If you purchased a digital upgrade, search your inbox for "Nero" or "Cleverbridge" (their common payment processor).
When users search for a "new" serial number for software this old, they are often looking for a way to reinstall the software after losing their original retail box or email confirmation.
(Just a fictional example, please do not use it)
A suite of over 20 applications, including Nero Burning ROM for disc creation, Nero Vision for video editing, and Nero BackItUp for system protection. Key Features of Version 7.10.1.0