A Dirichlet character $\chi$ of prime power modulus $N$ is minimal if the following conditions both hold:

The conductor of $\chi$ does not lie in the open interval $(\sqrt{N},N)$, and if $N$ is a square divisible by 16 then ${\rm cond}(\chi)\in \{\sqrt{N},N\}$.

Both the order and conductor of $\chi$ are minimal among the set of all Dirichlet character $\chi\psi^2$ for which ${\rm cond}(\psi){\rm cond}(\chi\psi)  N$.
This includes all primitive Dirichlet characters of prime power modulus, but not every minimal Dirichlet character of prime power modulus is primitive.
For a composite modulus $N$ with prime power factorization $N=p_1^{e_1}\cdots p_n^{e_n}$, a Dirichlet character $\chi$ of modulus $N$ is minimal if and only if every character in its unique factorization into Dirichlet characters of modulus $p_1^{e_1},\cdots,p_n^{e_n}$ is minimal. The trivial Dirichlet character is minimal.
 Review status: beta
 Last edited by Andrew Sutherland on 20200206 08:38:59