Higher level quadratically twisted Gauss sums and totally isotropic subspaces

We consider a generalized Gauss sum supported on matrices over a number field. We evaluate this Gauss sum and relate it to the number of totally isotropic subspaces of related quadratic spaces. Then we consider a further generalization of such a Gauss sum, realizing its value in terms of numbers of totally isotropic subspaces of related quadratic spaces.