Congruences involving S_k^(2)(2/7)

Prove the mod-p^2 congruences and p-adic integrality relations involving S_k^{(2)}(2/7) stated in Conjecture 4.18.

Background

The conjecture concerns quadratic forms associated with 57 and gives four residue-symbol cases, a weighted congruence, and a higher-power p-adic assertion.