Hecke generalized the argument that you mention to prove quadratic reciprocity relative to any given number field $K$ (see, e.g. his Lectures on the Theory of Algebraic Numbers).
In The Fourier-Analytic Proof of Quadratic Reciprocity Michael C. Berg describes the subsequent development of this line of research. Quoting from the book's summary:
The relative quadratic case was first settled by Hecke in 1923, then recast by Weil in 1964 into the language of unitary group representations. The analytic proof of the general n-th order case is still an open problem today, going back to the end of Hecke's famous treatise of 1923.