Transposases are important tools in genome engineering, and there is considerable interest in engineering more efficient ones. Here, we seek to understand the factors determining their activity using the Sleeping Beauty transposase. Recent work suggests that protein coevolutionary information can be used to classify groups of physically connected, coevolving residues into elements called "sectors", which have proven useful for understanding the folding, allosteric interactions, and enzymatic activity of proteins. Using extensive mutagenesis data, protein modeling and analysis of folding energies, we show that (i) The Sleeping Beauty transposase contains two sectors, which span across conserved domains, and are enriched in DNA-binding residues, indicating that the DNA binding and endonuclease functions of the transposase coevolve; (ii) Sector residues are highly sensitive to mutations, and most mutations of these residues strongly reduce transposition rate; (iii) Mutations with a strong effect on free energy of folding in the DDE domain of the transposase significantly reduce transposition rate. (iv) Mutations that influence DNA and protein-protein interactions generally reduce transposition rate, although most hyperactive mutants are also located on the protein surface, including residues with protein-protein interactions. This suggests that hyperactivity results from the modification of protein interactions, rather than the stabilization of protein fold.