  Recently, D. Choi obtained a description of the coefficients of the infinite
product expansions of meromorphic modular forms over $\Gamma_0(N)$. Using this
result, we provide some bounds on these infinite product coefficients for
holomorphic modular forms. We give an exponential upper bound for the growth of
these coefficients. We show that this bound is also a lower bound in the case
that the genus of the associated modular curve $X_0(N)$ is $0$ or $1$.
