A3 engine manage management (omissions control light) on

Hope someone can point me in the right direction here,
Just recently bought an 04 A3 TDI. Didn't realise during the test drive but during idle or tick over the car has a judder and the rev counter is erratic. Also, after a day of owning it the omissions control light came on which I think is the engine management light but please correct me if I'm wrong. It went out the day after that but has come back on today as my girlfriend was doing down the motorway and as well as that the 'COOLANT' sign came up on the in car display with an audible warning and the temperature dropped down to 60. From day 1 the juddering at idle hasn't cleared as I thought it might. Being a new car I thought it may have sat for a while at the garage and it might have been a fuel problem. Luckily I know a good mechanic who will be able to look at it for me and it is still within its 3 month warranty but I was hoping someone could give me a heads up before I take it to my friend.

Sorry for the essay!

Ok thanks mike.
The only problem with that is the 160 mile distance between me and the garage. I spoke to my mechanic friend tonight and he is sure he can sort the problems I'm having. We both also know an Audi technician so hopefully I can get it sorted without too much hassle. Bit annoying though for the money we spent and the fact we chopped in a perfectly good car!

From what I've read though, there could be multiple problems. I just hope it's not expensive or difficult.

Thanks warren.

Really gutted on this as we got the car primarily for my girlfriend and I told her to buy German and that this car would be ok for her. Obviously she is panicking a bit as she does a 60-70 mile round trip every day and needs a reliable and economical car. I won't write it off just yet but I doubt she will trust it now.

I took it out for a drive earlier and it drives lovely' Really quick really smooth it's just on tick over its lumpy as hell. It's always un nerving when the management light is on as well. I just hope it's a simple resolve.