Mathematical Journal of Okayama University 59巻 1号
2017-01 発行

Evaluation of convolution sums and some remarks on cusp forms of weight 4 and level 12

Ramakrishhan, B. Harish-Chandra Research Institute
Sahu, Brundaban School of Mathematical Sciences National Institute of Science Education and Research
Publication Date
2017-01
Abstract
In this note, we evaluate certain convolution sums and make some remarks about the Fourier coefficients of cusp forms of weight 4 for Γ0(12). We express the normalized newform of weight 4 on Γ0(12) as a linear combination of the (quasimodular) Eisenstein series (of weight 2) E2(dz), d|12 and their derivatives. Now, by comparing the work of Alaca-Alaca-Williams [1] with our results, as a consequence, we express the coefficients c1,12(n) and c3,4(n) that appear in [1, Eqs.(2.7) and (2.12)] in terms of linear combination of the Fourier coefficients of newforms of weight 4 on Γ0(6) and Γ0(12). The properties of c1,12(n) and c3,4(n) that are derived in [1] now follow from the properties of the Fourier coefficients of the newforms mentioned above. We also express the newforms as a linear combination of certain eta-quotients and obtain an identity involving eta-quotients.
Keywords
convolution sums of the divisor function
Fourier coeffificients
newforms of integral weight