How to implement a modular form

We present a model for Fourier expansions of arbitrary modular forms. This model takes precisions and symmetries of such Fourier expansions into account. The value of this approach is illustrated by studying a series of examples. An implementation of these ideas is provided by the author. We discuss the technical background of this implementation, and we explain how to implement arbitrary Fourier expansions and modular forms. The framework allows us to focus on the considerations of a mathematical nature during this procedure. We conclude with a list of currently available implementations and a discussion of possible computational research.

► Many implementations of modular forms depend on individual approaches. ► The unification of these implementations can make research easier. ► The negative impact on performance of such a unification is low, and the unification can even be advantageous. ► An implementation as a framework enables researchers to rapidly implement concrete types of modular forms.