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In this paper we find new maximal cliques of size $\frac{q+1}{2}$ or$\frac{q+3}{2}$, accordingly as $q\equiv 1(4)$ or $q\equiv 3(4)$, in Paleygraphs of order $q^2$, where $q$ is an odd prime power. After that we use newcliques to define a family of eigenfunctions corresponding to bothnon-principal eigenvalues and having the cardinality of support $q+1$, which isthe minimum by the weight-distribution bound.

On eigenfunctions and maximal cliques of Paley graphs of square order