Is there a way to access the sign of an integer, self conjugate, linear combination of roots of 1?

More precisely, is there an algorithm (fast is preferred :-) that, given rationals $q_1,q_2,\ldots,q_n$ and integers $a_1,a_2,\ldots,a_n$ returns the sign of $$ {\Large\sum}_j a_j(e^{2\pi i q_j}+e^{-2\pi i q_j})\qquad?$$ Of course, this is just the sign of $\sum_j a_jcos(2\pi q_j)$.

Deciding 0 can be done working in a cyclotomic field, so the actual question is to decide whether the sum is positive.

If such an algorithm is known, I would appreciate pointers to actual implementations, if extant.


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