LMFDB - Newform orbit 1512.2.a.a

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1512,2,Mod(1,1512)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1512, 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("1512.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$N$	$=$	$1512 = 2^{3} \cdot 3^{3} \cdot 7$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	1512.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:	$12.0733807856$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$1$
Twist minimal:	yes
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 - 4 q^{5} - q^{7} - 4 q^{11} - q^{13} - 3 q^{17} + 2 q^{19} + 3 q^{23} + 11 q^{25} + 9 q^{29} - q^{31} + 4 q^{35} + 10 q^{37} - 10 q^{41} - q^{43} + 10 q^{47} + q^{49} - 9 q^{53} + 16 q^{55} + 3 q^{59}+ \cdots - 8 q^{95}+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

−4.00000

−1.00000

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	1512.2.a.a	✓	1
3.b	odd	2	1		1512.2.a.l	yes	1
4.b	odd	2	1		3024.2.a.b		1
12.b	even	2	1		3024.2.a.bc		1

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
1512.2.a.a	✓	1	1.a	even	1	1	trivial
1512.2.a.l	yes	1	3.b	odd	2	1
3024.2.a.b		1	4.b	odd	2	1
3024.2.a.bc		1	12.b	even	2	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{2}^{\mathrm{new}}(\Gamma_0(1512))$ :

T_{5} + 4

T_{11} + 4

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T$ T
$3$	$T$ T
$5$	$T + 4$ T + 4
$7$	$T + 1$ T + 1
$11$	$T + 4$ T + 4
$13$	$T + 1$ T + 1
$17$	$T + 3$ T + 3
$19$	$T - 2$ T - 2
$23$	$T - 3$ T - 3
$29$	$T - 9$ T - 9
$31$	$T + 1$ T + 1
$37$	$T - 10$ T - 10
$41$	$T + 10$ T + 10
$43$	$T + 1$ T + 1
$47$	$T - 10$ T - 10
$53$	$T + 9$ T + 9
$59$	$T - 3$ T - 3
$61$	$T + 6$ T + 6
$67$	$T + 13$ T + 13
$71$	$T - 13$ T - 13
$73$	$T - 10$ T - 10
$79$	$T - 10$ T - 10
$83$	$T + 4$ T + 4
$89$	$T - 9$ T - 9
$97$	$T$ T
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