Let $f$ be a normalized newform of weight $k\geq 2$, $p$ a prime, and $V$ the associated Galois representation (with coefficients in a finite extension $K$ of $\mathbf{Q}_p$ with ring of integers $\mathscr{O}$). Assume the residual representation attached to $V$ is absolutely irreducible, so that the choice of a $G_\mathbf{Q}$-stable $\mathscr{O}$-lattice $T$ in $V$ is unique up to $\mathscr{O}$-scaling. Let $A=V/T$ be the quotient, which is a discrete $G_\mathbf{Q}$-module isomorphic as an $\mathscr{O}$-module to $(K/\mathscr{O})^2$.

My question: is the Cartier dual $A^*=\mathrm{Hom}_\mathscr{O}(A,(K/\mathscr{O})(1))$, a free $\mathscr{O}$-module of rank $2$, necessarily (isomorphic to) a $G_\mathbf{Q}$-lattice in a representation associated to a newform (of some weight and level)? Along these same lines, do the representations $V^\vee=\mathrm{Hom}_K(V,K)$ and $V(1)$ also come from a newform? I believe a positive answer to the second question would also give a positive answer to my first question, but I might be mistaken. If the answer to the first question is yes, and the original form $f$ is $p$-ordinary, can I always find a $p$-ordinary form giving rise to $A^*$?

The reason I ask is because I'm trying to prove some things that require me to know certain facts about the Galois cohomology of $A^*$ which I know for $A$ (because $A$ comes from a modular form). So if I knew $A^*$ also came from a modular form, I'd have what I need.