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

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