Siegel newform (awaiting review)

Let \(M_{\rho}(K(N))\) be the space of Siegel modular forms of weight $\rho$ and paramodular level $N$.

If $M$ is a proper divisor of $N$, then for every divisor $D \mid (N/M)$, there are maps from $M_{\rho}(K(M))$ to $M_{\rho}(K(N))$. Such modular forms are said to be old, and together they span a subspace $M_{\rho}^{\rm old}(K(N)) \subseteq M_{\rho}(K(N))$.

The orthogonal complement of the $S_{\rho}^{\rm old}(K(N))$ in $S_{\rho}(K(N))$ with respect to the Petersson scalar product is denoted $S_{\rho}^{\rm new}(K(N))$, and we have the decomposition \[ S_{\rho}(K(N))=S_{\rho}^{\rm old}(K(N))\oplus S_{\rho}^{\rm new}(K(N)). \] into the old subspace and new subspace.

A newform is a cusp form $f\in S_{\rho}^{\rm new}(K(N))$ that is also an eigenform of all Hecke operators, normalized so that the $q$-expansion $f(\tau)=\sum a_{T} q^T$, where $q=e^{2\pi i {\rm tr}(T \tau) }$, begins with the coefficient $a_1=1$. The newforms are a basis for the vector space $S_{\rho}^{\rm new}(K(N))$.