Sign changes of coefficients of half-integral weight Hecke eigenforms

Abstract

Let f \mathfrak {f} be a cusp form of half-weight k + 1 / 2 k+1/2 and at most quadratic nebentype character whose Fourier coefficients are denoted as a f ( n ) \mathfrak {a}_\mathfrak {f}(n) . We study sign changes of the family { a f ( t p 2 ) } p ∈ P \{\mathfrak {a}_\mathfrak {f}(tp^2)\}_{p\in \mathbb {P}} where t t is a square-free number and p p runs through the prime numbers. By taking use of the relationship between the cusp form f \mathfrak {f} of half-integral weight and the Shimura lift f t f_t , we establish that under certain conditions, the Hecke eigenforms of half-integral weight could be determined by sign changes of the sequence { a f ( t p 2 ) } p ∈ P \{\mathfrak {a}_\mathfrak {f}(tp^2)\}_{p\in \mathbb {P}} .