cusp form

A cusp form is a modular form whose first coefficient in any expansion around a cusp is zero. Another more general way to define cusp forms is to consider the forms orthogonal to Eisenstein series with respect to the Petersson scalar product.

The Weierstrass $\Delta$ function, also called modular discriminant is a weight 12 cusp form for the full modular group $\sldeuxz$ \begin{equation} \Delta(z)=q\underset{n=1}{\overset{\infty}{\prod}}(1-q^n)^{24} \end{equation} The vector space of weight $k$ cusp forms for the full modular group is finite dimensionnal and non-trivial for $k$ integral greater than 12 and not 14.