Moments of norm-counting functions over integer representations as a sum of eight squares

Let $ \ell, k \ge 2 $ be integers. In this article, we investigate the $ \ell $th power moments of the arithmetic function $ \tau_{k,\mathbb{K}}(n) $, which counts the number of ways $ n $ can be expressed as a product of norms of $k$ non-zero ideals in the ring of integers of a non-normal cubic field $ \mathbb{K} $, evaluated over integers represented as sums of eight squares.