The Hijri era is calculated according to the Islamic lunar calendar and not the Julian or Gregorian solar one. It thus does not begin on January 1, 1 CE, but on the first day of the month of Muharram which occurred in 622 CE. Its Julian equivalent was April 19 but it is sometimes mistakenly placed on July 16. The error derives from the tabular Islamic calendar which was devised by later Islamic astronomers. This reckons time backwards according to the lunar calendar, which causes it to miss the three intercalary months (about 88 days) added to the then-lunisolar calendar between the time of the Hijra and AH 10, when Muhammad is recorded as having received a revelation prohibiting their use.
The date of the Hijra itself did not form the Islamic New Year. Instead, the system continues the earlier ordering of the months with the Hijra occurring around the 8th day of Rabi al-Awwal, 66 days into the first year.