I am confused by the justification of Equation 7 from https://sites.math.washington.edu/~morrow/336_14/papers/austin.pdf, which states that (substituting in the definitions and slightly modifying the notation)
$$\sum_{\substack{p\text{ prime}\\p\equiv a\pmod{N}}} \frac{1}{p^s} = \sum_{\chi}\left(\frac{\overline{\chi}(a)}{\phi(N)}\sum_{p\text{ prime}} \frac{\chi(p)}{p^s}\right)$$
where the second sum is over all Dirichlet characters modulo $N$, and the notation $\overline{\chi}(a)$ denotes the inverse of $\chi$ in the group of Dirichlet characters.
According to the source, this follows from the fact that $P_a(s) = P_a(s)\Phi_a(n)$ where
$$\Phi_a(n) = \begin{cases}1,&\text{if }n\equiv a\pmod N\\0,&\text{otherwise}\end{cases}$$
and
$$P_a(s) = \sum_{\substack{p\text{ prime}\\p\equiv a\pmod{N}}} \frac{1}{p^s}$$
and the orthogonality relations of the Dirichlet characters. But the right-hand side of Equation 7 (the first equation I listed) depends only on $s$, not on $n$, but $P_a(s)\Phi_a(n)$ depends on $a$ and $n$, so how are they equal.
Perhaps this isn't the best way to phrase my question. I could also rephrase my questions as follows:
Can someone please give a full, step-by-step derivation for Equation 7 that provides more details than the proof given in Tran's work?