Automorphic forms and infinite matrices

In the present paper, we show that an infinite dimensional vector whose components are Fourier coefficients of an automorphic form is characterized as an infinite dimensional vector which is annihilated by an infinite matrix constructed by the values of a Bessel function. Results and methods are all simple and concrete.Although the idea in the present paper is applicable to more general cases, our investigation will be restricted to the case of automorphic forms of weight 0, i.e., automorphic functions, with respect to SL(2, Z) on the upper half plane, in order to explain the main idea distinctly.