I need help to compute the following integral
$$\int_{\mathbb{R}^n}\widehat{\varphi}(\xi)e^{-4\pi^2 i|\xi|^2 t}e^{2\pi i\xi \cdot x} \mathrm{d}\xi $$
where $\widehat{\varphi}$ is the Fourier transform of $\varphi \in H^1(\mathbb{R}^n)$ (or if you prefer $\varphi \in \mathcal{S}(\mathbb{R}^n)$) i.e. $$\widehat{\varphi}(\xi)=\int_{\mathbb{R}^n}\varphi(x)e^{-2\pi ix\cdot\xi}\mathrm{d}x. $$
Thank you.