LMFDB - Newform orbit 450.4.a.g

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:	yes
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} + 11 q^{7} - 8 q^{8} - 36 q^{11} + 17 q^{13} - 22 q^{14} + 16 q^{16} - 12 q^{17} - 91 q^{19} + 72 q^{22} + 60 q^{23} - 34 q^{26} + 44 q^{28} - 276 q^{29} + 191 q^{31} - 32 q^{32}+ \cdots + 444 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

11.0000

−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.g	yes	1
3.b	odd	2	1		450.4.a.s	yes	1
5.b	even	2	1		450.4.a.n	yes	1
5.c	odd	4	2		450.4.c.b		2
15.d	odd	2	1		450.4.a.d	✓	1
15.e	even	4	2		450.4.c.i		2

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
450.4.a.d	✓	1	15.d	odd	2	1
450.4.a.g	yes	1	1.a	even	1	1	trivial
450.4.a.n	yes	1	5.b	even	2	1
450.4.a.s	yes	1	3.b	odd	2	1
450.4.c.b		2	5.c	odd	4	2
450.4.c.i		2	15.e	even	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} - 11

T_{11} + 36

T_{17} + 12

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T + 2$ T + 2
$3$	$T$ T
$5$	$T$ T
$7$	$T - 11$ T - 11
$11$	$T + 36$ T + 36
$13$	$T - 17$ T - 17
$17$	$T + 12$ T + 12
$19$	$T + 91$ T + 91
$23$	$T - 60$ T - 60
$29$	$T + 276$ T + 276
$31$	$T - 191$ T - 191
$37$	$T - 254$ T - 254
$41$	$T + 60$ T + 60
$43$	$T + 49$ T + 49
$47$	$T + 600$ T + 600
$53$	$T - 612$ T - 612
$59$	$T + 744$ T + 744
$61$	$T - 167$ T - 167
$67$	$T + 457$ T + 457
$71$	$T + 588$ T + 588
$73$	$T + 970$ T + 970
$79$	$T - 164$ T - 164
$83$	$T - 696$ T - 696
$89$	$T + 1248$ T + 1248
$97$	$T + 1099$ T + 1099
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