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 $V\_{f \otimes \chi,p}$?



Local Galois representation associated to twist of modular form
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 $f\otimes\chi$). 


