Editing Cusp form (section)

==Introduction==
A cusp form is distinguished in the case of modular forms for the [[modular group]] by the vanishing of the constant coefficient ''a''<sub>0</sub> in the [[Fourier series]] expansion (see [[q-expansion|''q''-expansion]])

:<math>\sum a_n q^n.</math>

This Fourier expansion exists as a consequence of the presence in the modular group's action on the [[upper half-plane]] via the transformation

:<math>z\mapsto z+1.</math>

For other groups, there may be some translation through several units, in which case the Fourier expansion is in terms of a different parameter.  In all cases, though, the limit as ''q'' → 0 is the limit in the upper half-plane as the [[imaginary part]] of ''z'' → ∞. Taking the quotient by the modular group, this limit corresponds to a [[Cusp (singularity)|cusp]] of a [[modular curve]] (in the sense of a point added for [[compactification (mathematics)|compactification]]). So, the definition amounts to saying that a cusp form is a modular form that vanishes at a cusp. In the case of other groups, there may be several cusps, and the definition becomes a modular form vanishing at ''all'' cusps. This may involve several expansions.