LMFDB - Newform orbit 768.2.a.e

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [768,2,Mod(1,768)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(768, 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("768.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 768 = 2^{8} \cdot 3 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	768.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:	$6.13251087523$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 24)
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^{3} - 2 q^{5} - 2 q^{7} + q^{9}+O(q^{10}) $

$ q + q^{3} - 2 q^{5} - 2 q^{7} + q^{9} - 4 q^{13} - 2 q^{15} - 2 q^{17} - 4 q^{19} - 2 q^{21} + 4 q^{23} - q^{25} + q^{27} - 6 q^{29} - 2 q^{31} + 4 q^{35} - 8 q^{37} - 4 q^{39} - 2 q^{41} - 4 q^{43} - 2 q^{45} + 12 q^{47} - 3 q^{49} - 2 q^{51} - 6 q^{53} - 4 q^{57} + 4 q^{59} - 2 q^{63} + 8 q^{65} + 12 q^{67} + 4 q^{69} + 12 q^{71} + 6 q^{73} - q^{75} - 10 q^{79} + q^{81} - 16 q^{83} + 4 q^{85} - 6 q^{87} + 10 q^{89} + 8 q^{91} - 2 q^{93} + 8 q^{95} - 2 q^{97}+O(q^{100}) $

Display 100 coefficients 10 coefficients

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

−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	768.2.a.e		1
3.b	odd	2	1		2304.2.a.l		1
4.b	odd	2	1		768.2.a.a		1
8.b	even	2	1		768.2.a.d		1
8.d	odd	2	1		768.2.a.h		1
12.b	even	2	1		2304.2.a.o		1
16.e	even	4	2		96.2.d.a		2
16.f	odd	4	2		24.2.d.a	✓	2
24.f	even	2	1		2304.2.a.e		1
24.h	odd	2	1		2304.2.a.b		1
48.i	odd	4	2		288.2.d.b		2
48.k	even	4	2		72.2.d.b		2
80.i	odd	4	2		2400.2.d.b		2
80.j	even	4	2		600.2.d.c		2
80.k	odd	4	2		600.2.k.b		2
80.q	even	4	2		2400.2.k.a		2
80.s	even	4	2		600.2.d.b		2
80.t	odd	4	2		2400.2.d.c		2
112.j	even	4	2		1176.2.c.a		2
112.l	odd	4	2		4704.2.c.a		2
144.u	even	12	4		648.2.n.c		4
144.v	odd	12	4		648.2.n.k		4
144.w	odd	12	4		2592.2.r.g		4
144.x	even	12	4		2592.2.r.f		4
240.t	even	4	2		1800.2.k.a		2
240.z	odd	4	2		1800.2.d.i		2
240.bb	even	4	2		7200.2.d.g		2
240.bd	odd	4	2		1800.2.d.b		2
240.bf	even	4	2		7200.2.d.d		2
240.bm	odd	4	2		7200.2.k.d		2

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
24.2.d.a	✓	2	16.f	odd	4	2
72.2.d.b		2	48.k	even	4	2
96.2.d.a		2	16.e	even	4	2
288.2.d.b		2	48.i	odd	4	2
600.2.d.b		2	80.s	even	4	2
600.2.d.c		2	80.j	even	4	2
600.2.k.b		2	80.k	odd	4	2
648.2.n.c		4	144.u	even	12	4
648.2.n.k		4	144.v	odd	12	4
768.2.a.a		1	4.b	odd	2	1
768.2.a.d		1	8.b	even	2	1
768.2.a.e		1	1.a	even	1	1	trivial
768.2.a.h		1	8.d	odd	2	1
1176.2.c.a		2	112.j	even	4	2
1800.2.d.b		2	240.bd	odd	4	2
1800.2.d.i		2	240.z	odd	4	2
1800.2.k.a		2	240.t	even	4	2
2304.2.a.b		1	24.h	odd	2	1
2304.2.a.e		1	24.f	even	2	1
2304.2.a.l		1	3.b	odd	2	1
2304.2.a.o		1	12.b	even	2	1
2400.2.d.b		2	80.i	odd	4	2
2400.2.d.c		2	80.t	odd	4	2
2400.2.k.a		2	80.q	even	4	2
2592.2.r.f		4	144.x	even	12	4
2592.2.r.g		4	144.w	odd	12	4
4704.2.c.a		2	112.l	odd	4	2
7200.2.d.d		2	240.bf	even	4	2
7200.2.d.g		2	240.bb	even	4	2
7200.2.k.d		2	240.bm	odd	4	2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{2}^{\mathrm{new}}(\Gamma_0(768))$:

$ T_{5} + 2 $

$ T_{7} + 2 $

$ T_{11} $

$ T_{19} + 4 $

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T $ T
$3$	$ T - 1 $ T - 1
$5$	$ T + 2 $ T + 2
$7$	$ T + 2 $ T + 2
$11$	$ T $ T
$13$	$ T + 4 $ T + 4
$17$	$ T + 2 $ T + 2
$19$	$ T + 4 $ T + 4
$23$	$ T - 4 $ T - 4
$29$	$ T + 6 $ T + 6
$31$	$ T + 2 $ T + 2
$37$	$ T + 8 $ T + 8
$41$	$ T + 2 $ T + 2
$43$	$ T + 4 $ T + 4
$47$	$ T - 12 $ T - 12
$53$	$ T + 6 $ T + 6
$59$	$ T - 4 $ T - 4
$61$	$ T $ T
$67$	$ T - 12 $ T - 12
$71$	$ T - 12 $ T - 12
$73$	$ T - 6 $ T - 6
$79$	$ T + 10 $ T + 10
$83$	$ T + 16 $ T + 16
$89$	$ T - 10 $ T - 10
$97$	$ T + 2 $ T + 2
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Overview	Random
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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:	\( N \)	\(=\)	\( 768 = 2^{8} \cdot 3 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	768.a (trivial)

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