I am reading Jacques Martinet Perfect Lattices in Euclidean space and I have troubles to understand his definition of Spectrum of a lattice. It is possible to have an example or another way to think about it? Here is the definition:
Definition 1.4.3 Let $\Lambda$ be a lattice, and let $\mathcal{F}$ be a finite family of nonzero vectors in $\Lambda$, invariant under the symmetry $x \to -x$. The spectrum (with respect to $\mathcal{F}$) of a vector $x \in \Lambda$ is the sequence $(a_1,\alpha_1),\ldots,(a_k,\alpha_k),0\leq \alpha_1<\dots<\alpha_k)$ where the $\alpha_i$ are the absolute values of the scalar products $x\cdot y$, $y \in \mathcal{F}$, and $a_i$ is the number of pairs $\pm y, y \in \mathcal{F}$ with $|x\cdot y|=\alpha_i$. The spectrum of $\mathcal{F}$ is the sequence $(m_i,a_i,\alpha_i)$ where $m_i$ is the number of pairs $\pm x, x\in \mathcal{F}$ with spectrum $(a_i,\alpha_i)$. The minimal spectrum of $\Lambda$ (or simply the spectrum of $\Lambda$) is the spectrum of the set $S(\Lambda)$ of minimal vectors in $\Lambda$.