Thanks, that's helpful. One more question: is there a relationship between the level $N_f$ of $f$ and the conductor $N_E$ of $E$? For instance, if $f$ is one one of the two forms at level 15 with integer coefficients, I'm not really sure how to find the corresponding $E$.

I seem to get that $a_p(f)=a_p(E)^2-2p$ for all primes $p$ that split in the quadratic imaginary field $f$ has CM by. This led me to believe that there may be a base change involved. Is that the correct relationship between the Fourier coefficients of $f$ and $E$?

Thanks! I’m sure it’s probably obvious, but is there a quick way to see that my condition is equivalent to saying that $G$ has a character of degree equal to the square root of the index of the center? This seems to be the usual way groups of central type are defined.

Thank you! If I may ask a related question: I know that there is an automorphic rep with the same $L$-function as $Sym^2 f$. Why does it follow that $$\sum_{p} \frac{\lambda_f(p^2)}{p^s}$$ is also bounded when $s\to 1^{+}$?