Abstract: Let k be a positive even integer, and H_^*k be the set of all normalized Hecke primitive eigencuspforms of weight k for Γ=SL₂(Z). The Fourier expansion of f∈H_^*k at the cusp ∞ is defined by f(z)=λ_f(n)n^(k-1)/2e^2πinz, where λ_f(n) is the eigenvalue of the (normalized) Hecke operator T_n. The Omega result for the summatory function λ_f(nⁱ)λ_f(n^j) is investigated. Set
E_1,2(f,x)=λ_f(nⁱ)λ_f(n^j)-c_j-1x, i=1, j=2,3,
where c₁, c₂ is a suitable constant. Then it is proved that
E_1,2(f,x)=Ω(x^5/12),E_1,3(f,x)=Ω(x^7/16).

Key words: Omega theorem, Dirichlet series, automorphic L-functions