It was recently shown that the renormalization ol quantum Held theory is organized by the Hopf algebra of decorated rooted trees, whose coproduct identifies the divergences requiring subtraction and whose antipode achieves this. We automate this process in a few lines of recursive symbolic code, which deliver a finite renormalized expression for any Feynman diagram. We thus verify a representation of the operator product expansion, which generalizes Chen's Lemma for iterated integrals. The subset of diagrams whose forest struct lire entails a unique primitive siibdivergence provides a representation of the Hopf algebra H_R of undecorated rooted trees. Our undeoorated Hopf algebra program is designed to process the 24213878 BPHZ contributions to the renormalization of 7813 diagrams, with up to 12 loops. We consider 10 models, each in nine renormalization schemes. The two simplest models reveal a notable feature of the subalgebra of Connes and Moscovici, corresponding to the commutative part of the Hopf algebra H_T of the dilleomorphism group: it assigns to Feynman diagrams those weights which remove zeta values from the countcrtcrms of the minimal subtraction scheme. We devise a fast algorithm for these weights, whose squares are summed with a permutation factor, to give rational counterterms.