Certainly, you are allowed according to Jewish law to leave your radio, television, computer, heater, air conditioner, or any other electrical appliace on the entire shabbat. In fact, when I used to be religious, I would set my radio timer such that it would play about two hours during each shabbat. I did not want to leave it on the whole time or else I would not be able to sleep. Besides, I did want to spend at least some time studying the Torah. But sometimes I imagined myself owning eight radios, and setting each one to play two hours each, such that it would never, ever stop playing during my waking hours.
Turning the computer on or off, or printing things, on Shabbat is definitely prohibited by all Torah authorities. But there does seem to be some disagreement as to whether one can simply type on one's computer, since one is not really printing anything but rather creating images on a screen that are not really there. But I am skating on thin ice here, as I do not know this subject to comment in any more detail than that. Certainly just about every worthy Rabbi would suggest to you to just stay off the computer for that one day of the week.