We present an ultrahigh-precision numerical study of the spectrum of multifractal exponents Δq characterizing anomalous scaling of wave function moments ⟨|ψ|2q⟩ at the quantum Hall transition. The result reads Δq=2q(1−q)[b0+b1(q−1/2)2+⋯], with b0=0.1291±0.0002 and b1=0.0029±0.0003. The central finding is that the spectrum is not exactly parabolic: b1≠0. This rules out a class of theories of the Wess-Zumino-Witten type proposed recently as possible conformal field theories of the quantum Hall critical point.