It is possible to negotiate rent on an apartment after you have made the deposit?

I am planning a move to a new apartment, and have already signed the notice of disclosure and given the deposit. However, I am realizing I may have been a bit hasty in jumping on the offer without negotiating, as there are many vacancies in the area. So, my question is, is it too late to request a lower rent payment? Would I run the risk of the landlord getting peeved and rejecting my application altogether?