Thanks but in case 1, the local Galois representation of $f$ at $\ell$ is not irreducible ("principal series") and for $f \otimes \chi$, the local Galois representation at $\ell$ is becoming irreducible ("supercuspidal") where as after twisting by $\chi$ the local Galois representation at $\ell$, I was thinking, should still be diagonal ("ramified principal series")? Sorry I am still confused.