Congruences involving S_k^(2)(1/11)

Prove the mod-p^2 congruences and higher-power p-adic integrality relations involving S_k^{(2)}(1/11) stated in Conjecture 4.20.

Background

Conjecture 4.20 concerns the specialization S_k{(2)}(1/11), the quadratic forms associated with 78, and the weighted congruence related to the first series in Conjecture 1.2.

References

Conjecture 4.20. Let p > 3 be a prime with p ≠ 11.

— A family of polynomials and related congruences and series  (2505.02767 - Sun, 5 May 2025) in Conjecture 4.20, Section 4