LMFDB - Newform orbit 2940.2.a.k

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2940,2,Mod(1,2940)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2940, 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("2940.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2940 = 2^{2} \cdot 3 \cdot 5 \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2940.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:	$23.4760181943$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 420)
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} + q^{9}+O(q^{10}) $

$ q + q^{3} + q^{5} + q^{9} - 4 q^{11} - 7 q^{13} + q^{15} + 6 q^{17} - 3 q^{19} - 2 q^{23} + q^{25} + q^{27} - 2 q^{29} - 7 q^{31} - 4 q^{33} - 7 q^{37} - 7 q^{39} + 8 q^{41} + 5 q^{43} + q^{45} - 10 q^{47} + 6 q^{51} - 8 q^{53} - 4 q^{55} - 3 q^{57} - 10 q^{59} + 6 q^{61} - 7 q^{65} + 3 q^{67} - 2 q^{69} - 15 q^{73} + q^{75} + q^{79} + q^{81} - 8 q^{83} + 6 q^{85} - 2 q^{87} - 2 q^{89} - 7 q^{93} - 3 q^{95} + 10 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

Atkin-Lehner signs

$ p $	Sign
$2$	$-1$
$3$	$-1$
$5$	$-1$
$7$	$-1$

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	2940.2.a.k		1
3.b	odd	2	1		8820.2.a.l		1
7.b	odd	2	1		2940.2.a.b		1
7.c	even	3	2		2940.2.q.c		2
7.d	odd	6	2		420.2.q.b	✓	2
21.c	even	2	1		8820.2.a.bb		1
21.g	even	6	2		1260.2.s.a		2
28.f	even	6	2		1680.2.bg.j		2
35.i	odd	6	2		2100.2.q.d		2
35.k	even	12	4		2100.2.bc.d		4

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
420.2.q.b	✓	2	7.d	odd	6	2
1260.2.s.a		2	21.g	even	6	2
1680.2.bg.j		2	28.f	even	6	2
2100.2.q.d		2	35.i	odd	6	2
2100.2.bc.d		4	35.k	even	12	4
2940.2.a.b		1	7.b	odd	2	1
2940.2.a.k		1	1.a	even	1	1	trivial
2940.2.q.c		2	7.c	even	3	2
8820.2.a.l		1	3.b	odd	2	1
8820.2.a.bb		1	21.c	even	2	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{2}^{\mathrm{new}}(\Gamma_0(2940))$:

$ T_{11} + 4 $

$ T_{13} + 7 $

$ T_{17} - 6 $

$ T_{31} + 7 $

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T $ T
$3$	$ T - 1 $ T - 1
$5$	$ T - 1 $ T - 1
$7$	$ T $ T
$11$	$ T + 4 $ T + 4
$13$	$ T + 7 $ T + 7
$17$	$ T - 6 $ T - 6
$19$	$ T + 3 $ T + 3
$23$	$ T + 2 $ T + 2
$29$	$ T + 2 $ T + 2
$31$	$ T + 7 $ T + 7
$37$	$ T + 7 $ T + 7
$41$	$ T - 8 $ T - 8
$43$	$ T - 5 $ T - 5
$47$	$ T + 10 $ T + 10
$53$	$ T + 8 $ T + 8
$59$	$ T + 10 $ T + 10
$61$	$ T - 6 $ T - 6
$67$	$ T - 3 $ T - 3
$71$	$ T $ T
$73$	$ T + 15 $ T + 15
$79$	$ T - 1 $ T - 1
$83$	$ T + 8 $ T + 8
$89$	$ T + 2 $ T + 2
$97$	$ T - 10 $ T - 10
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Elliptic curves over $\Q$
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Level:	\( N \)	\(=\)	\( 2940 = 2^{2} \cdot 3 \cdot 5 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2940.a (trivial)

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