chi]$	$=$	2352.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:	$138.772492334$
Analytic rank:	$1$
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 - 3 q^{3} + 2 q^{5} + 9 q^{9} - 12 q^{11} + 66 q^{13} - 6 q^{15} + 70 q^{17} - 92 q^{19} - 16 q^{23} - 121 q^{25} - 27 q^{27} - 122 q^{29} + 64 q^{31} + 36 q^{33} - 306 q^{37} - 198 q^{39} - 50 q^{41}+ \cdots - 108 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

2.00000

9.00000

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	2352.4.a.k		1
4.b	odd	2	1		1176.4.a.l		1
7.b	odd	2	1		336.4.a.i		1
21.c	even	2	1		1008.4.a.k		1
28.d	even	2	1		168.4.a.b	✓	1
56.e	even	2	1		1344.4.a.u		1
56.h	odd	2	1		1344.4.a.h		1
84.h	odd	2	1		504.4.a.d		1

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
168.4.a.b	✓	1	28.d	even	2	1
336.4.a.i		1	7.b	odd	2	1
504.4.a.d		1	84.h	odd	2	1
1008.4.a.k		1	21.c	even	2	1
1176.4.a.l		1	4.b	odd	2	1
1344.4.a.h		1	56.h	odd	2	1
1344.4.a.u		1	56.e	even	2	1
2352.4.a.k		1	1.a	even	1	1	trivial

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{4}^{\mathrm{new}}(\Gamma_0(2352))$ :

T_{5} - 2

T_{11} + 12

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T$ T
$3$	$T + 3$ T + 3
$5$	$T - 2$ T - 2
$7$	$T$ T
$11$	$T + 12$ T + 12
$13$	$T - 66$ T - 66
$17$	$T - 70$ T - 70
$19$	$T + 92$ T + 92
$23$	$T + 16$ T + 16
$29$	$T + 122$ T + 122
$31$	$T - 64$ T - 64
$37$	$T + 306$ T + 306
$41$	$T + 50$ T + 50
$43$	$T + 20$ T + 20
$47$	$T + 176$ T + 176
$53$	$T - 526$ T - 526
$59$	$T - 540$ T - 540
$61$	$T - 818$ T - 818
$67$	$T - 228$ T - 228
$71$	$T + 864$ T + 864
$73$	$T + 106$ T + 106
$79$	$T + 736$ T + 736
$83$	$T + 588$ T + 588
$89$	$T + 146$ T + 146
$97$	$T - 1214$ T - 1214
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