I don't know much about Treks, but I would not resist the urge to look. I would be interested in knowing if the 600 group is original. I have my doubts along with the Normandy hubs. Based on what I could see of the seat cluster, it is not later than 84. @
Ex Pres is likely correct about the script. My 84 610 has the 600 series script on the chain stay, or did. BTW, the 500 family is has tube material as a common thread with variations in geometry and group configuration that determines the final two digits. 510 is a sport frame, 520 a touring frame, etc. A 560 would be the race version. See this page for more info:
Trek Bike Models by Year and Color
A full break down would be nice but I haven't seen one.
Edit: Just looked on the link above and found new cats with full descriptions. Yours might be a 510 and the script on the TT was there in 84.