Computations on Overconvergence Rates Related to the Eisenstein Family

We provide for primes $p\ge 5$ a method to compute valuations appearing in
the "formal" Katz expansion of the family
$\frac{E_{k}^{\ast}}{V(E_{k}^{\ast})}$ derived from the family of Eisenstein
series $E_{k}^{\ast}$. The overconvergence rates of the members of this family
go back to a conjecture from Coleman. We will describe two algorithms: the
first one to compute the Katz expansion of an overconvergent modular form and
the second one, which uses the first algorithm, to compute valuations appearing
in the "formal" Katz expansion. Based on data obtained using these algorithms
we make a precise conjecture about a constant appearing in the overconvergence
rates related to the classical Eisenstein series at level $p$.