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The optical absorption spectrum of the carbon (4,2) nanotube is computed using an ab initio many-body approach which takes into account excitonic effects. We develop a new method involving a local basis set which is symmetric with respect to the screw-symmetry of the tube. Such a method has the advantages of scaling faster than plane-wave methods and allowing for a precise determination of the symmetry character of the single-particle states, two-particle excitations, and selection rules. The binding energy of the lowest, optically active states is approximately 0.8 eV. The corresponding exciton wave functions are delocalized along the circumference of the tube and localized in the direction of the tube axis.