  In this paper, we proved generating functions of Gromov-Witten cycles of the
elliptic orbifold lines with weights (3,3,3), (4,4,2), and (6,3,2) are
cycle-valued quasi-modular forms. This is a generalization of Milanov and
Ruan's work on cycle-valued level. First we construct a global cohomology field
theory (CohFT) for simple elliptic singularities (modulo an extension problem)
and prove its modularity. Then, we apply Teleman's reconstruction theorem to
prove mirror theorems on cycled-valued level and match it with a CohFT from
Gromov-Witten theory of a corresponding orbifold.This solves the extension
property as well as inducing the modularity for a Gromov-Witten CohFT.
