Q. Abarr, J. Alfaro, P. Allison, J. Alvarez-Muñiz, T. Anderson, H. Barnett, A. Basharina-Freshville, J. J. Beatty, L. Beaufore, D. Z. Besson, M. Betts, R. Bose, D. Braun, B. Chamanbahar, P. Chen, Y. Chen, J. M. Clem, T. Coakley, A. Connolly, L. Cremonesi, A. Cummings, P. Dasgupta, C. Deaconu, J. Flaherty, C. Godden, P. W. Gorham, C. Hornhuber, J. Hoffman, K. Hughes, A. Hynous, A. Jung, A. M. Kofman, Y. Ku, D. Kullgren, C. -Y. Kuo, P. Linton, L. Lisalda, T. C. Liu, W. Luszczak, S. C. Mackey, Z. Martin, K. McBride, C. Miki, J. Nam, R. J. Nichol, A. Novikov, A. Nozdrina, E. Oberla, S. Prohira, H. Pumphrey, D. Radomski, B. F. Rauch, R. Scrandis, D. Seckel, M. F. H. Seikh, J. Shiao, G. Simburger, J. Tutt, A. G. Vieregg, S. -H. Wang, D. Washington, C. Welling, P. Windischhofer, S. A. Wissel, C. Xie, J. Yao, R. Young, E. Zas, A. Zeolla

The Payload for Ultrahigh Energy Observations (PUEO) is a NASA balloon-borne instrument for the detection of ultra-high energy (UHE) neutrinos with energies above $10^{17.5}~\textrm{eV}$ via either the Askaryan effect or geomagnetic emissions from an upward-going air shower. The main instrument trigger system for PUEO is a fully digital supersample rate beamformer based on 24 Xilinx Radio Frequency System-on-a-Chip (RFSoC) digitizers sampling 192 channels operating at $3~\textrm{GSa/s}$ and a system clock frequency of $375~\textrm{MHz}$. The trigger implements frequency band conditioning, dynamic radio-frequency interference (RFI) rejection, and matched filtering, with significant emphasis on optimization to reduce both the power and resource usage while maintaining sensitivity. The system implements 48 total synthetic antenna beams with up to 8 antennas each, covering a $\sim25^\circ$ range in zenith and $\sim60^\circ$ range in azimuth. Preflight testing demonstrated a trigger performance of a minimum signal-to-noise ratio (SNR) of $\sim1.5$ using simulated signals while consuming between $5-7~\textrm{W}$ in the trigger logic.