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

Abstract

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 ).

All modular forms of weight 2 can be expressed by Eisenstein series

“…Their insight provides a precise connection between the resulting expressions for cuspidal Hecke eigenforms and the special values of the associated L-functions. This connection also appeared in subsequent work by Borisov-Gunnells [4][5][6], who investigated specific modular forms associated with toric varieties, Kohnen-Martin [17], the first named author [23], and Dickson-Neururer [11], who investigated the case of higher levels. The nonvanishing of specific L-values was crucial in all cases.…”

“…We now explain the three key differences of the present paper compared to previous work [4][5][6]11,17,18]. The first key difference is the appearance of E k (N ) as opposed to E k (N ) ∞ .…”

“…The first lemma of this section is a variant of, for instance, Propositions 3.7 and 4.1 in [23] or Corollary 4.2 in [11]. It relates products of Eisenstein series to special values of Lfunctions, generalizing results by Rankin [21] and various other authors.…”

“…It relates products of Eisenstein series to special values of Lfunctions, generalizing results by Rankin [21] and various other authors. We omit the proof, which is mutatis mutandis the one of Corollary 4.2 of [11]. Then there is…”

“…A key difference between [23] and [11] is that the latter employs the products of Eisenstein series of varying weight, while the weight of Eisenstein series is fixed in [23] and the level of Eisenstein series varies. We follow [23] in this paper.…”