Let $S_k(N)$ be the space of weight $k$ cusp forms with respect to $\Gamma_0(N)$ for $1\leq N\leq 100$. We have a decomposition:
$$S_k(N)= S_k^{\text{new}}\oplus S_k^{\text{old}}$$ Suppose that we know the eigenbasis of new forms of $S_k(N)$, for all $1\leq N\leq 100$. How do we find eigenbasis of old forms?