Definition:Dirichlet Character/Primitive Character

Definition
Let $q \in \Z_{>1}$.

Let $\paren {\Z / q \Z}$ denote the ring of integers modulo $q$.

Let $G = \paren {\Z / q \Z}^\times$ be the group of units of $\paren {\Z / q \Z}$.

Let $\C^\times$ be the group of units of $\C$.

Let $\chi_0$ be the trivial (Dirichlet) character modulo $q$.

Let $q^*$ be the least divisor of $q$ such that:
 * $\chi = \chi_0 \chi^*$

where $\chi^*$ is some character modulo $q^*$.

If $q = q^*$ then $\chi$ is called primitive, otherwise $\chi$ is imprimitive.