LMFDB - Newform orbit 448.4.a.h

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [448,4,Mod(1,448)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(448, 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("448.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$N$	$=$	$448 = 2^{6} \cdot 7$
Weight:	$k$	$=$	$4$
Character orbit:	$[\chi]$	$=$	448.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:	$26.4328556826$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$1$
Twist minimal:	no (minimal twist has level 56)
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} + 16 q^{5} + 7 q^{7} - 23 q^{9} + 24 q^{11} + 68 q^{13} - 32 q^{15} + 54 q^{17} - 46 q^{19} - 14 q^{21} - 176 q^{23} + 131 q^{25} + 100 q^{27} + 174 q^{29} + 116 q^{31} - 48 q^{33} + 112 q^{35}+ \cdots - 552 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

16.0000

7.00000

−23.0000

Atkin-Lehner signs

$p$	Sign
$2$	$-1$
$7$	$-1$

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	448.4.a.h		1
4.b	odd	2	1		448.4.a.l		1
8.b	even	2	1		112.4.a.d		1
8.d	odd	2	1		56.4.a.a	✓	1
24.f	even	2	1		504.4.a.g		1
24.h	odd	2	1		1008.4.a.u		1
40.e	odd	2	1		1400.4.a.f		1
40.k	even	4	2		1400.4.g.f		2
56.e	even	2	1		392.4.a.c		1
56.h	odd	2	1		784.4.a.i		1
56.k	odd	6	2		392.4.i.e		2
56.m	even	6	2		392.4.i.d		2

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
56.4.a.a	✓	1	8.d	odd	2	1
112.4.a.d		1	8.b	even	2	1
392.4.a.c		1	56.e	even	2	1
392.4.i.d		2	56.m	even	6	2
392.4.i.e		2	56.k	odd	6	2
448.4.a.h		1	1.a	even	1	1	trivial
448.4.a.l		1	4.b	odd	2	1
504.4.a.g		1	24.f	even	2	1
784.4.a.i		1	56.h	odd	2	1
1008.4.a.u		1	24.h	odd	2	1
1400.4.a.f		1	40.e	odd	2	1
1400.4.g.f		2	40.k	even	4	2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{4}^{\mathrm{new}}(\Gamma_0(448))$ :

T_{3} + 2

T_{5} - 16

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T$ T
$3$	$T + 2$ T + 2
$5$	$T - 16$ T - 16
$7$	$T - 7$ T - 7
$11$	$T - 24$ T - 24
$13$	$T - 68$ T - 68
$17$	$T - 54$ T - 54
$19$	$T + 46$ T + 46
$23$	$T + 176$ T + 176
$29$	$T - 174$ T - 174
$31$	$T - 116$ T - 116
$37$	$T + 74$ T + 74
$41$	$T + 10$ T + 10
$43$	$T + 480$ T + 480
$47$	$T - 572$ T - 572
$53$	$T - 162$ T - 162
$59$	$T + 86$ T + 86
$61$	$T - 904$ T - 904
$67$	$T - 660$ T - 660
$71$	$T + 1024$ T + 1024
$73$	$T - 770$ T - 770
$79$	$T - 904$ T - 904
$83$	$T - 682$ T - 682
$89$	$T + 102$ T + 102
$97$	$T + 218$ T + 218
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