On the Up operator acting on p-adic overconvergent modular forms when X0(p) has genus 1

In this note we will show how to compute Up acting on spaces of overconvergent p-adic modular forms when X0(p) has genus 1. We first give a construction of Banach bases for spaces of overconvergent p-adic modular forms, and then give an algorithm to approximate both the characteristic power series of the Up operator and eigenvectors of finite slope for Up, and present some explicit examples. We will also relate this to the conjectures of Clay on the slopes of overconvergent modular forms.