(Sorry for my poor english..)
Let $\chi : (\mathbb{Z}/N\mathbb{Z})^{*}\to \mathbb{C}$ be a Dirichlet character and $p\mid N$ be a prime. I have some questions about the definition of $p$-primary component of $\chi$ and local component of $\chi$ at $p$, and their relations.
First, I think that the definition of $p$-primary component $\chi_p$ is $\chi_p: (\mathbb{Z}/p^{e}\mathbb{Z})^{*}\to \mathbb{C}$ with $\chi_p(a)=\chi(n_a)$ where $e=v_p(N)$ and \begin{equation} n_a\equiv a \pmod{p^e} \text{ and } n_a\equiv 1\pmod{\frac{N}{p^e}}. \end{equation} Is this definition correct?
Furthermore, are $p$-primary component $\chi_p$ and local component $\chi_{p,1}$ of $\chi$ at $p$ related?