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–Livné. It also generalizes a construction of Scholl to indefinite division quaternion algebras, and provides the first motivic construction of new-subspaces of modular forms.