Formulas for the coefficients of half-integral weight harmonic Maaß forms

Algebraic Formulas for the Coefficients of Half-integral Weight Harmonic Weak Maass Forms

We prove that the coefficients of certain weight −1/2 harmonic Maass forms are “traces” of singular moduli for weak Maass forms. To prove this theorem, we construct a theta lift from spaces of weight −2 harmonic weak Maass forms to spaces of weight −1/2 vectorvalued harmonic weak Maass forms on Mp2(Z), a result which is of independent interest. We then prove a general theorem which guarantees (...

Coefficients of Half-integral Weight Modular Forms

In this paper we study the distribution of the coefficients a(n) of half integral weight modular forms modulo odd integers M . As a consequence we obtain improvements of indivisibility results for the central critical values of quadratic twists of L-functions associated with integral weight newforms established in [O-S]. Moreover, we find a simple criterion for proving cases of Newman’s conject...

Sign Changes of Coefficients of Half Integral Weight Modular Forms

For a half integral weight modular form f we study the signs of the Fourier coefficients a(n). If f is a Hecke eigenform of level N with real Nebentypus character, and t is a fixed square-free positive integer with a(t) 6= 0, we show that for all but finitely many primes p the sequence (a(tp2m))m has infinitely many signs changes. Moreover, we prove similar (partly conditional) results for arbi...

Fourier Coefficients of Half - Integral Weight Modular Forms modulo `

S. Chowla conjectured that every prime p has the property that there are infinitely many imaginary quadratic fields whose class number is not a multiple of p. Gauss’ genus theory guarantees the existence of infinitely many such fields when p = 2, and the work of Davenport and Heilbronn [D-H] suffices for the prime p = 3. In addition, the DavenportHeilbronn result demonstrates that a positive pr...

MAT1200: Modular Forms of Half Integral Weight