Hecke-Siegel type threshold for square-free Fourier coefficients: an improvement

Abstract

We prove that if $f$ is a non zero cusp form of weight $k$ on ${\Gamma }_0(N)$ with character $\chi$ such that $N/(\text{conductor }\chi )$ square-free, then there exists a square-free $n{\ll }_{\epsilon }{k}^{3+\epsilon }{N}^{7/2+\epsilon }$ such that $a(f,n)\ne 0$. This significantly improves the already known existential and quantitative result from previous works.