Poisson summation formula is widely used in many parts of the litterature, its classical formulation for sums over integers as well as its adelic version. What is its corresponding form for more general number fields? I was expecting to find it used in proofs of the functional equation for Dedekind L-functions attached to a number field, but I only found Poisson formulas for elements summed over an ideal rather than sums over ideals.

Is there anything of the form, for $K$ a number field, $E$ its ring of integer, and $f$ a suitable real valued function, $$\sum_{a \subset E} f(a) = C \sum_{a' \in E'} \hat{f}(a')$$

where the sum runs over integer ideals $a$ of $E$? What are the notions of dual and Fourier transforms here? Should we rather have something with norms of ideals instead of ideals? That is to say, is $N$ is the norm of the number field $K$,

$$\sum_{a \subset E} f(N(a)) = C \sum_{a' \in E'} \hat{f}(N(a'))$$

Thanks in advance, I do not find anything suitable in the litterature.