Take whatever you think you deserve and multiply it by 1.5.
More often than not, they’ll go along with it (leaving you wondering if you should have doubled instead). If they complain, you can reduce it a bit and be sure you’re not going below your fair market value.
Also, increase that amount a bit with each new client. You’ll have an expanded portfolio and more experience, so you’re worth more.
Fresh out of college I thought I was worth $25/hr because I didn’t have as much real-world experience. I asked for $50 and they agreed on $40 — was very glad I didn’t start at $25!
The next job I asked for $60/hr and got it (smaller job, so fewer hours total, which probably helped).
My next job after that I asked for $75/hr. They offered $50 because there were some budget constraints. It was a very fun job, so I took it and made sure I never worked more hours beyond what they could pay me (except for occasional critical stuff).
They ended up just hiring me with a salary when they raised some more money. I definitely think they would not have done this if I hadn’t strictly stuck to what I thought my time was worth and always stayed within those boundaries. If I accepted $30/hr as a freelancer and was flexible when they had budgetary constraints, they probably would have kept me a freelancer and never fully hired me.
So whatever value you pick, ask for more than that. Increase the amount you ask as you continue to work and learn. Be very strict in sticking with the valuation you set for yourself (and adjust it if necessary). Only grant exceptions when there’s a good reason for it, like the job will give you great experience or it’s a very fun project, or charity, etc.
When people get an intuitive feeling for what you’re worth they’re less likely to screw you over. People don’t cajole their lawyer to take less pay or to sometimes work for free, because lawyers don’t put up with that. Programmers should be the same. I’ve run into a lot of companies that try to marginalize and underpay freelance programmers when that technology is at the core of the company. For some reason they don’t do the same with their lawyers.