LMFDB - Newform orbit 504.4.a.h

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [504,4,Mod(1,504)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(504, 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("504.1"); S:= CuspForms(chi, 4); N := Newforms(S);

Level:	$N$	$=$	$504 = 2^{3} \cdot 3^{2} \cdot 7$
Weight:	$k$	$=$	$4$
Character orbit:	$[\chi]$	$=$	504.a (trivial)

Newform invariants

comment:select newform

sage:traces = [1,0,0,0,16,0,7] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

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

Self dual:	yes
Analytic conductor:	$29.7369626429$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$1$
Twist minimal:	no (minimal twist has level 168)
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 + 16 q^{5} + 7 q^{7} + 18 q^{11} - 54 q^{13} + 128 q^{17} + 52 q^{19} + 202 q^{23} + 131 q^{25} - 302 q^{29} - 200 q^{31} + 112 q^{35} - 150 q^{37} - 172 q^{41} + 164 q^{43} + 460 q^{47} + 49 q^{49}+ \cdots - 554 q^{97}+O(q^{100})

q + 16 * q^5 + 7 * q^7 + 18 * q^11 - 54 * q^13 + 128 * q^17 + 52 * q^19 + 202 * q^23 + 131 * q^25 - 302 * q^29 - 200 * q^31 + 112 * q^35 - 150 * q^37 - 172 * q^41 + 164 * q^43 + 460 * q^47 + 49 * q^49 + 190 * q^53 + 288 * q^55 - 96 * q^59 + 622 * q^61 - 864 * q^65 + 744 * q^67 + 54 * q^71 + 742 * q^73 + 126 * q^77 - 92 * q^79 + 228 * q^83 + 2048 * q^85 + 116 * q^89 - 378 * q^91 + 832 * q^95 - 554 * 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

16.0000

7.00000

Atkin-Lehner signs

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

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	504.4.a.h		1
3.b	odd	2	1		168.4.a.d	✓	1
4.b	odd	2	1		1008.4.a.t		1
12.b	even	2	1		336.4.a.a		1
21.c	even	2	1		1176.4.a.h		1
24.f	even	2	1		1344.4.a.z		1
24.h	odd	2	1		1344.4.a.l		1
84.h	odd	2	1		2352.4.a.bj		1

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
168.4.a.d	✓	1	3.b	odd	2	1
336.4.a.a		1	12.b	even	2	1
504.4.a.h		1	1.a	even	1	1	trivial
1008.4.a.t		1	4.b	odd	2	1
1176.4.a.h		1	21.c	even	2	1
1344.4.a.l		1	24.h	odd	2	1
1344.4.a.z		1	24.f	even	2	1
2352.4.a.bj		1	84.h	odd	2	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{4}^{\mathrm{new}}(\Gamma_0(504))$ :

T_{5} - 16

T5 - 16

T_{11} - 18

T11 - 18

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T$ T
$3$	$T$ T
$5$	$T - 16$ T - 16
$7$	$T - 7$ T - 7
$11$	$T - 18$ T - 18
$13$	$T + 54$ T + 54
$17$	$T - 128$ T - 128
$19$	$T - 52$ T - 52
$23$	$T - 202$ T - 202
$29$	$T + 302$ T + 302
$31$	$T + 200$ T + 200
$37$	$T + 150$ T + 150
$41$	$T + 172$ T + 172
$43$	$T - 164$ T - 164
$47$	$T - 460$ T - 460
$53$	$T - 190$ T - 190
$59$	$T + 96$ T + 96
$61$	$T - 622$ T - 622
$67$	$T - 744$ T - 744
$71$	$T - 54$ T - 54
$73$	$T - 742$ T - 742
$79$	$T + 92$ T + 92
$83$	$T - 228$ T - 228
$89$	$T - 116$ T - 116
$97$	$T + 554$ T + 554
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