Being self-employed, you are a business and normally the Turbo Tax self-employed is sufficient. However, NY has a separate form that needs to be filed and the online version does not generate it. You would have to purchase Turbo Tax Business to do so.
An easier method is to print out the IT-204LL and mail it in. The IT-204-LL can be manually filled in and mailed. If you are disregarded entity, fill out the top, part 1 and part 3. If your NY source income for the entity is equal to or less than $100,000, then there is a $25 filing fee. I am attaching further instructions. The mailing address is on the form.