A Poisson-Jacobi-type transformation for the sum $\sum_{n=1}^\infty n^{-2m} \exp (-an^2}$ for positive integer $m$

We obtain an asymptotic expansion for the sum $$S(a;w)=\sum_{n=1}^\infty \frac{e^{-an^2}}{n^{w}}$$ as $a\rightarrow 0$ in $|\arg\,a|<\pi/2$ for arbitrary finite $w>0$. The result when $w=2m$, where $m$ is a positive integer, is the analogue of the well-known Poisson-Jacobi transformation for the sum with $m=0$. Numerical results are given to illustrate the accuracy of the expansion.