Let $\mathbb{F}_q$ be a finite field with $q$ elements. Let $L(x)=a x^{q^2} +b x^q + cx\in \mathbb{F}_{q^3}[x]$. Assume that there exists an element $\beta\in\mathbb{F}_{q^3}$ such that $$\{L(x) \mid x \in \mathbb{F}_{q^3} \}=\{ \beta e \mid e\in \mathbb{F}_q\},$$ where $\mathbb{F}_q$ is the subfield of $\mathbb{F}_{q^3}$.
Denote $\{ \beta e \mid e\in \mathbb{F}_q\}=S_1$.It is well known that $L(x)$ is a linear transformation from $\mathbb{F}_{q^3}$ to $S_1$. I known that all the elements in $S_1$ must satisfy the polynomial $$H(x)=x^q-\beta^{q-1}x.$$
Denote $T(e)=\{x\in \mathbb{F}_{q^3} \mid L(x)=\beta e\}$ for $e \in \mathbb{F}_{q^3}$. The question is, given $e$, what is the minimal polynomial $f_e(x)$ that all the elements in $T(e)$ must satisfy, i.e., $f_e(x)=0$ for all $x \in T(e)$?
When we say "minimal", we mean the polynomial with the minmal degree. Because $g(x)=f(x)h(x)$ for any $h(x)\in\mathbb{F}_{q^3}$ is a such polynomial if $f(x)$ is.