LMFDB - Newform orbit 440.6.a.a

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [440,6,Mod(1,440)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(440, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 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("440.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$N$	$=$	$440 = 2^{3} \cdot 5 \cdot 11$
Weight:	$k$	$=$	$6$
Character orbit:	$[\chi]$	$=$	440.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:	$70.5688807177$
Analytic rank:	$0$
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 + 23 q^{3} + 25 q^{5} + 211 q^{7} + 286 q^{9} + 121 q^{11} + 26 q^{13} + 575 q^{15} - 407 q^{17} + 1789 q^{19} + 4853 q^{21} - 6 q^{23} + 625 q^{25} + 989 q^{27} - 2387 q^{29} + 9453 q^{31} + 2783 q^{33}+ \cdots + 34606 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

23.0000

25.0000

211.000

286.000

Atkin-Lehner signs

$p$	Sign
$2$	$-1$
$5$	$-1$
$11$	$-1$

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	440.6.a.a	✓	1
4.b	odd	2	1		880.6.a.a		1

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
440.6.a.a	✓	1	1.a	even	1	1	trivial
880.6.a.a		1	4.b	odd	2	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $T_{3} - 23$ T3 - 23 acting on $S_{6}^{\mathrm{new}}(\Gamma_0(440))$ .

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T$ T
$3$	$T - 23$ T - 23
$5$	$T - 25$ T - 25
$7$	$T - 211$ T - 211
$11$	$T - 121$ T - 121
$13$	$T - 26$ T - 26
$17$	$T + 407$ T + 407
$19$	$T - 1789$ T - 1789
$23$	$T + 6$ T + 6
$29$	$T + 2387$ T + 2387
$31$	$T - 9453$ T - 9453
$37$	$T + 6917$ T + 6917
$41$	$T + 9774$ T + 9774
$43$	$T - 3108$ T - 3108
$47$	$T + 14290$ T + 14290
$53$	$T + 18665$ T + 18665
$59$	$T - 36646$ T - 36646
$61$	$T + 22945$ T + 22945
$67$	$T - 35848$ T - 35848
$71$	$T - 16647$ T - 16647
$73$	$T + 34642$ T + 34642
$79$	$T - 20554$ T - 20554
$83$	$T - 7674$ T - 7674
$89$	$T + 98879$ T + 98879
$97$	$T - 55764$ T - 55764
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