The modular forms of relevance will be holomorphic functions on the
upper half plane that transform a certain way under subgroups of
SL(2,Z) acting by linear fractional transformations. In particular
they will be invariant under the translation sending z to z+1,
and so have an expression as a Fourier series in powers of
q=exp(2 Pi i z)
f(z)= a1 q + a2 q^2 +a3 q^3 +.....
Two forms are congruent modulo a prime p if the coefficients of
like powers of q are congruent modulo p.