The product expansion and the weakly holomorphic (Jacobi) form it lifts from appear as Example 2.4 in Gritsenko, Nikulin - *Automorphic Forms and Lorentzian Kac--Moody Algebras II*, Internat. J. Math. 9(2), 201--275.

[(arXiv)][1].

The square root of $\chi_{10}$ is denoted there by $\Delta_5(Z)$.


  [1]: https://arxiv.org/abs/alg-geom/9611028