A note on the sum of cubic residues with even and odd index

Let p be an odd prime and ℓ∈ N. We prove that the sum of cubic residues of pℓ with even index equals the sum of cubic residues of pℓ with odd index if and only if p≡1(mod4). We also prove that for 2 pℓ, p≡1(mod4) is sufficient but not necessary for the two sums to be equal. We also present a closed-form expression for the sum and study some properties.