We formulate a Serre-type conjecture for $n$-dimensional Galois representations that are tamely ramified at $p$. The weights are predicted using a representation-theoretic recipe. For $n=3$, some of these weights were not predicted by the previous conjecture of Ash, Doud, Pollack, and Sinnott. Computational evidence for these extra weights is provided by calculations of Doud and Pollack. We obtain theoretical evidence for $n=4$ by using automorphic inductions of Hecke characters