Let $p$ be an odd prime and $q=p^n$ for some $n \geq 1$. If $\mathbb{F}_q$ is the unique, up to isomorphism, finite field with $q$ elements then the cuspidal representations of the group $\rm{GL}_2(\mathbb{F}_q)$ have dimension $q-1$ and their characters can be described explicitly. See, for example, the first displayed equation in Schein's paper: Reduction modulo p of cuspidal representations and weights in Serre's conjecture.
Given two such characters $\chi_1$ and $\chi_2$, viewed as representations over $\overline{\mathbb{Q}_p}$, we can take their reductions modulo $p$ which are typically denoted $\overline{\chi}_1$ and $\overline{\chi}_2$ respectively. Is it true that if $\chi_1 \neq \chi_2$ then $\overline{\chi}_1 \neq \overline{\chi}_2$? In the aforementioned work of Schein, and the reference of Diamond within that paper, the reductions of $\chi_1$ and $\chi_2$ are computed directly but I can't how to use these descriptions to answer my question. I might just be missing a fact from modular representation theory.
Thanks for any help in advance!
