Under GRH, moments of character-twisted sums of Fourier coefficients of a fixed modular form are bounded by the number of characters times Y^m times a logarithmic power, matching the character-sum bounds.
Davenport, Multiplicative Number Theory , Third edition, Graduate Texts in Mathematics, vol
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Bounds for Moments of Twisted Fourier coefficients of Modular Forms
Under GRH, moments of character-twisted sums of Fourier coefficients of a fixed modular form are bounded by the number of characters times Y^m times a logarithmic power, matching the character-sum bounds.