There are typically terms set in a decree of dissolution of marriage or decree of paternity that dictate who gets to claim a child for tax purposes each year. It is fairly common for parents to simply alternate years with Mom taking even numbered years, and Dad taking odd numbered years for instance.
Parties are also free to agree to other arrangements, and of course, the court is free to order whatever terms it finds equitable and in the child’s best interest.
It is unclear as to whether or not the fact that the child was in state custody for half the year would have an effect on who gets to claim the child. You could seek a motion to clarify from the court.
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This information is general in nature and should not be construed as tax advice. You should work with your attorney or tax professional to determine the tax advantages that will work best for your situation.