Let $k \ge 2$ and $N \ge 1$. The space $S_k(\Gamma_0(N))$ decomposes as a direct sum of old subspaces generated by degeneracy maps from lower levels and a new subspace
The new subspace has a basis of simultaneous eigenforms for all Hecke operators $T_\ell$ with $\ell \nmid N$ and all operators $U_p$ with $p \mid N$; these eigenforms are normalised by $a_1=1$ and are the newforms of level $N$.