On a variance of Hecke eigenvalues in arithmetic progressions

Let a(n)a(n) be the eigenvalue of a holomorphic Hecke eigenform f under the nth Hecke operator. We derive asymptotic formulae for the variance∑b=1q|∑n≤Xn≡b(modq)a(n)|2 when X1/4+ε≤q≤X1/2−εX1/4+ε≤q≤X1/2−ε or X1/2+ε≤q≤X1−εX1/2+ε≤q≤X1−ε, that exhibit distinct behavior. The analogous problem for the divisor function will be studied as well.