Formed at the state level like any corporation, but with no shareholders. State formation and federal tax exemption are two separate things: forming a nonprofit does not make it tax exempt. Exemption is a separate IRS application.
NY DOS fee schedule: "Not-For-Profit Corporation Incorporation: $75"
No recurring filing with the Secretary of State. Tax filings are separate and are not covered here.
The deadlines above are the rules. Your actual dates depend on when you form, so the calendar is built from your formation date — free, no account.
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