Modular Forms with Only Nonnegative Coefficients

Abstract We study modular forms for $${\text{ SL }}_2(\mathbb {Z})$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mrow> <mml:mspace/> <mml:mtext>SL</mml:mtext> <mml:mspace/> </mml:mrow> <mml:mn>2</mml:mn> </mml:msub> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>Z</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> 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 $${\text{ SL }}_2(\mathbb {Z})$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mrow> <mml:mspace/> <mml:mtext>SL</mml:mtext> <mml:mspace/> </mml:mrow> <mml:mn>2</mml:mn> </mml:msub> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>Z</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> 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 n th Fourier coefficient of any form with no negative Fourier coefficients.