LMFDB - Group of Dirichlet characters of modulus 160

Show commands: Pari/GP / SageMath

sage: H = DirichletGroup(160)

pari: g = idealstar(,160,2)

Character group

sage: G.order() pari: g.no
Order	=	64
sage: H.invariants() pari: g.cyc
Structure	=	$C_{8}\times C_{4}\times C_{2}$
sage: H.gens() pari: g.gen
Generators	=	$\chi_{160}(31,\cdot)$, $\chi_{160}(101,\cdot)$, $\chi_{160}(97,\cdot)$

First 32 of 64 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	$-1$	$1$	$3$	$7$	$9$	$11$	$13$	$17$	$19$	$21$	$23$	$27$
$\chi_{160}(1,\cdot)$	160.a	1	no	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
$\chi_{160}(3,\cdot)$	160.ba	8	yes	$1$	$1$	$e\left(\frac{7}{8}\right)$	$1$	$-i$	$e\left(\frac{3}{8}\right)$	$e\left(\frac{7}{8}\right)$	$i$	$e\left(\frac{5}{8}\right)$	$e\left(\frac{7}{8}\right)$	$1$	$e\left(\frac{5}{8}\right)$
$\chi_{160}(7,\cdot)$	160.s	4	no	$1$	$1$	$1$	$i$	$1$	$-i$	$-1$	$i$	$-i$	$i$	$-i$	$1$
$\chi_{160}(9,\cdot)$	160.q	4	no	$1$	$1$	$-i$	$1$	$-1$	$-i$	$-i$	$-1$	$i$	$-i$	$1$	$i$
$\chi_{160}(11,\cdot)$	160.w	8	no	$-1$	$1$	$e\left(\frac{3}{8}\right)$	$-i$	$-i$	$e\left(\frac{5}{8}\right)$	$e\left(\frac{3}{8}\right)$	$-1$	$e\left(\frac{7}{8}\right)$	$e\left(\frac{1}{8}\right)$	$i$	$e\left(\frac{1}{8}\right)$
$\chi_{160}(13,\cdot)$	160.v	8	yes	$-1$	$1$	$e\left(\frac{7}{8}\right)$	$-1$	$-i$	$e\left(\frac{3}{8}\right)$	$e\left(\frac{3}{8}\right)$	$i$	$e\left(\frac{5}{8}\right)$	$e\left(\frac{3}{8}\right)$	$-1$	$e\left(\frac{5}{8}\right)$
$\chi_{160}(17,\cdot)$	160.m	4	no	$-1$	$1$	$i$	$i$	$-1$	$-1$	$i$	$i$	$1$	$-1$	$-i$	$-i$
$\chi_{160}(19,\cdot)$	160.y	8	yes	$-1$	$1$	$e\left(\frac{5}{8}\right)$	$-i$	$i$	$e\left(\frac{7}{8}\right)$	$e\left(\frac{5}{8}\right)$	$1$	$e\left(\frac{5}{8}\right)$	$e\left(\frac{3}{8}\right)$	$i$	$e\left(\frac{7}{8}\right)$
$\chi_{160}(21,\cdot)$	160.x	8	no	$1$	$1$	$e\left(\frac{7}{8}\right)$	$i$	$-i$	$e\left(\frac{1}{8}\right)$	$e\left(\frac{3}{8}\right)$	$-1$	$e\left(\frac{3}{8}\right)$	$e\left(\frac{1}{8}\right)$	$-i$	$e\left(\frac{5}{8}\right)$
$\chi_{160}(23,\cdot)$	160.s	4	no	$1$	$1$	$1$	$-i$	$1$	$i$	$-1$	$-i$	$i$	$-i$	$i$	$1$
$\chi_{160}(27,\cdot)$	160.ba	8	yes	$1$	$1$	$e\left(\frac{5}{8}\right)$	$1$	$i$	$e\left(\frac{1}{8}\right)$	$e\left(\frac{5}{8}\right)$	$-i$	$e\left(\frac{7}{8}\right)$	$e\left(\frac{5}{8}\right)$	$1$	$e\left(\frac{7}{8}\right)$
$\chi_{160}(29,\cdot)$	160.z	8	yes	$1$	$1$	$e\left(\frac{5}{8}\right)$	$i$	$i$	$e\left(\frac{7}{8}\right)$	$e\left(\frac{1}{8}\right)$	$1$	$e\left(\frac{5}{8}\right)$	$e\left(\frac{7}{8}\right)$	$-i$	$e\left(\frac{7}{8}\right)$
$\chi_{160}(31,\cdot)$	160.b	2	no	$-1$	$1$	$-1$	$-1$	$1$	$-1$	$1$	$1$	$-1$	$1$	$-1$	$-1$
$\chi_{160}(33,\cdot)$	160.p	4	no	$-1$	$1$	$i$	$-i$	$-1$	$1$	$i$	$-i$	$-1$	$1$	$i$	$-i$
$\chi_{160}(37,\cdot)$	160.v	8	yes	$-1$	$1$	$e\left(\frac{1}{8}\right)$	$-1$	$i$	$e\left(\frac{5}{8}\right)$	$e\left(\frac{5}{8}\right)$	$-i$	$e\left(\frac{3}{8}\right)$	$e\left(\frac{5}{8}\right)$	$-1$	$e\left(\frac{3}{8}\right)$
$\chi_{160}(39,\cdot)$	160.k	4	no	$-1$	$1$	$-i$	$-1$	$-1$	$-i$	$i$	$-1$	$i$	$i$	$-1$	$i$
$\chi_{160}(41,\cdot)$	160.l	4	no	$1$	$1$	$i$	$-1$	$-1$	$-i$	$i$	$1$	$i$	$-i$	$-1$	$-i$
$\chi_{160}(43,\cdot)$	160.u	8	yes	$1$	$1$	$e\left(\frac{5}{8}\right)$	$-1$	$i$	$e\left(\frac{5}{8}\right)$	$e\left(\frac{5}{8}\right)$	$i$	$e\left(\frac{3}{8}\right)$	$e\left(\frac{1}{8}\right)$	$-1$	$e\left(\frac{7}{8}\right)$
$\chi_{160}(47,\cdot)$	160.o	4	no	$1$	$1$	$-i$	$-i$	$-1$	$1$	$i$	$i$	$-1$	$-1$	$i$	$i$
$\chi_{160}(49,\cdot)$	160.f	2	no	$1$	$1$	$1$	$-1$	$1$	$-1$	$1$	$-1$	$-1$	$-1$	$-1$	$1$
$\chi_{160}(51,\cdot)$	160.w	8	no	$-1$	$1$	$e\left(\frac{1}{8}\right)$	$i$	$i$	$e\left(\frac{7}{8}\right)$	$e\left(\frac{1}{8}\right)$	$-1$	$e\left(\frac{5}{8}\right)$	$e\left(\frac{3}{8}\right)$	$-i$	$e\left(\frac{3}{8}\right)$
$\chi_{160}(53,\cdot)$	160.bb	8	yes	$-1$	$1$	$e\left(\frac{1}{8}\right)$	$1$	$i$	$e\left(\frac{1}{8}\right)$	$e\left(\frac{5}{8}\right)$	$i$	$e\left(\frac{7}{8}\right)$	$e\left(\frac{1}{8}\right)$	$1$	$e\left(\frac{3}{8}\right)$
$\chi_{160}(57,\cdot)$	160.i	4	no	$-1$	$1$	$-1$	$-i$	$1$	$i$	$-1$	$i$	$i$	$i$	$i$	$-1$
$\chi_{160}(59,\cdot)$	160.y	8	yes	$-1$	$1$	$e\left(\frac{3}{8}\right)$	$i$	$-i$	$e\left(\frac{1}{8}\right)$	$e\left(\frac{3}{8}\right)$	$1$	$e\left(\frac{3}{8}\right)$	$e\left(\frac{5}{8}\right)$	$-i$	$e\left(\frac{1}{8}\right)$
$\chi_{160}(61,\cdot)$	160.x	8	no	$1$	$1$	$e\left(\frac{1}{8}\right)$	$-i$	$i$	$e\left(\frac{7}{8}\right)$	$e\left(\frac{5}{8}\right)$	$-1$	$e\left(\frac{5}{8}\right)$	$e\left(\frac{7}{8}\right)$	$i$	$e\left(\frac{3}{8}\right)$
$\chi_{160}(63,\cdot)$	160.n	4	no	$1$	$1$	$-i$	$i$	$-1$	$-1$	$i$	$-i$	$1$	$1$	$-i$	$i$
$\chi_{160}(67,\cdot)$	160.u	8	yes	$1$	$1$	$e\left(\frac{3}{8}\right)$	$-1$	$-i$	$e\left(\frac{3}{8}\right)$	$e\left(\frac{3}{8}\right)$	$-i$	$e\left(\frac{5}{8}\right)$	$e\left(\frac{7}{8}\right)$	$-1$	$e\left(\frac{1}{8}\right)$
$\chi_{160}(69,\cdot)$	160.z	8	yes	$1$	$1$	$e\left(\frac{7}{8}\right)$	$-i$	$-i$	$e\left(\frac{5}{8}\right)$	$e\left(\frac{3}{8}\right)$	$1$	$e\left(\frac{7}{8}\right)$	$e\left(\frac{5}{8}\right)$	$i$	$e\left(\frac{5}{8}\right)$
$\chi_{160}(71,\cdot)$	160.r	4	no	$-1$	$1$	$i$	$1$	$-1$	$-i$	$-i$	$1$	$i$	$i$	$1$	$-i$
$\chi_{160}(73,\cdot)$	160.i	4	no	$-1$	$1$	$-1$	$i$	$1$	$-i$	$-1$	$-i$	$-i$	$-i$	$-i$	$-1$
$\chi_{160}(77,\cdot)$	160.bb	8	yes	$-1$	$1$	$e\left(\frac{3}{8}\right)$	$1$	$-i$	$e\left(\frac{3}{8}\right)$	$e\left(\frac{7}{8}\right)$	$-i$	$e\left(\frac{5}{8}\right)$	$e\left(\frac{3}{8}\right)$	$1$	$e\left(\frac{1}{8}\right)$
$\chi_{160}(79,\cdot)$	160.e	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	$160$
Structure	\(C_{8}\times C_{4}\times C_{2}\)
Order	$64$

Character	Orbit	Order	Primitive	\(-1\)	\(1\)	\(3\)	\(7\)	\(9\)	\(11\)	\(13\)	\(17\)	\(19\)	\(21\)	\(23\)	\(27\)
\(\chi_{160}(1,\cdot)\)	160.a	1	no	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)
\(\chi_{160}(3,\cdot)\)	160.ba	8	yes	\(1\)	\(1\)	\(e\left(\frac{7}{8}\right)\)	\(1\)	\(-i\)	\(e\left(\frac{3}{8}\right)\)	\(e\left(\frac{7}{8}\right)\)	\(i\)	\(e\left(\frac{5}{8}\right)\)	\(e\left(\frac{7}{8}\right)\)	\(1\)	\(e\left(\frac{5}{8}\right)\)
\(\chi_{160}(7,\cdot)\)	160.s	4	no	\(1\)	\(1\)	\(1\)	\(i\)	\(1\)	\(-i\)	\(-1\)	\(i\)	\(-i\)	\(i\)	\(-i\)	\(1\)
\(\chi_{160}(9,\cdot)\)	160.q	4	no	\(1\)	\(1\)	\(-i\)	\(1\)	\(-1\)	\(-i\)	\(-i\)	\(-1\)	\(i\)	\(-i\)	\(1\)	\(i\)
\(\chi_{160}(11,\cdot)\)	160.w	8	no	\(-1\)	\(1\)	\(e\left(\frac{3}{8}\right)\)	\(-i\)	\(-i\)	\(e\left(\frac{5}{8}\right)\)	\(e\left(\frac{3}{8}\right)\)	\(-1\)	\(e\left(\frac{7}{8}\right)\)	\(e\left(\frac{1}{8}\right)\)	\(i\)	\(e\left(\frac{1}{8}\right)\)
\(\chi_{160}(13,\cdot)\)	160.v	8	yes	\(-1\)	\(1\)	\(e\left(\frac{7}{8}\right)\)	\(-1\)	\(-i\)	\(e\left(\frac{3}{8}\right)\)	\(e\left(\frac{3}{8}\right)\)	\(i\)	\(e\left(\frac{5}{8}\right)\)	\(e\left(\frac{3}{8}\right)\)	\(-1\)	\(e\left(\frac{5}{8}\right)\)
\(\chi_{160}(17,\cdot)\)	160.m	4	no	\(-1\)	\(1\)	\(i\)	\(i\)	\(-1\)	\(-1\)	\(i\)	\(i\)	\(1\)	\(-1\)	\(-i\)	\(-i\)
\(\chi_{160}(19,\cdot)\)	160.y	8	yes	\(-1\)	\(1\)	\(e\left(\frac{5}{8}\right)\)	\(-i\)	\(i\)	\(e\left(\frac{7}{8}\right)\)	\(e\left(\frac{5}{8}\right)\)	\(1\)	\(e\left(\frac{5}{8}\right)\)	\(e\left(\frac{3}{8}\right)\)	\(i\)	\(e\left(\frac{7}{8}\right)\)
\(\chi_{160}(21,\cdot)\)	160.x	8	no	\(1\)	\(1\)	\(e\left(\frac{7}{8}\right)\)	\(i\)	\(-i\)	\(e\left(\frac{1}{8}\right)\)	\(e\left(\frac{3}{8}\right)\)	\(-1\)	\(e\left(\frac{3}{8}\right)\)	\(e\left(\frac{1}{8}\right)\)	\(-i\)	\(e\left(\frac{5}{8}\right)\)
\(\chi_{160}(23,\cdot)\)	160.s	4	no	\(1\)	\(1\)	\(1\)	\(-i\)	\(1\)	\(i\)	\(-1\)	\(-i\)	\(i\)	\(-i\)	\(i\)	\(1\)
\(\chi_{160}(27,\cdot)\)	160.ba	8	yes	\(1\)	\(1\)	\(e\left(\frac{5}{8}\right)\)	\(1\)	\(i\)	\(e\left(\frac{1}{8}\right)\)	\(e\left(\frac{5}{8}\right)\)	\(-i\)	\(e\left(\frac{7}{8}\right)\)	\(e\left(\frac{5}{8}\right)\)	\(1\)	\(e\left(\frac{7}{8}\right)\)
\(\chi_{160}(29,\cdot)\)	160.z	8	yes	\(1\)	\(1\)	\(e\left(\frac{5}{8}\right)\)	\(i\)	\(i\)	\(e\left(\frac{7}{8}\right)\)	\(e\left(\frac{1}{8}\right)\)	\(1\)	\(e\left(\frac{5}{8}\right)\)	\(e\left(\frac{7}{8}\right)\)	\(-i\)	\(e\left(\frac{7}{8}\right)\)
\(\chi_{160}(31,\cdot)\)	160.b	2	no	\(-1\)	\(1\)	\(-1\)	\(-1\)	\(1\)	\(-1\)	\(1\)	\(1\)	\(-1\)	\(1\)	\(-1\)	\(-1\)
\(\chi_{160}(33,\cdot)\)	160.p	4	no	\(-1\)	\(1\)	\(i\)	\(-i\)	\(-1\)	\(1\)	\(i\)	\(-i\)	\(-1\)	\(1\)	\(i\)	\(-i\)
\(\chi_{160}(37,\cdot)\)	160.v	8	yes	\(-1\)	\(1\)	\(e\left(\frac{1}{8}\right)\)	\(-1\)	\(i\)	\(e\left(\frac{5}{8}\right)\)	\(e\left(\frac{5}{8}\right)\)	\(-i\)	\(e\left(\frac{3}{8}\right)\)	\(e\left(\frac{5}{8}\right)\)	\(-1\)	\(e\left(\frac{3}{8}\right)\)
\(\chi_{160}(39,\cdot)\)	160.k	4	no	\(-1\)	\(1\)	\(-i\)	\(-1\)	\(-1\)	\(-i\)	\(i\)	\(-1\)	\(i\)	\(i\)	\(-1\)	\(i\)
\(\chi_{160}(41,\cdot)\)	160.l	4	no	\(1\)	\(1\)	\(i\)	\(-1\)	\(-1\)	\(-i\)	\(i\)	\(1\)	\(i\)	\(-i\)	\(-1\)	\(-i\)
\(\chi_{160}(43,\cdot)\)	160.u	8	yes	\(1\)	\(1\)	\(e\left(\frac{5}{8}\right)\)	\(-1\)	\(i\)	\(e\left(\frac{5}{8}\right)\)	\(e\left(\frac{5}{8}\right)\)	\(i\)	\(e\left(\frac{3}{8}\right)\)	\(e\left(\frac{1}{8}\right)\)	\(-1\)	\(e\left(\frac{7}{8}\right)\)
\(\chi_{160}(47,\cdot)\)	160.o	4	no	\(1\)	\(1\)	\(-i\)	\(-i\)	\(-1\)	\(1\)	\(i\)	\(i\)	\(-1\)	\(-1\)	\(i\)	\(i\)
\(\chi_{160}(49,\cdot)\)	160.f	2	no	\(1\)	\(1\)	\(1\)	\(-1\)	\(1\)	\(-1\)	\(1\)	\(-1\)	\(-1\)	\(-1\)	\(-1\)	\(1\)
\(\chi_{160}(51,\cdot)\)	160.w	8	no	\(-1\)	\(1\)	\(e\left(\frac{1}{8}\right)\)	\(i\)	\(i\)	\(e\left(\frac{7}{8}\right)\)	\(e\left(\frac{1}{8}\right)\)	\(-1\)	\(e\left(\frac{5}{8}\right)\)	\(e\left(\frac{3}{8}\right)\)	\(-i\)	\(e\left(\frac{3}{8}\right)\)
\(\chi_{160}(53,\cdot)\)	160.bb	8	yes	\(-1\)	\(1\)	\(e\left(\frac{1}{8}\right)\)	\(1\)	\(i\)	\(e\left(\frac{1}{8}\right)\)	\(e\left(\frac{5}{8}\right)\)	\(i\)	\(e\left(\frac{7}{8}\right)\)	\(e\left(\frac{1}{8}\right)\)	\(1\)	\(e\left(\frac{3}{8}\right)\)
\(\chi_{160}(57,\cdot)\)	160.i	4	no	\(-1\)	\(1\)	\(-1\)	\(-i\)	\(1\)	\(i\)	\(-1\)	\(i\)	\(i\)	\(i\)	\(i\)	\(-1\)
\(\chi_{160}(59,\cdot)\)	160.y	8	yes	\(-1\)	\(1\)	\(e\left(\frac{3}{8}\right)\)	\(i\)	\(-i\)	\(e\left(\frac{1}{8}\right)\)	\(e\left(\frac{3}{8}\right)\)	\(1\)	\(e\left(\frac{3}{8}\right)\)	\(e\left(\frac{5}{8}\right)\)	\(-i\)	\(e\left(\frac{1}{8}\right)\)
\(\chi_{160}(61,\cdot)\)	160.x	8	no	\(1\)	\(1\)	\(e\left(\frac{1}{8}\right)\)	\(-i\)	\(i\)	\(e\left(\frac{7}{8}\right)\)	\(e\left(\frac{5}{8}\right)\)	\(-1\)	\(e\left(\frac{5}{8}\right)\)	\(e\left(\frac{7}{8}\right)\)	\(i\)	\(e\left(\frac{3}{8}\right)\)
\(\chi_{160}(63,\cdot)\)	160.n	4	no	\(1\)	\(1\)	\(-i\)	\(i\)	\(-1\)	\(-1\)	\(i\)	\(-i\)	\(1\)	\(1\)	\(-i\)	\(i\)
\(\chi_{160}(67,\cdot)\)	160.u	8	yes	\(1\)	\(1\)	\(e\left(\frac{3}{8}\right)\)	\(-1\)	\(-i\)	\(e\left(\frac{3}{8}\right)\)	\(e\left(\frac{3}{8}\right)\)	\(-i\)	\(e\left(\frac{5}{8}\right)\)	\(e\left(\frac{7}{8}\right)\)	\(-1\)	\(e\left(\frac{1}{8}\right)\)
\(\chi_{160}(69,\cdot)\)	160.z	8	yes	\(1\)	\(1\)	\(e\left(\frac{7}{8}\right)\)	\(-i\)	\(-i\)	\(e\left(\frac{5}{8}\right)\)	\(e\left(\frac{3}{8}\right)\)	\(1\)	\(e\left(\frac{7}{8}\right)\)	\(e\left(\frac{5}{8}\right)\)	\(i\)	\(e\left(\frac{5}{8}\right)\)
\(\chi_{160}(71,\cdot)\)	160.r	4	no	\(-1\)	\(1\)	\(i\)	\(1\)	\(-1\)	\(-i\)	\(-i\)	\(1\)	\(i\)	\(i\)	\(1\)	\(-i\)
\(\chi_{160}(73,\cdot)\)	160.i	4	no	\(-1\)	\(1\)	\(-1\)	\(i\)	\(1\)	\(-i\)	\(-1\)	\(-i\)	\(-i\)	\(-i\)	\(-i\)	\(-1\)
\(\chi_{160}(77,\cdot)\)	160.bb	8	yes	\(-1\)	\(1\)	\(e\left(\frac{3}{8}\right)\)	\(1\)	\(-i\)	\(e\left(\frac{3}{8}\right)\)	\(e\left(\frac{7}{8}\right)\)	\(-i\)	\(e\left(\frac{5}{8}\right)\)	\(e\left(\frac{3}{8}\right)\)	\(1\)	\(e\left(\frac{1}{8}\right)\)
\(\chi_{160}(79,\cdot)\)	160.e	2	no	\(-1\)	\(1\)	\(-1\)	\(1\)	\(1\)	\(1\)	\(1\)	\(-1\)	\(1\)	\(-1\)	\(1\)	\(-1\)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Properties

Learn more

Character group

First 32 of 64 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.