A paramodular group $\Gamma$ of degree $g$ preserves an integral alternating $2g \times 2g$ matrix $T=\left( \begin{array}{ll} 0 & d \\ -d & 0 \end{array}\right)$ for $d=\mathrm{diag}(d_1,d_2,\dots,d_g)$ with $d_j\mid d_{j+1}$ for $1\leqslant j \leqslant g-1$: $ \Gamma=\left\{\gamma \in \GL(2g,\mathbb{Z})\, : \, \gamma^t T \gamma=T\right\}. $ For $\delta= \left( \begin{array}{ll} 1_g & 0 \\ 0 & d \end{array}\right)$, the conjugate $\Gamma^{\mathrm{para}}(d)=\delta\Gamma\delta^{-1}\subseteq \mathrm{Sp}(2g,\Q)$ is also referred to as a paramodular group.

In degree $2$, $K(N)$$= \Gamma^{\mathrm{para}}(1,N)$.