LMFDB - Newform orbit 8670.2.a.v

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8670,2,Mod(1,8670)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8670, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("8670.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8670 = 2 \cdot 3 \cdot 5 \cdot 17^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8670.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:	$69.2302985525$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 510)
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^{2} + q^{3} + q^{4} - q^{5} + q^{6} + q^{8} + q^{9} - q^{10} - 4 q^{11} + q^{12} - 2 q^{13} - q^{15} + q^{16} + q^{18} + 4 q^{19} - q^{20} - 4 q^{22} + q^{24} + q^{25} - 2 q^{26} + q^{27}+ \cdots - 4 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

1.00000

−1.00000

Atkin-Lehner signs

$ p $	Sign
$2$	$ -1 $
$3$	$ -1 $
$5$	$ +1 $
$17$	$ +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	8670.2.a.v		1
17.b	even	2	1		510.2.a.e	✓	1
51.c	odd	2	1		1530.2.a.b		1
68.d	odd	2	1		4080.2.a.ba		1
85.c	even	2	1		2550.2.a.l		1
85.g	odd	4	2		2550.2.d.k		2
255.h	odd	2	1		7650.2.a.bx		1

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
510.2.a.e	✓	1	17.b	even	2	1
1530.2.a.b		1	51.c	odd	2	1
2550.2.a.l		1	85.c	even	2	1
2550.2.d.k		2	85.g	odd	4	2
4080.2.a.ba		1	68.d	odd	2	1
7650.2.a.bx		1	255.h	odd	2	1
8670.2.a.v		1	1.a	even	1	1	trivial

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{2}^{\mathrm{new}}(\Gamma_0(8670))$:

$ T_{7} $

$ T_{11} + 4 $

$ T_{13} + 2 $

$ T_{23} $

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T - 1 $ T - 1
$3$	$ T - 1 $ T - 1
$5$	$ T + 1 $ T + 1
$7$	$ T $ T
$11$	$ T + 4 $ T + 4
$13$	$ T + 2 $ T + 2
$17$	$ T $ T
$19$	$ T - 4 $ T - 4
$23$	$ T $ T
$29$	$ T - 2 $ T - 2
$31$	$ T + 8 $ T + 8
$37$	$ T + 6 $ T + 6
$41$	$ T - 6 $ T - 6
$43$	$ T + 4 $ T + 4
$47$	$ T $ T
$53$	$ T + 10 $ T + 10
$59$	$ T + 4 $ T + 4
$61$	$ T - 2 $ T - 2
$67$	$ T - 4 $ T - 4
$71$	$ T $ T
$73$	$ T - 6 $ T - 6
$79$	$ T + 8 $ T + 8
$83$	$ T + 12 $ T + 12
$89$	$ T + 6 $ T + 6
$97$	$ T - 14 $ T - 14
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Overview	Random
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Classical	Maass
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Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 8670 = 2 \cdot 3 \cdot 5 \cdot 17^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8670.a (trivial)

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