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Rankin and Swinnerton-Dyer (1970) prove that all zeros of the Eisenstein series E k in the standard fundamental domain for Γ lie on A:= {e iθ : π/2 ≤ θ ≤ 2π/3). In this paper we generalize their theorem, providing conditions under which the zeros of other modular forms lie only on the arc A. Using this result we prove a speculation of Ono, namely that the zeros of the unique gap function in M k , the modular form with the maximal number of consecutive zero coefficients in its q-expansion following the constant 1, has zeros only on A. In addition, we show that the j-invariant maps these zeros to totally real algebraic integers of degree bounded by a simple function of weight k.

A generalization of a theorem of Rankin and Swinnerton-Dyer on zeros of modular forms