On Davenport's Multiplicative Number Theory, Page $68$, It tried to derive a functional equation that relates $\psi(x,\chi)$ and $\psi(x^{-1},\overline{\chi})$. However I am stuck. I tried using the definitions but cannot understand.
How it writes the equality
$$\tau(\overline{\chi})\psi(x,\chi)=\sum_{m=1}^{q} \overline{\chi}(m) \sum_{n=-\infty}^{\infty} e^{-n^2 \pi x/q+ 2 \pi i mn/q}$$