Coefficient Bounds for Level 2 Cusp Forms and Modular Functions

Abstract

We give explicit upper bounds for the coefficients of arbitrary weight $k$, level 2 cusp forms, making Deligne's well-known $O(n^{\frac{k-1}{2}+\epsilon })$ bound precise. We also derive asymptotic formulas and explicit upper bounds for the coefficients of certain level 2 modular functions.