In the paper *Construction of some families of 2-dimensional crystalline representations* by Berger-Li-Zhu, a very explicit description is given for the $(\phi,\Gamma)$-module attached to some 2-dimensional crystalline representations. In particular, if one takes the (local) Galois representation arising from a modular form $f$ of level prime to $p$ with $a_p(f)=0$, then a very simple formula is given for the $\phi$ and $\Gamma$ action on the $(\phi,\Gamma)$-module $D(V)$.

Further, if one takes the $\psi=1$ invariants of $D(V)$ one should recover $H^1_{Iw}(V)$. I was hoping to see these $\psi=1$ invariants completely explicitly in the $a_p(f)=0$ case. In trying to do so, it appears to boil down to having to find a function $P$ in some ring like $B_{rig}$ satisfying $\psi^2(P) = -P$.

So here's my question: is there any way to find an explicit $P$ satisfying this equation?