LMFDB - Newform orbit 448.4.a.d

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:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$1$
Twist minimal:	no (minimal twist has level 28)
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 - 4 q^{3} - 6 q^{5} + 7 q^{7} - 11 q^{9} + 12 q^{11} + 82 q^{13} + 24 q^{15} - 30 q^{17} - 68 q^{19} - 28 q^{21} + 216 q^{23} - 89 q^{25} + 152 q^{27} - 246 q^{29} - 112 q^{31} - 48 q^{33} - 42 q^{35}+ \cdots - 132 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

−4.00000

−6.00000

7.00000

−11.0000

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.d		1
4.b	odd	2	1		448.4.a.m		1
8.b	even	2	1		28.4.a.b	✓	1
8.d	odd	2	1		112.4.a.c		1
24.f	even	2	1		1008.4.a.f		1
24.h	odd	2	1		252.4.a.c		1
40.f	even	2	1		700.4.a.e		1
40.i	odd	4	2		700.4.e.f		2
56.e	even	2	1		784.4.a.n		1
56.h	odd	2	1		196.4.a.b		1
56.j	odd	6	2		196.4.e.d		2
56.p	even	6	2		196.4.e.c		2
168.i	even	2	1		1764.4.a.k		1
168.s	odd	6	2		1764.4.k.k		2
168.ba	even	6	2		1764.4.k.e		2

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
28.4.a.b	✓	1	8.b	even	2	1
112.4.a.c		1	8.d	odd	2	1
196.4.a.b		1	56.h	odd	2	1
196.4.e.c		2	56.p	even	6	2
196.4.e.d		2	56.j	odd	6	2
252.4.a.c		1	24.h	odd	2	1
448.4.a.d		1	1.a	even	1	1	trivial
448.4.a.m		1	4.b	odd	2	1
700.4.a.e		1	40.f	even	2	1
700.4.e.f		2	40.i	odd	4	2
784.4.a.n		1	56.e	even	2	1
1008.4.a.f		1	24.f	even	2	1
1764.4.a.k		1	168.i	even	2	1
1764.4.k.e		2	168.ba	even	6	2
1764.4.k.k		2	168.s	odd	6	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} + 4

T_{5} + 6

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T$ T
$3$	$T + 4$ T + 4
$5$	$T + 6$ T + 6
$7$	$T - 7$ T - 7
$11$	$T - 12$ T - 12
$13$	$T - 82$ T - 82
$17$	$T + 30$ T + 30
$19$	$T + 68$ T + 68
$23$	$T - 216$ T - 216
$29$	$T + 246$ T + 246
$31$	$T + 112$ T + 112
$37$	$T + 110$ T + 110
$41$	$T + 246$ T + 246
$43$	$T - 172$ T - 172
$47$	$T - 192$ T - 192
$53$	$T + 558$ T + 558
$59$	$T + 540$ T + 540
$61$	$T + 110$ T + 110
$67$	$T + 140$ T + 140
$71$	$T + 840$ T + 840
$73$	$T + 550$ T + 550
$79$	$T + 208$ T + 208
$83$	$T + 516$ T + 516
$89$	$T + 1398$ T + 1398
$97$	$T - 1586$ T - 1586
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