Let $F=\mathbb{Q}(\sqrt{D})$ be a real quadratic field and consider the classical Hilbert modular forms over $F$. Let $\varepsilon_0>1$ be the fundamental unit of $F$ and write $d=\varepsilon_0\sqrt{D}$. For every $a\in F$, write $a^\prime$ as its Galois conjugate.
In the book by Paul Garrett, the Hecke action on the classical Hilbert modular forms with parallel weights is defined when the narrow class number of $F$ equals $1$. If I understand correctly, the definition for $T_\mathfrak{p}$ boils down to \begin{align*} (T_\mathfrak{p}f)(z)=\mathrm{N}(\mathfrak{p})^{k-1}f(pz)+\mathrm{N}(\mathfrak{p})^{-1}\sum_{\alpha\in\mathcal{O}_F/\mathfrak{p}}f\bigg(\frac{z+\alpha}{p}\bigg) \end{align*} where $p$ is a totally positive generator for $\mathfrak{p}$. And to compute the action of the Hecke operators $T_\mathfrak{p}$ on the Fourier coefficients, one should use a Gauss-like sum as the following: for any totally positive element $n\in\mathcal{O}_F$, \begin{align*} \sum_{\alpha\in\mathcal{O}_F/\mathfrak{p}}e^{2\pi i \big(\frac{n\alpha}{pd}+\frac{n^\prime \alpha^\prime}{p^\prime d^\prime}\big)}=\begin{cases} \mathrm{N}(\mathfrak{p})&\text{ if }p|n, \\ 0&\text{ if }p\nmid n. \end{cases} \end{align*} (One should compare this with the sum $\sum_{a=0}^{p-1}e^{2\pi ina/p}$ for rational prime $p$ and rational integer $n$, which is needed for the elliptic modular forms.)
However, this seems to be wrong when $F=\mathbb{Q}(\sqrt{2})$ and $\mathfrak{p}=(2-\sqrt{2})$. In this case, $p=2-\sqrt{2}$, $d=(1+\sqrt{2})\cdot\sqrt{2}=2+\sqrt{2}$, and $\mathcal{O}_F/\mathfrak{p}=\{0,1\}$. The sum then becomes \begin{align*} 1+e^{\pi i(n+n^\prime)}=2 \end{align*} for all $n$, since $2|n+n^\prime$ regardless of whether $p$ divides $n$ or not. I am thus not sure if I make any mistake along the way.
In any case, I would truly appreciate any comments on computing the Fourier coefficients of $T_\mathfrak{p}f$ in this scenario. It would be great if there are references where the explicit computations are written down. Thank you so much!