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We report the first observations of the first harmonic (directed flow, v1) and the fourth harmonic (v4), in the azimuthal distribution of particles with respect to the reaction plane in Au+Au collisions at the BNL Relativistic Heavy Ion Collider (RHIC). Both measurements were done taking advantage of the large elliptic flow (v2) generated at RHIC. From the correlation of v2 with v1 it is determined that v2 is positive, or in-plane. The integrated v4 is about a factor of 10 smaller than v2. For the sixth (v6) and eighth (v8) harmonics upper limits on the magnitudes are reported.