Let \(\textbf{f}\) be a primitive Hilbert cusp form of weight k and level \(\mathfrak {n}\) with fourier coefficients \(c_\textbf{f}(\mathfrak {m})\) . We prove a non-trivial upper bound for almost all Fourier coefficients \(c_\textbf{f}(\mathfrak {m})\) of \(\textbf{f}\) . This generalizes the bounds obtained by Luca, Radziwiłł and Shparlinski. We also prove the existence of infinitely many integral ideals \(\mathfrak {m}\) for which the Fourier coefficients \(c_\textbf{f}(\mathfrak {m})\) have the improved upper bound and further we obtain a refinement of these integral ideals in terms of prime powers. In particular, this enables us to deduce the bound for Fourier coefficients of elliptic cusp forms beyond the ‘typical size’. Moreover, we prove further improvements of the bound under the assumption of Littlewood’s conjecture. Finally, we study a lower bound for the Fourier coefficients at prime powers provided the corresponding Hecke eigen angle is badly approximable.
