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Infinite Product Exponents for Modular Forms

Recently, D. Choi obtained a description of the coefficients of the infinite product expansions of meromorphic modular forms over $Γ_0(N)$. 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 $1$.