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

We show, for levels of the form N=paqbN′ with N′ squarefree, that in weights k≥4 every cusp form f∈Sk(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)≠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.