LMFDB - Group of Dirichlet characters of modulus 40

Show commands: PariGP / SageMath

sage: H = DirichletGroup(40)

pari: g = idealstar(,40,2)

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	=	$\chi_{40}(31,\cdot)$, $\chi_{40}(21,\cdot)$, $\chi_{40}(17,\cdot)$

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	$-1$	$1$	$3$	$7$	$9$	$11$	$13$	$17$	$19$	$21$	$23$	$27$
$\chi_{40}(1,\cdot)$	40.a	1	no	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
$\chi_{40}(3,\cdot)$	40.k	4	yes	$1$	$1$	$i$	$i$	$-1$	$1$	$-i$	$-i$	$-1$	$-1$	$-i$	$-i$
$\chi_{40}(7,\cdot)$	40.j	4	no	$1$	$1$	$i$	$-i$	$-1$	$-1$	$-i$	$i$	$1$	$1$	$i$	$-i$
$\chi_{40}(9,\cdot)$	40.c	2	no	$1$	$1$	$-1$	$-1$	$1$	$1$	$-1$	$-1$	$1$	$1$	$-1$	$-1$
$\chi_{40}(11,\cdot)$	40.g	2	no	$-1$	$1$	$1$	$-1$	$1$	$1$	$-1$	$1$	$1$	$-1$	$-1$	$1$
$\chi_{40}(13,\cdot)$	40.i	4	yes	$-1$	$1$	$-i$	$-i$	$-1$	$-1$	$-i$	$-i$	$1$	$-1$	$i$	$i$
$\chi_{40}(17,\cdot)$	40.l	4	no	$-1$	$1$	$-i$	$i$	$-1$	$1$	$-i$	$i$	$-1$	$1$	$-i$	$i$
$\chi_{40}(19,\cdot)$	40.e	2	yes	$-1$	$1$	$-1$	$1$	$1$	$1$	$1$	$-1$	$1$	$-1$	$1$	$-1$
$\chi_{40}(21,\cdot)$	40.d	2	no	$1$	$1$	$-1$	$1$	$1$	$-1$	$-1$	$1$	$-1$	$-1$	$1$	$-1$
$\chi_{40}(23,\cdot)$	40.j	4	no	$1$	$1$	$-i$	$i$	$-1$	$-1$	$i$	$-i$	$1$	$1$	$-i$	$i$
$\chi_{40}(27,\cdot)$	40.k	4	yes	$1$	$1$	$-i$	$-i$	$-1$	$1$	$i$	$i$	$-1$	$-1$	$i$	$i$
$\chi_{40}(29,\cdot)$	40.f	2	yes	$1$	$1$	$1$	$-1$	$1$	$-1$	$1$	$-1$	$-1$	$-1$	$-1$	$1$
$\chi_{40}(31,\cdot)$	40.b	2	no	$-1$	$1$	$-1$	$-1$	$1$	$-1$	$1$	$1$	$-1$	$1$	$-1$	$-1$
$\chi_{40}(33,\cdot)$	40.l	4	no	$-1$	$1$	$i$	$-i$	$-1$	$1$	$i$	$-i$	$-1$	$1$	$i$	$-i$
$\chi_{40}(37,\cdot)$	40.i	4	yes	$-1$	$1$	$i$	$i$	$-1$	$-1$	$i$	$i$	$1$	$-1$	$-i$	$-i$
$\chi_{40}(39,\cdot)$	40.h	2	no	$-1$	$1$	$1$	$1$	$1$	$-1$	$-1$	$-1$	$-1$	$1$	$1$	$1$

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	$40$
Structure	\(C_{2}\times C_{2}\times C_{4}\)
Order	$16$

Character	Orbit	Order	Primitive	\(-1\)	\(1\)	\(3\)	\(7\)	\(9\)	\(11\)	\(13\)	\(17\)	\(19\)	\(21\)	\(23\)	\(27\)
\(\chi_{40}(1,\cdot)\)	40.a	1	no	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)
\(\chi_{40}(3,\cdot)\)	40.k	4	yes	\(1\)	\(1\)	\(i\)	\(i\)	\(-1\)	\(1\)	\(-i\)	\(-i\)	\(-1\)	\(-1\)	\(-i\)	\(-i\)
\(\chi_{40}(7,\cdot)\)	40.j	4	no	\(1\)	\(1\)	\(i\)	\(-i\)	\(-1\)	\(-1\)	\(-i\)	\(i\)	\(1\)	\(1\)	\(i\)	\(-i\)
\(\chi_{40}(9,\cdot)\)	40.c	2	no	\(1\)	\(1\)	\(-1\)	\(-1\)	\(1\)	\(1\)	\(-1\)	\(-1\)	\(1\)	\(1\)	\(-1\)	\(-1\)
\(\chi_{40}(11,\cdot)\)	40.g	2	no	\(-1\)	\(1\)	\(1\)	\(-1\)	\(1\)	\(1\)	\(-1\)	\(1\)	\(1\)	\(-1\)	\(-1\)	\(1\)
\(\chi_{40}(13,\cdot)\)	40.i	4	yes	\(-1\)	\(1\)	\(-i\)	\(-i\)	\(-1\)	\(-1\)	\(-i\)	\(-i\)	\(1\)	\(-1\)	\(i\)	\(i\)
\(\chi_{40}(17,\cdot)\)	40.l	4	no	\(-1\)	\(1\)	\(-i\)	\(i\)	\(-1\)	\(1\)	\(-i\)	\(i\)	\(-1\)	\(1\)	\(-i\)	\(i\)
\(\chi_{40}(19,\cdot)\)	40.e	2	yes	\(-1\)	\(1\)	\(-1\)	\(1\)	\(1\)	\(1\)	\(1\)	\(-1\)	\(1\)	\(-1\)	\(1\)	\(-1\)
\(\chi_{40}(21,\cdot)\)	40.d	2	no	\(1\)	\(1\)	\(-1\)	\(1\)	\(1\)	\(-1\)	\(-1\)	\(1\)	\(-1\)	\(-1\)	\(1\)	\(-1\)
\(\chi_{40}(23,\cdot)\)	40.j	4	no	\(1\)	\(1\)	\(-i\)	\(i\)	\(-1\)	\(-1\)	\(i\)	\(-i\)	\(1\)	\(1\)	\(-i\)	\(i\)
\(\chi_{40}(27,\cdot)\)	40.k	4	yes	\(1\)	\(1\)	\(-i\)	\(-i\)	\(-1\)	\(1\)	\(i\)	\(i\)	\(-1\)	\(-1\)	\(i\)	\(i\)
\(\chi_{40}(29,\cdot)\)	40.f	2	yes	\(1\)	\(1\)	\(1\)	\(-1\)	\(1\)	\(-1\)	\(1\)	\(-1\)	\(-1\)	\(-1\)	\(-1\)	\(1\)
\(\chi_{40}(31,\cdot)\)	40.b	2	no	\(-1\)	\(1\)	\(-1\)	\(-1\)	\(1\)	\(-1\)	\(1\)	\(1\)	\(-1\)	\(1\)	\(-1\)	\(-1\)
\(\chi_{40}(33,\cdot)\)	40.l	4	no	\(-1\)	\(1\)	\(i\)	\(-i\)	\(-1\)	\(1\)	\(i\)	\(-i\)	\(-1\)	\(1\)	\(i\)	\(-i\)
\(\chi_{40}(37,\cdot)\)	40.i	4	yes	\(-1\)	\(1\)	\(i\)	\(i\)	\(-1\)	\(-1\)	\(i\)	\(i\)	\(1\)	\(-1\)	\(-i\)	\(-i\)
\(\chi_{40}(39,\cdot)\)	40.h	2	no	\(-1\)	\(1\)	\(1\)	\(1\)	\(1\)	\(-1\)	\(-1\)	\(-1\)	\(-1\)	\(1\)	\(1\)	\(1\)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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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.