We study modular forms for \({\text{ SL }}_2(\mathbb {Z})\) with no negative Fourier coefficients. Let A(k) be the positive integer where if the first A(k) Fourier coefficients of a modular form of weight k for \({\text{ SL }}_2(\mathbb {Z})\) are nonnegative, then all of its Fourier coefficients are nonnegative, so that A(k) can be interpreted as a “nonnegativity Sturm bound”. We give upper and lower bounds for A(k), as well as an upper bound on the nth Fourier coefficient of any form with no negative Fourier coefficients.
