Cross sections of the process e+e− → π0 π0 J / ψ at center-of-mass energies between 3.808 and 4.600 GeV are measured with high precision by using 12.4 fb−1 of data samples collected with the BESIII detector operating at the BEPCII collider facility. A fit to the measured energy-dependent cross sections confirms the existence of the charmoniumlike state Y ( 4220 ) . The mass and width of the Y ( 4220 ) are determined to be ( 4220.4 ± 2.4 ± 2.3 ) MeV / c2 and ( 46.2 ± 4.7 ± 2.1 )     MeV , respectively, where the first uncertainties are statistical and the second systematic. The mass and width are consistent with those measured in the process e+e− → π+π− J / ψ. The neutral charmonium-like state Zc ( 3900 )0 is observed prominently in the π 0 J / ψ invariant-mass spectrum, and, for the first time, an amplitude analysis is performed to study its properties. The spin-parity of Zc ( 3900 )0 is determined to be JP = 1+, and the pole position is ( 3893.1 ± 2.2 ± 3.0 ) − i ( 22.2 ± 2.6 ± 7.0 )     MeV / c2, which is consistent with previous studies of electrically charged Zc ( 3900 )±. In addition, cross sections of e+e− → π0Zc ( 3900 )0 → π0π0 J / ψ are extracted, and the corresponding line shape is found to agree with that of the Y ( 4220 ).