Hecke operators, history (awaiting review)

Hecke [MR:1513122], building on Mordell, introduced operators acting on vector spaces of modular forms. The forms which were simultaneous eigenvalues of these operators have multiplicative Fourier coefficients so that their associated Dirichlet series have Euler products and functional equations. Hecke mainly worked with level 1 modular forms. The more subtle theory of Hecke operators for spaces of higher level and with character was developed by Atkin and Lehner, and Li [MR:268123].