First things first, what is GE? General Education (whatever that may be)? Second, do realize that this is more than just "wrong"; it is downright unethical. Copying in the workplace can get you into serious trouble and result in possible loss of your future work experiences/reputation. Also, don't tell the professor what to do. Don't even suggest the zero, let them handle it how they will. It was your fault, now it is up to them to make the decision.
On the lesser side of things, depending on the severity of cheating, you will definitely not be suspended. You may take an instant F in the class and have to re-take, but this is all should be outlined in their syllabus.
And finally, remove both your name and school from this posting. If a future employer sees this, they will not hire you, guaranteed.