May I ask you one more question? Here the root number $(-1)^{\max\{k,k_g\}+1} \eta_f(p)^2$ turns out to be independent of $\psi(q)$. If we assume the form $g$ to be of level 1 (instead of level $q$), is the root number becoming $(-1)^{\max\{k,k_g\}+1} \eta_f(p)^2 \psi(q)$ instead?

Fantastic! This is exactly what I want. Many thanks for your time!

Can you possibly tell me how do the local root numbers $\epsilon_p(\frac{1}{2},\pi_{f,p},\psi_p)$ and $\epsilon_q(\frac{1}{2},\pi_{g,q},\psi_q)$ exactly look like, in terms of the character $\psi$ and the pseudo-eigenvalues of $f$ and $g$?