LMFDB - Newform orbit 3840.2.a.b

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3840,2,Mod(1,3840)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3840, 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("3840.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3840 = 2^{8} \cdot 3 \cdot 5 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3840.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:	$30.6625543762$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1920)
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} - q^{5} - 2 q^{7} + q^{9}+O(q^{10}) $

$ q - q^{3} - q^{5} - 2 q^{7} + q^{9} - 2 q^{11} + 2 q^{13} + q^{15} - 8 q^{17} + 8 q^{19} + 2 q^{21} + 8 q^{23} + q^{25} - q^{27} - 10 q^{29} - 8 q^{31} + 2 q^{33} + 2 q^{35} - 2 q^{37} - 2 q^{39} - 6 q^{41} + 4 q^{43} - q^{45} - 4 q^{47} - 3 q^{49} + 8 q^{51} + 2 q^{53} + 2 q^{55} - 8 q^{57} + 6 q^{59} - 2 q^{63} - 2 q^{65} + 12 q^{67} - 8 q^{69} - 8 q^{71} + 6 q^{73} - q^{75} + 4 q^{77} - 8 q^{79} + q^{81} + 8 q^{83} + 8 q^{85} + 10 q^{87} + 6 q^{89} - 4 q^{91} + 8 q^{93} - 8 q^{95} + 10 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

−2.00000

1.00000

Atkin-Lehner signs

$ p $	Sign
$2$	$ -1 $
$3$	$ +1 $
$5$	$ +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	3840.2.a.b		1
4.b	odd	2	1		3840.2.a.s		1
8.b	even	2	1		3840.2.a.x		1
8.d	odd	2	1		3840.2.a.k		1
16.e	even	4	2		1920.2.k.g	yes	2
16.f	odd	4	2		1920.2.k.a	✓	2
48.i	odd	4	2		5760.2.k.j		2
48.k	even	4	2		5760.2.k.f		2

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
1920.2.k.a	✓	2	16.f	odd	4	2
1920.2.k.g	yes	2	16.e	even	4	2
3840.2.a.b		1	1.a	even	1	1	trivial
3840.2.a.k		1	8.d	odd	2	1
3840.2.a.s		1	4.b	odd	2	1
3840.2.a.x		1	8.b	even	2	1
5760.2.k.f		2	48.k	even	4	2
5760.2.k.j		2	48.i	odd	4	2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{2}^{\mathrm{new}}(\Gamma_0(3840))$:

$ T_{7} + 2 $

$ T_{11} + 2 $

$ T_{13} - 2 $

$ T_{17} + 8 $

$ T_{19} - 8 $

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T $ T
$3$	$ T + 1 $ T + 1
$5$	$ T + 1 $ T + 1
$7$	$ T + 2 $ T + 2
$11$	$ T + 2 $ T + 2
$13$	$ T - 2 $ T - 2
$17$	$ T + 8 $ T + 8
$19$	$ T - 8 $ T - 8
$23$	$ T - 8 $ T - 8
$29$	$ T + 10 $ T + 10
$31$	$ T + 8 $ T + 8
$37$	$ T + 2 $ T + 2
$41$	$ T + 6 $ T + 6
$43$	$ T - 4 $ T - 4
$47$	$ T + 4 $ T + 4
$53$	$ T - 2 $ T - 2
$59$	$ T - 6 $ T - 6
$61$	$ T $ T
$67$	$ T - 12 $ T - 12
$71$	$ T + 8 $ T + 8
$73$	$ T - 6 $ T - 6
$79$	$ T + 8 $ T + 8
$83$	$ T - 8 $ T - 8
$89$	$ T - 6 $ T - 6
$97$	$ T - 10 $ T - 10
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Level:	\( N \)	\(=\)	\( 3840 = 2^{8} \cdot 3 \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3840.a (trivial)

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