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 ${\\rm Leech}$ lattice, unique up to scalar, which invariant by orthogonal group of Leech}$. The harmonic Siegel theta series built from these forms are modular cuspforms weight $13$ for Sp}{2g}(\\mathbb{Z})$. We prove that they eigenforms, determine one their Fourier coefficients, give informations about standard L}$-functions. These interesting since, recent work authors, only Sp}{2n}(\\mathbb{Z})$, any $n\\geq 1$.