  We give a description of asymptotic quadratic growth rates for geodesic
segments on covers of Veech surfaces in terms of the modular fiber
parameterizing coverings of a fixed Veech surface. To make the paper self
contained we derive the necessary asymptotic formulas from the Gutkin-Judge
formula. As an application of the method we define and analyze d-symmetric
elliptic differentials and their modular fibers F^{sym}_d. For given genus g,
g-symmetric elliptic differentials (with fixed base lattice) provide a
2-dimensional family of translation surfaces. We calculate several asymptotic
constants, to establish their dependence on the translation geometry of
F^{sym}_d and their sensitivity as SL(2,Z)-orbit invariants.
