Galois representation associated to CM-newforms

Question

Let $f(z)=\sum_{n\ge 1}a(n)e(nz)$, be a newform of CM-type, and let $\psi_f$ be the associated Hecke character, so that, $$ f(z)=\sum_{\mathfrak{a}}\psi_f(\mathfrak{a})e(N(\mathfrak{a})z), $$ and let $\rho_{\lambda,f}$ be the associated Galois representation. Let $\frak{p}$ be a prime ideal of the field by which $f$ has CM. My question is:

How to prove that the characteristic polynomial of $\rho_{\lambda,f}(\mathrm{Frob}_{\mathfrak{p}})$ satisfying $$ (x-\psi_f(\mathfrak{p}))(x-\psi(\mathfrak{p'}))? $$ where $\mathfrak{p'}$ the conjugate of $\mathfrak{p}$.

Answer 1

Let me abbreviate $\rho_{\lambda,f}$ as $\rho$, and $\psi_f$ as $\psi$. By definition, $L(s,\rho)=L(s,f)=L(s,\psi)$. The equality of the Euler factors of $L(s,\rho)$ and $L(s,\psi)$ at the split prime $p=\mathfrak{p}\mathfrak{p}'$ means that $$\det(1-\rho(\mathrm{Frob}_{\mathfrak{p}})p^{-s})=(1-\psi(\mathfrak{p})p^{-s})(1-\psi(\mathfrak{p}')p^{-s}).$$ Multiplying both sides by $p^{2s}$ and renaming $p^s$ to $x$, we get $$\det(x-\rho(\mathrm{Frob}_{\mathfrak{p}}))=(x-\psi(\mathfrak{p}))(x-\psi(\mathfrak{p}')).$$