Let $p$ be a prime number, $\psi:G_\mathbb{Q} \rightarrow \bar{\mathbb{Q}}_p$ be an odd character of conductor $N$ prime to $p$, with finite image and such that $\psi(p)=1$. Let $\mathcal{W}$ be the weight space representing the homomorphism $\mathbb{Z}_p^{\times} \rightarrow \mathbb{G}_m$ and $\epsilon_p$ is the $p$-adic cyclotomic character.

Let $k$ be a character of $\mathcal{W}$ corresponding to some integer at least equal to two. Can we vary $Ext^{1}_{G_\mathbb{Q}}(1,\psi \epsilon_p^k)$ $p$-adically with respect to $k$ (here $Ext^{1}_{G_\mathbb{Q}}(1,\psi \epsilon_p^k)=Ext^{1}_{f}(1,\psi \epsilon_p^k)$ and it is of dimension one since $k\geq 2$).

Let $p$ be a prime number, $\psi:G_\mathbb{Q} \rightarrow \bar{\mathbb{Q}}_p$ be an odd character of conductor $N$ prime to $p$, with finite image and such that $\psi(p)=1$. Let $\mathcal{W}$ be the weight space representing the homomorphism $\mathbb{Z}_p^{\times} \rightarrow \mathbb{G}_m$ and $\epsilon_p$ denotes the $p$-adic cyclotomic character.

Let $k$ be a character of $\mathcal{W}$ corresponding to some integer at least equal to two. Can we vary a non trivial $1$-co cycle of $Ext^{1}_{G_\mathbb{Q}}(1,\psi \epsilon_p^k)$ $p$-adically with respect to $k$ (here $Ext^{1}_{G_\mathbb{Q}}(1,\psi \epsilon_p^k)=Ext^{1}_{f}(1,\psi \epsilon_p^k)$ and it is of dimension one since $k\geq 2$).