LMFDB - Newform orbit 392.8.a.c

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [392,8,Mod(1,392)] 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("392.1"); S:= CuspForms(chi, 8); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(392, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 8, names="a")

Level:	$ N $	$=$	$ 392 = 2^{3} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	392.a (trivial)

Newform invariants

comment:select newform

sage:traces = [1,0,18] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)

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

Self dual:	yes
Analytic conductor:	$122.454929990$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 56)
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 + 18 q^{3} - 160 q^{5} - 1863 q^{9} + 5704 q^{11} - 1388 q^{13} - 2880 q^{15} + 31434 q^{17} + 19966 q^{19} - 77136 q^{23} - 52525 q^{25} - 72900 q^{27} - 193374 q^{29} + 26356 q^{31} + 102672 q^{33}+ \cdots - 10626552 q^{99}+O(q^{100}) $

q + 18 * q^3 - 160 * q^5 - 1863 * q^9 + 5704 * q^11 - 1388 * q^13 - 2880 * q^15 + 31434 * q^17 + 19966 * q^19 - 77136 * q^23 - 52525 * q^25 - 72900 * q^27 - 193374 * q^29 + 26356 * q^31 + 102672 * q^33 + 204346 * q^37 - 24984 * q^39 + 663050 * q^41 - 335920 * q^43 + 298080 * q^45 - 1119812 * q^47 + 565812 * q^51 + 112782 * q^53 - 912640 * q^55 + 359388 * q^57 - 536154 * q^59 + 1170264 * q^61 + 222080 * q^65 + 3890660 * q^67 - 1388448 * q^69 + 2505344 * q^71 + 1435070 * q^73 - 945450 * q^75 + 176536 * q^79 + 2762181 * q^81 + 6211622 * q^83 - 5029440 * q^85 - 3480732 * q^87 + 4729062 * q^89 + 474408 * q^93 - 3194560 * q^95 + 2129562 * q^97 - 10626552 * 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

18.0000

−160.000

−1863.00

Atkin-Lehner signs

$ p $	Sign
$2$	$ -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	392.8.a.c		1
7.b	odd	2	1		56.8.a.a	✓	1
21.c	even	2	1		504.8.a.a		1
28.d	even	2	1		112.8.a.b		1
56.e	even	2	1		448.8.a.e		1
56.h	odd	2	1		448.8.a.f		1

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
56.8.a.a	✓	1	7.b	odd	2	1
112.8.a.b		1	28.d	even	2	1
392.8.a.c		1	1.a	even	1	1	trivial
448.8.a.e		1	56.e	even	2	1
448.8.a.f		1	56.h	odd	2	1
504.8.a.a		1	21.c	even	2	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $ T_{3} - 18 $ T3 - 18 acting on $S_{8}^{\mathrm{new}}(\Gamma_0(392))$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T $ T
$3$	$ T - 18 $ T - 18
$5$	$ T + 160 $ T + 160
$7$	$ T $ T
$11$	$ T - 5704 $ T - 5704
$13$	$ T + 1388 $ T + 1388
$17$	$ T - 31434 $ T - 31434
$19$	$ T - 19966 $ T - 19966
$23$	$ T + 77136 $ T + 77136
$29$	$ T + 193374 $ T + 193374
$31$	$ T - 26356 $ T - 26356
$37$	$ T - 204346 $ T - 204346
$41$	$ T - 663050 $ T - 663050
$43$	$ T + 335920 $ T + 335920
$47$	$ T + 1119812 $ T + 1119812
$53$	$ T - 112782 $ T - 112782
$59$	$ T + 536154 $ T + 536154
$61$	$ T - 1170264 $ T - 1170264
$67$	$ T - 3890660 $ T - 3890660
$71$	$ T - 2505344 $ T - 2505344
$73$	$ T - 1435070 $ T - 1435070
$79$	$ T - 176536 $ T - 176536
$83$	$ T - 6211622 $ T - 6211622
$89$	$ T - 4729062 $ T - 4729062
$97$	$ T - 2129562 $ T - 2129562
show more
show less

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 392 = 2^{3} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	392.a (trivial)

\(n\):		e.g. 2-40 or 990-1000
Significant digits:
Format:

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more