LMFDB - Newform orbit 416.4.a.c

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [416,4,Mod(1,416)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(416, 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("416.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$N$	$=$	$416 = 2^{5} \cdot 13$
Weight:	$k$	$=$	$4$
Character orbit:	$[\chi]$	$=$	416.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:	$24.5447945624$
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 + q^{3} - q^{5} + 5 q^{7} - 26 q^{9} + 10 q^{11} - 13 q^{13} - q^{15} + 93 q^{17} - 82 q^{19} + 5 q^{21} - 192 q^{23} - 124 q^{25} - 53 q^{27} - 106 q^{29} + 172 q^{31} + 10 q^{33} - 5 q^{35} + 379 q^{37}+ \cdots - 260 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

1.00000

−1.00000

5.00000

−26.0000

Atkin-Lehner signs

$p$	Sign
$2$	$-1$
$13$	$+1$

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	416.4.a.c	yes	1
4.b	odd	2	1		416.4.a.b	✓	1
8.b	even	2	1		832.4.a.h		1
8.d	odd	2	1		832.4.a.k		1

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
416.4.a.b	✓	1	4.b	odd	2	1
416.4.a.c	yes	1	1.a	even	1	1	trivial
832.4.a.h		1	8.b	even	2	1
832.4.a.k		1	8.d	odd	2	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $T_{3} - 1$ T3 - 1 acting on $S_{4}^{\mathrm{new}}(\Gamma_0(416))$ .

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T$ T
$3$	$T - 1$ T - 1
$5$	$T + 1$ T + 1
$7$	$T - 5$ T - 5
$11$	$T - 10$ T - 10
$13$	$T + 13$ T + 13
$17$	$T - 93$ T - 93
$19$	$T + 82$ T + 82
$23$	$T + 192$ T + 192
$29$	$T + 106$ T + 106
$31$	$T - 172$ T - 172
$37$	$T - 379$ T - 379
$41$	$T + 148$ T + 148
$43$	$T + 329$ T + 329
$47$	$T + 631$ T + 631
$53$	$T - 160$ T - 160
$59$	$T + 478$ T + 478
$61$	$T - 300$ T - 300
$67$	$T + 722$ T + 722
$71$	$T - 335$ T - 335
$73$	$T - 90$ T - 90
$79$	$T + 788$ T + 788
$83$	$T - 96$ T - 96
$89$	$T + 866$ T + 866
$97$	$T + 998$ T + 998
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