For an odd prime \(p\), a symplectic space \(V\) of dimension \(2r\), and fixed-point-free \(g\in{\operatorname{Sp}}(V)\), the declared finite Weil normalization gives
\[
\Phi_p(g,V)=\left(\frac{-1}{p}\right)^r
\left(\frac{\det(g-I)}p\right).
\]
With \(P_g(X)=\det(XI-g)\), even dimension gives \(\det(g-I)=P_g(1)\). The phase is therefore read from a characteristic polynomial and a square class.
For a hyperbolic space \(U\oplus U^*\) with \(g=(a,(a^{-1})^*)\) and \(a-I\) invertible,
\[
\Phi_p(g,U\oplus U^*)=\left(\frac{\det a}{p}\right).
\]
If \(g\) has order prime to \(p\), then
\[
V=\ker(g-I)\perp\operatorname{im}(g-I).
\]
The moving summand is nondegenerate and fixed-point-free. Orthogonal block phases multiply.
Scope
The fixed-point-free formula applies to the moving part. The full trace with a fixed part requires its declared normalization data.