The paper asks how many leading Fourier coefficients of a modular form of weight k for SL2(Z) must be checked as nonnegative to guarantee that all coefficients are nonnegative. This quantity, A(k), is a 'nonnegativity Sturm bound'.

The set of normalized forms with nonnegative coefficients is shown to be a bounded finite convex polytope, which proves that A(k) is well-defined. A lower bound comes from positivity of the Poincaré series coefficients, using Bessel function estimates. An upper bound comes from bounds on cusp form coefficients in terms of the first floor(k/12) coefficients. For small weights, A(k) is computed exactly; for 36 ≤ k ≤ 88, Lean's linarith tactic finds the smallest N whose first N coefficient inequalities imply the remaining ones up to a computed bound B(k).

The paper proves (k-1)^2/(16π^2) < A(k) ≤ k^4(log k + log log k)^2/7316. It also gives an upper bound on the nth coefficient of any form with nonnegative coefficients. Exact values of A(k) are computed for 12 ≤ k ≤ 88.