In this thesis, we perform a review of the theory of overconvergent modular forms, then we explore the distribution of the eigenvalues of the Hecke operator T_p by considering their p-adic valuations. We begin by covering algebraic and geometric definitions of modular forms, then expanding these definitions to overconvergent modular forms. We then introduce algorithms, from "Computations with classical and p-adic modular forms" by Alan G. B. Lauder, which provide a method for calculating the p-adic valuations of the aforementioned eigenvalues. In order to implement these algorithms, programs were written for the Sagemath computer algebra program to perform the necessary calculations. These programs were used to collect lists of p-adic valuations, for various values of p and for spaces of modular forms of various weights and of various levels. The collected data confirms the fact that the Gouvea-Mazur conjecture is false, but also indicates that it may be a useful approximation of the true behavior at large weights or at large values of p, at least for the first few slopes. It shows the existence of  "plateaus" of weights which have the same slopes, up to the precision used, even at low values of p and k. The reason for the existence of these "plateaus" is unknown.