We study the slopes of the Atkin’s U operator acting on overconvergent p-adic modular forms. In the case of tame level 1 and for p =5,7,13; we compute a quadratic lower bound for the Newton polygon of U. The methods of proof are explicit and rely on a certain deformation of the U operator and its characteristic power series.

This gives us the possibility to compute the smallest possible slope for p=5,7 and to prove necessary and sufficient conditions on the weight such that the dimension of the cuspidal space is one. This result allows us to exhibit some p-adic analytic families of modular forms in the framework of Coleman’s theory.

We then formulate a conjecture that would allow us to extend our analysis to all the congruence classes modulo p−1.