Local Newforms for GSp(4) A Summary Representation Theory Paramodular Vectors Zeta Integrals Non supercuspidal Representations Hecke Operators Proofs of the Main Theorems. Local Newforms for GSp(4) describes a theory of new and oldforms for representations of GSp(4) over a non archimedean local field. This theory considers vectors fixed by the paramodular groups and singles out certain vectors that encode canonical information such as L factors and epsilon factors through their Hecke and Atkin Lehner eigenvalues. While there are analogies to the GL(2) case this theory is novel and unanticipated by the existing framework of conjectures. An appendix includes extensive tables about the results and the representation theory of GSp(4).