LMFDB - Group of Dirichlet characters of modulus 60

Show commands: PariGP / SageMath

sage: H = DirichletGroup(60)

pari: g = idealstar(,60,2)

Character group

sage: G.order() pari: g.no
Order	=	16
sage: H.invariants() pari: g.cyc
Structure	=	$C_{2}\times C_{2}\times C_{4}$
sage: H.gens() pari: g.gen
Generators	=	$\chi_{60}(31,\cdot)$, $\chi_{60}(41,\cdot)$, $\chi_{60}(37,\cdot)$

Characters

Each row describes a character. When available, the columns show the orbit label, order of the character, whether the character is primitive, and several values of the character.

Character	Orbit	Order	Primitive	$-1$	$1$	$7$	$11$	$13$	$17$	$19$	$23$	$29$	$31$	$37$	$41$
$\chi_{60}(1,\cdot)$	60.a	1	no	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
$\chi_{60}(7,\cdot)$	60.j	4	no	$1$	$1$	$-i$	$-1$	$-i$	$i$	$1$	$i$	$-1$	$-1$	$i$	$1$
$\chi_{60}(11,\cdot)$	60.e	2	no	$1$	$1$	$-1$	$1$	$1$	$-1$	$-1$	$1$	$-1$	$-1$	$1$	$-1$
$\chi_{60}(13,\cdot)$	60.k	4	no	$-1$	$1$	$-i$	$1$	$i$	$-i$	$-1$	$i$	$-1$	$1$	$-i$	$1$
$\chi_{60}(17,\cdot)$	60.i	4	no	$1$	$1$	$i$	$-1$	$-i$	$-i$	$-1$	$i$	$1$	$1$	$i$	$-1$
$\chi_{60}(19,\cdot)$	60.f	2	no	$-1$	$1$	$1$	$-1$	$-1$	$-1$	$-1$	$1$	$1$	$-1$	$-1$	$1$
$\chi_{60}(23,\cdot)$	60.l	4	yes	$-1$	$1$	$i$	$1$	$i$	$i$	$1$	$i$	$1$	$-1$	$-i$	$-1$
$\chi_{60}(29,\cdot)$	60.b	2	no	$-1$	$1$	$-1$	$-1$	$-1$	$1$	$1$	$1$	$-1$	$1$	$-1$	$-1$
$\chi_{60}(31,\cdot)$	60.c	2	no	$-1$	$1$	$-1$	$-1$	$1$	$1$	$-1$	$-1$	$1$	$-1$	$1$	$1$
$\chi_{60}(37,\cdot)$	60.k	4	no	$-1$	$1$	$i$	$1$	$-i$	$i$	$-1$	$-i$	$-1$	$1$	$i$	$1$
$\chi_{60}(41,\cdot)$	60.g	2	no	$-1$	$1$	$1$	$-1$	$1$	$-1$	$1$	$-1$	$-1$	$1$	$1$	$-1$
$\chi_{60}(43,\cdot)$	60.j	4	no	$1$	$1$	$i$	$-1$	$i$	$-i$	$1$	$-i$	$-1$	$-1$	$-i$	$1$
$\chi_{60}(47,\cdot)$	60.l	4	yes	$-1$	$1$	$-i$	$1$	$-i$	$-i$	$1$	$-i$	$1$	$-1$	$i$	$-1$
$\chi_{60}(49,\cdot)$	60.d	2	no	$1$	$1$	$-1$	$1$	$-1$	$-1$	$1$	$-1$	$1$	$1$	$-1$	$1$
$\chi_{60}(53,\cdot)$	60.i	4	no	$1$	$1$	$-i$	$-1$	$i$	$i$	$-1$	$-i$	$1$	$1$	$-i$	$-1$
$\chi_{60}(59,\cdot)$	60.h	2	yes	$1$	$1$	$1$	$1$	$-1$	$1$	$-1$	$-1$	$-1$	$-1$	$-1$	$-1$

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 $\F_{q}$

Modulus	$60$
Structure	\(C_{2}\times C_{2}\times C_{4}\)
Order	$16$

Character	Orbit	Order	Primitive	\(-1\)	\(1\)	\(7\)	\(11\)	\(13\)	\(17\)	\(19\)	\(23\)	\(29\)	\(31\)	\(37\)	\(41\)
\(\chi_{60}(1,\cdot)\)	60.a	1	no	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)
\(\chi_{60}(7,\cdot)\)	60.j	4	no	\(1\)	\(1\)	\(-i\)	\(-1\)	\(-i\)	\(i\)	\(1\)	\(i\)	\(-1\)	\(-1\)	\(i\)	\(1\)
\(\chi_{60}(11,\cdot)\)	60.e	2	no	\(1\)	\(1\)	\(-1\)	\(1\)	\(1\)	\(-1\)	\(-1\)	\(1\)	\(-1\)	\(-1\)	\(1\)	\(-1\)
\(\chi_{60}(13,\cdot)\)	60.k	4	no	\(-1\)	\(1\)	\(-i\)	\(1\)	\(i\)	\(-i\)	\(-1\)	\(i\)	\(-1\)	\(1\)	\(-i\)	\(1\)
\(\chi_{60}(17,\cdot)\)	60.i	4	no	\(1\)	\(1\)	\(i\)	\(-1\)	\(-i\)	\(-i\)	\(-1\)	\(i\)	\(1\)	\(1\)	\(i\)	\(-1\)
\(\chi_{60}(19,\cdot)\)	60.f	2	no	\(-1\)	\(1\)	\(1\)	\(-1\)	\(-1\)	\(-1\)	\(-1\)	\(1\)	\(1\)	\(-1\)	\(-1\)	\(1\)
\(\chi_{60}(23,\cdot)\)	60.l	4	yes	\(-1\)	\(1\)	\(i\)	\(1\)	\(i\)	\(i\)	\(1\)	\(i\)	\(1\)	\(-1\)	\(-i\)	\(-1\)
\(\chi_{60}(29,\cdot)\)	60.b	2	no	\(-1\)	\(1\)	\(-1\)	\(-1\)	\(-1\)	\(1\)	\(1\)	\(1\)	\(-1\)	\(1\)	\(-1\)	\(-1\)
\(\chi_{60}(31,\cdot)\)	60.c	2	no	\(-1\)	\(1\)	\(-1\)	\(-1\)	\(1\)	\(1\)	\(-1\)	\(-1\)	\(1\)	\(-1\)	\(1\)	\(1\)
\(\chi_{60}(37,\cdot)\)	60.k	4	no	\(-1\)	\(1\)	\(i\)	\(1\)	\(-i\)	\(i\)	\(-1\)	\(-i\)	\(-1\)	\(1\)	\(i\)	\(1\)
\(\chi_{60}(41,\cdot)\)	60.g	2	no	\(-1\)	\(1\)	\(1\)	\(-1\)	\(1\)	\(-1\)	\(1\)	\(-1\)	\(-1\)	\(1\)	\(1\)	\(-1\)
\(\chi_{60}(43,\cdot)\)	60.j	4	no	\(1\)	\(1\)	\(i\)	\(-1\)	\(i\)	\(-i\)	\(1\)	\(-i\)	\(-1\)	\(-1\)	\(-i\)	\(1\)
\(\chi_{60}(47,\cdot)\)	60.l	4	yes	\(-1\)	\(1\)	\(-i\)	\(1\)	\(-i\)	\(-i\)	\(1\)	\(-i\)	\(1\)	\(-1\)	\(i\)	\(-1\)
\(\chi_{60}(49,\cdot)\)	60.d	2	no	\(1\)	\(1\)	\(-1\)	\(1\)	\(-1\)	\(-1\)	\(1\)	\(-1\)	\(1\)	\(1\)	\(-1\)	\(1\)
\(\chi_{60}(53,\cdot)\)	60.i	4	no	\(1\)	\(1\)	\(-i\)	\(-1\)	\(i\)	\(i\)	\(-1\)	\(-i\)	\(1\)	\(1\)	\(-i\)	\(-1\)
\(\chi_{60}(59,\cdot)\)	60.h	2	yes	\(1\)	\(1\)	\(1\)	\(1\)	\(-1\)	\(1\)	\(-1\)	\(-1\)	\(-1\)	\(-1\)	\(-1\)	\(-1\)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Learn more

Character group

Characters

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.