Let $\mathbb{Z}_p(1) :=T_p(\overline{\mathbb{Q}}_p^\times) =\mathbb{Z}_pt$ be the Tate module of $\overline{\mathbb{Q}}_p^\times$, i.e. the Galois group $G_{\mathbb{Q}_p}= \operatorname{Gal}(\overline{\mathbb{Q}}_p/\mathbb{Q}_p)$ acts $\mathbb{Z}_p$-linearly on $t$ via the $p$-adic cyclotomic character $\chi:G_{\mathbb{Q}_p}\to\mathbb{Z}_p^\times$. Let $V := \mathbb{Q}_p\otimes_{\mathbb{Z}_p}\mathbb{Z}_p(1) =: \mathbb{Q}_p(1) = \mathbb{Q}_pt$.
Let $C$ be the completion of $\overline{\mathbb{Q}}_p$ which is algebraically closed. The action of $G_{\mathbb{Q}_p}$ on $\overline{\mathbb{Q}}_p$ extends to an action on $C$ by density and continuity. Let $C(i)$ be the $i$-th Tate twist, i.e. $C(i) := C\otimes_{\mathbb{Z}_p}\mathbb{Z}_p(i) = C t^i$ where $g\in G_{\mathbb{Q}_p}$ acts on $ct^i\in C(i)$ via $\chi^i(g)(gc)t^i$.
The $i$-th Hodge-Tate number of $V$ is defined to be $h_i = \dim (C(-i)\otimes_{\mathbb{Q}_p} V)^{G_{\mathbb{Q}_p}}$.
For any integer $i\in\mathbb{Z}$, I think there is an isomorphism of $G_{\mathbb{Q}_p}$-modules $$ C(-i)\otimes_{\mathbb{Q}_p} V = Ct^{-i}\otimes_{\mathbb{Q}_p}\mathbb{Q}_pt = Ct^{1-i} $$ such that $g\in G_{\mathbb{Q}_p}$ acts on $ct^{1-i}$ via $\chi^{1-i}(g)(gc)t^{1-i}$. So in particular, if $i=1$, then $$ C(-1)\otimes_{\mathbb{Q}_p}V = C $$ so that $$ h_1 = \dim(C^{G_{\mathbb{Q}_p}}) = \dim(\mathbb{Q}_p) = 1. $$ When $i\neq 1$, how can I show that $h_i = 0$?