Quaternionic Manin symbols, Brandt matrices, and Hilbert modular forms

In this paper, we propose a generalization of the algorithm we developed previously. Along the way, we also develop a theory of quaternionic $M$-symbols whose definition bears some resemblance to the classical $M$-symbols, except for their combinatorial nature. The theory gives a more efficient way to compute Hilbert modular forms over totally real number fields, especially quadratic fields, and we have illustrated it with several examples. Namely, we have computed all the newforms of prime levels of norm less than 100 over the quadratic fields $\mathbb {Q}(\sqrt {29})$ and $\mathbb {Q}(\sqrt {37})$, and whose Fourier coefficients are rational or are defined over a quadratic field.