Let f denote a normalized Hecke eigenform of an even integral weight κ ≥ 2 on the group SL2 (ℤ). We study power moments of the coefficients of an L-function associated with the eigenform f over square-free positive integers represented by the polynomial  g(x1, x2, x3, x4, x5, x6) = x21 + x22 + S3≤j≤6 xj (xj + 1) ∈ ℤ[x1, x2, x3, x4, x5, x6].