To expand a bit on my comment on @KateGregory’s answer, the problem you face with using your own computer is “who owns what?”. How is a line drawn between things that you do on that computer for the company vs. for yourself? If you work on other projects off the clock, can the company make any claim to those? Does the computer have to comply with all corporate standards & policies, including antivirus, web filtering, remote scanning for unapproved software, etc.? Who owns the software you’re using? Who’s responsible for the licensing?

You are going to have to wait a couple weeks before you can make any kind of pitch to get a usable computer. Otherwise, you come across as the new guy who wants everything “just so”, and that will not earn you any points.