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

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 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 Γ₀(6) and Γ₀(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.