OBD-II Code P0513 is defined as a Incorrect Immobilizer Key
The key in use has not been programmed to the vehicle. New vehicle's uses keys that have a resistor chip inside them to tell the vehicle it is okay to start. Replacement keys need to be programmed to the vehicle's control module before they can be used to start the vehicle. This is part of the vehicle's Anti-Theft system. All vehicle keys are needed to program the new key. Use of an unprogrammed key will set code P0513 and illuminated the anit-theft light