A modification of Szász–Mirakyan operators is presented that reproduces the functions 1 and e2 a x, a> 0 fixed. We prove uniform convergence, order of approximation via a certain weighted modulus of continuity, and a quantitative Voronovskaya-type theorem. A comparison with the classical Szász–Mirakyan operators is given. Some shape preservation properties of the new operators are discussed as well. Using a natural transformation, we also present a uniform error estimate for the operators in terms of the first- and second-order moduli of smoothness.