j (τ) = \frac{E _{4} ( τ ) ^{3}}{Δ ( τ )}

Γ (1)

, holomorphic on

h

and meromorphic at infinity. Let

γ = (a c b d) \in Γ_{1} (2) .

Since

c

is even,

γ^{'} = (a c /2 2 b d) \in Γ (1), 2 γ τ = γ^{'} (2 τ) .

j

therefore gives

\frac{j ( γ τ )}{j ( 2 γ τ )} = \frac{j ( τ )}{j ( 2 τ )} .

The quotient is meromorphic on

h

and at the cusps, so it is a weight-zero modular function of level

Γ_{1} (2)

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