State tax filing

And if you do have a lot of other income, outside of the NY wages, then you may be in a situation where you need to pay CT quarterly estimated taxes thru the year to make up for any excess  taxes you eventually do have to pay to CT. 

 

But you'll find that out the first tax year you file in CT....and can start doing so after you find out what your situation is  

 

A few small amounts of interest and dividends wouldn't require those quarterly estimates, but if you have a lot of investment income, or a lot of investment capital gains, or a lot of self-employment income...you might need to start paying those estimates to CT.   I  wouldn't start being concerned until you start getting 10or20 thousand of other non-NY income for which no CT withholding is done.

____________*Answers are correct to the best of my knowledge when posted, but should not be considered to be legal or official tax advice.*