In the year 2002, Lin detected a nontrivial family in the stable homotopy groups of spheres $\pi_{t-6}S$ which is represented by $h_ng_0\tilde{\gamma}_{3} \in {\rm Ext}_A^{6,t}(Z_p,Z_p)$ in the Adams spectral sequence, where t=2pⁿ(p-1)+6(p²+p+1)(p-1) and p≥7 is a prime number. This article generalizes the result and proves the existence of a
new nontrivial family of filtration s+6 in the stable homotopy groups of spheres $\pi_{t_1-s-6}S$ which is represented by $h_ng_0\tilde{\gamma}_{s+3}\in {\rm Ext}_A^{s+6,t_1}(Z_p,Z_p)$
in the Adams spectral sequence, where n≥4, 0≤ s< p-4, t₁=2pⁿ(p-1)+2(p-1)((s+3)p²+(s+3)p+(s+3))+s.