Jacquet-Langlands correspondence for non-Eichler orders

In this note, we give a concrete realization of the Jacquet-Langlands correspondence for non-Eichler orders of indefinite quaternion algebras defined over $\mathbb{Q}$. To be more precise, we consider a special type of index-two suborder of the Eichler order of level $N$ in the quaternion algebra with an even discriminant $D$.