  The main purpose of this article is to give the integral cohomology of
classical principal congruence subgroups in SL(2,Z) as well as their analogues
in the third braid group with local coefficients in symmetric powers of the
natural symplectic representation. The resulting answers (1) correspond to
certain modular forms in characteristic zero, and (2) the cohomology of certain
spaces in homotopy theory in characteristic p. The torsion is given in terms of
the structure of a "p-divided power algebra".
  The work is an extension of the work in arXiv:1204.5390v1 as well as
extensions of a classical computation of Shimura to integral coefficients. The
results here contrast the local coefficients such as that in [Looijenga, J.
Alg. Geom., 5, '96] and [Tillmann, Q. J. Math., 61, '10].
