You don't! The reason you are being punished is probably because you did something wrong. I highly doubt that she will let you go out. Because letting you go out is like rewarding you. If you did...
it kind of depends on why you have to convince her.- its your life if you want to go to college.... THEN GO.Dont let people hold you back and if she loves you she will understand your doing something...
To let your mom get you do anything, make sure she trusts you. Do the chores and things you're supposed to do. Maybe a little more (a little buttering-up can't hurt, eh?). Don't go over budget...