I am working on Tate's thesis, and I have some problems with computations, yet the result seems to be a good natural motivation for introducing the arithmetic conductor of a character.

Let $F$ be a non-archimedean local field, and $\psi$ a non-trivial additive character. Define the $\psi$-Fourier transform for a locally constant and compactly supported function on $F$ to be $$\widehat{\Phi}(x) = \int_F \Phi(y)\psi(xy)dy$$

We normalize the involved Haar measure so that the Fourier inversion formula holds, that is to say $$\widehat{\widehat{\Phi}}(x) = \Phi(-x)$$

I would like to understand why this implies that the ring of integers $\mathcal{O}$ have volume $q^{c(\psi)/2}$?

Recall that $c(\psi)$ is defined by $\psi$ trivial on $\mathfrak{p}^{c(\psi)}$ and not on $\mathfrak{p}^{c(\psi)-1}$, where $\mathfrak{p}$ is the maximal ideal of $\mathcal{O}$. I tried manipulating the integrals splitting into classes modulo $\mathfrak{p}^{c(\psi)}$, but I cannot make it work properly.