Dirac operators in tensor categories and the motive of quaternionic modular forms

We define a motive whose realizations afford modular forms (of arbitrary weight) on an indefinite division quaternion algebra. This generalizes work of Iovita-Spiess to odd weights in the spirit of Jordan-Livnè. It also generalizes a construction of Scholl to indefinite division quaternion algebras, and provides the first motivic construction of new-subspaces of modular forms.