In your example, “gotta” represents the usual pronunciation of “got to” (as in has / have got to, meaning has / have to). Note that like many words in English, there is a difference between the pronunciation and the correct spelling — “gotta” is not the correct spelling, but simply a way of representing the pronunciation. (It’s actually rather silly, since that is in fact the usual pronunciation — it's not slang.) There's nothing unusual about contracting an auxiliary verb in English; what's unusual is that the contraction "gotta" is not an acceptable spelling (and is used only to represent normal speech). Note that the phrase “got a” can also be pronounced “gotta” (again, the correct spelling is “got a”). For example: He’s gotta new car = he’s got a new car.