Space of paramodular newforms (awaiting review)

For each level $N$, weight $k$, and character $\chi$ the space $S_{k,j}(K(N),\chi)$ of cuspidal modular forms can be decomposed as an internal direct sum \[ S_{k,j}(K(N),\chi) = S_{k,j}^{\rm old}(K(N),\chi) \oplus S_{k,j}^{\rm new}(K(N),\chi). \] The old subspace $S_{k,j}^{\rm old}(K(N),\chi)$ is generated by all elements of $S_{k,j}(K(N),\chi)$ which are in the image of the level-raising operators $\eta, \theta, \theta'$.

The new subspace $S_{k,j}^{\rm new}(K(N),\chi)$ is the orthogonal complement of $S_{k,j}^{\rm old}(K(N),\chi)$ with respect to the Petersson inner product on $S_{k,j}(K(N),\chi)$. The newforms in $S_{k,j}(K(N),\chi)$ are a canonical basis for this subspace.