Sparse arrays provide a large aperture with few elements. Through an exhaustive search of all possible thinning patterns for small linear sparse arrays, the Fourier properties of arrays with optimal coarray properties are explored. In addition coarray properties of sparse arrays with minimum peak sidelobe level are given. Minimum hole arrays have optimal low peak sidelobe level, while minimum redundancy arrays have a peak sidelobe level that is close to the global minimum. Other arrays with minimum peak sidelobe level have coarray properties similar to those of minimum hole and minimum redundancy arrays. Sparse arrays with optimal coarray qualities also possess optimal Fourier qualities.