  We consider the fourteen families $W$ of Calabi-Yau threefolds with one
complex structure parameter and Picard-Fuchs equation of hypergeometric type,
like the mirror of the quintic in $\mathbb{P}^4$. Mirror symmetry identifies
the masses of even--dimensional D--branes of the mirror Calabi-Yau $M$ with
four periods of the holomorphic $(3,0)$-form over a symplectic basis of
$H_3(W,\mathbb{Z})$. It was discovered by Chad Schoen that the singular fiber
at the conifold of the quintic gives rise to a Hecke eigenform of weight four
under $\Gamma_0(25)$, whose Hecke eigenvalues are determined by the Hasse-Weil
zeta function which can be obtained by counting points of that fiber over
finite fields. Similar features are known for the thirteen other cases. In two
cases we further find special regular points, so called rank two attractor
points, where the Hasse-Weil zeta function gives rise to modular forms of
weight four and two. We numerically identify entries of the period matrix at
these special fibers as periods and quasiperiods of the associated modular
forms. In one case we prove this by constructing a correspondence between the
conifold fiber and a Kuga-Sato variety. We also comment on simpler applications
to local Calabi-Yau threefolds.
