Evaluation of the convolution sums Σl+20m=n σ(l)σ(m), Σ4l+5m=n σ(l)σ(m) and Σ2l+5m=n σ(l)σ(m).

The theory of quasimodular forms is used to evaluate the convolution sums ... σ(l) σ(m), ... σ(l) σ(m) and ... σ(l) σ(m) for all positive integers n. As a consequence, the number of representations of a positive integer n by the octonary quadratic form x12 + x22 + x32 + x42 + 5(x52 + x62 + x72 + x82 is determined. [ABSTRACT FROM AUTHOR]