Quartic congruences and eta products

Let a15(n), a20(n) and let p 3 be a prime. In this paper, for p ≡ 3 (mod 4) we reveal the connection between a20(p) and residue-counts of x4 − 4x2 + 4x modulo p as x runs over 0, 1, . . . , p − 1, and the connection between a24(p) and residue-counts of x3 + c/x modulo p as x runs over 1, 2, . . . , p −1, where c is an integer not divisible by p. We also deduce the congruences for a15(p), a24(p) modulo 16 and a20(p) modulo 4, and pose some analogous conjectures.