LMFDB - Newform orbit 2368.2.a.m

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2368,2,Mod(1,2368)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2368, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("2368.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$N$	$=$	$2368 = 2^{6} \cdot 37$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	2368.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:	$18.9085751986$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$1$
Twist minimal:	no (minimal twist has level 37)
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} + q^{7} - 2 q^{9} + 3 q^{11} + 4 q^{13} + 6 q^{17} + 2 q^{19} + q^{21} - 6 q^{23} - 5 q^{25} - 5 q^{27} + 6 q^{29} + 4 q^{31} + 3 q^{33} - q^{37} + 4 q^{39} - 9 q^{41} + 8 q^{43} - 3 q^{47}+ \cdots - 6 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

1.00000

−2.00000

Atkin-Lehner signs

$p$	Sign
$2$	$-1$
$37$	$+1$

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	2368.2.a.m		1
4.b	odd	2	1		2368.2.a.d		1
8.b	even	2	1		592.2.a.a		1
8.d	odd	2	1		37.2.a.b	✓	1
24.f	even	2	1		333.2.a.b		1
24.h	odd	2	1		5328.2.a.k		1
40.e	odd	2	1		925.2.a.b		1
40.k	even	4	2		925.2.b.e		2
56.e	even	2	1		1813.2.a.b		1
88.g	even	2	1		4477.2.a.a		1
104.h	odd	2	1		6253.2.a.b		1
120.m	even	2	1		8325.2.a.p		1
296.h	odd	2	1		1369.2.a.c		1
296.j	even	4	2		1369.2.b.a		2

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
37.2.a.b	✓	1	8.d	odd	2	1
333.2.a.b		1	24.f	even	2	1
592.2.a.a		1	8.b	even	2	1
925.2.a.b		1	40.e	odd	2	1
925.2.b.e		2	40.k	even	4	2
1369.2.a.c		1	296.h	odd	2	1
1369.2.b.a		2	296.j	even	4	2
1813.2.a.b		1	56.e	even	2	1
2368.2.a.d		1	4.b	odd	2	1
2368.2.a.m		1	1.a	even	1	1	trivial
4477.2.a.a		1	88.g	even	2	1
5328.2.a.k		1	24.h	odd	2	1
6253.2.a.b		1	104.h	odd	2	1
8325.2.a.p		1	120.m	even	2	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{2}^{\mathrm{new}}(\Gamma_0(2368))$ :

T_{3} - 1

T_{5}

T_{7} - 1

T_{11} - 3

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T$ T
$3$	$T - 1$ T - 1
$5$	$T$ T
$7$	$T - 1$ T - 1
$11$	$T - 3$ T - 3
$13$	$T - 4$ T - 4
$17$	$T - 6$ T - 6
$19$	$T - 2$ T - 2
$23$	$T + 6$ T + 6
$29$	$T - 6$ T - 6
$31$	$T - 4$ T - 4
$37$	$T + 1$ T + 1
$41$	$T + 9$ T + 9
$43$	$T - 8$ T - 8
$47$	$T + 3$ T + 3
$53$	$T - 3$ T - 3
$59$	$T - 12$ T - 12
$61$	$T + 8$ T + 8
$67$	$T + 4$ T + 4
$71$	$T - 15$ T - 15
$73$	$T - 11$ T - 11
$79$	$T - 10$ T - 10
$83$	$T - 9$ T - 9
$89$	$T - 6$ T - 6
$97$	$T - 8$ T - 8
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