Primitive Dirichlet character

A Dirichlet character modulo $N$ is primitive if its character on $(\mathbb{Z}/N\mathbb{Z}{)}^{\times }$ does not factor through reduction to the unit group for any proper divisor of $N$. Its least inducing modulus is called its conductor. Primitivity is essential in the usual formula expressing its finite Fourier transform as a conjugate character times a Gauss sum of a Dirichlet character.