I'm reading Bump's automorphic form, chapter 4.1, and this note written by Garrett. The later note said that there are $q(q-1)/2$ different supercuspidal representations of $\mathrm{GL}_{2}(\mathbb{F}_{q})$, which comes from anisotropic torus $T_{a}(F) = E^{\times}$ ($E$ is an extension field of $F$ with $[E:F]=2$.) In Bump, it said that for each character $\chi:E^{\times} \to \mathbb{C}^{\times}$ that does not factor through the norm map $N:E^{\times}\to F^{\times}$, we have a Weil representation $\pi = (\pi(\chi), W(\chi))$ of $\mathrm{GL}_{2}(F)$. However, there are $(q-1)^{2} - (q-1) = (q-1)(q-2)$ such characters, so I think there are some characters $\chi, \chi'$ that gives same representation $\pi(\chi)\simeq \pi(\chi')$. However, I don't know how to decide whether the representations are isomorphic or not, and I think both articles don't explain about this case. Do we have to compute character (trace) of the representation $W(\chi)$ and compare it, or is there any other methods?
I found that there's one way to do it by using a Whittaker model: Theorem 4.1.2 of Bump claims that the induced representation $\mathrm{Ind}_{N}^{\mathrm{GL}_{2}}\psi_{N}$ contains every irreducible representation of dimension $>1$ exactly once, and dimension counting gives an answer. But I'm sure that there's a more elementary way to do this.