I unplugged and cleaned everything I could, still no luck.
An other detail I found out, if that can help to identify the problem : If I leave it off for some time, it will turn on an work for a 30 to 60 seconds. After that, it won't turn on at all, the fan will do the stop and start that I mentioned in the first post, etc.. If I leave it off for an hour, it will work for an other 30 to 60 seconds.