  This article starts a computational study of congruences of modular forms and
modular Galois representations modulo prime powers. Algorithms are described
that compute the maximum integer modulo which two monic coprime integral
polynomials have a root in common in a sense that is defined. These techniques
are applied to the study of congruences of modular forms and modular Galois
representations modulo prime powers. Finally, some computational results with
implications on the (non-)liftability of modular forms modulo prime powers and
possible generalisations of level raising are presented.
