The newform subspace of a newform $f$ in $S_k^{\rm new}(N,\chi)$ is the subspace generated by $T_p(f)$ for all Hecke operators $T_p$ for $p$ coprime to the level, equivalently, the subspace generated by the Galois conjugates of $f$.

Every newspace has a canonical decomposition into newform subspaces.