Virago.......Just got through Completely (and I mean Completely) making an 83 920 roadworthy again. Two things to look at. 1 , if you are sure it's the front cylinder , that little mixture control valve on the left side of the frame up under the steering neck can go bad and lean out the front cylinder to the point where it won't fire. If your bike uses that little contraption , plug the 2 vacuum ports it works off of , and toss it in the trash. 2 , the TCI (computer under the gas tank bolted to the top of the frame..). Just had one rebuilt for the 920 , and it is like a dream. Cost is 99 bucks , and it is totally worth it (if that is your problem).
Here's a link to it:http://homepage.mac.com/ddenowh/TCI/index.html

If yours has the second generation Hitachi carbs on it , the little orifice in the bowl (that runs the idle circuit) has to be spotless (what a terrible design) in order to run properly. Hope this helps. BB