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 $a(f,n)\neq 0$. This significantly improves the already known existential and quantitative result from previous works.