Shimura curves of discriminant 14 and 15 and associated Heun Functions

The Shimura curve of discriminant $D$ for $D=14, 15$ is uniformized by a subgroup of an arithmetic quadrilateral Fuchsian group $(2, 2, 2, q)$, where $q=4, 6$. We relate the generator of the ring of quaternionic modular forms on this Shimura curve to explicit Heun functions for the quadrilateral group. We also discuss how the Picard-Fuchs equation of the associated family of abelian surfaces has solutions that are modular forms on $X^{D}(1) / W_{D}$, where $W_D$ is the full group of Atkin-Lehner involutions. This leads us to completely describe the rational exceptional sets of the associated Heun functions, and the algebraic values attained by the Heun function on these points, for example $${\rm He}\left( 81, \frac{1}{2}; \frac{1}{3}, \frac{1}{6}, \frac{1}{2}, \frac{1}{2};-\frac{729}{112}\right)= \left( \frac{2^2 \cdot 3^3 \cdot 5^3}{7^5} \right)^{\frac{1}{6}}$$.