A domain containing all zeros of the partial theta function

We consider the partial theta function, i.e., the sum of the bivariate series $\theta(q,z):=\sum_{j=0}^{\infty}q^{j(j+1)/2}z^j$ for $q\in(-1,1)$, $z\in\mathbb{C}$. We show that for any value of the parameter $q\in(0,1)$, all zeros of the function $\theta(q,.)$ belong to the domain $\{\operatorname{Re}~z < 0,|\operatorname{Im}~z|<132\}\cup\{\operatorname{Re}~z\geq 0,|z|<18\}$. For $q\in(-1,0)$, all zeros belong to the strip $\{|\operatorname{Im}~z|<132\}$.