Divisors of modular forms on Γ0(4)

There is a relationship between the values of a sequence of modular functions at points in the divisor of a meromorphic modular form and the exponents of its infinite product expansion. We make this relationship explicit for the case of modular forms on the congruence subgroup Γ0(4). We also consider some applications to classical number theoretic functions such as the number of representations of an integer as a sum of squares.