Congruences involving S_k^(2)(−4) and S_k^(2)(9)

Establish the mod-p^2 congruences and p-adic integrality relations involving S_k^{(2)}(−4) and S_k^{(2)}(9) stated in Conjecture 4.15.

Conjecture 4.15 concerns quadratic forms involving 33 and proposes several residue-symbol cases together with a weighted p-adic congruence.

Conjecture 4.15. Let p > 3 be a prime.

— A family of polynomials and related congruences and series (2505.02767 - Sun, 5 May 2025) in Conjecture 4.15, Section 4

Conjecture 4.17. Let p be an odd prime.

— A family of polynomials and related congruences and series (2505.02767 - Sun, 5 May 2025) in Conjecture 4.17, Section 4