  In this paper we study certain families of motives, which arise as direct
summands of the cohomology of the Dwork family. We computationally find
examples of interesting families with the following three properties. Firstly,
their geometric monodromy group is Zariski dense in $\operatorname{SL}_n$.
Secondly, they realise many different unipotent operators as the monodromy
operator at $t = \infty$. Thirdly, all their Hodge numbers are $\leq 1$.
  This has consequences for Galois representations. Namely, if a nilpotent
operator $N$ appears as the monodromy at $t = \infty$ in one of our families,
we can construct potentially automorphic representations with $\ell$-adic
monodromy given by $N$ at a fixed prime $p$. As another application, we obtain
a new proof of some cases of the recent local-global compatibility theorem of
Matsumoto.
