I'm struck by the preconceived notions atheists who have posted here so far have about gods. Nearly all qualities assigned to a god in this thread so far are that of an Abrahamic deity...almost so much as I'm lead to think that many atheists don't lack belief in gods in general; they lack the belief in the God of Abraham, and that many of them have very limited knowledge of gods of any religions outside the Abrahamic paradigm.
I agree with most of that. When somebody refers to a god in Western culture, they usually mean the god of Abraham, the one that get the capital-G. Likewise, and the word religion generally refers to Abrahamic religion. I tend to think that that's what they mean until something they write or say suggests otherwise, and it's usually correct. When it's not, it's because somebody wrote capital-G "God" when what they meant was some other concept of a deity such as when Einstein did just that. That's the default meaning of the word "God," although that didn't appear in the OP. Still, as worded, the OP is suggesting monotheism with "a god" rather than "a god or gods."
Where I disagree is that atheists are only rejecting the Abrahamic god.
I do agree, however, that few atheists seem to know much about the other religions and gods that people adopt. I don't. I can tell you a little bit about the dharmic religions and almost nothing about the various forms of polytheism and their gods, but that's because that's not useful or interesting to me.
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To answer the OP in the language of monotheism, if the deity is noninterventionist like the deist deity, we would expect to find no evidence for that - just a clockwork universe obeying regular laws as we find the world, and having that knowledge would be useless. It would affect no decisions I make and thus the answer regarding its existence would be irrelevant - the position called apatheism.
Regarding interventionist gods like the Abrahamic god, it would depend how it intervened. That one is said to be benevolent, to have produced a revelation, perform miracles, and answer prayer, so we ought to see revelation that appears transhuman in origin - words no man would be expected to have written and which transform people positively. We ought to see evidence for supernaturalism in daily life - magic, or the suspension of the known laws of nature.
For other interventionist gods, we ought to see something similar - some compelling manifestation of supernaturalism, something nature wouldn't do without intelligent oversight.
And even with all of that, we still wouldn't know that we were detecting a supernatural entity. We shouldn't be surprised to learn that some transhuman extraterrestrial race that itself arose naturalistically via abiogenesis and biological evolution could appear to suspend the laws of nature, which is why the Abrahamic creation apologist are working in vain trying to falsify evolution theory, since even if they could successfully demonstrate that the theory was wrong, the new paradigm would become that a deceptive, transhuman with immense power arranged the earth to look like Darwin was correct but was found out, and we shouldn't assume supernaturalism to account for that, either.
In the end, the whole concept of the supernaturalism adds nothing even if the word has some meaning, which is another discussion.