How to find all Dirichlet characters

Question

I want to know all the Dirichlet characters modulo $m$. I know that the number of such characters are $\phi(m)$. But how do find each and every character. for small moduli I could do it using some guessing. But how do we do it if $m$ is large?

Answer 1

The usual strategy is to factor $m$ into prime powers using the FTA, then noting that the Dirichlet characters modulo those prime powers are the ones that really matter.

From there you reduce to the case of prime powers. You know that

$$\left(\Bbb Z/p^e\Bbb Z\right)^*\cong \Bbb Z/\varphi(p^e)\Bbb Z$$

when $p$ is odd and is congruent to

$$\Bbb Z/2\Bbb Z\times\Bbb Z/2^{e-2}\Bbb Z$$

when $p=2$.

Then these are cyclic groups so you know their characters easily. Then you stitch them back together by noting that all characters modulo $m$ are of the form

$$\chi(x)=\prod_{p|m}\chi_p(x)$$

and $\chi_p$ is any character of $\left(\Bbb Z/p^e\Bbb Z\right)^*$.