Eichler orders, trees and Representation Fields

If H and D are two orders in a central simple algebra A with D of maximal rank, the representation field F(D|H) is a subfield of the spinor class field of the genus of D which determines the set of spinor genera of orders in that genus representing the order H. Previous work have focused on two cases, maximal orders D and commutative orders H. In this work, we show how to compute the representation field F(D|H) when D is the intersection of a finite family of maximal orders, e.g., an Eichler order, and H is arbitrary. Examples are provided.