Lattice invariants from the heat kernel (II)

Given an integral lattice $\Lambda$ of rank $n$ and a finite sequence $m_1 \leq m_2 \leq ... \leq m_k$ of natural numbers we construct a modular form $\Theta_{m_1,m_2,...,m_k,\Lambda}$ of level $N=N(\Lambda)$. The weight of this modular form is $nk/2+\sum_{i=1}^k m_k$. This construction generalizes the theta series $\Theta_\Lambda$ of integral lattices, because $\Theta_\Lambda = \Theta_{0,\Lambda}$. We give the $q$-expansions of the modular forms $\Theta_{m,m,\Lambda}$, and $\Theta_{1,1,1,\Lambda}$ and show that (up to some scaling) they are given by power series with integer coefficients.