Thanks for the details. If the Coleman family F has tame level N, tame nebentype $\chi$, is specialization commutes with twist by $\chi^{-1}$? If so, then let f be a classical specialization, then the twist f by $\chi^{-1}$ is forced to have tame level N, which seems a bit surprising.

Thanks very much for the classification. Would you please provide more details on the construction of $f^c$(and $F^c$)? Thanks. One way I can imagine is first twist by the inverse of the Neben-type, and then consider the associated newform (new except for p), but I dont know whether this method works in family.

Thanks very much. This seems much more plausible.

Thanks for the details. I think the norm form in your comment means trace form. But what do you mean by "fixed by ()^*", it would strange if $\{x\in{D}:\ x^*=x\}$ has dimension three.

Thanks very much for the answer. But would you please explain a little bit more on the construction? We do can find some F splits B and is stable under $*$, but how to construct the desired $e\in{B\otimes{F}}$ ?And how to guarantee that we can make p splits in F?

If we take $z=(z_n) $ with $Tr_{n+1/n}^{LT}(z_{n+1})=z_n$ and $x=(x_n)$ with $x_n=[q/\pi^n](z_n)$, then we have $x\in{S}$. Take $f(T)\in{B_{rig,F}}^+$ such that $f(u_n)=\log_{LT}(x_n)$ and $f(T)\in{B_{rig,F}^{+,\psi_q=1/\pi}}$, $V=F(\chi_{\pi})$, then we have $h_{F_n,V}^1(\partial{f(T)}u)=\delta(z_n)$ and the sequence $\{\delta(z_n)\}$ can be viewed as an element in $H^1_{Iw}(F,V)$. In this situation, do we have that $\{h_{F_n,V(\chi_{\pi}^j)}^1(\partial{f(T)}u\otimes e_{j})\}_{n\geq1}$ is the twist a la soule of $\{\delta(z_n)\}_{n\geq1}$ by $\chi_{\pi}^j$?

Thanks for pointing out the typos. In the special case of $V=K(\chi_{\pi})$, as in Thm 3.4.5, $y=f(T)\otimes{t_{\pi}^{-1}u}$ and $\nabla(y)=\partial{f(T)}u$. Doesn't the fact $h_{F_n,V}^1(\partial {f(T)}u)=(q/\pi)^{-n}\delta(x_n)$ and the general construction of ${h_{F_n,-}}$ implies that $h_{F_n, V(\chi_\pi^j)}(\partial{f(T)}u\otimes{e_j})$ is the twist (a la Soule) of $(q/\pi)^{-n}\delta(x_n)$ by $\chi^j_{\pi}$?