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Let  f  be a Maass cusp form for SL 3 (ℤ) with Fourier coefficients   . We consider the sum   , where   . A bound better than    is proved to be valid for certain transcendental numbers   . This bound improves Miller's result (2006) for these α. For α close to a rational number    with   , the smooth sum    is further proved to decay rapidly. This extends a result of Booker (2000 and 2005) on a smooth sum of    without the exponential function. These bounds manifest a strange vibration nature of Maass cusp forms. The main techniques include rational approximation of real numbers, a Voronoi summation formula for SL 3 (ℤ), and its asymptotic expansion.

Sums of Fourier coefficients of a Maass form for SL 3 (ℤ) twisted by exponential functions