Performance Engineering of Modular Symbols

Abstract

We present a new program MFSplit which computes information about newform subspaces for modular forms of weight 2 and trivial character. Modular forms are certain functions in mathematics that appear in many different subfields of mathematics, including number theory and complex analysis; newform subspaces are spaces spanned by a special type of modular forms and are, in some sense, building blocks of spaces of modular forms. Our program MFSplit is based on modular symbols, which is a formalism commonly used to compute modular forms. Existing computer algebra systems such as Sage and Magma include implementations of modular symbols. Our implementation applies the principles of performance engineering to this computational number theory problem, and MFSplit is at least 3 times faster than existing implementations. Consequently, we were able to compute information about newform subspaces for level N ≤ 50000, extending previous efforts that computed this information up to N ≤ 16000. Based on this computation, we analyze the performance characteristics of our program and generate more data related to certain conjectures in mathematics.