Let $(\rho_{f,\lambda})_\lambda$ be the system of Deligne's $\ell$-adic representations attached to a modular newform $f$ (where $\lambda$ runs over the finite places of the number field $K$ generated by the Fourier coefficients of $f$), and take a character $\chi\colon\mathrm G_{\mathbb Q}\rightarrow\overline{\mathbb Q}^\times$ of finite order (assume for simplicity it takes values in $K$). Fix embeddings of $\overline{\mathbb Q}$ into each $\overline{\mathbb Q}_\ell$. Then we can look at the system of $\lambda$-adic realizations $(\rho_{f,\lambda}\otimes\chi)_\lambda$.

Let me call a system $(\rho_\lambda)_\lambda$ of $\ell$-adic representations *compatible* if for each prime $p$ the polynomial
$$ P_p(T)=\begin{cases}\det(1-\mathrm{Frob_p}T,\rho_\lambda^{I_p}), & \lambda\not\mid p,\\ \det(1-\varphi T,\mathrm D_{\mathrm{cris}}(\rho_\lambda)), & \lambda\mid p\end{cases} $$ does not depend on the choice of a place $\lambda$ and has coefficients in $\overline{\mathbb Q}$.

It is known that the system $(\rho_{f,\lambda})_\lambda$ is compatible in this sense (using results of Scholl and Saito), and it seems to be well-known that this holds also for the system $(\rho_{f,\lambda}\otimes\chi)_\lambda$. Probably this can be seen somehow using Weil-Deligne representations, but I haven't yet worked out the details (any hints or references on this are welcome!).

But my actual question is the following. If we define the $L$-function of such a compatible system as usual as the Euler product $$ L((\rho_\lambda)_\lambda,s)=\!\!\!\prod_{p\text{ any prime}}\!\!\! P_p(p^{-s}),$$ then the $L$-function associated to $(\rho_{f,\lambda})_\lambda$ is just the $L$-function of $f$. But what is the $L$-function of $(\rho_{f,\lambda}\otimes\chi)_\lambda$? If we view $\chi$ as a Dirichlet character of $(\mathbb{Z}/N)^\times$ for the minimal possible $N$ via class field theory, then one often considers the $L$-function defined by $$ L(f,\chi,s)=\sum_{n=1}^\infty \chi(n)a_nn^{-s}, $$ where the $a_n$ are the Fourier coefficients. But in general the $L$-function of the system $(\rho_{f,\lambda}\otimes\chi)_\lambda$ *differs* from this one. Indeed, D. Loeffler's answer to my question How large is Dcris of certain twists of modular forms? shows that we get at least an extra Euler factor at $p$ if $f$ is $p$-ordinary and the $p$-part of $\chi$ cancels the $p$-part of the nebentype of $f$. Can we in general tell which additional Euler factors (compared to $L(f,\chi,s)$) this $L$-function has?