||Evaluation of convolution sums and some remarks on cusp forms of weight 4 and level 12
Ramakrishhan, B.Sahu, Brundaban
Mathematical Journal of Okayama University
79 , 2017-01 , Department of Mathematics, Faculty of Science, Okayama University
In this note, we evaluate certain convolution sums and make some remarks about the Fourier coefficients of cusp forms of weight 4 for Γ<sub>0</sub>(12). We express the normalized newform of weight 4 on Γ<sub>0</sub>(12) as a linear combination of the (quasimodular) Eisenstein series (of weight 2) <i>E<sub>2</sub>(dz)</i>, <i>d</i>|12 and their derivatives. Now, by comparing the work of Alaca-Alaca-Williams  with our results, as a consequence, we express the coefficients <i>c<sub>1,12</sub>(n)</i> and <i>c<sub>3,4</sub>(n)</i> 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<sub>0</sub>(6) and Γ<sub>0</sub>(12). The properties of <i>c<sub>1,12</sub>(n)</i> and <i>c<sub>3,4</sub>(n)</i> 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.