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.

**Authors:**

**Knowl status:**

- Review status: beta
- Last edited by Eran Assaf on 2022-08-31 09:55:03

**Referred to by:**

Not referenced anywhere at the moment.

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