Motorcycle Repair: bike wont start, harley dealership, minute intervals

Question
I have a 1998 Custom 1200 Harley Davidson Sportster. I have a problem that started about 6 weeks ago and has been intermittent but is happening more often now.  After I've ridden the bike between  a good half hour in city traffic and about  1 1/2 hours on country roads and let it sit for more than 45 minutes it wil not start, it rolls over and over but will not start. If I let it sit for  2 or more hours it will start.  The second to last time this happened I let it sit about  2 hours but as soon as I got down the road it started popping (not sputtering) like crazy, jerking and then died. I got it to restart right away and it did the same thing. I went ahead and had it towed to a friends house. He told me the next day that it started and ran fine. Ok, I went ahead and had him replace the battery new plugs, he checked the plug wires (fine) and he also put in a known working coil (not a new one). I took it home and it was fine. The next day I rode 30 minutes to work, it was fine.  I rode home (which took about 40 minutes in 92 degree temp). I was home for 1 hour, came out to start it, wouldn't start, after about 25 minutes of  trying to start it at 3-5 minute intervals I just gave it lots of gas it started kind of roughly but smoothed out, I took off and as soon as it got down the road into 3rd gear it started the popping, jerking and dying.  I coasted home, started it the next day (ran fine) and rode it to the Harley dealership about 10 miles away.  They couldn't find anything wrong with it. I told them to recreate the situation, they did, they said it was getting spark and then it wasn't, they decided to put a new coil in there and try that. I told them that I had put a working coil in there and it still did the same thing and that's where it stands now. Haven't heard anything from them since.  I have gotten different suggestions from bad gas to stator and I don't know what to think. Any ideas? I'm of course no mechanic, I'm just a woman who wants to have her bike in riding condition for the summer. Thanks so much for listening.
Sindey

Answer
After reading your problem I would say it sounds heat related. Your coil and ignition can break down due to heat. Since you already tried a new coil and that didnt solve the problem, I would next check the ignition. The dealer should be able to test the ignition. I have done it by using a blow dryer to create the heat. Another thing you can check is to make sure the fuel line isnt getting vapor locked.
One thing I wouldnt do is pay for the coil they put on. There are ways to test coils without just putting one on.
Good luck and happy riding
Mike