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We describe a scheme that characterizes the main beam and sidelobe in allStokes parameters employing parameters that allow reconstruction of thecomplete beam patterns and, also, afford an easy way to see how the beamchanges with azimuth, zenith angle, and time. For the main beam in Stokes I theparameters include the beam width, ellipticity and its orientation, coma andits orientation, the point-source gain, the integrated gain (or, equivalently,the main beam efficiency); for the other Stokes parameters the beam parametersinclude beam squint and beam squash. For the first sidelobe ring in Stokes Ithe parameters include an 8-term Fourier series describing the height, radius,and radial width; for the other Stokes parameters they include only thesidelobe's fractional polarization.  We illustrate the technique by applying it to the Arecibo telescope. The mainbeam width is smaller and the sidelobe levels higher than for auniformly-illuminated aperture of the same effective area. These effects aremodeled modestly well by a blocked aperture, with the blocked area equal toabout 10% of the effective area (this corresponds to 5% physical blockage). Inpolarized emission, the effects of beam squint (difference in pointingdirection between orthogonal polarizations) and squash (difference in beamwidthbetween orthogonal polarizations) do not correspond to theoretical expectationand are higher than expected; these effects are almost certainly caused by theblockage. The first sidelobe is highly polarized because of blockage.Comment: 42 pages, 18 figures, accepted by PAS

All‐Stokes Parameterization of the Main Beam and First Sidelobe for the Arecibo Radio Telescope