LMFDB - Newform orbit 800.4.a.d

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [800,4,Mod(1,800)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(800, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("800.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$N$	$=$	$800 = 2^{5} \cdot 5^{2}$
Weight:	$k$	$=$	$4$
Character orbit:	$[\chi]$	$=$	800.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:	$47.2015280046$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$1$
Twist minimal:	no (minimal twist has level 160)
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^{3} + 6 q^{7} - 23 q^{9} - 60 q^{11} - 50 q^{13} + 30 q^{17} - 40 q^{19} - 12 q^{21} + 178 q^{23} + 100 q^{27} + 166 q^{29} - 20 q^{31} + 120 q^{33} - 10 q^{37} + 100 q^{39} - 250 q^{41} + 142 q^{43}+ \cdots + 1380 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

−2.00000

6.00000

−23.0000

Atkin-Lehner signs

$p$	Sign
$2$	$+1$
$5$	$+1$

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	800.4.a.d		1
4.b	odd	2	1		800.4.a.h		1
5.b	even	2	1		160.4.a.b	yes	1
5.c	odd	4	2		800.4.c.e		2
8.b	even	2	1		1600.4.a.bj		1
8.d	odd	2	1		1600.4.a.r		1
15.d	odd	2	1		1440.4.a.n		1
20.d	odd	2	1		160.4.a.a	✓	1
20.e	even	4	2		800.4.c.f		2
40.e	odd	2	1		320.4.a.i		1
40.f	even	2	1		320.4.a.f		1
60.h	even	2	1		1440.4.a.o		1
80.k	odd	4	2		1280.4.d.f		2
80.q	even	4	2		1280.4.d.k		2

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
160.4.a.a	✓	1	20.d	odd	2	1
160.4.a.b	yes	1	5.b	even	2	1
320.4.a.f		1	40.f	even	2	1
320.4.a.i		1	40.e	odd	2	1
800.4.a.d		1	1.a	even	1	1	trivial
800.4.a.h		1	4.b	odd	2	1
800.4.c.e		2	5.c	odd	4	2
800.4.c.f		2	20.e	even	4	2
1280.4.d.f		2	80.k	odd	4	2
1280.4.d.k		2	80.q	even	4	2
1440.4.a.n		1	15.d	odd	2	1
1440.4.a.o		1	60.h	even	2	1
1600.4.a.r		1	8.d	odd	2	1
1600.4.a.bj		1	8.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 $S_{4}^{\mathrm{new}}(\Gamma_0(800))$ :

T_{3} + 2

T_{11} + 60

T_{13} + 50

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T$ T
$3$	$T + 2$ T + 2
$5$	$T$ T
$7$	$T - 6$ T - 6
$11$	$T + 60$ T + 60
$13$	$T + 50$ T + 50
$17$	$T - 30$ T - 30
$19$	$T + 40$ T + 40
$23$	$T - 178$ T - 178
$29$	$T - 166$ T - 166
$31$	$T + 20$ T + 20
$37$	$T + 10$ T + 10
$41$	$T + 250$ T + 250
$43$	$T - 142$ T - 142
$47$	$T - 214$ T - 214
$53$	$T + 490$ T + 490
$59$	$T - 800$ T - 800
$61$	$T - 250$ T - 250
$67$	$T + 774$ T + 774
$71$	$T + 100$ T + 100
$73$	$T - 230$ T - 230
$79$	$T - 1320$ T - 1320
$83$	$T - 982$ T - 982
$89$	$T - 874$ T - 874
$97$	$T - 310$ T - 310
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