Congruences involving S_k^(2)(−4) and S_k^(2)(81)

Prove the mod-p^2 congruences and higher-power p-adic assertions for the specializations and weighted sums involving S_k^{(2)}(−4) and S_k^{(2)}(81) specified in Conjecture 4.14.

Background

The conjecture gives several cases associated with representations by quadratic forms of discriminant parameter 30 and includes a weighted congruence and p-adic lifting statement.