Looking at the character table below of $\operatorname{GL_2}(F_q)$ for some prime power $q$ in Fulton-Harris (Representation Theory, A First Course), there are $\frac{1}{2}q(q-1)$ Cuspidal representations of dimension $q-1$ For which the irreducible character $X_{\varphi}$ is given in the last line of the table. Quoting from the book:
These $\frac{1}{2}q(q - 1)$ representations, for $\varphi\ne\varphi^q$, and with $X_{\varphi}= X_{\varphi^q}$, therefore complete the list of irreducible representations for $\operatorname{GL_2}(F_q)$ .
I am missing an explicit description of the $\varphi_i$ maps for $i \in \{1,\dots,\frac{1}{2}q(q-1)\}$ in the characters $X_{\varphi_i}$.
All possible maps $\varphi_i$ are completely determinated by the image of a generator of the cyclic group $\Bbb F_{q^2}^*$, say:
\begin{equation}\label{key} \begin{aligned} \varphi_i: \Bbb F_{q^2}^* &\to \Bbb C^* \\ \xi_{q^2-1} &\to (\varepsilon)^i \end{aligned} \end{equation}
where $\xi_{q^2-1},\varepsilon$ are primitive $(q^2-1)^{th}$ roots of unity in $\Bbb F_{q^2}^*$ and $\Bbb C^*$ respectively.
Let's keep only the $\varphi_i$ for which $\varphi\ne\varphi^q$:
\begin{equation} \begin{aligned} \varphi_i = \varphi_i^q &\Leftrightarrow \forall z=\xi_{q^2-1}^k ,\ \varphi_i(z) = \varphi_i(z^q) \\ &\Leftrightarrow \forall k\in \{1, \dots, q^2-1\},\ \varepsilon^{ik} = \varepsilon^{ikq} \\ &\Leftrightarrow \forall k\in \{1, \dots, q^2-1\},\ 1 = \varepsilon^{ik(q-1)} \end{aligned} \end{equation}
We can see that for $i \in \{j(q-1)\}_{j\in\{ {1,\dots, q+1}\}}$ gives maps $\varphi_i$ such that $\varphi=\varphi^q$.
We obtained $q^2-1 - (q+1)=q(q-1)$ maps, but we need to restrict more in order to have only $\frac{1}{2}q(q-1)$ irreducible characters $X_{\varphi_i}$. Any help in the right direction is much appreciated.