We evaluate the convolution sum $\displaystyle W_{ a,b}(n):= \sum_{ al+bm=n} \hspace{ -3mm} \sigma(l) \sigma(m)$ for $(a,b)=(1,28), (4,7), (2,7)$ for all positive integers $n$. We use a modular form approach. We also re-evaluate the known sums $W_{ 1,14}(n)$ and $W_{ 1,7}(n)$ with our method. We then use these evaluations to determine the number of representations of $n$ by the octonary quadratic form $x_1^2 + x_2^2 +x_3^2 + x_4^2 + 7(x_5^2 + x_6^2 + x_7^2 + x_8^2)$. Finally we compare our evaluations of the sums $W_{ 1,7}(n)$ and $W_{ 1,14}(n)$ with the evaluations of Lemire and Williams [10] and Royer [13] to express the modular forms $\Delta_{ 4,7}(z)$, $\Delta_{ 4,14, 1}(z)$ and $\Delta_{ 4,14, 2}(z)$ (given in [10, 13]) as linear combinations of eta quotients.
Categories: math.NT