Question about uncorrelated random variables that are not independent

[StatStudent3]

I understand that if random variables are uncorrelated, then Cov(X, Y) = 0 and Corr(X, Y) = 0 are among the 2 conditions.

However, what is the condition that it can be uncorrelated, but NOT independent at the same time?

[Dason]

The condition is just what you said. The variables are uncorrelated but they not independent. I don't think there is a single defining characteristic other than both conditions are met at the same time.

For example if you take X to be a uniform random variable on [-1, 1] then X and X^2 are uncorrelated but they are definitely not independent.