Let $\pi$ be an irreducible cuspidal automorphic representation of $GL(2)$ over a global field with factorisation $\pi = \otimes \pi_v$.
Then at most finitely many $\pi_v$ are not spherical.
Can it happen that $\pi_v(g) = \chi_v(\det g)$ for a unitary character of $F_v^\times$? Can it happen infinitely often?
Is the Steinberg spherical? Can it occur as $\pi_v$? Can it appear infinitely often?