LMFDB - Newform orbit 960.4.a.q

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [960,4,Mod(1,960)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(960, 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("960.1"); S:= CuspForms(chi, 4); N := Newforms(S);

Level:	$N$	$=$	$960 = 2^{6} \cdot 3 \cdot 5$
Weight:	$k$	$=$	$4$
Character orbit:	$[\chi]$	$=$	960.a (trivial)

Newform invariants

comment:select newform

sage:traces = [1,0,-3,0,5,0,20,0,9,0,-16] 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:	$56.6418336055$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$1$
Twist minimal:	no (minimal twist has level 120)
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} + 5 q^{5} + 20 q^{7} + 9 q^{9} - 16 q^{11} - 58 q^{13} - 15 q^{15} + 38 q^{17} - 4 q^{19} - 60 q^{21} - 80 q^{23} + 25 q^{25} - 27 q^{27} - 82 q^{29} - 8 q^{31} + 48 q^{33} + 100 q^{35} - 426 q^{37}+ \cdots - 144 q^{99}+O(q^{100})

q - 3 * q^3 + 5 * q^5 + 20 * q^7 + 9 * q^9 - 16 * q^11 - 58 * q^13 - 15 * q^15 + 38 * q^17 - 4 * q^19 - 60 * q^21 - 80 * q^23 + 25 * q^25 - 27 * q^27 - 82 * q^29 - 8 * q^31 + 48 * q^33 + 100 * q^35 - 426 * q^37 + 174 * q^39 - 246 * q^41 + 524 * q^43 + 45 * q^45 - 464 * q^47 + 57 * q^49 - 114 * q^51 + 702 * q^53 - 80 * q^55 + 12 * q^57 + 592 * q^59 - 574 * q^61 + 180 * q^63 - 290 * q^65 + 172 * q^67 + 240 * q^69 + 768 * q^71 - 558 * q^73 - 75 * q^75 - 320 * q^77 + 408 * q^79 + 81 * q^81 - 164 * q^83 + 190 * q^85 + 246 * q^87 - 510 * q^89 - 1160 * q^91 + 24 * q^93 - 20 * q^95 + 514 * q^97 - 144 * 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

5.00000

20.0000

9.00000

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	960.4.a.q		1
4.b	odd	2	1		960.4.a.bd		1
8.b	even	2	1		120.4.a.e	✓	1
8.d	odd	2	1		240.4.a.a		1
24.f	even	2	1		720.4.a.s		1
24.h	odd	2	1		360.4.a.m		1
40.e	odd	2	1		1200.4.a.bj		1
40.f	even	2	1		600.4.a.a		1
40.i	odd	4	2		600.4.f.f		2
40.k	even	4	2		1200.4.f.h		2
120.i	odd	2	1		1800.4.a.e		1
120.w	even	4	2		1800.4.f.k		2

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
120.4.a.e	✓	1	8.b	even	2	1
240.4.a.a		1	8.d	odd	2	1
360.4.a.m		1	24.h	odd	2	1
600.4.a.a		1	40.f	even	2	1
600.4.f.f		2	40.i	odd	4	2
720.4.a.s		1	24.f	even	2	1
960.4.a.q		1	1.a	even	1	1	trivial
960.4.a.bd		1	4.b	odd	2	1
1200.4.a.bj		1	40.e	odd	2	1
1200.4.f.h		2	40.k	even	4	2
1800.4.a.e		1	120.i	odd	2	1
1800.4.f.k		2	120.w	even	4	2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{4}^{\mathrm{new}}(\Gamma_0(960))$ :

T_{7} - 20

T7 - 20

T_{11} + 16

T11 + 16

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T$ T
$3$	$T + 3$ T + 3
$5$	$T - 5$ T - 5
$7$	$T - 20$ T - 20
$11$	$T + 16$ T + 16
$13$	$T + 58$ T + 58
$17$	$T - 38$ T - 38
$19$	$T + 4$ T + 4
$23$	$T + 80$ T + 80
$29$	$T + 82$ T + 82
$31$	$T + 8$ T + 8
$37$	$T + 426$ T + 426
$41$	$T + 246$ T + 246
$43$	$T - 524$ T - 524
$47$	$T + 464$ T + 464
$53$	$T - 702$ T - 702
$59$	$T - 592$ T - 592
$61$	$T + 574$ T + 574
$67$	$T - 172$ T - 172
$71$	$T - 768$ T - 768
$73$	$T + 558$ T + 558
$79$	$T - 408$ T - 408
$83$	$T + 164$ T + 164
$89$	$T + 510$ T + 510
$97$	$T - 514$ T - 514
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