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Let f , g be two distinct primitive holomorphic cusp forms of even integral weights \(k_{1},k_{2}\) for the full modular group \(\Gamma =SL(2,\mathbb {Z})\) , respectively. In this paper, we are interested in the upper bounds for the integral mean square of error terms involving \(\lambda _{f}^{2i_{1}}(n), \lambda _{f}^{2}(n^{j_{1}})\) and \(\lambda _{f}^{2}(n^{i_{2}})\lambda _{g}^{2}(n^{j_{2}})\) on average in terms of f , g respectively. Here \(i_{1}\ge 2, j_{1}\ge 3\) and \(i_{2},j_{2}\ge 1\) are positive integers.
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