Lovasz theta number for $n$-cyle is $\frac{ncos(\frac{\pi}{n})}{1+cos(\frac{\pi}{n})}$.

Is the $0$-error capacity of $7$-cycle:

$(1)$ known not to be of form $\frac{7cos(\frac{\pi}{7})}{1+cos(\frac{\pi}{7})}$?

$(2)$ known to be of form $7^q$ for some $q\in \mathbb Q$?

Is the $0$-error capacity of $7$-cycle:

$(1)$ known to be of form $7^q$ for some $q\in \mathbb Q$?