I worked remotely from my home for a NY employer for a 3 months summer job out of necessity as due to COVID they did not want anyone coming to NY. That was my only income for 2021. During that time I lived in New Mexico though my permanent address is in Washington D.C. The NY employer withheld NY state income taxes.
Can I recover a refund of all my NY state income taxes withheld? IF so, how do I proceed?
Then do I put zero when TTpremiere asks for NY Income Allocation?
Do I have to file three state forms?
If so, is it as a non-resident NY, part-time NM and part-time D.C.?
Probably not. New York taxes remote employers who are assigned to a NY office. If sounds like you were supposed to work in NY but didn't due Covid-19.
New York adopted pandemic rules that said: "If you are a nonresident whose primary office is in New York State, your days telecommuting during the pandemic are considered days worked in the state unless your employer has established a bona fide employer office at your telecommuting location.
"There are a number of factors that determine whether your employer has established a bona fide employer office at your telecommuting location. In general, unless your employer specifically acted to establish a bona fide employer office at your telecommuting location, you will continue to owe New York State income tax on income earned while telecommuting."
Your income should be 100% New York on a New York nonresident return.
You are a bit unclear about the other two states. If your are a Washington, DC resident, then you'd file a resident DC return and nonresident New Mexico and New York returns.
Your NY income would be taxed by DC and NM and you can claim a credit for NY and NM tax on DC so you'd owe no DC tax. Your NY income would be taxed twice by NY and NM since you were a nonresident of both states when you earned the income.
If you are a part-year NM resident, then you can claim a credit for NY tax. You are supposed to file a part-year DC return even if you have no income in DC.