Mathematical Journal of Okayama University 59巻 1号

2017-01 発行

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 Γ_{0}(12). We express the normalized newform of weight 4 on Γ_{0}(12) as a linear combination of the (quasimodular) Eisenstein series (of weight 2) *E*_{2}(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 *c*_{1,12}(n) and *c*_{3,4}(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 Γ_{0}(6) and Γ_{0}(12). The properties of *c*_{1,12}(n) and *c*_{3,4}(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