Congruences involving S_k^(2)(5/16)

Establish the mod-p^2 congruences and weighted congruence involving S_k^{(2)}(5/16) specified in Conjecture 4.19.

Background

Conjecture 4.19 predicts several cases associated with quadratic forms involving 70 and gives a separate weighted mod-p2 identity.

References

Conjecture 4.19. Let p > 3 be a prime.

A family of polynomials and related congruences and series  (2505.02767 - Sun, 5 May 2025) in Conjecture 4.19, Section 4