# 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 $\textrm{SL}_{2}(\mathbb{Z})$

 $\Delta(z)=q\underset{n=1}{\overset{\infty}{\prod}}(1-q^{n})^{24}$ (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.

Title cusp form CuspForm 2013-03-22 14:07:43 2013-03-22 14:07:43 olivierfouquetx (2421) olivierfouquetx (2421) 9 olivierfouquetx (2421) Definition msc 11F11