Consider the sum
$$\sum_{n\ne 0, n\in \mathbb Z}|n|^{-1}(|n|+Y)^2|W_s(nz)|^2$$
where $W_s(nz)=2(ny)^{1/2}K_{s−1/2}(2πny) e(x) $ is the Whittaker function, where $K_{s−1/2}$ is the K-Bessel function.
Based on what I have seen in a book (Iwaniec's Spectral methods in automorphic forms, page 56), it seems that this sum can be bounded by a constant multiple of
$$e^{\pi |s|} |s| y^{−1} (y^{−1}|s| + Y )^2.$$
But I am not sure what estimate for whittaker/K-Bessel function we are using here. Any effort will be appreciated!