Page "Modularity theorem" Paragraph 11

we see that we have written the Fourier expansion of a function of the complex variable τ, so the coefficients of the q-series are also thought of as the Fourier coefficients of.

The function obtained in this way is, remarkably, a cusp form of weight two and level N and is also an eigenform ( an eigenvector of all Hecke operators ); this is the Hasse – Weil conjecture, which follows from the modularity theorem.