Nothing found on the usual ser# list sites for the 77 other than a generic years made and toltal mfg'd.
But,,,If you go to the Calguns' forum at this address:
Have an Old Winchester? - Calguns.net
There is a long running thread ,,several years worth, of nothing but people posting Winchester firearms info (ser#,model, caliber,ect) and requesting date of manufacture.
The threads author, 'TRAPP55' has access to the Winchester Polishing Room records (PR records) as he explains in the opening post and how they are much more useful in searching for the info.
Many sites use the (George) Madis comprised data which was published years ago but has serious flaws in it. The problems with it have been known for a long time, but it continues to be published still and like anything else,,repeated enough times, it becomes hard to dispute or change peoples minds. It's in Madis's book after all!
Anyway,,go to that link,,sign up for the forum and log on to the thread and ask away about your 77.
There are at least a couple other inquirys about Model 77's in the 20+ pages of the thread.
They are very helpful over there with inquiries,info and any talk about Winchester.