= Fourier expansion of a character-twisted Eisenstein series
{c}
{title2=$c_n=2(-2\pi i)^k((k-1)!N^k)^{-1}\sum_{r\mid n}r^{k-1}S_\chi(r)$}
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 are
$$
c_n=\frac{2(-2\pi i)^k}{(k-1)!N^k}\sum_{r\mid n}r^{k-1}S_\chi(r).
$$
For a <primitive Dirichlet character> this simplifies to $2(-2\pi i)^kg(\chi)((k-1)!N^k)^{-1}\sum_{r\mid n}\overline\chi(r)r^{k-1}$. Without primitivity the simplified formula is generally false.
Back to article page
