LMFDB - Newform orbit 5220.2.a.o

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [5220,2,Mod(1,5220)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(5220, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("5220.1"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$N$	$=$	$5220 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 29$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	5220.a (trivial)

Newform invariants

comment:select newform

sage:traces = [1,0,0,0,1,0,0,0,0,0,6,0,6,0,0,0,-4,0,2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(19)] == traces)

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

Self dual:	yes
Analytic conductor:	$41.6819098551$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$1$
Twist minimal:	no (minimal twist has level 1740)
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^{5} + 6 q^{11} + 6 q^{13} - 4 q^{17} + 2 q^{19} + 8 q^{23} + q^{25} - q^{29} - 2 q^{31} + 2 q^{41} - 7 q^{49} + 14 q^{53} + 6 q^{55} - 4 q^{59} + 6 q^{61} + 6 q^{65} - 12 q^{67} + 12 q^{71} - 16 q^{73}+ \cdots - 12 q^{97}+O(q^{100})

q + q^5 + 6 * q^11 + 6 * q^13 - 4 * q^17 + 2 * q^19 + 8 * q^23 + q^25 - q^29 - 2 * q^31 + 2 * q^41 - 7 * q^49 + 14 * q^53 + 6 * q^55 - 4 * q^59 + 6 * q^61 + 6 * q^65 - 12 * q^67 + 12 * q^71 - 16 * q^73 - 2 * q^79 - 12 * q^83 - 4 * q^85 + 2 * q^89 + 2 * q^95 - 12 * q^97

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
$2$	$-1$
$3$	$-1$
$5$	$-1$
$29$	$+1$

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	5220.2.a.o		1
3.b	odd	2	1		1740.2.a.a	✓	1
12.b	even	2	1		6960.2.a.z		1
15.d	odd	2	1		8700.2.a.q		1
15.e	even	4	2		8700.2.g.b		2

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
1740.2.a.a	✓	1	3.b	odd	2	1
5220.2.a.o		1	1.a	even	1	1	trivial
6960.2.a.z		1	12.b	even	2	1
8700.2.a.q		1	15.d	odd	2	1
8700.2.g.b		2	15.e	even	4	2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{2}^{\mathrm{new}}(\Gamma_0(5220))$ :

T_{7}

T7

T_{11} - 6

T11 - 6

T_{19} - 2

T19 - 2

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T$ T
$3$	$T$ T
$5$	$T - 1$ T - 1
$7$	$T$ T
$11$	$T - 6$ T - 6
$13$	$T - 6$ T - 6
$17$	$T + 4$ T + 4
$19$	$T - 2$ T - 2
$23$	$T - 8$ T - 8
$29$	$T + 1$ T + 1
$31$	$T + 2$ T + 2
$37$	$T$ T
$41$	$T - 2$ T - 2
$43$	$T$ T
$47$	$T$ T
$53$	$T - 14$ T - 14
$59$	$T + 4$ T + 4
$61$	$T - 6$ T - 6
$67$	$T + 12$ T + 12
$71$	$T - 12$ T - 12
$73$	$T + 16$ T + 16
$79$	$T + 2$ T + 2
$83$	$T + 12$ T + 12
$89$	$T - 2$ T - 2
$97$	$T + 12$ T + 12
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