LMFDB - Dirichlet character \(\chi

Show commands: PariGP / SageMath

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(560, base_ring=CyclotomicField(2))

M = H._module

chi = DirichletCharacter(H, M([0,0,1,1]))

pari: [g,chi] = znchar(Mod(209,560))

Basic properties

Modulus:	$560$
Conductor:	$35$	sage: chi.conductor() pari: znconreyconductor(g,chi)
Order:	$2$	sage: chi.multiplicative_order() pari: charorder(g,chi)
Real:	yes
Primitive:	no, induced from $\chi_{35}(34,\cdot)$	sage: chi.is_primitive() pari: #znconreyconductor(g,chi)==1
Minimal:	no
Parity:	odd	sage: chi.is_odd() pari: zncharisodd(g,chi)

Galois orbit 560.p

$\chi_{560}(209,\cdot)$

sage: chi.galois_orbit()

order = charorder(g,chi)

[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]

Related number fields

Field of values:	$\Q$
Fixed field:	$\Q(\sqrt{-35})$

Values on generators

$(351,421,337,241)$ → $(1,1,-1,-1)$

First values

$a$	$-1$	$1$	$3$	$9$	$11$	$13$	$17$	$19$	$23$	$27$	$29$	$31$
$\chi_{ 560 }(209, a)$	$-1$	$1$	$1$	$1$	$1$	$1$	$1$	$-1$	$-1$	$1$	$1$	$-1$

sage: chi.jacobi_sum(n)

Gauss sum

sage: chi.gauss_sum(a)

pari: znchargauss(g,chi,a)

Jacobi sum

sage: chi.jacobi_sum(n)

Kloosterman sum

sage: chi.kloosterman_sum(a,b)

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 $\F_{q}$

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more