  This paper aims to study the Betti homology and de Rham cohomology of twisted
symmetric powers of the Kloosterman connection of rank two on the torus. We
compute the period pairing and, with respect to certain bases, interpret these
associated period numbers in terms of the Bessel moments. Via the rational
structures on Betti homology and de Rham cohomology, we prove the
$\mathbb{Q}$-linear and quadratic relations among these Bessel moments.
