Congruences involving S_k^(2)(2) and S_k^(2)(9)

Establish the mod-p^2 congruences and p-adic integrality relations involving S_k^{(2)}(2) and S_k^{(2)}(9) stated in Conjecture 4.11.

Background

Conjecture 4.11 concerns quadratic forms associated with 21 and gives several residue-symbol cases together with a weighted p-adic congruence.