By considering a bridge between Bram-Halmos and Embry characterizations for the subnormality of cyclic operators, we extend the Curto-Fialkow and Embry truncated complex moment problem, and solve the problem finding the finitely atomic representing measure ${mu}$ such that ${gamma}_{ij}={int}ar{z}^iz^jd{mu},;(0{le}i+j{le}2n,;|i-j|{le}n+s,;0{le}s{le}n);$ the cases of s = n and s = 0 are induced by Bram-Halmos and Embry characterizations, respectively. The former is the Curto-Fialkow truncated complex moment problem and the latter is the Embry truncated complex moment problem.