LMFDB - Newform orbit 200.8.a.a

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [200,8,Mod(1,200)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(200, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 8, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("200.1"); S:= CuspForms(chi, 8); N := Newforms(S);

Level:	$ N $	$=$	$ 200 = 2^{3} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	200.a (trivial)

Newform invariants

comment:select newform

sage:traces = [1,0,-69,0,0,0,174] 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:	$62.4770050968$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
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 - 69 q^{3} + 174 q^{7} + 2574 q^{9} + 7111 q^{11} - 468 q^{13} - 9555 q^{17} - 42601 q^{19} - 12006 q^{21} - 77526 q^{23} - 26703 q^{27} - 61312 q^{29} + 251710 q^{31} - 490659 q^{33} - 83462 q^{37}+ \cdots + 18303714 q^{99}+O(q^{100}) $

q - 69 * q^3 + 174 * q^7 + 2574 * q^9 + 7111 * q^11 - 468 * q^13 - 9555 * q^17 - 42601 * q^19 - 12006 * q^21 - 77526 * q^23 - 26703 * q^27 - 61312 * q^29 + 251710 * q^31 - 490659 * q^33 - 83462 * q^37 + 32292 * q^39 + 363477 * q^41 - 34188 * q^43 - 708812 * q^47 - 793267 * q^49 + 659295 * q^51 - 891762 * q^53 + 2939469 * q^57 + 2809152 * q^59 - 3211510 * q^61 + 447876 * q^63 + 1372033 * q^67 + 5349294 * q^69 + 4508308 * q^71 + 628179 * q^73 + 1237314 * q^77 + 6130474 * q^79 - 3786831 * q^81 + 9921981 * q^83 + 4230528 * q^87 + 1806599 * q^89 - 81432 * q^91 - 17367990 * q^93 + 11676482 * q^97 + 18303714 * 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

−69.0000

174.000

2574.00

Atkin-Lehner signs

$ p $	Sign
$2$	$ -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	200.8.a.a	✓	1
4.b	odd	2	1		400.8.a.r		1
5.b	even	2	1		200.8.a.h	yes	1
5.c	odd	4	2		200.8.c.b		2
20.d	odd	2	1		400.8.a.c		1
20.e	even	4	2		400.8.c.c		2

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
200.8.a.a	✓	1	1.a	even	1	1	trivial
200.8.a.h	yes	1	5.b	even	2	1
200.8.c.b		2	5.c	odd	4	2
400.8.a.c		1	20.d	odd	2	1
400.8.a.r		1	4.b	odd	2	1
400.8.c.c		2	20.e	even	4	2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $ T_{3} + 69 $ T3 + 69 acting on $S_{8}^{\mathrm{new}}(\Gamma_0(200))$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T $ T
$3$	$ T + 69 $ T + 69
$5$	$ T $ T
$7$	$ T - 174 $ T - 174
$11$	$ T - 7111 $ T - 7111
$13$	$ T + 468 $ T + 468
$17$	$ T + 9555 $ T + 9555
$19$	$ T + 42601 $ T + 42601
$23$	$ T + 77526 $ T + 77526
$29$	$ T + 61312 $ T + 61312
$31$	$ T - 251710 $ T - 251710
$37$	$ T + 83462 $ T + 83462
$41$	$ T - 363477 $ T - 363477
$43$	$ T + 34188 $ T + 34188
$47$	$ T + 708812 $ T + 708812
$53$	$ T + 891762 $ T + 891762
$59$	$ T - 2809152 $ T - 2809152
$61$	$ T + 3211510 $ T + 3211510
$67$	$ T - 1372033 $ T - 1372033
$71$	$ T - 4508308 $ T - 4508308
$73$	$ T - 628179 $ T - 628179
$79$	$ T - 6130474 $ T - 6130474
$83$	$ T - 9921981 $ T - 9921981
$89$	$ T - 1806599 $ T - 1806599
$97$	$ T - 11676482 $ T - 11676482
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:	\( N \)	\(=\)	\( 200 = 2^{3} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	200.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