The main goal of the paper is to prove that the Hecke modular functions are in some sense rare objects. A geometric approach is applied. Two topologies in the space of complex sequences with polynomial growth are denned, and in both cases we prove that the set of Fourier coefficients of Hecke modular functions form a discrete subset. A quantitative version of this statement is also provided. The proof of the main result depends on non-linear twists of degree two L-functions.
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