Evaluation of Some Convolution Sums and Representation Numbers of Quadratic Forms of Discriminant -135

We evaluate the convolution
using the theory of quasimodular forms and determine the number of representations of a positive integer n by some direct sum of the forms

discriminant -135 by

modular forms. Moreover, we explain how to determine the number of representations of Q + tP, t > 1, t|135 for various sums Q , P of the above quadratic forms.