  One can assign to four-dimensional N=2 supersymmetric Heterotic string vacua
a set of classification invariants including a lattice $\Lambda_S$ and
vector-valued modular forms. Some of the classification invariants are
constrained by the condition that the Coulomb branch monodromy matrices should
be integer-valued. We computed numerically the period polynomials of
meromorphic cusp forms for some rank-1 $\Lambda_S$; we then computed the
monodromy matrices and extracted general patterns of the constraints on the
invariants. The constraints we got imply that a large fraction of the Heterotic
string vacua we studied satisfy the necessary conditions for a non-linear sigma
model interpretation in the dual Type IIA description. Our computation can also
be used to identify diffeomorphism classes of real six-dimensional manifolds
that cannot be realized by K3-fibred Calabi--Yau threefolds.
