Given modular forms $f, g$ for $SL_{2}(\mathbb{Z})$ of weights $l, k$ resp. where $l < k$, is there a way to see if $f \mid g$? That is, is there a way to see if $g = fh$ for some modular form $h$ of weight $k-l$?
At least in terms of $q$-expansions over $\mathbb{F}_{p}$, there are some examples: $E_{p-1} \mid f$ for any reduced form $f$ since $E_{p-1} \equiv 1 \pmod{p}$. So perhaps the above question can be more easily attacked in the mod $p$ setting? Would there be a way to extend the $E_{p-1}$ case to the mod $p^{m}$ setting?
I'm not sure how one would approach answering this question other than literally dividing $q$-expansions and writing the result as a linear combination of basis elements from the appropriate space of forms. If we only look at Eisenstein series, then some results on Bernoulli numbers could be useful.