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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Overview	Random
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Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 588 = 2^{2} \cdot 3 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	588.a (trivial)

\(n\):		e.g. 2-40 or 990-1000
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