Regularized inner products and weakly holomorphic Hecke eigenforms**The research of the first author is supported by the Alfried Krupp Prize for Young University Teachers of the Krupp foundation and the research leading to these results receives funding from the European Research Council under the European Union’s Seventh Framework Programme (FP/2007-2013) / ERC Grant agreement n. 335220—AQSER. The research of the

Abstract

We show that the image of repeated differentiation on weak cusp forms is precisely the subspace which is orthogonal to the space of weakly holomorphic modular forms. This gives a new interpretation of weakly holomorphic Hecke eigenforms.