Modular forms — the shape of |f(τ)|

A modular form is a holomorphic function on the upper half-plane that barely moves when you act on its input by $S L_{2} (Z)$ : for weight $k$ ,

f (\frac{a τ + b}{c τ + d}) = (c τ + d)^{k} f (τ), (a c b d) \in S L_{2} (Z) .

Two of those matrices generate the rest: $T : τ \mapsto τ + 1$ (so $f$ is periodic and has a q-expansion in $q = e^{2 π i τ}$ ) and $S : τ \mapsto - 1/ τ$ . Because $∣ q ∣ = e^{- 2 π Im τ}$ , the series converge fast, and the whole function is pinned down by its values on one tile — the fundamental domain. Below, $∣ f (τ) ∣$ is a landscape you can rotate.

The surface

Form

3D surface — log₁₀|f(τ)| over the upper half-plane · drag to rotate

Filled contour — the same landscape, from directly above

The discriminant, weight 12 — the first cusp form, a pure eta product Δ = q ∏(1−qⁿ)²⁴. It never vanishes on the upper half-plane, so the landscape is a smooth basin: |Δ| decays toward the cusp (Im τ → ∞) and climbs as you approach the real axis.

Where it lives — the fundamental domain

The shaded tile $D$ is $∣ Re τ ∣ \leq \frac{1}{2}$ above the unit circle. Its corners are the special points $ρ = e^{2 π i /3}$ and $i$ — exactly where $E_{4}$ and $E_{6}$ vanish. The map $T$ slides $D$ left and right (the two neighbours shown), and $S$ folds the region below the arc back up — together they tile the whole half-plane with copies of $D$ .

A congruence subgroup like $Γ_{0} (N)$ keeps only some of those symmetries, so its fundamental domain is several copies of $D$ glued along the arcs — the setting where newforms and elliptic curves of level $N$ live. Drawing those glued domains (congruence vs non-congruence) in 3D is the next build.

Reading the landscape — what works, what doesn't

What the picture gets right

The q-periodicity is visible — the surface repeats every step of 1 in Re τ. The cusp (Im τ → ∞) shows as |f| flattening. The zeros of E₄ (at ρ) and E₆ (at i) are honest wells; Δ, which never vanishes, is a smooth basin. Distances are real: this is a faithful plot of |f| from its definition.

What it doesn't show — yet

Only the modulus. A phase portrait (colour = arg f) would expose the orderof each zero by its winding — the valence formula made visible, the way the polynomial demo shows the argument principle. The weight-

k

stretch under

S

also isn't on this strip. Those, plus congruence domains, are the honest to-do.

Why it matters here. A modular form's q-expansion coefficients

a_{n}

are arithmetic — for a weight-2 newform they count points on an elliptic curve,

a_{p} = p + 1 - # E (F_{p})

. See that identity run live in the eigenbook.