  By employing the theory of vector-valued automorphic forms for
non-unitarizable representations, we provide a new bound for the number of
linear relations with algebraic coefficients between the periods of an
algebraic Riemann surface with many automorphisms. The previous best-known
general bound for this number was the genus of the Riemann surface, a result
due to Wolfart. Our new bound significantly improves on this estimate, and it
can be computed explicitly from the canonical representation of the Riemann
surface. As observed by Shiga and Wolfart, this bound may then be used to
estimate the dimension of the endomorphism algebra of the Jacobian of the
Riemann surface. We demonstrate with a few examples how this improved bound
allows one, in some instances, to actually compute the dimension of this
endomorphism algebra, and to determine whether the Jacobian has complex
multiplication.
