Modular Forms — Fundamental Domain Explorer

Fundamental domain of SL(2,Z)

Click inside the upper half-plane to drop a point, then apply the generators T and S to trace its orbit across the tiling.

Generators

Current point

click canvas

word (from z₀)

–

orbit length

Legend

fundamental domain F

γF, |word|=1

γF, |word|=2

orbit point

How it works

The modular group SL(2,Z) acts on the upper half-plane ℍ = {z : Im(z) > 0} by γz = (az+b)/(cz+d) for integer matrices with ad−bc=1. Every orbit meets the classical fundamental domain F = {z : |z| ≥ 1, |Re(z)| ≤ ½} exactly once (up to its boundary identifications). Translated and inverted copies γF, generated by words in T (z→z+1) and S (z→−1/z), tile the rest of ℍ without gaps or overlaps.

T = [1 1; 0 1]   z → z+1
S = [0 −1; 1 0]  z → −1/z
SL(2,Z) = ⟨S, T⟩

The bottom strip plots a truncated Eisenstein series E₄(z) = 1 + 240·Σ σ₃(n)qⁿ, q = e^(2πiz), sampled live along the boundary contour of F — the moving dot sweeps the same path the domain's edge traces in the plot above.

Re(E₄(z)) along ∂F — q-expansion, live