Using 24 million ψ decays in CLEO-c, we have searched for higher multipole admixtures in electric-dipole-dominated radiative transitions in charmonia. Let b2 and a2 denote the normalized magnetic quadrupole (M2) amplitudes in the transitions ψ → γχ(c1,c2) and χ(c1,c2) → γJ/ψ, respectively. Previous measurements found ratios aJ=1 /aJ=2 and aJ=1 /bJ=1 in significant disagreement with theoretical 2 2 2 2 predictions, where the J in the superscript refers to the angular momentum of the χcJ . By performing unbinned maximum likelihood fits to the full five-parameter angular distributions, we found the following values of M2 admixtures for Jχ = 1: aJ=1 = (−6.26 ± 0.63 ± 0.24) × 10−2 and 2 bJ=1 = (2.76 ± 0.73 ± 0.23) × 10−2 , 2 which agree well with theoretical expectations for a vanishing anomalous magnetic moment of the charm quark. For Jχ = 2, if we fix the electric octupole (E3) amplitudes to zero as theory predicts for transitions between S and P states of charmonium, we find: aJ=2 = (−9.3 ± 1.6 ± 0.3) × 10−2 , and 2 bJ=2 = (1.0 ± 1.3 ± 0.3) × 10−2 . 2 If we allow for E3 amplitudes we find, with a four-parameter fit, aJ=2 = (−7.9 ± 1.9 ± 0.3) × 10−2 , 2 bJ=2 = (0.2 ± 1.4 ± 0.4) × 10−2 , 2 aJ=2 = (1.7 ± 1.4 ± 0.3) × 10−2 , and 3 bJ=2 = (−0.8 ± 1.2 ± 0.2) × 10−2 . 3