Technical would be a better place as the Stardock admins look at that more, but anyways.
If you didn't get Diplomacy through trinity, you must have bought it online via impulse. If that's the case it should just be a matter of installing impulse on your new computer, signing in with your account and hitting install. It shouldn't ask for a serial number, so I'm not sure what you were doing.
That said, if you ever want to view a serial number, just right click on the game from the "My Games" menu in impulse and click "View Serial".
