The next generation of multi-function weather and air surveillance radar systems will make use of phased array antennas to provide unprecedented scanning flexibility and reduced timelines for observations.  The weather application demands precise matching between co-polarized patterns (+/-0.05 dB) and low cross polarization (below -35 dBc, and preferably lower) over the entire scan range, while simultaneously delivering two-way sidelobes no higher than -80 dBc (far out) and -54 dBc (close in) with a sub-1$\degreesym$ beam.  This poses significant antenna design and system-level calibration challenges that require new mathematical analysis and synthesis techniques.  While low-sidelobe synthesis and large array analysis techniques are available for planar arrays, they will almost certainly require scan angle-dependent polarimetric calibration to meet the requirements for weather.  They also suffer from variation in beam width with scan angle, limiting the fidelity of scientific observations.  Cylindrical arrays offer the potential for very low cross polarization and minimal pattern variation with scan angle, but there are few available large-array (unit cell) and pattern synthesis techniques for complex antenna element geometries.  This paper details a finite element phase mode pattern analysis technique that has been developed in support of the Cylindrical Polarimetric Phased Array Radar (CPPAR) demonstrator, which makes use of series-fed, aperture-coupled, stacked patch antennas.  This analysis has been combined with an alternating projections pattern synthesis technique to show that despite the presence of significant surface/creeping wave and grating lobe effects in the horizontal polarization, low sidelobe synthesis is possible with a sacrifice in beam efficiency.  Early measurements will be presented to demonstrate the utility of these techniques.