LMFDB - Newform orbit 9408.2.a.df

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [9408,2,Mod(1,9408)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(9408, 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("9408.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 9408 = 2^{6} \cdot 3 \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	9408.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:	$75.1232582216$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 84)
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} + 4 q^{5} + q^{9}+O(q^{10}) $

$ q + q^{3} + 4 q^{5} + q^{9} + 2 q^{11} - 6 q^{13} + 4 q^{15} + 4 q^{17} + 4 q^{19} - 2 q^{23} + 11 q^{25} + q^{27} + 2 q^{29} + 2 q^{33} - 2 q^{37} - 6 q^{39} - 4 q^{43} + 4 q^{45} + 12 q^{47} + 4 q^{51} + 6 q^{53} + 8 q^{55} + 4 q^{57} + 8 q^{59} + 6 q^{61} - 24 q^{65} - 8 q^{67} - 2 q^{69} - 14 q^{71} + 2 q^{73} + 11 q^{75} - 12 q^{79} + q^{81} + 4 q^{83} + 16 q^{85} + 2 q^{87} + 16 q^{95} + 2 q^{97} + 2 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

4.00000

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	9408.2.a.df		1
4.b	odd	2	1		9408.2.a.bn		1
7.b	odd	2	1		1344.2.a.a		1
8.b	even	2	1		2352.2.a.a		1
8.d	odd	2	1		588.2.a.d		1
21.c	even	2	1		4032.2.a.bn		1
24.f	even	2	1		1764.2.a.k		1
24.h	odd	2	1		7056.2.a.cd		1
28.d	even	2	1		1344.2.a.k		1
56.e	even	2	1		84.2.a.a	✓	1
56.h	odd	2	1		336.2.a.f		1
56.j	odd	6	2		2352.2.q.b		2
56.k	odd	6	2		588.2.i.d		2
56.m	even	6	2		588.2.i.e		2
56.p	even	6	2		2352.2.q.z		2
84.h	odd	2	1		4032.2.a.bm		1
112.j	even	4	2		5376.2.c.q		2
112.l	odd	4	2		5376.2.c.p		2
168.e	odd	2	1		252.2.a.a		1
168.i	even	2	1		1008.2.a.a		1
168.v	even	6	2		1764.2.k.a		2
168.be	odd	6	2		1764.2.k.k		2
280.c	odd	2	1		8400.2.a.e		1
280.n	even	2	1		2100.2.a.r		1
280.y	odd	4	2		2100.2.k.i		2
504.be	even	6	2		2268.2.j.a		2
504.co	odd	6	2		2268.2.j.n		2
840.b	odd	2	1		6300.2.a.w		1
840.bm	even	4	2		6300.2.k.g		2

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
84.2.a.a	✓	1	56.e	even	2	1
252.2.a.a		1	168.e	odd	2	1
336.2.a.f		1	56.h	odd	2	1
588.2.a.d		1	8.d	odd	2	1
588.2.i.d		2	56.k	odd	6	2
588.2.i.e		2	56.m	even	6	2
1008.2.a.a		1	168.i	even	2	1
1344.2.a.a		1	7.b	odd	2	1
1344.2.a.k		1	28.d	even	2	1
1764.2.a.k		1	24.f	even	2	1
1764.2.k.a		2	168.v	even	6	2
1764.2.k.k		2	168.be	odd	6	2
2100.2.a.r		1	280.n	even	2	1
2100.2.k.i		2	280.y	odd	4	2
2268.2.j.a		2	504.be	even	6	2
2268.2.j.n		2	504.co	odd	6	2
2352.2.a.a		1	8.b	even	2	1
2352.2.q.b		2	56.j	odd	6	2
2352.2.q.z		2	56.p	even	6	2
4032.2.a.bm		1	84.h	odd	2	1
4032.2.a.bn		1	21.c	even	2	1
5376.2.c.p		2	112.l	odd	4	2
5376.2.c.q		2	112.j	even	4	2
6300.2.a.w		1	840.b	odd	2	1
6300.2.k.g		2	840.bm	even	4	2
7056.2.a.cd		1	24.h	odd	2	1
8400.2.a.e		1	280.c	odd	2	1
9408.2.a.bn		1	4.b	odd	2	1
9408.2.a.df		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 $S_{2}^{\mathrm{new}}(\Gamma_0(9408))$:

$ T_{5} - 4 $

$ T_{11} - 2 $

$ T_{13} + 6 $

$ T_{17} - 4 $

$ T_{19} - 4 $

$ T_{31} $

Hecke characteristic polynomials

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

Level:	\( N \)	\(=\)	\( 9408 = 2^{6} \cdot 3 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	9408.a (trivial)

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