Assume $p$ be prime number ($p>2$), and let $u$ be any topological generator of the group $1 + p \mathbb{Z}_p$ (an open subgroup of the group of units $\mathbb{Z}_p^\times$ of the ring of $p$-adic integers $\mathbb{Z}_p$). For example, $u$ could be $1 + p$ or $\exp(p)$.
Let $G_u$ be the uniform pro-$p$ group of rank $2$, with generators $x,y$ satisfying the relation $xyx^{-1} = y^u$. Let $\varphi$ be the automorphism of $G_u$ given by $\varphi(g) = xgx^{-1}$.
Can you calculate the Mahler Coefficients of $\varphi$, with respect to the ordered basis $\{x,y\}$ of $G$?
