LMFDB - Newform orbit 504.4.a.e

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)

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);

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")

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 = [1,0,0,0,10,0,-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 + 10 q^{5} - 7 q^{7} + 52 q^{11} - 10 q^{13} + 54 q^{17} - 52 q^{19} - 48 q^{23} - 25 q^{25} + 186 q^{29} + 224 q^{31} - 70 q^{35} + 94 q^{37} + 478 q^{41} - 316 q^{43} - 256 q^{47} + 49 q^{49} + 66 q^{53}+ \cdots + 1554 q^{97}+O(q^{100})

q + 10 * q^5 - 7 * q^7 + 52 * q^11 - 10 * q^13 + 54 * q^17 - 52 * q^19 - 48 * q^23 - 25 * q^25 + 186 * q^29 + 224 * q^31 - 70 * q^35 + 94 * q^37 + 478 * q^41 - 316 * q^43 - 256 * q^47 + 49 * q^49 + 66 * q^53 + 520 * q^55 - 420 * q^59 + 342 * q^61 - 100 * q^65 + 668 * q^67 + 272 * q^71 - 86 * q^73 - 364 * q^77 + 1360 * q^79 - 188 * q^83 + 540 * q^85 + 366 * q^89 + 70 * q^91 - 520 * q^95 + 1554 * 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

10.0000

−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.e		1
3.b	odd	2	1		168.4.a.e	✓	1
4.b	odd	2	1		1008.4.a.q		1
12.b	even	2	1		336.4.a.b		1
21.c	even	2	1		1176.4.a.g		1
24.f	even	2	1		1344.4.a.x		1
24.h	odd	2	1		1344.4.a.k		1
84.h	odd	2	1		2352.4.a.bh		1

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
168.4.a.e	✓	1	3.b	odd	2	1
336.4.a.b		1	12.b	even	2	1
504.4.a.e		1	1.a	even	1	1	trivial
1008.4.a.q		1	4.b	odd	2	1
1176.4.a.g		1	21.c	even	2	1
1344.4.a.k		1	24.h	odd	2	1
1344.4.a.x		1	24.f	even	2	1
2352.4.a.bh		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} - 10

T5 - 10

T_{11} - 52

T11 - 52

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T$ T
$3$	$T$ T
$5$	$T - 10$ T - 10
$7$	$T + 7$ T + 7
$11$	$T - 52$ T - 52
$13$	$T + 10$ T + 10
$17$	$T - 54$ T - 54
$19$	$T + 52$ T + 52
$23$	$T + 48$ T + 48
$29$	$T - 186$ T - 186
$31$	$T - 224$ T - 224
$37$	$T - 94$ T - 94
$41$	$T - 478$ T - 478
$43$	$T + 316$ T + 316
$47$	$T + 256$ T + 256
$53$	$T - 66$ T - 66
$59$	$T + 420$ T + 420
$61$	$T - 342$ T - 342
$67$	$T - 668$ T - 668
$71$	$T - 272$ T - 272
$73$	$T + 86$ T + 86
$79$	$T - 1360$ T - 1360
$83$	$T + 188$ T + 188
$89$	$T - 366$ T - 366
$97$	$T - 1554$ T - 1554
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