Sign changes of coefficients of half-integral weight Hecke eigenforms

Abstract:

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