Congruences involving S_k^(2)(1/4)

Establish the mod-p^2 congruences and higher-power p-adic integrality assertions involving S_k^{(2)}(1/4) stated in Conjecture 4.9.

Background

Conjecture 4.9 proposes congruences controlled by representations by x^2+10y^2 and 2x^2+5y^2, together with weighted identities and p-adic extensions.