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This article is a research exposition based on the author's talk at theInternational Colloquium on Automorphic Representations and L-Functions, 2012,held at TIFR, Mumbai. We consider some special cases of the following question:when is a natural subset of the Fourier coefficients sufficient to uniquelydetermine a "modular form"? Two kinds of modular forms are considered in thisarticle: a) classical modular forms of half-integral weight, and b) Siegelmodular forms of genus 2 and integral weight. These two apparently differentscenarios turn out to be closely related. Our results were motivated by, andhave several interesting connections to, automorphic L-functions and Besselmodels.

Determination of modular forms by fundamental Fourier coefficients