The average value of Fourier coefficients of cusp forms in arithmetic progressions

Recently Blomer showed that if α(n)α(n) denote the normalized Fourier coefficients of any holomorphic cusp form f with integral weight, then∑b=1q|∑n⩽Xn≡b(modq)α(n)|2≪f,εX1+ε holds uniformly in q⩽Xq⩽X. By an elementary argument we show that independent of q,∑b=1q|∑n⩽Xn≡b(modq)α(n)|2≪fX(logX)2, where α(n)α(n) could be the normalized Fourier coefficients of any reasonable cusp forms, including Maass cusp forms, holomorphic cusp forms with half-integral or integral weights.