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1. Historical introduction. In 1989 Knopp [6] found explicit formulas for the Fourier coefficients of an arbitrary cusp form and more generally, but conditionally, of a holomorphic modular form (with a possible pole at i∞) on the full modular group, Γ (1), of weight k, 4/3 < k < 2, and multiplier system v. He assumed that there are no nontrivial cusp forms on Γ (1) of complementary weight 2−k and conjugate multiplier system v. In our initial paper we remove this assumption and capture the Fourier coefficients of an arbitrary “Niebur modular integral” on Γ (1) of weight k, 1 < k < 2. En route we also obtain expressions for the Fourier coefficients of an arbitrary cusp form on Γ (1) of weight k, 0 < k < 1. In particular we present formulas for the Fourier coefficients of η(τ), 0 < r < 2, where η(τ) is the Dedekind eta-function. An actual formula for the Fourier coefficients of an arbitrary modular form, even in the case of the full modular group, is not always available. For forms of weight greater than two the problem was solved by Petersson [11], who introduced the (parabolic) Poincaré series. Additionally, by considering a nonanalytic version of this series, he derived the coefficients of certain forms of weight two [12]. By integrating one of these forms, Petersson [12, p. 202] was the first to find the coefficients of the absolute modular invariant J(τ). For forms of negative weight Rademacher and Zuckerman [18] discovered expressions for the coefficients by relying on the circle method. Furthermore, Rademacher [15] employed a sharpened version of this method to rediscover Petersson’s formula for J(τ). We remark that both approaches

The Fourier coefficients of modular forms and Niebur modular integrals having small positive weight, I