"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 $L_p(s,f \otimes \psi) \neq 1$.

If $f(z)=\sum_{n \geq 1} a_n q^n$ is a newform of level $\Gamma_1(N)$ and $\chi$ is a Dirichlet character modulo $m$, then the naïve twist of $f$ by $\chi$ is the modular form $f_\chi(z) = \sum_{n \geq 1} a_n \chi(n) q^n$. As was already pointed out $f_\chi$ is not always a newform, but there is a unique newform $f \otimes \chi$ sharing the same Hecke eigenvalues at primes $p$ not dividing $m$. If $N$ and $m$ are coprime then $f_\chi = f \otimes \chi$ is a newform, but this is not always the case in general. A criterion for $f_\chi$ being a newform (equivalently $f_\chi = f \otimes \chi$) has been worked out by Atkin--Li in their article Twists of newforms and pseudo-eigenvalues of $W$-operators (see Corollary 3.1).

The question of determining the Euler factor of $f \otimes \chi$ at $p$ is clearly a local one, so we may assume that $\chi$ is a primitive Dirichlet character of conductor $p^\alpha$ with $\alpha \geq 1$, and that $p$ divides $N$. In general the Euler factor of $f \otimes \chi$ at $p$ can be determined from the local automorphic representation associated to $f$, as explained by Peter Humphries. There is however a special case which is easy, namely when $f$ is $p$-primitive, meaning that $f$ has minimal level among its twists by characters of $p$-power conductor. If $f$ is $p$-primitive and $a_p \neq 0$ then we have the formula $$L_p(f \otimes \chi,s)^{-1} = 1- \bar{a}_p \cdot (\psi \chi)_0(p) p^{-s}$$ where $\psi$ is the Nebentypus character of $f$, and $(\psi \chi)_0$ is the primitive Dirichlet character associated to $\psi \chi$. This is explained in Merel's article Symboles de Manin et valeurs de fonctions $L$ (Section 2.6).

The L-series of $\rho_{f, \lambda} \otimes \chi$ is the $L$-series of $f \otimes \chi$, where $f \otimes \chi$ is the unique newform such that $a_\ell(f \otimes \chi) = \chi(\ell) a_\ell(f)$ for all but finitely many $\ell$. So your Galois-theoretic question reduces to a purely automorphic one, namely determining the Hecke eigenvalues of $f \otimes \chi$ at the bad primes (the ones dividing the conductor of $\chi$).

In general, this is a bit fiddly to do algorithmically, if your starting point is just the $q$-expansion of $f$; but of course if you know the local factors of the automorphic representation associated to $\chi$ then you can read off the corresponding data for $f \otimes \chi$ immediately.