Superconducting La 1.937 Sr 0.063 CuO 4 crystals grown by the travelling-solvent floating-zone technique were thermally treated under various temperatures and oxygen pressures for moderately adjusting the oxygen content.The response of intrinsic electronic property of the crystals to the change of hole density in La 2 x Sr x CuO 4 in the vicinity of the magic doping of x=1/16 (= 0.0625) is studied in detail by magnetic measurements under various fields up to 1 T.It is found that when the superconducting critical temperature (T C) increases with the oxygen content,there appears also a new subtle electronic state that can be detected from the differential curves of diamagnetic susceptibility dχ/dT of the crystal sample.In contrast with the intrinsic state,the new subtle electronic state is very fragile under the magnetic fields.Our results indicate that a moderate change in oxygen doping does not significantly modify the intrinsic electronic state originally existing at the magic doping level. Superconducting La 1.937 Sr 0.063 CuO 4 crystals grown by the traveling-solvent floating-zone technique were thermally treated under various temperatures and oxygen pressures for moderately adjusting the oxygen content. The response of intrinsic electronic property of the crystals to the change of hole density in La 2 x Sr x CuO 4 in the vicinity of the magic doping of x = 1/16 (= 0.0625) is studied in detail by magnetic measurements under various fields up to 1 T.It is found that when the superconducting critical temperature (TC ) increases with the oxygen content, there appears also a new subtle electronic state that can be detected from the differential curves of diamagnetic susceptibility dχ / dT of the crystal sample. contrast with the intrinsic state, the new subtle electronic state is very fragile under the magnetic fields. Our results that that a moderate change in oxygen doping does not significantly modify the intrinsic electronic state originally existing at the magic doping level.