In this paper, the following questions for computing coefficients of Fourier series are discussed: n-order Filon quadrature formula and its partial cases, some features of applying the Filon method in computing coefficients when the adaptive integration strategy is employed, the program implementation of 3-order and 5-order Filon quadrature formulas, using the adaptive integration strategy, and the experimental results of applying them in computing coefficients of Fourier series.