Descent for projective twists of modular curves

Let F be a number field and [Formula: see text] a rational prime. We obtain a simple descent criterion characterizing those projective Galois representations [Formula: see text] for which the corresponding twist [Formula: see text] of the principal modular curve of level p is defined over [Formula: see text]. We also give a more concrete version of this result for representations which arise from elliptic curves over cyclic number fields.