# Exponential sums related to cusp forms

Let $$f(z)=\sum_{n\geq 1} a_f(n) e^{2\pi n i z}$$ be a holomorphic newform on the upper half-plane of weight $k$ for $\Gamma_0(N)$ and of trivial character which is normalized so that $a_f(1)=1$.

In Jutila's book "A method in the theory of exponential sums" he proves an estimate of the form:

$$\sum_{n\leq x} a_f(n) e^{2\pi i n \frac{p}{q}} \ll q^{2/3}x^{k/2-1/6+\varepsilon}$$

where $a_f(n)$ are the coefficients of a cusp form of weight $k$ for the full modular group ($N=1)$, and $p$ and $q$ are coprime integers. Conceivably, a similar estimate holds for coefficients of holomorphic cuspforms for congruence subgroups. Does anyone know a reference for an estimate of the form:

$$\sum_{n\leq x} a_f(n) e^{2\pi i n\frac{p}{q}} \ll_{f,q} \ \ x^{k/2-\delta}$$

where $a_f(n)$ are the coefficients of a holomorphic cusp form of weight $k$ for $\Gamma_0(N)$, $p$ and $q$ are coprime integers, and $\delta>0$?

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On MathSciNet, some of the papers which cite Jutila's book apparently have results about cusp forms of arbitrary level. You might look at those. – Stopple Jul 26 '11 at 21:34
For some reason, my connection to MathSciNet isn't working today. That is (part of) the reason I asked the question. – Micah Milinovich Jul 26 '11 at 21:52
For example, Blomer and Harcos, "Hybrid bounds for twisted L-functions" J. Reine Angew. Math. 621 (2008), 53–79. – Stopple Jul 26 '11 at 22:04