Does such a function exist?

I am looking for a function with the following property:

Let $v_1,v_2$ be two linearly independent vectors in $mathbb{R}^2.$

I am given a smooth function $g:(0,1) rightarrow (0,infty).$

I am trying to understand if there exists a smooth (non-constant) function $f:(0,1) times mathbb R^2 rightarrow mathbb R^2$ with the property that for all $n,m in mathbb Z$ such that $g(t)=n/m$, where $n/m$ is a reduced fraction, we have for all $x in mathbb R^2$

$$f(t,mv_1+x)=f(t,x)=f(t,nv_2+x)$$ and $n,m$ are the minimal periods of the function $f$, i.e. for all natural numbers $1le n_0 <n$ we do not have

$$f(t,nv_2+x)=f(t,x)$$ and the same for $m.$