Modular Forms

Visualizing the fundamental domain of SL(2,Z) and the action of modular transformations on the upper half-plane

4
4
Argument
2.0
f(γτ) = (cτ+d)^k · f(τ), γ = [[a,b],[c,d]] ∈ SL(2,Z)

The Eisenstein series E_k(τ) = 1 − (2k/B_k) Σ σ_{k-1}(n) q^n where q = e^{2πiτ}. The colored region shows the fundamental domain |τ| ≥ 1, |Re(τ)| ≤ 1/2, with domain coloring revealing the complex structure of the modular form.