For a nontrivial additive character \(\psi\) of \({\mathbb F}_q\) and a nontrivial multiplicative character \(\chi\) of \({\mathbb F}_q^\times\), define
\[
G(\chi,\psi)=\sum_{x\in{\mathbb F}_q^\times}\chi(x)\psi(x),\qquad
\gamma(\chi,\psi)=q^{-1/2}G(\chi,\psi).
\]
The exact identities are
\[
|G(\chi,\psi)|^2=q,\qquad
\gamma(\chi,\psi)\gamma(\chi^{-1},\psi)=\chi(-1).
\]
The magnitude calculation substitutes \(x=ty\) in the double sum. The inner sum is \(q-1\) for \(t=1\) and \(-1\) otherwise; multiplicative-character orthogonality supplies the result.
The normalized scalar therefore lies on the unit circle. Its value depends on the declared additive character and lifting normalization. Finite Gauss phases capture a residue-field part of normalization.
Scope
A full wild local factor also contains ramification depth, additive-character conductor, measure normalization, and pair interaction data. One finite phase is not that full factor.