Computing the local $2$-component of a non-selfdual automorphic representation of ${\mathrm{G}\mathrm{L}}_3$

In this paper, we explicitly determine the local $2$-adic component of a non-selfdual automorphic representation $\mathrm{\Pi }$ of ${\mathrm{G}\mathrm{L}}_3$ constructed by van Geemen and Top. We prove that ${\mathrm{\Pi }}_2$ is a parabolically induced representation of ${\mathrm{G}\mathrm{L}}_3({\mathbb{Q}}_2)$ given by ${\mathrm{\Pi }}_2={\mathrm{I}\mathrm{n}\mathrm{d}}_P^{{\mathrm{G}\mathrm{L}}_3({\mathbb{Q}}_2)}(\pi ⊠\chi )$, where $P$ is the standard parabolic subgroup of ${\mathrm{G}\mathrm{L}}_3$ with Levi subgroup ${\mathrm{G}\mathrm{L}}_2\times {\mathrm{G}\mathrm{L}}_1$, $\chi$ is an unramified character of ${\mathbb{Q}}_2^{\times }$ satisfying $\chi (2)=-2\sqrt{-1}$, and $\pi$ is a supercuspidal representation of ${\mathrm{G}\mathrm{L}}_2({\mathbb{Q}}_2)$. Furthermore, we describe $\pi$ explicitly as a compactly induced representation $\pi ={\mathrm{c}-\mathrm{I}\mathrm{n}\mathrm{d}}_{J_{\alpha }}^{{\mathrm{G}\mathrm{L}}_2({\mathbb{Q}}_2)}\mathrm{\Lambda }$ and determine the representation $\mathrm{\Lambda }$ explicitly. The proof relies on explicit computations of Hecke eigenvalues using computer calculations. The automorphic representation $\mathrm{\Pi }$ is realized in the cuspidal cohomology of the congruence subgroup ${\mathrm{\Gamma }}_0(128)\subset {\mathrm{S}\mathrm{L}}_3(\mathbb{Z})$. By computing the Hecke eigenvalues of an associated Hecke eigenvector, we are able to uniquely identify the local structure of ${\mathrm{\Pi }}_2$. As an application, we obtain an explicit description of the $2$-adic local component of the Galois representation ${\rho }_{\mathrm{v}\mathrm{G}\mathrm{T},\mathrm{\ell }}$ associated with $\mathrm{\Pi }$.