Do some research, but I think you need a steering wheel from a car with a production date earlier than 4/99. One -stage vs two stage?
This topic has been covered many times over all the BMW forums. Have a search around. Your E39 has a build date of prior March 1999, so it's a single stage airbag setup and you'll need a single stage version of the M steering wheel that you have - they came in E36 M3's for instance. You'll not be able to put on a dual stage airbag steering wheel from a Post March 1999 build date E39 i.e the wheel you've just purchased.
Some owners have fitted the later 2 stage steering wheel and connected the airbag to one of the two on the back of the facelift airbag unit, so you could do that, but in the case of any incident it's found the car was changed like this, your insurance company may not like it and could refuse a claim. This is one of the main reasons hardly anyone goes this way, the ones who have usually have their cars as weekend/show cars etc.