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).

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

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) (1998), 201--275.

(arXiv).

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