We show, for levels of the form N = p a q b N ′ with N ′ squarefree, that in weights k ≥ 4 every cusp form f ∈ 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) = 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.1 When χ = 1N is the principal character modulo N we write M k (N ) for M k (N, 1N ).
We prove two principal results. Firstly, we characterise Maass forms in terms of functional equations for Dirichlet series twisted by primitive characters. The key point is that the twists are allowed to be meromorphic. This weakened analytic assumption applies in the context of our second theorem, which shows that the quotient of the symmetric square L-function of a Maass newform and the Riemann zeta function has infinitely many poles.
This was originally an appendix to our paper 'Fourier expansions at cusps' [1]. The purpose of this note is to give a proof of a theorem of Shimura on the action of Aut(C) on modular forms for Γ(N ) from the perspective of algebraic modular forms. As the theorem is well-known, we do not intend to publish this note but want to keep it available as a preprint.
A interpretation of the Rogers-Zudilin approach to the Boyd conjectures is established. This is based on a correspondence of modular forms which is of independent interest. We use the reinterpretation for two applications to values of L-series and values of their derivatives.
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