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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, thenthere 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 knownexistential and quantitative result from previous works.

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