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How do I find the image of an element of a differential algebra in the cohomlogy?

Following the documentation on Commutative Differential Graded Algebras, I have defined a differential graded algebra $C$. I have some element $x \in C$, in degree $4$. I can get a basis for the cohomology at degree 4 by


and generators for cocycles and coboundaries by


How do I check if $x$ is a cocycle, and if it is, what it is in terms of the basis of the cohomology above?

I'm not sure I used the right tags, feel free to edit.