Problem with my E90 330I

It slams 5th gear in auto mode and it misses it at anything over 5000 rpm in sport mode.. is this a simple problem (like the computer needs an update or tranny fluid change) or something worse.. also lurches going into first but that’s a common problem I’ve been reading with these cars.

Background- I’ve had the car for a little under a month, it’s got 132915 miles on it, and it’s a 2006. Just did an oil change. I have paper work that says the transmission was checked out at 107xxx miles..