Whole numbers are among the most practical and most important mathematical objects. Humans have studied them for millennia. Number theory aims to understand patterns possessed by whole numbers. Fundamental questions revolve around multiplication: how often are numbers in some sequence even (i.e. divisible by two)? Divisible by three? Or five? Nineteenth century researchers introduced symmetry actions to reveal hidden patterns in numbers. And, in the 1970's, Robert Langlands made far-reaching conjectures on symmetry. These conjectures have occupied number theorists ever since. They predict patterns seen by symmetry actions will arise equally from the calculus of complex numbers ("modular forms"). A pattern appearing in two places is an example of a mathematical reciprocity. This project will refine Langlands' reciprocity prediction. The new tool is geometric spaces of symmetry actions, constructed by Emerton and Gee over the past fifteen years. These spaces are believed to convert reciprocity questions into geometrical ones. This project establishes instances of this belief. It will connect divisibility patterns from the world of modular forms to geometrical theorems on Emerton and Gee's spaces. The project has substantial broader impacts. Computational data will be included in the widely-used L-functions and Modular Forms Database. The project also develops computational tools for teaching. Open education resources (OERs) are learning materials placed in the public domain.