Siegel modular forms of weight 13 and the Leech lattice

For $g=8,12,16$ and $24$, there is a nonzero alternating $g$-multilinear form on the $\mathrm{L}\mathrm{e}\mathrm{e}\mathrm{c}\mathrm{h}$ lattice, unique up to a scalar, which is invariant by the orthogonal group of $\mathrm{L}\mathrm{e}\mathrm{e}\mathrm{c}\mathrm{h}$. The harmonic Siegel theta series built from these alternating forms are Siegel modular cuspforms of weight $13$ for ${\rm Sp}_{2g}($\mathbb{Z}$)$. We prove that they are nonzero eigenforms, determine one of their Fourier coefficients, and give informations about their standard ${\rm L}$-functions. These forms are interesting since, by a recent work of the authors, they are the only nonzero Siegel modular forms of weight $13$ for ${\mathrm{S}\mathrm{p}}_{2n}(\mathbb{Z})$, for any $n\ge 1$.