LMFDB - Newform orbit 392.4.a.c

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [392,4,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, 4); 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, 4, names="a")

Level:	$ N $	$=$	$ 392 = 2^{3} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	392.a (trivial)

Newform invariants

comment:select newform

sage:traces = [1,0,2,0,16,0,0,0,-23] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(9)] == traces)

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

Self dual:	yes
Analytic conductor:	$23.1287487223$
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 + 2 q^{3} + 16 q^{5} - 23 q^{9} + 24 q^{11} + 68 q^{13} + 32 q^{15} - 54 q^{17} + 46 q^{19} + 176 q^{23} + 131 q^{25} - 100 q^{27} - 174 q^{29} + 116 q^{31} + 48 q^{33} + 74 q^{37} + 136 q^{39} + 10 q^{41}+ \cdots - 552 q^{99}+O(q^{100}) $

q + 2 * q^3 + 16 * q^5 - 23 * q^9 + 24 * q^11 + 68 * q^13 + 32 * q^15 - 54 * q^17 + 46 * q^19 + 176 * q^23 + 131 * q^25 - 100 * q^27 - 174 * q^29 + 116 * q^31 + 48 * q^33 + 74 * q^37 + 136 * q^39 + 10 * q^41 - 480 * q^43 - 368 * q^45 + 572 * q^47 - 108 * q^51 - 162 * q^53 + 384 * q^55 + 92 * q^57 + 86 * q^59 + 904 * q^61 + 1088 * q^65 + 660 * q^67 + 352 * q^69 + 1024 * q^71 - 770 * q^73 + 262 * q^75 - 904 * q^79 + 421 * q^81 - 682 * q^83 - 864 * q^85 - 348 * q^87 + 102 * q^89 + 232 * q^93 + 736 * q^95 + 218 * q^97 - 552 * 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

2.00000

16.0000

−23.0000

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	392.4.a.c		1
4.b	odd	2	1		784.4.a.i		1
7.b	odd	2	1		56.4.a.a	✓	1
7.c	even	3	2		392.4.i.d		2
7.d	odd	6	2		392.4.i.e		2
21.c	even	2	1		504.4.a.g		1
28.d	even	2	1		112.4.a.d		1
35.c	odd	2	1		1400.4.a.f		1
35.f	even	4	2		1400.4.g.f		2
56.e	even	2	1		448.4.a.h		1
56.h	odd	2	1		448.4.a.l		1
84.h	odd	2	1		1008.4.a.u		1

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
56.4.a.a	✓	1	7.b	odd	2	1
112.4.a.d		1	28.d	even	2	1
392.4.a.c		1	1.a	even	1	1	trivial
392.4.i.d		2	7.c	even	3	2
392.4.i.e		2	7.d	odd	6	2
448.4.a.h		1	56.e	even	2	1
448.4.a.l		1	56.h	odd	2	1
504.4.a.g		1	21.c	even	2	1
784.4.a.i		1	4.b	odd	2	1
1008.4.a.u		1	84.h	odd	2	1
1400.4.a.f		1	35.c	odd	2	1
1400.4.g.f		2	35.f	even	4	2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{4}^{\mathrm{new}}(\Gamma_0(392))$:

$ T_{3} - 2 $

T3 - 2

$ T_{5} - 16 $

T5 - 16

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T $ T
$3$	$ T - 2 $ T - 2
$5$	$ T - 16 $ T - 16
$7$	$ T $ T
$11$	$ T - 24 $ T - 24
$13$	$ T - 68 $ T - 68
$17$	$ T + 54 $ T + 54
$19$	$ T - 46 $ T - 46
$23$	$ T - 176 $ T - 176
$29$	$ T + 174 $ T + 174
$31$	$ T - 116 $ T - 116
$37$	$ T - 74 $ T - 74
$41$	$ T - 10 $ T - 10
$43$	$ T + 480 $ T + 480
$47$	$ T - 572 $ T - 572
$53$	$ T + 162 $ T + 162
$59$	$ T - 86 $ T - 86
$61$	$ T - 904 $ T - 904
$67$	$ T - 660 $ T - 660
$71$	$ T - 1024 $ T - 1024
$73$	$ T + 770 $ T + 770
$79$	$ T + 904 $ T + 904
$83$	$ T + 682 $ T + 682
$89$	$ T - 102 $ T - 102
$97$	$ T - 218 $ T - 218
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:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	392.a (trivial)

\(n\):		e.g. 2-40 or 80-90
Significant digits:
Format:

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more