LMFDB - Newform orbit 99.9.c.a

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [99,9,Mod(10,99)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(99, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1]))

N = Newforms(chi, 9, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("99.10");

S:= CuspForms(chi, 9);

N := Newforms(S);

Level:	$ N $	$=$	$ 99 = 3^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 9 $
Character orbit:	$[\chi]$	$=$	99.c (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	yes
Analytic conductor:	$40.3304823961$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 11)
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$

$=$

$ q + 256 q^{4} - 1151 q^{5}+O(q^{10}) $

$ q + 256 q^{4} - 1151 q^{5} - 14641 q^{11} + 65536 q^{16} - 294656 q^{20} + 531793 q^{23} + 934176 q^{25} - 1541233 q^{31} + 716447 q^{37} - 3748096 q^{44} + 6080638 q^{47} + 5764801 q^{49} + 15265438 q^{53} + 16851791 q^{55} + 4101553 q^{59} + 16777216 q^{64} + 19806767 q^{67} - 7043087 q^{71} - 75431936 q^{80} + 84100993 q^{89} + 136139008 q^{92} - 81155713 q^{97}+O(q^{100}) $

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/99\mathbb{Z}\right)^\times$.

$n$	$46$	$56$
$\chi(n)$	$-1$	$1$

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} $

10.1

256.000

−1151.00

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
11.b	odd	2	1	CM by $\Q(\sqrt{-11}) $

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	99.9.c.a		1
3.b	odd	2	1		11.9.b.a	✓	1
11.b	odd	2	1	CM	99.9.c.a		1
12.b	even	2	1		176.9.h.a		1
33.d	even	2	1		11.9.b.a	✓	1
132.d	odd	2	1		176.9.h.a		1

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
11.9.b.a	✓	1	3.b	odd	2	1
11.9.b.a	✓	1	33.d	even	2	1
99.9.c.a		1	1.a	even	1	1	trivial
99.9.c.a		1	11.b	odd	2	1	CM
176.9.h.a		1	12.b	even	2	1
176.9.h.a		1	132.d	odd	2	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $ T_{2} $ T2 acting on $S_{9}^{\mathrm{new}}(99, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T $ T
$3$	$ T $ T
$5$	$ T + 1151 $ T + 1151
$7$	$ T $ T
$11$	$ T + 14641 $ T + 14641
$13$	$ T $ T
$17$	$ T $ T
$19$	$ T $ T
$23$	$ T - 531793 $ T - 531793
$29$	$ T $ T
$31$	$ T + 1541233 $ T + 1541233
$37$	$ T - 716447 $ T - 716447
$41$	$ T $ T
$43$	$ T $ T
$47$	$ T - 6080638 $ T - 6080638
$53$	$ T - 15265438 $ T - 15265438
$59$	$ T - 4101553 $ T - 4101553
$61$	$ T $ T
$67$	$ T - 19806767 $ T - 19806767
$71$	$ T + 7043087 $ T + 7043087
$73$	$ T $ T
$79$	$ T $ T
$83$	$ T $ T
$89$	$ T - 84100993 $ T - 84100993
$97$	$ T + 81155713 $ T + 81155713
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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 99 = 3^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 9 \)
Character orbit:	\([\chi]\)	\(=\)	99.c (of order \(2\), degree \(1\), minimal)

\(n\)	\(46\)	\(56\)
\(\chi(n)\)	\(-1\)	\(1\)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more