LMFDB - Newform orbit 1568.2.a.e

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1568,2,Mod(1,1568)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1568, 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("1568.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1568 = 2^{5} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1568.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:	$12.5205430369$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 32)
Fricke sign:	$+1$
Sato-Tate group:	$N(\mathrm{U}(1))$

$q$-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

−3.00000

Atkin-Lehner signs

$ p $	Sign
$2$	$ -1 $
$7$	$ -1 $

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
4.b	odd	2	1	CM by $\Q(\sqrt{-1}) $

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	1568.2.a.e		1
4.b	odd	2	1	CM	1568.2.a.e		1
7.b	odd	2	1		32.2.a.a	✓	1
7.c	even	3	2		1568.2.i.f		2
7.d	odd	6	2		1568.2.i.g		2
8.b	even	2	1		3136.2.a.m		1
8.d	odd	2	1		3136.2.a.m		1
21.c	even	2	1		288.2.a.d		1
28.d	even	2	1		32.2.a.a	✓	1
28.f	even	6	2		1568.2.i.g		2
28.g	odd	6	2		1568.2.i.f		2
35.c	odd	2	1		800.2.a.d		1
35.f	even	4	2		800.2.c.e		2
56.e	even	2	1		64.2.a.a		1
56.h	odd	2	1		64.2.a.a		1
63.l	odd	6	2		2592.2.i.t		2
63.o	even	6	2		2592.2.i.e		2
77.b	even	2	1		3872.2.a.f		1
84.h	odd	2	1		288.2.a.d		1
91.b	odd	2	1		5408.2.a.g		1
105.g	even	2	1		7200.2.a.v		1
105.k	odd	4	2		7200.2.f.m		2
112.j	even	4	2		256.2.b.b		2
112.l	odd	4	2		256.2.b.b		2
119.d	odd	2	1		9248.2.a.f		1
140.c	even	2	1		800.2.a.d		1
140.j	odd	4	2		800.2.c.e		2
168.e	odd	2	1		576.2.a.c		1
168.i	even	2	1		576.2.a.c		1
224.v	odd	8	4		1024.2.e.j		4
224.x	even	8	4		1024.2.e.j		4
252.s	odd	6	2		2592.2.i.e		2
252.bi	even	6	2		2592.2.i.t		2
280.c	odd	2	1		1600.2.a.n		1
280.n	even	2	1		1600.2.a.n		1
280.s	even	4	2		1600.2.c.l		2
280.y	odd	4	2		1600.2.c.l		2
308.g	odd	2	1		3872.2.a.f		1
336.v	odd	4	2		2304.2.d.j		2
336.y	even	4	2		2304.2.d.j		2
364.h	even	2	1		5408.2.a.g		1
420.o	odd	2	1		7200.2.a.v		1
420.w	even	4	2		7200.2.f.m		2
476.e	even	2	1		9248.2.a.f		1
616.g	odd	2	1		7744.2.a.v		1
616.o	even	2	1		7744.2.a.v		1

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
32.2.a.a	✓	1	7.b	odd	2	1
32.2.a.a	✓	1	28.d	even	2	1
64.2.a.a		1	56.e	even	2	1
64.2.a.a		1	56.h	odd	2	1
256.2.b.b		2	112.j	even	4	2
256.2.b.b		2	112.l	odd	4	2
288.2.a.d		1	21.c	even	2	1
288.2.a.d		1	84.h	odd	2	1
576.2.a.c		1	168.e	odd	2	1
576.2.a.c		1	168.i	even	2	1
800.2.a.d		1	35.c	odd	2	1
800.2.a.d		1	140.c	even	2	1
800.2.c.e		2	35.f	even	4	2
800.2.c.e		2	140.j	odd	4	2
1024.2.e.j		4	224.v	odd	8	4
1024.2.e.j		4	224.x	even	8	4
1568.2.a.e		1	1.a	even	1	1	trivial
1568.2.a.e		1	4.b	odd	2	1	CM
1568.2.i.f		2	7.c	even	3	2
1568.2.i.f		2	28.g	odd	6	2
1568.2.i.g		2	7.d	odd	6	2
1568.2.i.g		2	28.f	even	6	2
1600.2.a.n		1	280.c	odd	2	1
1600.2.a.n		1	280.n	even	2	1
1600.2.c.l		2	280.s	even	4	2
1600.2.c.l		2	280.y	odd	4	2
2304.2.d.j		2	336.v	odd	4	2
2304.2.d.j		2	336.y	even	4	2
2592.2.i.e		2	63.o	even	6	2
2592.2.i.e		2	252.s	odd	6	2
2592.2.i.t		2	63.l	odd	6	2
2592.2.i.t		2	252.bi	even	6	2
3136.2.a.m		1	8.b	even	2	1
3136.2.a.m		1	8.d	odd	2	1
3872.2.a.f		1	77.b	even	2	1
3872.2.a.f		1	308.g	odd	2	1
5408.2.a.g		1	91.b	odd	2	1
5408.2.a.g		1	364.h	even	2	1
7200.2.a.v		1	105.g	even	2	1
7200.2.a.v		1	420.o	odd	2	1
7200.2.f.m		2	105.k	odd	4	2
7200.2.f.m		2	420.w	even	4	2
7744.2.a.v		1	616.g	odd	2	1
7744.2.a.v		1	616.o	even	2	1
9248.2.a.f		1	119.d	odd	2	1
9248.2.a.f		1	476.e	even	2	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{2}^{\mathrm{new}}(\Gamma_0(1568))$:

$ T_{3} $

$ T_{5} - 2 $

$ T_{11} $

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T $ T
$3$	$ T $ T
$5$	$ T - 2 $ T - 2
$7$	$ T $ T
$11$	$ T $ T
$13$	$ T + 6 $ T + 6
$17$	$ T + 2 $ T + 2
$19$	$ T $ T
$23$	$ T $ T
$29$	$ T + 10 $ T + 10
$31$	$ T $ T
$37$	$ T + 2 $ T + 2
$41$	$ T + 10 $ T + 10
$43$	$ T $ T
$47$	$ T $ T
$53$	$ T - 14 $ T - 14
$59$	$ T $ T
$61$	$ T - 10 $ T - 10
$67$	$ T $ T
$71$	$ T $ T
$73$	$ T - 6 $ T - 6
$79$	$ T $ T
$83$	$ T $ T
$89$	$ T + 10 $ T + 10
$97$	$ T + 18 $ T + 18
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Overview	Random
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Classical	Maass
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Elliptic curves over $\Q$
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Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 1568 = 2^{5} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1568.a (trivial)

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