It seems that the prevailing sentiment is that you shouldn't ever try to contact the game admins AT ALL if you were banned, and I don't agree with that.
"I'm sorry - I screwed up and I hope you'll give me another chance" or "I think there's been a big misunderstanding and I'd like to clear the air" seem like perfectly constructive and reasonable ways to reach out.
The key is to accept no (or silence) as an answer. Badgering someone after they've made it clear they don't want to talk to you is annoying at best, harassment at worst. Extend the olive branch and leave it at that.