Abstract: Classical Modular forms are certain holomorphic functions on the complex upper half space H with invariance under the congruence subgroups of SL(2,Z), in particular they are periodic. Among them is the subspace of cusp forms and Eisenstein series. The Fourier coefficients in the Fourier expansion of these two types of objects grow differently, those of cusp forms grow slower than those of (individual) Eisenstein series. We will discuss a recent work on a question which asks the converse in a more general context: if the growth as above is `slow', does the form necessarily come from the space of cusp forms?