On a general divisor problem related to a certain Dedekind zeta-function over a specific sequence of positive integers

We investigate the average behavior of coefficients of the Dirichlet series of positive integral power of the Dedekind zeta-function ζK₃(s) of a non-normal cubic extension K₃ of Q over a certain sequence of positive integers. More precisely, we prove an asymptotic formula with an error term for the sum\[ ∑_{{a₁²+a₂²+a₃²+a₄²+a₅²+a₆²≤ {x}}{(a₁,a₂,a₃,a₄,a₅,a₆)⁶}}ak,K_3 (a₁²+a₂²+a₃²+a₄²+a₅²+a₆²),\]where (ζK₃(s))ᵏ:=∑ₙ₌₁∞ak,K₃(n)nˢ.