LMFDB - Newform orbit 450.4.a.o

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [450,4,Mod(1,450)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(450, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("450.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$N$	$=$	$450 = 2 \cdot 3^{2} \cdot 5^{2}$
Weight:	$k$	$=$	$4$
Character orbit:	$[\chi]$	$=$	450.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:	$26.5508595026$
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 + 2 q^{2} + 4 q^{4} - q^{7} + 8 q^{8} - 42 q^{11} - 67 q^{13} - 2 q^{14} + 16 q^{16} - 54 q^{17} - 115 q^{19} - 84 q^{22} + 162 q^{23} - 134 q^{26} - 4 q^{28} + 210 q^{29} - 193 q^{31} + 32 q^{32} - 108 q^{34}+ \cdots - 684 q^{98}+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

4.00000

−1.00000

8.00000

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

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
150.4.a.a	✓	1	3.b	odd	2	1
150.4.a.h	yes	1	15.d	odd	2	1
150.4.c.e		2	15.e	even	4	2
450.4.a.f		1	5.b	even	2	1
450.4.a.o		1	1.a	even	1	1	trivial
450.4.c.a		2	5.c	odd	4	2
1200.4.a.i		1	60.h	even	2	1
1200.4.a.bb		1	12.b	even	2	1
1200.4.f.c		2	60.l	odd	4	2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{4}^{\mathrm{new}}(\Gamma_0(450))$ :

T_{7} + 1

T_{11} + 42

T_{17} + 54

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T - 2$ T - 2
$3$	$T$ T
$5$	$T$ 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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