Local Orders, Root Numbers, and Modular Curves

Let F be a locally compact field, and let wr be an irreducible, admissible representation of GL2(F) on an infinite dimensional complex vector space. Let c be the central quasicharacter of wx. Let K be a separable quadratic field extension of F, and let X be a quasicharacter of K* such that xw = 1 on F*. The character X gives a 1-dimensional representation of the Weil group WK of K, via the isomorphism Wab K* of local classfield theory. Let -xw be the admissible irreducible representation of GL2(F) which corresponds (in the sense of Langlands) to the 2-dimensional induced representation IndWFX of the Weil group of F. The representation