The Convolution Sum ∑al+bm=n σ(l)σ(m) for (a, b) = (1, 28), (4, 7), (1, 14), (2, 7), (1, 7)
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We evaluate the convolution sum Wa,b(n) := ∑al+bm=n σ(l)σ(m) for (a, b) = (1, 28), (4, 7), (1, 14), (2, 7) for all positive integers n. We use a modular form approach. We also reevaluate the known sums W1,14(n) and W1,7(n) with our method. We then use these evaluations to determine the number of representations of n by the octonary quadratic form x21 + x22 + x23 + x24 + 7(x25 + x26 + x27 + x28 ). Finally we express the modular forms 4,7(z), 4,14,1(z) and 4,14,2(z) (given in [10, 14]) as linear combinations of eta quotients
Abstract
We evaluate the convolution sum Wa,b(n) := ∑al+bm=n σ(l)σ(m) for (a, b) = (1, 28), (4, 7), (1, 14), (2, 7) for all positive integers n. We use a modular form approach. We also reevaluate the known sums W1,14(n) and W1,7(n) with our method. We then use these evaluations to determine the number of representations of n by the octonary quadratic form x21 + x22 + x23 + x24 + 7(x25 + x26 + x27 + x28 ). Finally we express the modular forms 4,7(z), 4,14,1(z) and 4,14,2(z) (given in [10, 14]) as linear combinations of eta quotients
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