On the number of representations by certain octonary quadratic forms with coefficients 1, 2, 3, 4 and 6

In this paper, we find formulas for the number of representations of certain diagonal octonary quadratic forms with coefficients $1,2,3,4$ and $6$. We obtain these formulas by constructing explicit bases of the space of modular forms of weight $4$ on ${\mathrm{\Gamma }}_0(48)$ with character.