Higher congruences between modular forms

It is well-known that two modular forms on the same congruence subgroup and of the same weight, with coefficients in the integer ring of a number field, are congruent modulo a prime ideal in this integer ring, if the first B coefficients of the forms are congruent modulo this prime ideal, where B is an effective bound depending only on the congruence subgroup and the weight of the forms. In this thesis, we generalize this result to congruences modulo powers of prime ideals and to modular forms of distinct weights. We also determine necessary conditions on the weights for there to be congruences between cusp forms modulo powers of prime ideals, with special emphasis on congruences between eigenforms because of their connection to Galois representations. Additionally, we investigate the maximal congruences between newforms on Γ0(N), and also between newforms in case of level-lowering from Γ0(Np) to Γ0(N). This investigation leads to a very interesting set of conjectures, and we include all computed numerical evidence supporting these conjectures.