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Abstract In our recent work, we proposed a differential operator for the evaluation of the multi-dimensional residues on isolated (zero-dimensional) poles. In this paper we discuss some new insight on evaluating the (generalized) Cachazo-He-Yuan (CHY) forms of the scattering amplitudes using this differential operator. We introduce a tableau represen-tation for the coefficients appearing in the proposed differential operator. Combining the tableaux with the polynomial form of the scattering equations, the evaluation of the gen-eralized CHY form becomes a simple combinatoric problem. It is thus possible to obtain the coefficients arising in the differential operator in a straightforward way. We present the procedure for a complete solution of the n-gon amplitudes at one-loop level in a generalized CHY form. We also apply our method to fully evaluate the one-loop five-point amplitude in the maximally supersymmetric Yang-Mills theory; the final result is identical to the one obtained by Q-Cut.