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?](https://mathoverflow.net/questions/271142/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?



"Extra Euler factors" in one definition of the L-function of a twist of a modular form
Let me translate this into a problem purely about automorphic forms:


Take a newform $f \in \mathcal{S}\_k^{\ast}(q,\chi)$, and a primitive Dirichlet character $\psi$ modulo $q'$. Then there exists a newform $f \otimes \psi$ of weight $k$, level dividing $q {q'}^2$, and nebentypus induced by the primitive character inducing $\chi \psi^2$, such that whenever $(n,q') = 1$, the $n$-th Hecke eigenvalue $\lambda\_{f \otimes \psi}(n)$ of $f \otimes \psi$ is $\lambda\_f(n) \psi(n)$.


Since $f \otimes \psi$ is a newform, it has an $L$-function
\[L(s,f \otimes \psi) = \prod\_p L\_p(s,f \otimes \psi),\]
where for $p \nmid (q,q')$,
\[L\_p(s,f \otimes \psi) = \frac{1}{1 - \lambda\_f(p) \psi(p) p^{-s} + \chi(p) \psi^2(p) p^{-2s}},\]
while for $p \mid (q,q')$, this may be something more complicated.


On the other hand, we may define the "naïve" $L$-function associated to $f \otimes \psi$ as the analytic continuation of the Dirichlet series
\[\sum\_{n = 1}^{\infty} \frac{\lambda\_f(n) \psi(n)}{n^s} = \prod\_p \frac{1}{1 - \lambda\_f(p) \psi(p) p^{-s} + \chi(p) \psi^2(p) p^{-2s}},\]
where the Euler product identity holds by multiplicativity.


So I believe the question you are asking is when $L\_p(s,f \otimes \psi)$ is *not* equal to $(1 - \lambda\_f(p) \psi(p) p^{-s} + \chi(p) \psi^2(p) p^{-2s})^{-1}$ (or equivalently when $\lambda\_{f \otimes \psi}(p)$ is *not* equal to $\lambda\_f(p) \psi(p)$). This can only occur when $p \mid (q,q')$, in which case $(1 - \lambda\_f(p) \psi(p) p^{-s} + \chi(p) \psi^2(p) p^{-2s})^{-1} = 1$, and the answer depends sensitively on the local component $\pi\_{f,p}$ of $f$ (which is a ramified representation of $\mathrm{GL}\_2(\mathbb{Q}\_p)$) as well as the local components $\chi\_p,\psi\_p$ of $\chi,\psi$ (which are characters of $\mathbb{Q}\_p^{\times}$, the latter of which is ramified). This can be dealt with via a case-by-case approach.


* If $\pi\_{f,p}$ is supercuspidal, then
\[L\_p(s,f \otimes \psi) = 1.\]
This is also true if $\pi\_{f,p} = \omega\_p \mathrm{St}\_p$ is a special representation (so that the central character of $\pi\_{f,p}$ is $\chi\_p = \omega\_p^2$) with $\omega\_p$ and $\omega\_p \psi\_p$ both ramified, or if $\omega\_p$ is unramified. Note that $\pi\_{f,p} \otimes \psi\_p = \omega\_p \psi\_p \mathrm{St}\_p$ for special representations.
* If $\pi\_{f,p} = \omega\_p \mathrm{St}\_p$ is a special representation with $\omega\_p$ ramified but $\omega\_p \psi\_p$ unramified (so that $\omega\_p \psi\_p(p) \in \{\pm 1\}$), then
\[L\_p(s,f \otimes \psi) = \frac{1}{1 - \omega\_p \psi\_p(p) p^{-s-1/2}} \neq 1.\]
* Finally, if $\pi\_{f,p} = \omega\_{1,p} \boxplus \omega\_{2,p}$ is a ramified principal series representation (so that the central character of $\pi\_{f,p}$ is $\chi\_p = \omega\_{1,p} \omega\_{2,p}$), then $\pi\_{f,p} \otimes \psi\_p = \omega\_{1,p} \psi\_p \boxplus \omega\_{2,p} \psi\_p$, and
\[L\_p(s,f \otimes \psi) = \frac{1}{(1 - \omega\_{1,p} \psi\_p(p) p^{-s})(1 - \omega\_{2,p} \psi\_p(p) p^{-s})}.\]
If $\omega\_{1,p} \psi\_p$ and $\omega\_{2,p} \psi\_p$ are both ramified, then $\omega\_{1,p} \psi\_p(p) = \omega\_{2,p} \psi\_p(p) = 0$, and so $L\_p(s,f \otimes \psi) = 1$. However, if either one is unramified, then one of $\omega\_{1,p} \psi\_p(p), \omega\_{2,p} \psi\_p(p)$ is nonzero, and so $L\_p(s,f \otimes \psi) \neq 1$.


