The wake of a streamwise oscillating cylinder is presently investigated. The Reynolds number investigated is 300, based on the cylinder diameter d. The cylinder oscillates at an amplitude of 0.5 d and a frequency f$_e$/f$_s$=1.8, where f$_e$ is the cylinder oscillating frequency and f$_s$ is the natural vortex shedding frequency of a stationary cylinder. Under these conditions the flow is essentially two dimensional. A two-dimensional direct numerical simulation (DNS) scheme has been developed to calculate the flow. The DNS results display a street of binary vortices, each containing two counter-rotating vortical structures, symmetrical about the centerline, which is in excellent agreement with measurements. The drag and lift on the cylinder have been examined. The time averaged drag and lift are 1.4 and 0, respectively, which are the same as those on a stationary cylinder at the same Re. However, the fluctuating drag was high, about 2.68. It has been found that, being symmetrically formed about the centerline, the binary vortices induce an essentially zero fluctuating lift, which may have a profound implication in flow control and engineering.