LMFDB - Newform orbit 9408.2.a.a

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:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 672)
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} - 6 q^{11} + 5 q^{13} + 4 q^{15} - 2 q^{17} + q^{19} - 6 q^{23} + 11 q^{25} - q^{27} + 3 q^{31} + 6 q^{33} - 3 q^{37} - 5 q^{39} + 6 q^{41} - 5 q^{43} - 4 q^{45} + 4 q^{47} + 2 q^{51} + 6 q^{53} + 24 q^{55} - q^{57} - 6 q^{59} - 2 q^{61} - 20 q^{65} - 7 q^{67} + 6 q^{69} + 16 q^{71} + 3 q^{73} - 11 q^{75} + 11 q^{79} + q^{81} + 12 q^{83} + 8 q^{85} - 4 q^{89} - 3 q^{93} - 4 q^{95} + 6 q^{97} - 6 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.a		1
4.b	odd	2	1		9408.2.a.bs		1
7.b	odd	2	1		9408.2.a.dd		1
7.d	odd	6	2		1344.2.q.a		2
8.b	even	2	1		4704.2.a.bh		1
8.d	odd	2	1		4704.2.a.p		1
28.d	even	2	1		9408.2.a.bp		1
28.f	even	6	2		1344.2.q.l		2
56.e	even	2	1		4704.2.a.r		1
56.h	odd	2	1		4704.2.a.a		1
56.j	odd	6	2		672.2.q.j	yes	2
56.m	even	6	2		672.2.q.e	✓	2
168.ba	even	6	2		2016.2.s.a		2
168.be	odd	6	2		2016.2.s.b		2

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
672.2.q.e	✓	2	56.m	even	6	2
672.2.q.j	yes	2	56.j	odd	6	2
1344.2.q.a		2	7.d	odd	6	2
1344.2.q.l		2	28.f	even	6	2
2016.2.s.a		2	168.ba	even	6	2
2016.2.s.b		2	168.be	odd	6	2
4704.2.a.a		1	56.h	odd	2	1
4704.2.a.p		1	8.d	odd	2	1
4704.2.a.r		1	56.e	even	2	1
4704.2.a.bh		1	8.b	even	2	1
9408.2.a.a		1	1.a	even	1	1	trivial
9408.2.a.bp		1	28.d	even	2	1
9408.2.a.bs		1	4.b	odd	2	1
9408.2.a.dd		1	7.b	odd	2	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{2}^{\mathrm{new}}(\Gamma_0(9408))$:

$ T_{5} + 4 $

$ T_{11} + 6 $

$ T_{13} - 5 $

$ T_{17} + 2 $

$ T_{19} - 1 $

$ T_{31} - 3 $

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 + 6 $ T + 6
$13$	$ T - 5 $ T - 5
$17$	$ T + 2 $ T + 2
$19$	$ T - 1 $ T - 1
$23$	$ T + 6 $ T + 6
$29$	$ T $ T
$31$	$ T - 3 $ T - 3
$37$	$ T + 3 $ T + 3
$41$	$ T - 6 $ T - 6
$43$	$ T + 5 $ T + 5
$47$	$ T - 4 $ T - 4
$53$	$ T - 6 $ T - 6
$59$	$ T + 6 $ T + 6
$61$	$ T + 2 $ T + 2
$67$	$ T + 7 $ T + 7
$71$	$ T - 16 $ T - 16
$73$	$ T - 3 $ T - 3
$79$	$ T - 11 $ T - 11
$83$	$ T - 12 $ T - 12
$89$	$ T + 4 $ T + 4
$97$	$ T - 6 $ T - 6
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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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