LMFDB - Newform orbit 2112.4.a.s

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2112,4,Mod(1,2112)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2112, 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("2112.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$N$	$=$	$2112 = 2^{6} \cdot 3 \cdot 11$
Weight:	$k$	$=$	$4$
Character orbit:	$[\chi]$	$=$	2112.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:	$124.612033932$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$1$
Twist minimal:	no (minimal twist has level 66)
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} - 14 q^{7} + 9 q^{9} + 11 q^{11} - 80 q^{13} + 30 q^{17} + 56 q^{19} - 42 q^{21} + 126 q^{23} - 125 q^{25} + 27 q^{27} + 222 q^{29} + 16 q^{31} + 33 q^{33} + 106 q^{37} - 240 q^{39} + 114 q^{41}+ \cdots + 99 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

−14.0000

9.00000

Atkin-Lehner signs

$p$	Sign
$2$	$-1$
$3$	$-1$
$11$	$-1$

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	2112.4.a.s		1
4.b	odd	2	1		2112.4.a.g		1
8.b	even	2	1		528.4.a.d		1
8.d	odd	2	1		66.4.a.a	✓	1
24.f	even	2	1		198.4.a.f		1
24.h	odd	2	1		1584.4.a.i		1
40.e	odd	2	1		1650.4.a.h		1
40.k	even	4	2		1650.4.c.g		2
88.g	even	2	1		726.4.a.h		1
264.p	odd	2	1		2178.4.a.g		1

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
66.4.a.a	✓	1	8.d	odd	2	1
198.4.a.f		1	24.f	even	2	1
528.4.a.d		1	8.b	even	2	1
726.4.a.h		1	88.g	even	2	1
1584.4.a.i		1	24.h	odd	2	1
1650.4.a.h		1	40.e	odd	2	1
1650.4.c.g		2	40.k	even	4	2
2112.4.a.g		1	4.b	odd	2	1
2112.4.a.s		1	1.a	even	1	1	trivial
2178.4.a.g		1	264.p	odd	2	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{4}^{\mathrm{new}}(\Gamma_0(2112))$ :

T_{5}

T_{7} + 14

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T$ T
$3$	$T - 3$ T - 3
$5$	$T$ T
$7$	$T + 14$ T + 14
$11$	$T - 11$ T - 11
$13$	$T + 80$ T + 80
$17$	$T - 30$ T - 30
$19$	$T - 56$ T - 56
$23$	$T - 126$ T - 126
$29$	$T - 222$ T - 222
$31$	$T - 16$ T - 16
$37$	$T - 106$ T - 106
$41$	$T - 114$ T - 114
$43$	$T + 52$ T + 52
$47$	$T + 246$ T + 246
$53$	$T - 264$ T - 264
$59$	$T - 264$ T - 264
$61$	$T + 92$ T + 92
$67$	$T + 796$ T + 796
$71$	$T + 426$ T + 426
$73$	$T + 1174$ T + 1174
$79$	$T + 842$ T + 842
$83$	$T - 852$ T - 852
$89$	$T + 1062$ T + 1062
$97$	$T + 1282$ T + 1282
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