I've asked the background question here, which still left unanswered.
Now I have a more precise question. In my homework I've been asked to prove that
$$\left| \sum_{1\leq n \leq N} a_f (n)e^{2\pi i n \alpha}\right| \leq c_f N^{k\over 2}\log N $$
for any $ f \in S_k $ where $ f(\tau) = \sum\limits_{n=1}^\infty a_f (n)q^n $, any real $ \alpha $ and any $ N \geq 10 $.
That one I have proved.
Now I have to deduce that we have the same bound for the coefficients restricted to any arithmetic progression - that is for any $ 1 \leq q \in \mathbb Z $ and $ a \pmod q$ , we have: $$\left| \sum_{1 \leq n \leq N , n \equiv a \pmod q} a_f (n)\right| \leq c_f N^{k \over 2} \log N .$$
Can someone give me a hint on that one? I know that coefficients may change signs and I don't really know when, so a subset of them may sum to something larger.
