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
In addition to classical modular forms, the concept of a q‑expansion extends to mock modular forms, Jacobi