Not-for-profit corporation without 501c3

I'm filing taxes for an organization I volunteer for.  It is registered as a domestic not-for-profit corporation in the state of New York, which ultimately makes it a C corporation.  It doesn't have a federal 501c3 status and uses a fiscal sponsor in order to receive donations.

 

I'm guessing because it's not a federal 501c3, I should file the taxes like a normal C corporation, but has anybody encountered this before?  It's not really a for-profit business, even though it is a C corp, but maybe that only matters for the state it's registered in, and not on the federal returns.  Anybody know which forms it should file with?