A short note on sign changes and non-vanishing of Fourier coefficients of half-integral weight cusp forms

Non-vanishing and sign changes of Hecke eigenvalues for half-integral weight cusp forms

In this paper, we consider three problems about signs of the Fourier coefficients of a cusp form f with half-integral weight: – The first negative coefficient of the sequence {af(tn)}n∈N, – The number of coefficients af(tn ) of same signs, – Non-vanishing of coefficients af(tn ) in short intervals and in arithmetic progressions, where af(n) is the n-th Fourier coefficient of f and t is a square...

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...

Changes of Fourier Coefficients 6 of Half - Integral Weight Cusp Forms

Let k be a positive integer. Suppose that f is a modular form of weight k + 1/2 on Γ0(4). The Shimura correspondence defined in [12] maps f to a modular form F of integral weight 2k. In addition, if f is an eigenform of the Hecke operator Tk+1/2(p), then F is also an eigenform of the Hecke operators T2k(p) with the same eigenvalue. For more on half-integral weight modular forms, see [12]. Let t...

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...

Non-vanishing Modulò of Fourier Coeecients of Half-integral Weight Modular Forms