When starting there is no crank or sound?

I replaced the battery after the old one sat in a car for a few months without starting the car. The alarm went off when I was trying to jump the old battery.
After the battery was replaced, the windows, headlights and dash lights work, but the map lights and radio do not work and when I turn the key, there is no crank or sound. I can hear something under the hood like a computer whirring or something, then the sound goes away after 10 seconds or so.