On Fourier coefficients of elliptic modular forms $\bmod \, \ell$ with applications to Siegel modular forms

We study several aspects of nonvanishing Fourier coefficients of elliptic modular forms $\bmod \ell$, partially answering a question of Bella\"iche-Soundararajan concerning the asymptotic formula for the count of the number of Fourier coefficients upto $x$ which do not vanish $\bmod \ell$. We also propose a precise conjecture as a possible answer to this question. Further, we prove several results related to the nonvanishing of arithmetically interesting (e.g., primitive or fundamental) Fourier coefficients $\bmod \ell$ of a Siegel modular form with integral algebraic Fourier coefficients provided $\ell$ is large enough. We also make some efforts to make this "largeness" of $\ell$ effective.