  Let F be a totally real field and p an odd prime. If r is a continuous,
semisimple, totally odd mod p representation of the absolute Galois group of F
which is tamely ramified at all places of F dividing p, then we formulate a
conjecture specifying the weights for which r is modular. This extends the
conjecture of Diamond, Buzzard, and Jarvis, which supposed that p was
unramified in F. We also prove a theorem towards the conjecture and provide
some computational evidence.
