LMFDB - Newform orbit 4704.2.a.t

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4704,2,Mod(1,4704)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4704, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("4704.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4704 = 2^{5} \cdot 3 \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4704.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:	$37.5616291108$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 96)
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} + q^{9}+O(q^{10}) $

$ q + q^{3} - 2 q^{5} + q^{9} - 4 q^{11} + 2 q^{13} - 2 q^{15} + 6 q^{17} - 4 q^{19} - q^{25} + q^{27} + 2 q^{29} + 4 q^{31} - 4 q^{33} - 2 q^{37} + 2 q^{39} - 2 q^{41} - 4 q^{43} - 2 q^{45} + 8 q^{47} + 6 q^{51} + 10 q^{53} + 8 q^{55} - 4 q^{57} - 4 q^{59} - 6 q^{61} - 4 q^{65} - 4 q^{67} + 16 q^{71} + 6 q^{73} - q^{75} - 4 q^{79} + q^{81} + 12 q^{83} - 12 q^{85} + 2 q^{87} - 10 q^{89} + 4 q^{93} + 8 q^{95} + 14 q^{97} - 4 q^{99}+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 $
$7$	$ -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	4704.2.a.t		1
4.b	odd	2	1		4704.2.a.e		1
7.b	odd	2	1		96.2.a.a	✓	1
8.b	even	2	1		9408.2.a.bj		1
8.d	odd	2	1		9408.2.a.ct		1
21.c	even	2	1		288.2.a.c		1
28.d	even	2	1		96.2.a.b	yes	1
35.c	odd	2	1		2400.2.a.r		1
35.f	even	4	2		2400.2.f.a		2
56.e	even	2	1		192.2.a.a		1
56.h	odd	2	1		192.2.a.c		1
63.l	odd	6	2		2592.2.i.b		2
63.o	even	6	2		2592.2.i.q		2
84.h	odd	2	1		288.2.a.b		1
105.g	even	2	1		7200.2.a.e		1
105.k	odd	4	2		7200.2.f.x		2
112.j	even	4	2		768.2.d.h		2
112.l	odd	4	2		768.2.d.a		2
140.c	even	2	1		2400.2.a.q		1
140.j	odd	4	2		2400.2.f.r		2
168.e	odd	2	1		576.2.a.g		1
168.i	even	2	1		576.2.a.h		1
252.s	odd	6	2		2592.2.i.w		2
252.bi	even	6	2		2592.2.i.h		2
280.c	odd	2	1		4800.2.a.f		1
280.n	even	2	1		4800.2.a.co		1
280.s	even	4	2		4800.2.f.bh		2
280.y	odd	4	2		4800.2.f.e		2
336.v	odd	4	2		2304.2.d.s		2
336.y	even	4	2		2304.2.d.c		2
420.o	odd	2	1		7200.2.a.bx		1
420.w	even	4	2		7200.2.f.f		2

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
96.2.a.a	✓	1	7.b	odd	2	1
96.2.a.b	yes	1	28.d	even	2	1
192.2.a.a		1	56.e	even	2	1
192.2.a.c		1	56.h	odd	2	1
288.2.a.b		1	84.h	odd	2	1
288.2.a.c		1	21.c	even	2	1
576.2.a.g		1	168.e	odd	2	1
576.2.a.h		1	168.i	even	2	1
768.2.d.a		2	112.l	odd	4	2
768.2.d.h		2	112.j	even	4	2
2304.2.d.c		2	336.y	even	4	2
2304.2.d.s		2	336.v	odd	4	2
2400.2.a.q		1	140.c	even	2	1
2400.2.a.r		1	35.c	odd	2	1
2400.2.f.a		2	35.f	even	4	2
2400.2.f.r		2	140.j	odd	4	2
2592.2.i.b		2	63.l	odd	6	2
2592.2.i.h		2	252.bi	even	6	2
2592.2.i.q		2	63.o	even	6	2
2592.2.i.w		2	252.s	odd	6	2
4704.2.a.e		1	4.b	odd	2	1
4704.2.a.t		1	1.a	even	1	1	trivial
4800.2.a.f		1	280.c	odd	2	1
4800.2.a.co		1	280.n	even	2	1
4800.2.f.e		2	280.y	odd	4	2
4800.2.f.bh		2	280.s	even	4	2
7200.2.a.e		1	105.g	even	2	1
7200.2.a.bx		1	420.o	odd	2	1
7200.2.f.f		2	420.w	even	4	2
7200.2.f.x		2	105.k	odd	4	2
9408.2.a.bj		1	8.b	even	2	1
9408.2.a.ct		1	8.d	odd	2	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{2}^{\mathrm{new}}(\Gamma_0(4704))$:

$ T_{5} + 2 $

$ T_{11} + 4 $

$ T_{13} - 2 $

$ T_{19} + 4 $

$ T_{31} - 4 $

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T $ T
$3$	$ T - 1 $ T - 1
$5$	$ T + 2 $ T + 2
$7$	$ T $ T
$11$	$ T + 4 $ T + 4
$13$	$ T - 2 $ T - 2
$17$	$ T - 6 $ T - 6
$19$	$ T + 4 $ T + 4
$23$	$ T $ T
$29$	$ T - 2 $ T - 2
$31$	$ T - 4 $ T - 4
$37$	$ T + 2 $ T + 2
$41$	$ T + 2 $ T + 2
$43$	$ T + 4 $ T + 4
$47$	$ T - 8 $ T - 8
$53$	$ T - 10 $ T - 10
$59$	$ T + 4 $ T + 4
$61$	$ T + 6 $ T + 6
$67$	$ T + 4 $ T + 4
$71$	$ T - 16 $ T - 16
$73$	$ T - 6 $ T - 6
$79$	$ T + 4 $ T + 4
$83$	$ T - 12 $ T - 12
$89$	$ T + 10 $ T + 10
$97$	$ T - 14 $ T - 14
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Overview	Random
Universe	Knowledge

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:	\( N \)	\(=\)	\( 4704 = 2^{5} \cdot 3 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4704.a (trivial)

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