LMFDB - Newform orbit 3800.2.a.a

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3800,2,Mod(1,3800)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3800, 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("3800.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$N$	$=$	$3800 = 2^{3} \cdot 5^{2} \cdot 19$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	3800.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:	$30.3431527681$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$1$
Twist minimal:	no (minimal twist has level 760)
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} + q^{7} + 6 q^{9} + 4 q^{11} - q^{13} + 7 q^{17} - q^{19} - 3 q^{21} + 5 q^{23} - 9 q^{27} + 7 q^{29} - 2 q^{31} - 12 q^{33} + 6 q^{37} + 3 q^{39} + 6 q^{41} - 10 q^{43} + 8 q^{47} - 6 q^{49}+ \cdots + 24 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

1.00000

6.00000

Atkin-Lehner signs

$p$	Sign
$2$	$-1$
$5$	$+1$
$19$	$+1$

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	3800.2.a.a		1
4.b	odd	2	1		7600.2.a.t		1
5.b	even	2	1		760.2.a.e	✓	1
5.c	odd	4	2		3800.2.d.a		2
15.d	odd	2	1		6840.2.a.c		1
20.d	odd	2	1		1520.2.a.a		1
40.e	odd	2	1		6080.2.a.w		1
40.f	even	2	1		6080.2.a.a		1

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
760.2.a.e	✓	1	5.b	even	2	1
1520.2.a.a		1	20.d	odd	2	1
3800.2.a.a		1	1.a	even	1	1	trivial
3800.2.d.a		2	5.c	odd	4	2
6080.2.a.a		1	40.f	even	2	1
6080.2.a.w		1	40.e	odd	2	1
6840.2.a.c		1	15.d	odd	2	1
7600.2.a.t		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 $S_{2}^{\mathrm{new}}(\Gamma_0(3800))$ :

T_{3} + 3

T_{7} - 1

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T$ T
$3$	$T + 3$ T + 3
$5$	$T$ T
$7$	$T - 1$ T - 1
$11$	$T - 4$ T - 4
$13$	$T + 1$ T + 1
$17$	$T - 7$ T - 7
$19$	$T + 1$ T + 1
$23$	$T - 5$ T - 5
$29$	$T - 7$ T - 7
$31$	$T + 2$ T + 2
$37$	$T - 6$ T - 6
$41$	$T - 6$ T - 6
$43$	$T + 10$ T + 10
$47$	$T - 8$ T - 8
$53$	$T - 3$ T - 3
$59$	$T - 5$ T - 5
$61$	$T + 8$ T + 8
$67$	$T + 11$ T + 11
$71$	$T + 12$ T + 12
$73$	$T - 9$ T - 9
$79$	$T - 6$ T - 6
$83$	$T + 14$ T + 14
$89$	$T + 6$ T + 6
$97$	$T - 2$ T - 2
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