LMFDB - Newform orbit 588.6.a.a

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [588,6,Mod(1,588)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(588, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("588.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$N$	$=$	$588 = 2^{2} \cdot 3 \cdot 7^{2}$
Weight:	$k$	$=$	$6$
Character orbit:	$[\chi]$	$=$	588.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:	$94.3056860500$
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 - 9 q^{3} - 68 q^{5} + 81 q^{9} - 388 q^{11} + 316 q^{13} + 612 q^{15} + 1056 q^{17} - 1052 q^{19} + 624 q^{23} + 1499 q^{25} - 729 q^{27} + 7250 q^{29} - 2296 q^{31} + 3492 q^{33} + 12426 q^{37} - 2844 q^{39}+ \cdots - 31428 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

−9.00000

−68.0000

81.0000

Atkin-Lehner signs

$p$	Sign
$2$	$-1$
$3$	$+1$
$7$	$-1$

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	588.6.a.a	✓	1
7.b	odd	2	1		588.6.a.f	yes	1
7.c	even	3	2		588.6.i.g		2
7.d	odd	6	2		588.6.i.a		2

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
588.6.a.a	✓	1	1.a	even	1	1	trivial
588.6.a.f	yes	1	7.b	odd	2	1
588.6.i.a		2	7.d	odd	6	2
588.6.i.g		2	7.c	even	3	2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $T_{5} + 68$ T5 + 68 acting on $S_{6}^{\mathrm{new}}(\Gamma_0(588))$ .

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T$ T
$3$	$T + 9$ T + 9
$5$	$T + 68$ T + 68
$7$	$T$ T
$11$	$T + 388$ T + 388
$13$	$T - 316$ T - 316
$17$	$T - 1056$ T - 1056
$19$	$T + 1052$ T + 1052
$23$	$T - 624$ T - 624
$29$	$T - 7250$ T - 7250
$31$	$T + 2296$ T + 2296
$37$	$T - 12426$ T - 12426
$41$	$T + 5376$ T + 5376
$43$	$T - 14164$ T - 14164
$47$	$T + 4712$ T + 4712
$53$	$T - 3782$ T - 3782
$59$	$T + 25244$ T + 25244
$61$	$T + 20668$ T + 20668
$67$	$T - 49012$ T - 49012
$71$	$T - 4760$ T - 4760
$73$	$T - 65264$ T - 65264
$79$	$T + 49736$ T + 49736
$83$	$T + 7788$ T + 7788
$89$	$T + 36904$ T + 36904
$97$	$T - 98264$ T - 98264
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