Absolute convergence of cusp-form L-series from square coefficients (${\sum }_{n\ge 1}|a_n|n^{-\sigma }<\infty (\sigma >(k+1)/2)$)

By positive unfolding of a cusp-form square, $\sum |a_n{|}^2{n}^{-\beta }<\infty$ for $\beta >k$. Choose $k<\beta <2\sigma -1$. The Cauchy-Schwarz inequality bounds the absolute sum for the L-function of a cusp form by