Coefficient Bounds for Level 2 Cusp Forms and Modular Functions

TL;DR

This paper provides explicit bounds and asymptotic formulas for the Fourier coefficients of level 2 cusp forms and modular functions, refining existing theoretical bounds with precise estimates.

Contribution

It introduces explicit upper bounds for coefficients of level 2 cusp forms and modular functions, making Deligne's bounds more precise and applicable.

Findings

Explicit upper bounds for cusp form coefficients

Asymptotic formulas for modular function coefficients

Refinement of Deligne's bounds

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.