I've come across the following polynomials in my research and I am wondering if they have a name or if there is very much known about them:

\begin{equation} F_{\chi}(T) = \sum_{a = 1}^{n-1} \chi(a)T^a \end{equation}

where $\chi$ is a Dirichlet character of conductor $n$. The closest thing I can find on these are the Fekete polynomials (this is the special case where $\chi$ is the Legendre symbol and $n$ is a prime number) but nothing for general Dirichlet characters. Thanks in advance for any information you might be able give me.