Galois orbit of a Dirichlet character (reviewed)

The Galois orbit of a Dirichlet character $\chi$ of modulus $q$ and order $n$ is the set $[\chi]:=\{\sigma(\chi): \sigma\in \Gal(\Q(\zeta_n)/\Q)\}$, where $\sigma(\chi)$ denotes the Dirichlet character of modulus $q$ defined by $k \mapsto \sigma(\chi(k))$. The map $\chi\to \sigma(\chi)$ defines a faithful action of the Galois group $\Gal(\Q(\zeta_n)/\Q)$ on the set of Dirichlet characters of modulus $q$ and order $n$, each of which has $\Q(\zeta_n)$ as its field of values.