Let \(f:(\mathbb{R}^{d})^{k} \rightarrow \mathbb{R}\) be a compactly supported continuous function. Then for \(k \in \{1,\dots,d-1\}\), we get that \[\begin{align} & \int_{SL_d(\mathbb{R})/SL_d(\mathbb{Z})}\sum_{(x_1,x_2,\dots,x_k) \in (g \mathbb{Z}^{d})^k}f(x_1,x_2,\dots,x_k) \\ = & f(0,0,\dots,0) + \sum_{m=1}^{{n}}\sum_{\substack{D \in M_{m \times k }(\mathbb{Q}) \\ \mathop{\mathrm{rank}}(D) = m \\ D \text{ is row reduced echelon}}} N(D)^{-d} \int_{x \in K_\mathbb{R}^{d \times m }} g(x D ) dx, \end{align}\] where \(N(D)\) is the index of the sublattice \(\{ C \in M_{1 \times m }(\mathbb{Z}) \ | \ C \cdot D \in M_{1 \times n}(\mathbb{Z})\}\) in \(M_{1 \times m}(\mathbb{Z})\).