Only one parent may claim the child as a dependent. Only the parent who claims the child as a dependent can qualify to use an FSA or claim the child and dependent tax credit. Only the parent who claims the child as a dependent can use the child to qualify for EIC and head of household status. In short, you can't "split" any of the tax benefits of claiming a dependent.
Usually, the tax benefit will be larger when the parent who earns more claims the child, because they can also claim head of household status. Sometimes, if the parents who earns less will qualify for EIC, it will be better for the parent who earns less to claim the child (although they can't file as head of household if they pay less than half the overall household expenses, and the other parent must file as single if they don't claim a dependent.). The only way to know in your case is to test different combinations.
Be aware that in Turbotax, the parent who will claim the child should answer "no" to the question about a custody agreement, because this only applies to a court order between parents who are separated and live apart. And the parent who will not be claiming the child should not even list them on their return. The Turbotax interview questions are designed for parents who live apart and share custody and can lead to incorrect tax returns for parents who live together unmarried. Make sure the parent who will claim the child answers that the child lived with them "all year" (which is correct for a newborn) and don't file until you have the child's social security number.
