Let $\bar{\rho}$ be a residual ordinary and locally split galois representation associated to a weight $k$ level $1$ form (more specifically a mod $p$ companion form). In the sense of deformation theory, let $p$ be an unobstructed prime. Then dim $ H^{1}(G_{S}, dim H^{1}(G_{S}, Ad(\bar{\rho})) = 3$. Further $H^{1}(G_{S}, Ad(\bar{\rho})) = H^{1}(G_{S}, \mathbb{F}{p}) mathbb{F}_{p}) \oplus H^{1}(G{S}, H^{1}(G_{S}, Ad^{0}(\bar{\rho}))$.
What is the dimension of image of the map defined by
$$ H^{1}(G_{S}, Ad(\bar{\rho})) = H^{1}(G_{S}, \mathbb{F}{p}) mathbb{F}_{p}) \oplus H^{1}(G{S}, H^{1}(G_{S}, Ad^{0}(\bar{\rho})) \rightarrow H^{1}(I_{p}, Ad^{0}(\bar{\rho})) \rightarrow H^{1}(I_{p}, \mathbb{F}_{p}(\omega^{k-1}))$mathbb{F}_{p}(\omega^{k-1}))$$
$Ad^{0}(\bar{\rho}) $Ad^{0}(\bar{\rho}) \cong \mathbb{F}{p} mathbb{F}_{p} \oplus \mathbb{F}{p}(\omega^{k-1}) mathbb{F}_{p}(\omega^{k-1}) \oplus \mathbb{F}_{p}(\omega^{1-k})$ mathbb{F}_{p}(\omega^{1-k})$$
as an $I_p$ module.
