Divisibility Properties of Coefficients of Level $p$ Modular Functions for Genus Zero Primes

Abstract: Lehner's 1949 results on the $j$-invariant showed high divisibility of the function's coefficients by the primes $p\in{2,3,5,7}$. Expanding his results, we examine a canonical basis for the space of level $p$ modular functions holomorphic at the cusp 0. We show that the Fourier coefficients of these functions are often highly divisible by these same primes.