On the vanishing of coefficients of the powers of a theta function

A result on the Galois theory of $q$-difference equations leads to the following question: if $0<|q|<1$ and if one sets
$$ \theta_q(z):=\sum\limits_{m\in\mathbb{Z}} q^{m(m-1)/2} z^m, $$
can some coefficients of the Laurent series expansion of $\theta_q^n(z)$, $n \in \mathbb{N}^*$, vanish? We give a partial answer. This is a joint work with Jacques Sauloy (see arXiv:2007.16092[math.DS]).