Shaped line-source antenna beams can be synthesised by introducing complex roots in a Taylor line source distribution to achieve null filling in the shaped region. Optimisation of these roots and others allows individual control of the amplitude of ripples and the heights of unshaped sidelobes up to any specified order. The corresponding continuous aperture distribution can be made pure real by appropriate root-pairing, which makes the results applicable, via sampling, to resonantly spaced linear arrays.