Lattice invariants from the heat kernel (II)

Given an integral lattice $\mathrm{\Lambda }$ of rank $n$ and a finite sequence $m_1 \leq m_2 \leq … \leq m_k$ofnaturalnumbersweconstructamodularform$\Theta_{m_1,m_2,…,m_k,\Lambda}$oflevel$N=N(\Lambda)$. The weight of this modular form is $nk\mathrm{/}2+{\sum }_{i=1}^k{m}_k$. This construction generalizes the theta series ${\mathrm{\Theta }}_{\mathrm{\Lambda }}$ of integral lattices, because $\Theta_\Lambda = \Theta_{0,\Lambda}$\mathrm{.}Wegivethe$q$-expansionsofthemodularforms$\Theta_{m,m,\Lambda}$,and$\Theta_{1,1,1,\Lambda}$ and show that (up to some scaling) they are given by power series with integer coefficients.