Hecke operators, Hecke Eigensystems, and Formal Modular Forms over Number Fields

We develop an explicit theory of formal modular forms over arbitrary number fields $K$, as functions of modular points. We define modular points for $\Gamma_0({\mathfrak n})$ and $\Gamma_1({\mathfrak n})$, where the level ${\mathfrak n}$ is an integral ideal of $K$; Hecke operators and generalized Atkin-Lehner operators as functions of modular points; and associated Hecke eigensystems. We show how complete eigensystems may be recovered, uniquely up to unramified quadratic twist, from their restrictions to principal Hecke operators, and we give explicit formulas for principal operators suitable for machine computation. These have been implemented by the author in the case of imaginary quadratic fields, and used in his systematic computation of Bianchi cusp forms, which are available in the L-functions and modular forms database (LMFDB). While our description incorporates the classical theory for $K={\mathbb Q}$, and also extends work of the author and his students for imaginary quadratic fields, it applies to arbitrary number fields, and may be useful in the computation of spaces of automorphic forms for GL$(2,K)$ over number fields, whether via modular symbols or other methods.