WOLFRAM|DEMONSTRATIONS PROJECT

Table of Dirichlet Characters

modulus

11

χ(n)n	0	1	2	3	4	5	6	7	8	9	10
χ 1 (n)	0	1	1	1	1	1	1	1	1	1	1
χ 2 (n)	0	1	π 5 	- 2π 5 	2π 5 	4π 5 	- π 5 	- 3π 5 	3π 5 	- 4π 5 	-1
χ 3 (n)	0	1	2π 5 	- 4π 5 	4π 5 	- 2π 5 	- 2π 5 	4π 5 	- 4π 5 	2π 5 	1
χ 4 (n)	0	1	3π 5 	4π 5 	- 4π 5 	2π 5 	- 3π 5 	π 5 	- π 5 	- 2π 5 	-1
χ 5 (n)	0	1	4π 5 	2π 5 	- 2π 5 	- 4π 5 	- 4π 5 	- 2π 5 	2π 5 	4π 5 	1
χ 6 (n)	0	1	-1	1	1	1	-1	-1	-1	1	-1
χ 7 (n)	0	1	- 4π 5 	- 2π 5 	2π 5 	4π 5 	4π 5 	2π 5 	- 2π 5 	- 4π 5 	1
χ 8 (n)	0	1	- 3π 5 	- 4π 5 	4π 5 	- 2π 5 	3π 5 	- π 5 	π 5 	2π 5 	-1
χ 9 (n)	0	1	- 2π 5 	4π 5 	- 4π 5 	2π 5 	2π 5 	- 4π 5 	4π 5 	- 2π 5 	1
χ 10 (n)	0	1	- π 5 	2π 5 	- 2π 5 	- 4π 5 	π 5 	3π 5 	- 3π 5 	4π 5 	-1

Produce tables of Dirichlet characters. A Dirichlet character

χ

modulo

m

is a multiplicative function that satisfies:

1.

χ(n)

is periodic with period

m

.

2.

χ(n)

is zero when

n

is not coprime to

m

.

3.

χ(n)

is an

th

m

root of unity when

n

is relatively prime to

m

.

4. There are

ϕ(m)

Dirichlet characters for a given

m

.