Some extremal properties of positive trigonometric polynomials

For $n=8$ an upper bound is given for the functional
$$ V_n=\inf_{t_n}\frac{a_1+a_2+\dots+a_n}{(\sqrt{a_q}-\sqrt{a_0})^2}, $$
which is defined on the class of even, nonnegative, trigonometric polynomials $t_n(\varphi)=\sum_{k=0}^na_k\cos k\varphi$, such that $a_k\ge0$ ($k=0,\dots,n$), $a_1>a_0:V_8\le34,\!54461566$.