= Character large sieve
For coefficients supported on $N$ consecutive <integers>,
$$
\sum_{q\leq Q}\frac q{\phi(q)}\sum_{\chi\bmod q}^{*}\left|\sum_na_n\chi(n)\right|^2\leq(Q^2+2\pi N)\sum_n|a_n|^2.
$$
The star restricts to <primitive Dirichlet characters>. Their <finite Fourier transform of a primitive Dirichlet character> has normalization of <absolute value> $\sqrt q$. <Orthogonality of Dirichlet characters> therefore bounds the weighted contribution for one modulus by $\sum_{\substack{a\bmod q\\(a,q)=1}}|\sum_na_ne(an/q)|^2$. Sum over $q$ and use the <exponential-sum large sieve> on the distinct <reduced fractions> of denominators at most $Q$.
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