Local Galois representation associated to twist of modular form

Question

Let $f$ be a modular newform of weight $k \geq 2$, level $N$ (square free) and trivial nebentypus. Let $V_{f}$ be the $p$-adic (p odd) Galois representation associated $f$. We denote by $V_{f,l}:= V_{f}|_{G_{l}}$. Let $\chi$ be a quadratic character of conductor $l$. Suppose that $(N,l)=1$. Then $f \otimes \chi$ is a newform of weight $k$, level $Nl^2$ and trivial nebentypus.

Case I: $l \neq p$

In this case $V_{f,l} \sim \pmatrix{ \chi_{1} & 0 \\ 0 & \chi_2 }$.

But for $f \otimes \chi$, I think the representation $V_{f\otimes \chi, l}$ is irreducible. (Reference: Tilouine - Modular forms and Galois representation, Section 3.2, Bull Greek Math Soc. Vol 46)

Can some one explain why it should be irreducible? And how does $V_{f,l} \otimes \chi$ is related to $V_{f \otimes \chi,l}$?

Case II: $l=p$

Suppose that $f$ is ordinary at $p$. Is it true that $f \otimes \chi$ is ordinary at $p$? In this case how is $V_{f,p} \otimes \chi$ is related to $V_{f \otimes \chi,p}$?

Answer 1

I think it helps to put things in a larger perspective.

To an eigencuspform $f$ and a prime number $\ell$ is attached on the one hand an irreducible, admissible representation $\pi(f)_{\ell}$ of $\operatorname{GL}_{2}(\mathbb Q_{\ell})$ which is either

1) An irreducible principal series $\pi(\chi,\psi)$,

2) A special representation $\mu\cdot\operatorname{St}$,

3) A (super)cuspidal representation

and on the other hand a local $\operatorname{Gal}(\bar{\mathbb Q}_\ell/{\mathbb Q}_\ell)$-representation $\rho_f|G_{\mathbb Q_\ell}$ with coefficients in ${\mathbb Q}_p$ ($p\neq\ell$) which is either

1) Reducible and potentially unramified ($I_\ell$ acts through a finite quotient),

2) Reducible and not potentially unramified ($I_\ell$ acts through an infinite quotient),

3) Irreducible (and $I_\ell$ acts through a finite quotient).

If $\chi$ is a finite order character, then $\pi(f\otimes\chi)_\ell=\pi(f)_\ell\otimes\chi$, $\rho_{f\otimes\chi}|G_{\mathbb Q_\ell}=(\rho_f|G_{\mathbb Q_\ell})\otimes\chi$ and the two classifications above are stable by twisting by $\chi$.

Finally, the local-global compatibility property of the Langlands Reciprocity Conjecture (for $\operatorname{GL}_2$) (which is a theorem of Deligne and Carayol) states that $\pi(f)_\ell$ belongs to class $i$ if and only if $\rho_f|G_{\mathbb Q_\ell}$ belongs to class $i$.

In Tilouine's survey, it seems that section 3.1 is only valid for $f$ with non-zero eigenvalue if not supercuspidal at $\ell$ (an hypothesis which is not satisfied by your $f\otimes\chi$).