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 $V_{f \otimes \chi,p}$?

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 $V_{f \otimes \chi,p}$?