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From this verse, the sole reference to a new moon in the New Testament, it appears as though the early church observed them in some way, and they were castigated by their neighbors for doing so. Exactly how they kept them is not clear. Surely they did not observe them as Israel did, that is, with the assigned sacrifices and the blowing of trumpets. Hebrews 5 through 10 make abundantly clear that those sacrifices typify Christ's many-sided work and are no longer required as part of the worship of God.
If it were not for their link to God's festivals, the new moons could almost be taken as an anachronism today. But since the festivals are still to be kept and dating them is tied to the new moons, and in the absence of a direct command to celebrate or convene on them, it seems right to mark them by paying attention to their coming and going, at the very least.
John W. Ritenbaugh
The New Moons