Interchange of order of integration for $\int_0^\infty y^{s-1} e^{-ay} \int_0^\infty \frac{\sin(2yx)\cos(\pi x^2)}{\sinh(\pi x)}dx dy $

Question

I have been trying to interchange the order of integration for the integral: $$\int_0^\infty y^{s-1} e^{-ay} \int_0^\infty \frac{\sin(2yx)\cos(\pi x^2)}{\sinh(\pi x)}dx dy $$

I am unable to find a proof for the same. Using Mathematica I was able to see that the values are the same even when we interchange the order of integration upto some $s,a \in \mathbb{R}$ with $a>0$.

But for larger values of $s$, mathematica says it is divergent, so I am interested in the condition on $s,a$ such that we can interchange the order of integration.

Any help is highly appreciated.