Security Logout

I went out to my 01 GT the other day and the battery was completely dead. After trying to jump it, it started up right away and after two seconds, shut off. The Battery ended up being bad and I got a new one, but the same problem. After checking some of the posts, I tried leaving the key in the on position for at least 10 minutes and the same thing. The security light would come on, but not flash. Any ideas?