qexpansions

qexpansions refers to the series expansion of a modular form or related analytic object in powers of the variable q=exp(2πiτ), where τ lies in the complex upper half‑plane. Because modular forms exhibit periodicity under the action of the modular group, they can be expressed as Fourier series in q. The resulting q‑series, often called the q‑expansion, encodes significant arithmetic information about the form. For example, the Fourier coefficients in a q‑expansion of the discriminant modular form Δ(τ) are the Ramanujan τ‑function values, and these coefficients satisfy numerous congruence relations discovered over the course of the twentieth century.

The q‑expansion principle, a fundamental theorem of modular forms, states that a modular form is uniquely determined by its q‑expansion. As a consequence, many questions about modular forms can be reduced to properties of their coefficients. In practice, q‑expansions are computed using explicit formulas for Eisenstein series, theta series, or via recursive relations from differential equations. Computer algebra systems such as SAGE, MAGMA, and PARI/GP have built‑in routines to produce q‑expansions to arbitrary precision, making them useful tools in both theoretical investigations and computational experimentation.

In addition to classical modular forms, the concept of a q‑expansion extends to mock modular forms, Jacobi forms, and overconvergent modular symbols. Across these areas, the q‑series representation remains a central computational and conceptual device, bridging complex analysis, number theory, and algebraic geometry.