Sufficient statistic from characteristic function?

I have a density function $f(;\theta):{\mathbb R}\rightarrow{\mathbb R}_+$, where $\theta\in{\mathbb R}^d$. I know that there is a sufficient statistic $T$ of dimension $d$. If $\varphi_n$ is the characteristic function for a sample of size $n$, is it possible to create a grid $x_1,\dots,x_m$, $n > m\geq d$ such that $\varphi_n(x_1),\dots,\varphi_n(x_m)$ is a sufficient statistic too?