LMFDB - Newform orbit 2496.4.a.k

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2496,4,Mod(1,2496)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2496, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 4, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2496.1"); S:= CuspForms(chi, 4); N := Newforms(S);

Level:	$ N $	$=$	$ 2496 = 2^{6} \cdot 3 \cdot 13 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	2496.a (trivial)

Newform invariants

comment:select newform

sage:traces = [1,0,3,0,-6,0,20,0,9,0,-24] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$147.268767374$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 78)
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 + 3 q^{3} - 6 q^{5} + 20 q^{7} + 9 q^{9} - 24 q^{11} - 13 q^{13} - 18 q^{15} - 30 q^{17} + 16 q^{19} + 60 q^{21} - 72 q^{23} - 89 q^{25} + 27 q^{27} + 282 q^{29} + 164 q^{31} - 72 q^{33} - 120 q^{35}+ \cdots - 216 q^{99}+O(q^{100}) $

q + 3 * q^3 - 6 * q^5 + 20 * q^7 + 9 * q^9 - 24 * q^11 - 13 * q^13 - 18 * q^15 - 30 * q^17 + 16 * q^19 + 60 * q^21 - 72 * q^23 - 89 * q^25 + 27 * q^27 + 282 * q^29 + 164 * q^31 - 72 * q^33 - 120 * q^35 - 110 * q^37 - 39 * q^39 - 126 * q^41 - 164 * q^43 - 54 * q^45 - 204 * q^47 + 57 * q^49 - 90 * q^51 + 738 * q^53 + 144 * q^55 + 48 * q^57 - 120 * q^59 - 614 * q^61 + 180 * q^63 + 78 * q^65 - 848 * q^67 - 216 * q^69 + 132 * q^71 + 218 * q^73 - 267 * q^75 - 480 * q^77 - 1096 * q^79 + 81 * q^81 - 552 * q^83 + 180 * q^85 + 846 * q^87 + 210 * q^89 - 260 * q^91 + 492 * q^93 - 96 * q^95 - 1726 * q^97 - 216 * q^99

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

3.00000

−6.00000

20.0000

9.00000

Atkin-Lehner signs

$ p $	Sign
$2$	$ +1 $
$3$	$ -1 $
$13$	$ +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	2496.4.a.k		1
4.b	odd	2	1		2496.4.a.b		1
8.b	even	2	1		78.4.a.e	✓	1
8.d	odd	2	1		624.4.a.i		1
24.f	even	2	1		1872.4.a.e		1
24.h	odd	2	1		234.4.a.b		1
40.f	even	2	1		1950.4.a.c		1
104.e	even	2	1		1014.4.a.b		1
104.j	odd	4	2		1014.4.b.c		2

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
78.4.a.e	✓	1	8.b	even	2	1
234.4.a.b		1	24.h	odd	2	1
624.4.a.i		1	8.d	odd	2	1
1014.4.a.b		1	104.e	even	2	1
1014.4.b.c		2	104.j	odd	4	2
1872.4.a.e		1	24.f	even	2	1
1950.4.a.c		1	40.f	even	2	1
2496.4.a.b		1	4.b	odd	2	1
2496.4.a.k		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 $S_{4}^{\mathrm{new}}(\Gamma_0(2496))$:

$ T_{5} + 6 $

T5 + 6

$ T_{7} - 20 $

T7 - 20

$ T_{11} + 24 $

T11 + 24

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T $ T
$3$	$ T - 3 $ T - 3
$5$	$ T + 6 $ T + 6
$7$	$ T - 20 $ T - 20
$11$	$ T + 24 $ T + 24
$13$	$ T + 13 $ T + 13
$17$	$ T + 30 $ T + 30
$19$	$ T - 16 $ T - 16
$23$	$ T + 72 $ T + 72
$29$	$ T - 282 $ T - 282
$31$	$ T - 164 $ T - 164
$37$	$ T + 110 $ T + 110
$41$	$ T + 126 $ T + 126
$43$	$ T + 164 $ T + 164
$47$	$ T + 204 $ T + 204
$53$	$ T - 738 $ T - 738
$59$	$ T + 120 $ T + 120
$61$	$ T + 614 $ T + 614
$67$	$ T + 848 $ T + 848
$71$	$ T - 132 $ T - 132
$73$	$ T - 218 $ T - 218
$79$	$ T + 1096 $ T + 1096
$83$	$ T + 552 $ T + 552
$89$	$ T - 210 $ T - 210
$97$	$ T + 1726 $ T + 1726
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Elliptic curves over $\Q$
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Level:	\( N \)	\(=\)	\( 2496 = 2^{6} \cdot 3 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	2496.a (trivial)

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