LMFDB - Newform orbit 864.2.a.b

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [864,2,Mod(1,864)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(864, 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("864.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 864 = 2^{5} \cdot 3^{3} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	864.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:	$6.89907473464$
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)

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

−2.00000

−1.00000

Atkin-Lehner signs

$ p $	Sign
$2$	$ -1 $
$3$	$ -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	864.2.a.b	✓	1
3.b	odd	2	1		864.2.a.j	yes	1
4.b	odd	2	1		864.2.a.c	yes	1
8.b	even	2	1		1728.2.a.v		1
8.d	odd	2	1		1728.2.a.w		1
9.c	even	3	2		2592.2.i.u		2
9.d	odd	6	2		2592.2.i.f		2
12.b	even	2	1		864.2.a.k	yes	1
24.f	even	2	1		1728.2.a.g		1
24.h	odd	2	1		1728.2.a.f		1
36.f	odd	6	2		2592.2.i.s		2
36.h	even	6	2		2592.2.i.d		2

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
864.2.a.b	✓	1	1.a	even	1	1	trivial
864.2.a.c	yes	1	4.b	odd	2	1
864.2.a.j	yes	1	3.b	odd	2	1
864.2.a.k	yes	1	12.b	even	2	1
1728.2.a.f		1	24.h	odd	2	1
1728.2.a.g		1	24.f	even	2	1
1728.2.a.v		1	8.b	even	2	1
1728.2.a.w		1	8.d	odd	2	1
2592.2.i.d		2	36.h	even	6	2
2592.2.i.f		2	9.d	odd	6	2
2592.2.i.s		2	36.f	odd	6	2
2592.2.i.u		2	9.c	even	3	2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{2}^{\mathrm{new}}(\Gamma_0(864))$:

$ T_{5} + 2 $

$ T_{7} + 1 $

$ T_{11} - 2 $

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T $ T
$3$	$ T $ T
$5$	$ T + 2 $ T + 2
$7$	$ T + 1 $ T + 1
$11$	$ T - 2 $ T - 2
$13$	$ T - 1 $ T - 1
$17$	$ T + 6 $ T + 6
$19$	$ T - 5 $ T - 5
$23$	$ T + 6 $ T + 6
$29$	$ T + 8 $ T + 8
$31$	$ T + 8 $ T + 8
$37$	$ T + 5 $ T + 5
$41$	$ T + 8 $ T + 8
$43$	$ T - 4 $ T - 4
$47$	$ T - 10 $ T - 10
$53$	$ T + 4 $ T + 4
$59$	$ T + 14 $ T + 14
$61$	$ T - 3 $ T - 3
$67$	$ T - 13 $ T - 13
$71$	$ T - 4 $ T - 4
$73$	$ T - 9 $ T - 9
$79$	$ T + 11 $ T + 11
$83$	$ T - 12 $ T - 12
$89$	$ T - 2 $ T - 2
$97$	$ T - 1 $ T - 1
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Elliptic curves over $\Q$
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Level:	\( N \)	\(=\)	\( 864 = 2^{5} \cdot 3^{3} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	864.a (trivial)

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