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The Airbag management light comes on to inform you that the airbag system is not working correctly and it stores a fault code in the ECU with information on what is wrong.
You can only access this information via a diagnostic device.
I advise you to take your car to a Fiat main dealership or specialist to have a real time diagnostic check done so a full diagnose can be given (this will have a set charge of approx 20 mins labour) cost depends on labour rate.
It could be a failed part on the airbag system or just a software glitch that will require an update.