Newform subspace (awaiting review)

The newform subspace of a newform $f$ in $S_{k,j}^{\rm new}(K(N),\chi)$ is the subspace generated by $T_{(1,1,p,p)}(f)$ and $T_{(1,p,p,p^2)}(f)$ for all Hecke operators $T_{(1,1,p,p)}, T_{(1,p,p,p^2)}$ 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.