LMFDB - Newform orbit 2160.4.a.r

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2160,4,Mod(1,2160)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2160, 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("2160.1"); S:= CuspForms(chi, 4); N := Newforms(S);

Level:	$N$	$=$	$2160 = 2^{4} \cdot 3^{3} \cdot 5$
Weight:	$k$	$=$	$4$
Character orbit:	$[\chi]$	$=$	2160.a (trivial)

Newform invariants

comment:select newform

sage:traces = [1,0,0,0,5,0,22,0,0,0,-12] 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:	$127.444125612$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$1$
Twist minimal:	no (minimal twist has level 270)
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 + 5 q^{5} + 22 q^{7} - 12 q^{11} + 38 q^{13} + 105 q^{17} + 157 q^{19} - 117 q^{23} + 25 q^{25} - 66 q^{29} + 25 q^{31} + 110 q^{35} + 314 q^{37} + 504 q^{41} - 380 q^{43} - 252 q^{47} + 141 q^{49} - 3 q^{53}+ \cdots + 1328 q^{97}+O(q^{100})

q + 5 * q^5 + 22 * q^7 - 12 * q^11 + 38 * q^13 + 105 * q^17 + 157 * q^19 - 117 * q^23 + 25 * q^25 - 66 * q^29 + 25 * q^31 + 110 * q^35 + 314 * q^37 + 504 * q^41 - 380 * q^43 - 252 * q^47 + 141 * q^49 - 3 * q^53 - 60 * q^55 - 318 * q^59 + 293 * q^61 + 190 * q^65 + 322 * q^67 - 120 * q^71 + 44 * q^73 - 264 * q^77 - 917 * q^79 + 309 * q^83 + 525 * q^85 - 1272 * q^89 + 836 * q^91 + 785 * q^95 + 1328 * 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

5.00000

22.0000

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	2160.4.a.r		1
3.b	odd	2	1		2160.4.a.i		1
4.b	odd	2	1		270.4.a.c	✓	1
12.b	even	2	1		270.4.a.g	yes	1
20.d	odd	2	1		1350.4.a.z		1
20.e	even	4	2		1350.4.c.l		2
36.f	odd	6	2		810.4.e.s		2
36.h	even	6	2		810.4.e.k		2
60.h	even	2	1		1350.4.a.l		1
60.l	odd	4	2		1350.4.c.i		2

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
270.4.a.c	✓	1	4.b	odd	2	1
270.4.a.g	yes	1	12.b	even	2	1
810.4.e.k		2	36.h	even	6	2
810.4.e.s		2	36.f	odd	6	2
1350.4.a.l		1	60.h	even	2	1
1350.4.a.z		1	20.d	odd	2	1
1350.4.c.i		2	60.l	odd	4	2
1350.4.c.l		2	20.e	even	4	2
2160.4.a.i		1	3.b	odd	2	1
2160.4.a.r		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 $S_{4}^{\mathrm{new}}(\Gamma_0(2160))$ :

T_{7} - 22

T7 - 22

T_{11} + 12

T11 + 12

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T$ T
$3$	$T$ T
$5$	$T - 5$ T - 5
$7$	$T - 22$ T - 22
$11$	$T + 12$ T + 12
$13$	$T - 38$ T - 38
$17$	$T - 105$ T - 105
$19$	$T - 157$ T - 157
$23$	$T + 117$ T + 117
$29$	$T + 66$ T + 66
$31$	$T - 25$ T - 25
$37$	$T - 314$ T - 314
$41$	$T - 504$ T - 504
$43$	$T + 380$ T + 380
$47$	$T + 252$ T + 252
$53$	$T + 3$ T + 3
$59$	$T + 318$ T + 318
$61$	$T - 293$ T - 293
$67$	$T - 322$ T - 322
$71$	$T + 120$ T + 120
$73$	$T - 44$ T - 44
$79$	$T + 917$ T + 917
$83$	$T - 309$ T - 309
$89$	$T + 1272$ T + 1272
$97$	$T - 1328$ T - 1328
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