  We calculate the Jacobi Eisenstein series of weight $k \ge 3$ for a certain
representation of the Jacobi group, and evaluate these at $z = 0$ to give
coefficient formulas for a family of modular forms $Q_{k,m,\beta}$ of weight $k
\ge 5/2$ for the (dual) Weil representation on an even lattice. The forms we
construct always contain all cusp forms within their span. We explain how to
compute the representation numbers in the coefficient formulas for
$Q_{k,m,\beta}$ and the Eisenstein series of Bruinier and Kuss $p$-adically to
get an efficient algorithm. The main application is in constructing automorphic
products.
