Recently when I want to understand the construction of triple product p-adic L-function, I am really confused by the notion of dual form. To be precise, assume $f^\circ\in{S_k(N,\chi)}$ is an eigenform of weight k, level N (not necessary new) with nebentype $\chi$ and f be (one of the) p-stabilization of $f^{\circ}$, we can construct several modular forms from f: assume the $q$-expansion of f is $\sum_{n\geq1}a_nq^n$,

$(i)$ The conjugate form $f^*$ defined by $f^*(\tau)=\overline{f(-\bar{\tau})}$, the q-expansion of $f^*$ is $\sum_{n\geq1}\bar{a}_nq^n$;

$(ii)$ The twisted form $f_{\chi^{-1}}$ by $\chi^{-1}$ whose q-expansion is $\sum_{n\geq1}\chi^{-1}(n)a_nq^n$;

$(iii)$ and moreover as in Def.2.4 of the paper a note on p-adic Rankin-Selberg L-functions, the form $f^c$ which is the unique form which has level $Np^r$ for some r and whose Hecke eigenvalue away from N is same as $f_{\chi^{-1}}$.

What is the relation of the three forms? Would anyone provide some reference? Thanks

If F is a Coleman family of tame level N and nebentype $\chi$, could we define the corresponding family $F^*$, $F_{\chi^{-1}}$ or $F^c$ satisfying similar description on $q$-expansion? The paper A p-adic Gross-Zagier formula for diagoanl cycles(page 41) uses $F^*$ (for Hida family), but no details provided there and Lem.3.4 of 1 claims the existence of $F^c$ (at least for F is new at N) but I could not spell out the details. Would anyone please provide me some reference containing more details? Thanks