There are a few ways. The first way is to make sure that it weighs the correct amount. A genuine double eagle should weigh 33.431 grams.
A coin that is overweight or underweight is a big red flag, especially if it is overweight, or underweight by a lot (keep in mind that if you have a well worn coin, it is natural for it to be slightly underweight), since gold is a heavy metal, a reproduction will most likely be made out of a cheaper material and will be lighter or heavier. Next, measure it. A genuine coin should have a diameter of 34.1 mm and a thickness of 2 MM.
A too thick or too thin coin or a coin in the wrong diameter is another huge red flag. If your coin ends up matching the specifications of the genuine coin and the design matches what genuine coins look like, I would take it to a coin dealer and have them send it in for grading if he also believes it to be genuine. Very few 1907 double eagles exist and they bring large premiums over the spot price of gold, even in lesser grades.
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