For prime moduli p, the smoothed m-th moment of quadratic twists of a fixed modular form's coefficients grows like X Y^{m/2} (log X)^{m(m-3)/2}, with matching even-m lower bounds, conditional on GRH.
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Bounds for moments of twisted quadratic characters of prime modulus
For prime moduli p, the smoothed m-th moment of quadratic twists of a fixed modular form's coefficients grows like X Y^{m/2} (log X)^{m(m-3)/2}, with matching even-m lower bounds, conditional on GRH.