Values of a Dirichlet character on generators (reviewed)

A Dirichlet character $\chi\colon \Z\to \C$ of modulus $q$ is determined (via multiplicativity) by the values $\chi(g_1), \ldots, \chi(g_r)$ it takes on any list of generators $g_1,\ldots,g_r$ for the group $(\mathbb{Z}/q\mathbb{Z})^\times$. The generators $g_i$ can each be specified as integers in the interval $[1,q-1]$, and the values $\chi(g_i)$ are roots of unity $e(r):=\exp(2\pi i r)$, where $r$ is a rational number.