The operator basis of an effective field theory (EFT) is the set of independent operators which contribute to scattering processes. We embark on the first systematic studies of operator bases, aiming to elucidate the structure underlying what is meant by "independent operators". We show that operator bases are organized by the conformal algebra, allowing us to systematically account for redundancies associated with the use of equations of motion and integration by parts. As a means to study the operator basis, we introduce a partition function defined to count operators weighted by their field content. We provide a matrix integral formula that allows us to compute this partition function. This allows us to solve an outstanding problem in EFTs: determining the number of independent higher dimension operators in a given EFT. This solution is applied to the Standard Model EFT, where we enumerate the operator content up to dimension fifteen. The physical definition and rich structure underlying operator bases is suggestive that more physical information can be pulled from the operator basis, and we give a few speculative thoughts along these lines.
