Atkin-Lehner operator on supercuspidals

Suppose $$f$$ is a normalized cuspidal eigenform of level $$p^2N$$ ($$p\nmid N$$) and trivial character, such that the corresponding representation at $$p$$ is supercuspidal. Now suppose $$\chi$$ is primitive Hecke character of conductor $$p$$. We can apply the usual twisting operator by $$\chi$$ or $$\chi^{-1}$$ to $$f$$ to obtain normalized eigenforms $$f_\chi$$ and $$f_{\chi^{-1}}$$ with character $$\chi^2$$ and $$\chi^{-2}$$ respectively. Now we consider the adelic Atkin-Lahner operator given by the matrix $$\begin{pmatrix} 0 & 1 \\ p^2 & 0\end{pmatrix}$$ at $$p$$. Then one sees that it maps $$f_\chi$$ to $$f_{\chi^{-1}}\otimes(\chi^2\circ\det)$$ with some scalar multiple. My question is what is this scalar multiple? In particular is it a $$p$$-adic unit? (edited based on comment)

What you are asking for is a formula for the local epsilon-factors $$\varepsilon(\pi \otimes \chi)$$ where $$\pi = \pi_{f, p}$$ is the local component of $$f$$ at $$p$$. This is a deep question: it has to be, in some sense, since you can recover $$\pi$$ uniquely if you know the epsilon-factors of all its twists (Jacquet's local converse theorem).
Anyway, since you are assuming that $$f$$ has level $$Np^2$$ and trivial character, the representation $$\pi$$ is not too nasty: it's a "depth 0 supercuspidal", arising from a character $$\eta: \mathbf{F}_{p^2}^\times \to \mathbf{C}^\times$$ trivial on $$\mathbf{F}_p^\times$$. There is a paper by Jared Weinstein and me which gives an algorithm for computing $$\eta$$. Once you have this, there is a formula for the epsilon-factors in terms of Gauss sums over $$\mathbf{F}_{p^2}^\times$$. Up to some normalisation factor, $$\varepsilon(\pi \otimes \chi)$$ will be something like $$\sum_{x \in \mathbf{F}_{p^2}^\times} \eta(x) \chi(\mathrm{Nm}(x)) e^{2\pi i \mathrm{Tr}(x)/p}$$.