Does this trigonometric pattern continue for all primes $p=6m+1$? (Revised from its original form.) Consider primes $p=6m+1$, and $p>13$: 

$p=19=3\times\color{blue}{6}+1$:

Let $\beta = 2\pi/19.\,$ A root of $x^3+x^2-\color{blue}{6}x-7=0$ is $$x=2\big(\cos(2\beta)+\cos(3\beta)+\cos(5\beta)\big)$$ 
Note that $2+3=5$, or the last multiplier is the sum of the first two.

$p=31=3\times\color{blue}{10}+1$

Let $\beta = 2\pi/31.\,$ A root of $x^3+x^2-\color{blue}{10}x-8=0$ is $$x=2\big(\cos(2\beta)+\cos(4\beta)+\cos(8\beta)+\cos(16\beta)+\cos(30\beta)\big)$$
Similarly, $2+4+8+16=30$. Note that $\cos(30\beta) =\cos(32\beta)$.

$p=37=3\times\color{blue}{12}+1$

Let $\beta = 2\pi/37.\,$ A root of $x^3+x^2-\color{blue}{12}x+11=0$ is $$x=2\big(\cos(2\beta)+\cos(9\beta)+\cos(12\beta)+\cos(15\beta)+\cos(16\beta)+\cos(54\beta)\big)$$
Again, $2+9+12+15+16=54$.

$p=43=3\times\color{blue}{14}+1$

Let $\beta = 2\pi/43.\,$ A root of $x^3+x^2-\color{blue}{14}x+8=0$ is $$x=2\sum_{k=1}^7\cos\big(2^k\beta)$$
But alternatively,
$$x=2\big(\cos(\beta)+\cos(4\beta)+\cos(11\beta)+\cos(16\beta)+\cos(21\beta)+\cos(35\beta)+\cos(88\beta)\big)$$
and, $1+4+11+16+21+35=88$.
Question: Is it true that argument multipliers of the cubic roots obey $\displaystyle\sum_{k=1}^{m-1}a_k=a_m$ for any $p=6m+1$?

This situation can be summarized as follows (I take  the case p = 19 as an example):
Let $\zeta$ be a $19$-th root of unity then we can form the field of cyclotomic numbers $\mathbb{Q}(\zeta)$. The galois group of this extension (of $\mathbb{Q}$) is the cyclic group $C_{18}$ of order $18$. In order to construct a sub-extension  $K$ of dimension $3$, lets chose a subgroup $H$ of $G$ of index $3$, i.e. the cyclic subgroup of order $18/3 = 6$. The orbit of $\zeta^2$ gives $\{\zeta^2, \zeta^3, \zeta^5, \ldots \text{(complex conjugates)}\}$. The sum $x$ of these elements is invariant under $H$ so belongs to $K$, so it's minimal polynomial is cubic : $x^3 + x^2 -6x-7$.The same situation reproduces for the other values of $p$, so the problem rather concerns the powers of a generator in orbits of cosets of subgroups in the automorphism group of the cyclic group $C_p$.
