Depends on your contract. I've seen cases where they took 50% and cases where they took 10%. It all depends on the amount of risk involved and the amount of time and effort. Generally, they can take as much as you agree to. However, the bar association or court will slap their fee down if they feel the lawyer has taken a "disproportionate sum". Maybe NY has an exact % figure for that, but I don't know of any state that does.
In the UK, personal injury solicitors are allowed to charge their client a fee but most simply take their charges from the set Court Fees, which, in the event of the claimant winning, are paid by the defendant! Therefore, no charge is made to the claimant and they keep all their compensation awarded by the Court. Hence, why we have a lot of "No Win, No Fee" companies who only take PI cases they know they will win Do you not have them in the US?
I've seen 40% as the norm right off the bat. If it's a case that won't go to trial, I'd try and negotiate a 33% fee.