Congruences involving S_k^(2)(1/18)

Establish the mod-p^2 congruences and p-adic integrality assertion involving S_k^{(2)}(1/18) stated in Conjecture 4.21.

Background

Conjecture 4.21 treats the specialization S_k^{(2)}(1/18), predicts cases associated with quadratic forms involving 130, and includes the weighted congruence related to the second series in Conjecture 1.2.