Yeah, tell your mom not to worry about the peacoat. You can tell her you know a girl in Winnipeg Manitoba Canada that wore one for so many winters that it fell apart. One suggestion, since you haven't bought it yet, remember to try it on with something fairly bulky underneath like a sweater or hoody or something. As long as you can wear layers under the coat if it gets unseasonably cold you should be fine.
Also, I majorly agree with those suggesting you get it used. As far as I'm concerned anything over $75 is highway robbery. There are always tons of peacoats at used clothing places that are in great condition - basically old men who take meticulous care of their clothing, they die, and you win. You'll probably be much more likely to find a good one where you're going to though, so why not wait until you get there? ... other than that your mom has mocked you for wanting to do this to match your peers.
Having worked for Mastercard for several years, I definitly agree with the people suggesting you get a credit card and use it only a small amount and pay it off everytime it comes due. But only, only, do this if you know you will not surrender to temptation. If there is even the slightest chance you are the sort to surrender to tempation, you might want to get the card, but leave it at home with your parents and only use it when you're at home and otherwise financially supported.
Other things you might want in your dorm:
A small fridge. If you have room and are allow to. They make really small ones these days, and it's always nice to be able to keep some fruit and drinks in your room. If you get one with a tiny little freezer compartment, even better.
A signifiantly larger amount of socks and undies than you think you will need. You will be surprised at how rarely you have time for laundry, and depending on the facilities, you may find that the rare time you do find to do laundry is the same time as when everyone else has decided to do their laundry too.
A good set of ear plugs. Seriously. If you can block out the noise of the world, you can study or sleep just about anywhere.