As I said in the original post, I checked the W2 after it was imported, and everything was correct. I just checked the file again and it's correct in TurboTax. However, when the state (California) received it, the state taxes paid box was set at $0.
They said it's something about the way that TurboTax does the import and the state can't accept the information because it can be changed by the user (?).
And yes, I'm wondering if the same problem could have happened with the federal return.
