Fourier expansion of a character-twisted Eisenstein series ($c_n=2(-2\pi i{)}^k((k-1)!N^k{)}^{-1}{\sum }_{r\mid n}{r}^{k-1}{S}_{\chi }(r)$)

With $S_{\chi }(r)$ the finite Fourier transform in Gauss sum of a Dirichlet character, the constant coefficient of $G_k(\chi ,z)$ in $e^{2\pi iz/N}$ is $2L(\chi ,k)$, and its positive coefficients areFor a primitive Dirichlet character this simplifies to $2(-2\pi i{)}^{k}g(\chi )((k-1)!N^k{)}^{-1}{\sum }_{r\mid n}\overline{\chi }(r)r^{k-1}$. Without primitivity the simplified formula is generally false.