Mathematical Journal of Okayama University 59巻 1号
Harish-Chandra Research Institute
School of Mathematical Sciences National Institute of Science Education and Research
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  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  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.
convolution sums of the divisor function
newforms of integral weight