Well, I have 2 different formulas. And no idea how to prove any of them. And no way to compute using computer which of them is true. That is exactly a reason why i asked this question in the first place)

Even though I made a mistake in the description of my algebra(which Ifixed already), the problem didn't become any simpler. The complexity is that i have infinitely many such explicit algebraic problems. And i want to solve all of them in less than a year. So I clearly need some more sophisticated way of solving them. For example cohomologies of degree $(i,0)$ are just the cohomologies of $L_1$ which were computed in a paper by Goncharova.