Let $\bar{\rho}$ be a residual ordinary and locally split modular Galois Representation (into $GL_{2}(\mathbb{F}_p))$ associated to a weight $k$ and level $1$ form. In the sense of Deformation Theory, let $p$ be an unobstructed prime. What is the dimension of the image of the linear map
$ H^{1}(G_{S}, Ad(\bar{\rho})) \rightarrow H^{1}(I_{p},\mathbb{F}_{p}(\omega^{k-1})) $
where $S=\lbrace p,\infty \rbrace$ and $\omega$ is the mod $p$ cyclotomic character.