  We describe a method to determine all the isomorphism classes of principal
polarizations of the modular abelian surfaces $A_f$ with quaternionic
multiplication attached to a normalized newform $f$ without complex
multiplication. We include an example of $A_f$ with quaternionic multiplication
for which we find numerically a curve $C$ whose Jacobian is $A_f$ up to
numerical approximation, and we prove that it has quaternionic multiplication
and is isogenous to $A_f$.
