Eisenstein Series and Convolution Sums

Abstract: We compute Fourier series expansions of weight $2$ and weight $4$ Eisenstein series at various cusps. Then we use results of these computations to give formulas for the convolution sums $ \sum_{a+p b=n}\sigma(a)\sigma(b)$, $ \sum_{p_1a+p_2 b=n}\sigma(a)\sigma(b)$ and $ \sum_{a+p_1 p_2 b=n}\sigma(a)\sigma(b)$ where $p, p_1, p_2$ are primes.