LMFDB - Newform orbit 44.6.a.a

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [44,6,Mod(1,44)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(44, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0]))

N = Newforms(chi, 6, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("44.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$N$	$=$	$44 = 2^{2} \cdot 11$
Weight:	$k$	$=$	$6$
Character orbit:	$[\chi]$	$=$	44.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:	$7.05688807177$
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 + 7 q^{3} - 79 q^{5} - 50 q^{7} - 194 q^{9} + 121 q^{11} - 380 q^{13} - 553 q^{15} - 1154 q^{17} - 1824 q^{19} - 350 q^{21} + 3591 q^{23} + 3116 q^{25} - 3059 q^{27} + 8032 q^{29} - 2945 q^{31} + 847 q^{33}+ \cdots - 23474 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

7.00000

−79.0000

−50.0000

−194.000

Atkin-Lehner signs

$p$	Sign
$2$	$-1$
$11$	$-1$

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	44.6.a.a	✓	1
3.b	odd	2	1		396.6.a.e		1
4.b	odd	2	1		176.6.a.a		1
5.b	even	2	1		1100.6.a.a		1
5.c	odd	4	2		1100.6.b.a		2
8.b	even	2	1		704.6.a.d		1
8.d	odd	2	1		704.6.a.g		1
11.b	odd	2	1		484.6.a.b		1

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
44.6.a.a	✓	1	1.a	even	1	1	trivial
176.6.a.a		1	4.b	odd	2	1
396.6.a.e		1	3.b	odd	2	1
484.6.a.b		1	11.b	odd	2	1
704.6.a.d		1	8.b	even	2	1
704.6.a.g		1	8.d	odd	2	1
1100.6.a.a		1	5.b	even	2	1
1100.6.b.a		2	5.c	odd	4	2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $T_{3} - 7$ T3 - 7 acting on $S_{6}^{\mathrm{new}}(\Gamma_0(44))$ .

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T$ T
$3$	$T - 7$ T - 7
$5$	$T + 79$ T + 79
$7$	$T + 50$ T + 50
$11$	$T - 121$ T - 121
$13$	$T + 380$ T + 380
$17$	$T + 1154$ T + 1154
$19$	$T + 1824$ T + 1824
$23$	$T - 3591$ T - 3591
$29$	$T - 8032$ T - 8032
$31$	$T + 2945$ T + 2945
$37$	$T - 6979$ T - 6979
$41$	$T + 520$ T + 520
$43$	$T + 2486$ T + 2486
$47$	$T + 6920$ T + 6920
$53$	$T + 13718$ T + 13718
$59$	$T + 31779$ T + 31779
$61$	$T - 34156$ T - 34156
$67$	$T + 61503$ T + 61503
$71$	$T + 14971$ T + 14971
$73$	$T + 36444$ T + 36444
$79$	$T + 28538$ T + 28538
$83$	$T - 77482$ T - 77482
$89$	$T - 36271$ T - 36271
$97$	$T + 49799$ T + 49799
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