Is there any reference where I can find a complete description of the suspension homomorphisms $\Sigma:\pi_{n+k}(S^n)\to \pi_{n+k+1}(S^{n+1})$ between homotopy groups of spheres, for small values of $k$ and $n$?
More specifically, I am looking for the following formulation of $\Sigma$. The groups are finitely presented, so I would like a description of $\Sigma$ that expresses the image of each generator of $\pi_{n+k}(S^n)$ in terms of the generators of $\pi_{n+k+1}(S^{n+1})$.