LMFDB - Newform orbit 1200.4.a.bb

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1200,4,Mod(1,1200)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1200, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("1200.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$N$	$=$	$1200 = 2^{4} \cdot 3 \cdot 5^{2}$
Weight:	$k$	$=$	$4$
Character orbit:	$[\chi]$	$=$	1200.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:	$70.8022920069$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$1$
Twist minimal:	no (minimal twist has level 150)
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} + 9 q^{9} - 42 q^{11} - 67 q^{13} + 54 q^{17} + 115 q^{19} + 3 q^{21} + 162 q^{23} + 27 q^{27} - 210 q^{29} + 193 q^{31} - 126 q^{33} - 286 q^{37} - 201 q^{39} + 12 q^{41} - 263 q^{43}+ \cdots - 378 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

9.00000

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	1200.4.a.bb		1
4.b	odd	2	1		150.4.a.a	✓	1
5.b	even	2	1		1200.4.a.i		1
5.c	odd	4	2		1200.4.f.c		2
12.b	even	2	1		450.4.a.o		1
20.d	odd	2	1		150.4.a.h	yes	1
20.e	even	4	2		150.4.c.e		2
60.h	even	2	1		450.4.a.f		1
60.l	odd	4	2		450.4.c.a		2

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
150.4.a.a	✓	1	4.b	odd	2	1
150.4.a.h	yes	1	20.d	odd	2	1
150.4.c.e		2	20.e	even	4	2
450.4.a.f		1	60.h	even	2	1
450.4.a.o		1	12.b	even	2	1
450.4.c.a		2	60.l	odd	4	2
1200.4.a.i		1	5.b	even	2	1
1200.4.a.bb		1	1.a	even	1	1	trivial
1200.4.f.c		2	5.c	odd	4	2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{4}^{\mathrm{new}}(\Gamma_0(1200))$ :

T_{7} - 1

T_{11} + 42

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T$ T
$3$	$T - 3$ T - 3
$5$	$T$ T
$7$	$T - 1$ T - 1
$11$	$T + 42$ T + 42
$13$	$T + 67$ T + 67
$17$	$T - 54$ T - 54
$19$	$T - 115$ T - 115
$23$	$T - 162$ T - 162
$29$	$T + 210$ T + 210
$31$	$T - 193$ T - 193
$37$	$T + 286$ T + 286
$41$	$T - 12$ T - 12
$43$	$T + 263$ T + 263
$47$	$T + 414$ T + 414
$53$	$T + 192$ T + 192
$59$	$T + 690$ T + 690
$61$	$T + 733$ T + 733
$67$	$T + 299$ T + 299
$71$	$T - 228$ T - 228
$73$	$T - 938$ T - 938
$79$	$T - 160$ T - 160
$83$	$T - 462$ T - 462
$89$	$T + 240$ T + 240
$97$	$T + 511$ T + 511
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