As is known to all, the complex nonlinear Fokas-Lenells equation is widely used to describe the propagation of short pulses in optical fibers. In this work, we aim to study its time-space fractional modified form. An ancient Chinese algorithm called the Ying Bu Zu Shu (盈不足术), which is from an ancient Chinese mathematics monograph-The Nine Chapters on the Mathematical Art (九章算术) in about second century AD, is used to find its periodic solution. The solution obtained by our proposed method is the same as the one obtained via the variational method, which strongly proves the effectiveness and reliability of the proposed method. Finally, we plot the solution with different fractional orders in the form of 2-dimensional and 3-dimensional surfaces. The finding in this work is expected to be helpful for the study of the periodic solution in physics.