  We investigate certain finiteness questions that arise naturally when
studying approximations modulo prime powers of p-adic Galois representations
coming from modular forms. We link these finiteness statements with a question
by K. Buzzard concerning p-adic coefficient fields of Hecke eigenforms.
Specifically, we conjecture that for fixed N, m, and prime p with p not
dividing N, there is only a finite number of reductions modulo p^m of
normalized eigenforms on \Gamma_1(N). We consider various variants of our basic
finiteness conjecture, prove a weak version of it, and give some numerical
evidence.
