Suppose that No denotes the set of positive integers, r, s, n, α and β are elements of N_0 such that α and β are relatively prime, r and s are odd. For all levels αβ and r + s = 4, the evaluation of the convolution sums are carried out using in particular modular forms. Convolution sums belonging to this class of levels are then applied to determine formulae for the number of representations of n by the quadratic forms in sixteen squares ∑_(i=1)~(16)x_i~2 if the level αβ is a multiple of 4,and in sixteen variables ∑_(i=1)~8(x_(2i-1)~2+x_(2i-1)x_(2i) + x_(2i)~2)if the level αβ is a multiple of 3. By evaluating the convolution sums for αβ = 3,5, 6,8,9, 10,12,16,18, 20,25,27, 32, this approach is illustrated and explicit formulae for the number of representations of n by quadratic forms in sixteen squares and in sixteen variables are determined. The confluence of the determination of a formula for the number of representations of a positive integer is used to bring new identities into existence.
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