5-adic order of $\zeta-f(\overline 2)\zeta^2+f(\overline 2)\zeta^3+\zeta^4$

Question

Let $\pi:$(the 5-adic integers)$\to \mathbb{Z}/5\mathbb{Z}$ be the reduction map.

Let $f:\mathbb{Z}/5 \mathbb{Z} \to $ (the 5-adic integers) have the following properties $\forall x,y\in \mathbb{Z}/5\mathbb{Z}$

1. $f(x)f(y)=f(xy)$

2. $f(x)^{p-1}=1$ unless $f(x)=0$

3. $\pi(f(x))=x$

Also, let us adjoin a primitive $5$th root of unity, call it $\zeta_5=\zeta$, to the 5-adic integers.

I want to find the 5-adic order of $\zeta-f(\overline 2)\zeta^2+f(\overline 2)\zeta^3+\zeta^4$

I see that the order of $1-\zeta^i$ is $\frac 1 4$ for $i\not\equiv0$ (mod 5).

I also see that $f(\overline 4)=-1$ and that therefore $f(\overline 2)^2=-1$ and $f(\overline 3)^2=-1$. Perhaps this can be used to rewrite the sum $\zeta-f(\overline 2)\zeta^2+f(\overline 2)\zeta^3+\zeta^4$ in a "better" way.