The 10,000 is prorated, so it would depend on the date of death.
According to the State of New Jersey:
“The fiduciary of every resident estate or trust must file a New Jersey Gross Income Tax Fiduciary Return (Form NJ-1041) if gross income, before exemptions or deductions, was more than $10,000 (prorated for the number of months covered by a part-year return) during the tax year. “
NJ-1041
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