We prove that a compact quantum group is coamenable if and only if its corepresentation ring is amenable. We further propose a Følner condition for compact quantum groups and prove it to be equivalent to coamenability. Using this Følner condition, we prove that for a coamenable compact quantum group with tracial Haar state, the enveloping von Neumann algebra is dimension flat over the Hopf algebra of matrix coefficients. This generalizes a theorem of Lueck from the group case to the quantum group case, and provides examples of compact quantum groups with vanishing L²-Betti numbers.