LMFDB - Newform orbit 4056.2.a.m

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4056,2,Mod(1,4056)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4056, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("4056.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$N$	$=$	$4056 = 2^{3} \cdot 3 \cdot 13^{2}$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	4056.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:	$32.3873230598$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$1$
Twist minimal:	no (minimal twist has level 312)
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 + q^{3} - 2 q^{5} + q^{9} - 2 q^{15} + 2 q^{17} + 4 q^{19} - q^{25} + q^{27} + 6 q^{29} + 2 q^{37} - 6 q^{41} - 12 q^{43} - 2 q^{45} + 4 q^{47} - 7 q^{49} + 2 q^{51} + 6 q^{53} + 4 q^{57} + 8 q^{59}+ \cdots + 6 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

1.00000

−2.00000

1.00000

Atkin-Lehner signs

$p$	Sign
$2$	$+1$
$3$	$-1$
$13$	$+1$

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	4056.2.a.m		1
4.b	odd	2	1		8112.2.a.f		1
13.b	even	2	1		312.2.a.f	✓	1
13.d	odd	4	2		4056.2.c.h		2
39.d	odd	2	1		936.2.a.b		1
52.b	odd	2	1		624.2.a.d		1
65.d	even	2	1		7800.2.a.d		1
104.e	even	2	1		2496.2.a.c		1
104.h	odd	2	1		2496.2.a.s		1
156.h	even	2	1		1872.2.a.e		1
312.b	odd	2	1		7488.2.a.bs		1
312.h	even	2	1		7488.2.a.br		1

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
312.2.a.f	✓	1	13.b	even	2	1
624.2.a.d		1	52.b	odd	2	1
936.2.a.b		1	39.d	odd	2	1
1872.2.a.e		1	156.h	even	2	1
2496.2.a.c		1	104.e	even	2	1
2496.2.a.s		1	104.h	odd	2	1
4056.2.a.m		1	1.a	even	1	1	trivial
4056.2.c.h		2	13.d	odd	4	2
7488.2.a.br		1	312.h	even	2	1
7488.2.a.bs		1	312.b	odd	2	1
7800.2.a.d		1	65.d	even	2	1
8112.2.a.f		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 $S_{2}^{\mathrm{new}}(\Gamma_0(4056))$ :

T_{5} + 2

T_{7}

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T$ T
$3$	$T - 1$ T - 1
$5$	$T + 2$ T + 2
$7$	$T$ T
$11$	$T$ T
$13$	$T$ T
$17$	$T - 2$ T - 2
$19$	$T - 4$ T - 4
$23$	$T$ T
$29$	$T - 6$ T - 6
$31$	$T$ T
$37$	$T - 2$ T - 2
$41$	$T + 6$ T + 6
$43$	$T + 12$ T + 12
$47$	$T - 4$ T - 4
$53$	$T - 6$ T - 6
$59$	$T - 8$ T - 8
$61$	$T + 2$ T + 2
$67$	$T + 4$ T + 4
$71$	$T - 12$ T - 12
$73$	$T - 14$ T - 14
$79$	$T$ T
$83$	$T + 8$ T + 8
$89$	$T - 18$ T - 18
$97$	$T - 6$ T - 6
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