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 $\mathrm{\Delta }$ function, also called modular discriminant is a weight 12 cusp form for the full modular group ${\text{SL}}_2(\mathbb{Z})$

$$\mathrm{\Delta }(z)=q\prod _{n=1}^{\infty }{(1-q^n)}^{24}$$(1)

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.