Character covering number of PSL₂ (q)

For a group G and a character χ of G, let c(χ) denote the set of all irreducible characters of G, occurring in χ. We prove that whenever q≥ 8, all non-trivial irreducible character χ of PSL₂(q) satisfies c(χ⁴)=Irr(PSL₂(q)) if q=2²ᵐ⁺¹ and c(χ³)=Irr(PSL₂(q)) otherwise.