Omega theorems for the coefficients of automorphic L-functions

Let λf (n) denote the normalized Fourier coefficients of a primitive holomorphic cusp form f(z) of even integral weight k for the full modular group. In this paper, we investigate the error terms of the summatory function $${\sum }_{n\le x}{\lambda }_{f}^{2}\left({n}^{j}\right),j=\text{3,4},$$ and establish the following Ω-results: $$\sum_{n\le x}{\lambda }_{f}^{2}\left({n}^{3}\right)={c}_{3}x+\Omega \left({x}^{15/32}\right), \sum_{n\le x}{\lambda }_{f}^{2}\left({n}^{4}\right)={c}_{4}x+\Omega \left({x}^{12/25}\right),$$ where c3 and c4 are suitable constants.