A note on integral Euclidean lattices in dimension 3 (Q818738)

Given an integral lattice \(\Lambda\) in Euclidean \(n\)-space, its theta series is defined to be \(\Theta_\Lambda=\sum_{k=0}^\infty a_kq^k\) where \(a_k\) denotes the number of lattice points at a distance \(\sqrt{k}\) from the origin. In even dimension, many examples of such lattices are known where \(\Theta_\Lambda\equiv1\text{ (mod }p\))