Let $f(z)=\sum_{n=0}^{\infty}a_f(n)e^{2\pi inz}$ and $g(z)=\sum_{n=0}^{\infty}a_g(n)e^{2\pi inz}$ be two modular forms of integer weight $k$ for $SL_2(\mathbb{Z})$. Consider $$ h(z) := \sum_{n=0}^{\infty}a_f(n)a_g(n)e^{2\pi inz}. $$ Question: Is $h(z)$ again a modular form (possibly of different weight, $2k$)?
I guess one can use Hecke's converse theorem, but I'm unable to check the analytic continuation of the completed Gamma function $$ \Lambda_{h}(s) := (2\pi)^{-s}\Gamma(s) \sum_{n=1}^{\infty}a_f(n)a_g(n)n^{-s}, $$ nor if it satisfies the functional equation of the form $$ \Lambda_{h}(s) = i^{2k} \Lambda_{h}(2k-s). $$
If this is not true, I could not come up with any easy counter example also.