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We study several aspects of nonvanishing Fourier coefficients of ellipticmodular forms $\bmod \ell$, partially answering a question ofBella\"iche-Soundararajan concerning the asymptotic formula for the count ofthe 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 arithmeticallyinteresting (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.

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