Congruences involving S_k^(2)(−4) and S_k^(2)(1/6)

Prove the mod-p^2 congruences and p-adic integrality assertions involving S_k^{(2)}(−4) and S_k^{(2)}(1/6) specified in Conjecture 4.12.

Background

The conjecture predicts congruences associated with the quadratic forms x^2+22y^2 and 2x^2+11y^2, along with corresponding weighted sums.