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arXiv · 1603.00774

Products of Eisenstein series and Fourier expansions of modular forms at cusps

Martin Dickson·Michael Neururer
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Abstract

We show, for levels of the form $N = p^a q^b N'$ with $N'$ squarefree, that in weights $k \geq 4$ every cusp form $f \in \mathcal{S}_k(N)$ is a linear combination of products of certain Eisenstein series of lower weight. In weight $k=2$ we show that the forms $f$ which can be obtained in this way are precisely those in the subspace generated by eigenforms $g$ with $L(g, 1) \neq 0$. As an application of such representations of modular forms we can calculate Fourier expansions of modular forms at arbitrary cusps and we give several examples of such expansions in the last section.

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BibTeXRIS

Martin Dickson, Michael Neururer. 2018-03-01. Products of Eisenstein series and Fourier expansions of modular forms at cusps. https://doi.org/10.1016/j.jnt.2017.12.013

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