  We introduce a sequence of families of lattice polarized $K3$ surfaces. This
sequence is closely related to complex reflection groups of exceptional type.
Namely, we obtain modular forms coming from the inverse correspondences of the
period mappings attached to our sequence. We study a non-trivial relation
between our modular forms and invariants of complex reflection groups.
Especially, we consider a family concerned with the Shepherd-Todd group of
No.34 based on arithmetic properties of lattices and algebro-geometric
properties of the period mappings.
