Spectral properties of generalized Paley graphs and their associated irreducible cyclic codes

For $q=p^m$ with $p$ prime and $k\mid q-1$, we consider the generalized Paley graph $Γ(k,q) = Cay(\mathbb{F}_q, R_k)$, with $R_k=\{ x^k : x \in \mathbb{F}_q^* \}$, and the irreducible $p$-ary cyclic code $\mathcal{C}(k,q) = \{(\textrm{Tr}_{q/p}(γω^{ik})_{i=0}^{n-1})\}_{γ\in \mathbb{F}_q}$, with $ω$ a primitive element of $\mathbb{F}_q$ and $n=\tfrac{q-1}{k}$. We first express the spectra of $Γ(k,q)$ in terms of Gaussian periods. Then, we show that the spectra of $Γ(k,q)$ and $\mathcal{C}(k,q)$ are mutually determined by each other if further $k\mid \tfrac{q-1}{p-1}$. We give $Spec(Γ(k,q))$ explicitly for those graphs associated with irreducible 2-weight cyclic codes in the semiprimitive and exceptional cases. We also compute $Spec(Γ(3,q))$ and $Spec(Γ(4,q))$.