Galois representations associated with a non-selfdual automorphic representation of GL(3)

In 1994, van Geemen and Top constructed a non-selfdual motive of rank three over $\mathbb{Q}$ conjecturally associated with a cuspidal non-selfdual automorphic representation of ${\mathrm{G}\mathrm{L}}_3({\mathbb{A}}_{\mathbb{Q}})$ of level ${\mathrm{\Gamma }}_0(128)$. They experimentally confirmed the coincidence of the local $L$-factors at finitely many primes using computer. In this paper, we shall prove the coincidence of the local $L$-factors at every prime. To show this, we use the recent results of Harris-Lan-Taylor-Thorne and Scholze on the construction of Galois representations, and Greni'e’s results to compare three-dimensional $2$-adic Galois representations. We also prove the local-global compatibility at $p=2$, including the case $p=\mathrm{\ell }$.