This article starts a computational study of congruences of modular forms and
modular Galois representations modulo prime powers. With two integral
polynomials we associate an integer which we call the congruence number. It has
the virtue that it can be very quickly computed and that -- in many cases -- it
is the product of all prime powers modulo which the polynomials have roots in
common. 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 will be
presented.