In this paper, we investigate the distribution of the Fourier coefficients of integral weight modular forms on \(\textrm{SL}_{2}(\mathbb {Z})\) modulo an odd integer M. Let f be an integral weight modular form on \(\textrm{SL}_{2}(\mathbb {Z})\) with integral Fourier coefficients \(a_{f}(n)\) and M be an odd integer. For each r, under conditions that depend on the prime divisors of M, we prove that there are infinitely many integers n such that \(\begin{aligned} a_{f}(n)\equiv r\pmod {M}, \end{aligned}\) and we also provide an asymptotic lower bound on the number of such n. For certain modular forms f, our results allow us to completely determine the set of all odd integers M for which the Fourier coefficients of f take every residue class modulo M infinitely often. In particular, we explicitly calculate the set of such M for the theta series of the Niemeier lattices.
