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Took my 2005 Acura TL to discount tire for tire balance, they came and told me that my current lug nuts were bad and they have switched it out. Problem is I don't know what kind of lug nuts these are.
You should go back to discount tires and ask them what lug nut to use. I can't tell from the picture but they may have put a locking lug nut on one of the studs and that means you would need a special socket to take it off.
Either way they should be able to steer you in the right direction.
I have tried using a spline socket. is it specific? or can any work?
I've never used one, but I'm willing to bet that they come in different sizes just like other lug nuts. But like Ron A said, just go back to the shop or call them up and ask about the specific socket needed. Easiest way.