LMFDB - Newform orbit 3600.2.a.bb

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3600,2,Mod(1,3600)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3600, 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("3600.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$N$	$=$	$3600 = 2^{4} \cdot 3^{2} \cdot 5^{2}$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	3600.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:	$28.7461447277$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$1$
Twist minimal:	no (minimal twist has level 40)
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 + 2 q^{7} - 4 q^{11} - 4 q^{13} + 4 q^{19} + 2 q^{23} - 2 q^{29} - 4 q^{37} - 2 q^{41} - 6 q^{43} + 6 q^{47} - 3 q^{49} - 4 q^{53} - 12 q^{59} - 10 q^{61} + 14 q^{67} + 8 q^{71} - 8 q^{73} - 8 q^{77}+ \cdots - 16 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

2.00000

Atkin-Lehner signs

$p$	Sign
$2$	$+1$
$3$	$-1$
$5$	$-1$

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	3600.2.a.bb		1
3.b	odd	2	1		400.2.a.g		1
4.b	odd	2	1		1800.2.a.j		1
5.b	even	2	1		3600.2.a.k		1
5.c	odd	4	2		720.2.f.e		2
12.b	even	2	1		200.2.a.b		1
15.d	odd	2	1		400.2.a.b		1
15.e	even	4	2		80.2.c.a		2
20.d	odd	2	1		1800.2.a.s		1
20.e	even	4	2		360.2.f.c		2
24.f	even	2	1		1600.2.a.v		1
24.h	odd	2	1		1600.2.a.d		1
40.i	odd	4	2		2880.2.f.i		2
40.k	even	4	2		2880.2.f.h		2
60.h	even	2	1		200.2.a.d		1
60.l	odd	4	2		40.2.c.a	✓	2
84.h	odd	2	1		9800.2.a.bf		1
120.i	odd	2	1		1600.2.a.u		1
120.m	even	2	1		1600.2.a.f		1
120.q	odd	4	2		320.2.c.c		2
120.w	even	4	2		320.2.c.b		2
240.z	odd	4	2		1280.2.f.f		2
240.bb	even	4	2		1280.2.f.b		2
240.bd	odd	4	2		1280.2.f.a		2
240.bf	even	4	2		1280.2.f.e		2
420.o	odd	2	1		9800.2.a.d		1
420.w	even	4	2		1960.2.g.b		2

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
40.2.c.a	✓	2	60.l	odd	4	2
80.2.c.a		2	15.e	even	4	2
200.2.a.b		1	12.b	even	2	1
200.2.a.d		1	60.h	even	2	1
320.2.c.b		2	120.w	even	4	2
320.2.c.c		2	120.q	odd	4	2
360.2.f.c		2	20.e	even	4	2
400.2.a.b		1	15.d	odd	2	1
400.2.a.g		1	3.b	odd	2	1
720.2.f.e		2	5.c	odd	4	2
1280.2.f.a		2	240.bd	odd	4	2
1280.2.f.b		2	240.bb	even	4	2
1280.2.f.e		2	240.bf	even	4	2
1280.2.f.f		2	240.z	odd	4	2
1600.2.a.d		1	24.h	odd	2	1
1600.2.a.f		1	120.m	even	2	1
1600.2.a.u		1	120.i	odd	2	1
1600.2.a.v		1	24.f	even	2	1
1800.2.a.j		1	4.b	odd	2	1
1800.2.a.s		1	20.d	odd	2	1
1960.2.g.b		2	420.w	even	4	2
2880.2.f.h		2	40.k	even	4	2
2880.2.f.i		2	40.i	odd	4	2
3600.2.a.k		1	5.b	even	2	1
3600.2.a.bb		1	1.a	even	1	1	trivial
9800.2.a.d		1	420.o	odd	2	1
9800.2.a.bf		1	84.h	odd	2	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{2}^{\mathrm{new}}(\Gamma_0(3600))$ :

T_{7} - 2

T_{11} + 4

T_{13} + 4

T_{17}

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T$ T
$3$	$T$ T
$5$	$T$ T
$7$	$T - 2$ T - 2
$11$	$T + 4$ T + 4
$13$	$T + 4$ T + 4
$17$	$T$ T
$19$	$T - 4$ T - 4
$23$	$T - 2$ T - 2
$29$	$T + 2$ T + 2
$31$	$T$ T
$37$	$T + 4$ T + 4
$41$	$T + 2$ T + 2
$43$	$T + 6$ T + 6
$47$	$T - 6$ T - 6
$53$	$T + 4$ T + 4
$59$	$T + 12$ T + 12
$61$	$T + 10$ T + 10
$67$	$T - 14$ T - 14
$71$	$T - 8$ T - 8
$73$	$T + 8$ T + 8
$79$	$T + 16$ T + 16
$83$	$T + 2$ T + 2
$89$	$T + 6$ T + 6
$97$	$T + 16$ T + 16
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