Shimura decomposition (awaiting review)

Let $k$ be an odd integer. Let $S_0(N, \chi)$ denote the subspace of $S_{3/2}(N, \chi)$ spanned by theta series if $k=3$, and let $S_0(N, \chi)=0$ if $k>3$. Let $S_0'(N, \chi)$ denote the orthogonal complement of $S_0(N, \chi)$ in $S_{3/2}(N, \chi)$. Then \[ S_0'(N, \chi) = \oplus_F S_{k/2}(N, \chi, F), \] where $F$ runs over all newforms of weight $k-1$, level dividing $N/2$ and character $\chi^2$, and the space $S_{k/2}(N, \chi, F)$ is defined by \[ S_{k/2}(N, \chi, F) =\{ g \in S_0'(N, \chi): T_{p^2} g = \lambda_p(F) g \quad \forall\, p\nmid N\}. \]