k

and level

Γ (1)

f : h \to C

satisfying

f (γ τ) = (c τ + d)^{k} f (τ)

for every

γ = (a c b d) \in Γ (1)

, and having a meromorphic Fourier expansion at the cusp infinity. Equivalently,

f ∣_{k} γ = f

h

and holomorphic at infinity. Its Fourier expansion therefore has the form

f (τ) = \sum_{n \geq 0} a_{n} q^{n}, q = e^{2 πi τ} .

Since

- I \in Γ (1)

, a nonzero level-one form must have even weight.

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