Characterization of nuclearity for Beurling-Bj\"{o}rck spaces

We characterize the nuclearity of the Beurling-Bj"{o}rck spaces ${\mathcal{S}}_{(\eta )}^{(\omega )}({\mathbb{R}}^d)$ and ${\mathcal{S}}_{\{\eta \}}^{\{\omega \}}({\mathbb{R}}^d)$ in terms of the defining weight functions $\omega$ and $\eta$.