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As part of the Large Hadron Collider Luminosity upgrade (HiLumi) program, the US LARP collaboration and CERN are working together to design and build a 150 mm aperture Nb3Sn quadrupole magnet that aims at providing a nominal gradient of 140 T/m. In this paper we describe the optimization process yielding the selected 2D coil cross-section and the 3D coil ends design. For the 2D optimization a sector-coil model that allows fast computation of field harmonics is used to identify, among a large number of cases, those cross-sections that provide an acceptable field quality. A more detailed analysis of these solutions is then performed and it led to the selection of an optimized cross-section from which a real coil is built by approximating sectors with blocks of cable. A 3D design of the coil ends is then realized with the Roxie software. Optimization constraints are set on the integrated multipoles, the peak field, and the coil head length.