Minimal Polynomial Thing

Let $\zeta$ be a primitive $34$ th root of unity. The (Galois) conjugates of $\sin(\pi/17) = \frac{\zeta-1/\zeta}{2i}$ are of the form $\pm(\zeta^k-1/\zeta^k)/2i$ for $k$ odd and not divisible by $17$ ; that is, $\pm \sin(\pi k/17).$ There are sixteen distinct such conjugates, and we get a complete set if we take the plus sign and $-8 \le k \le 8$ ( $k \ne 0$ ). Then the minimal polynomial is just $\prod_{k=1}^8 (x-\sin(k\pi/17))(x+\sin(k\pi/17)) = \prod_{k=1}^8 (x^2-\sin^2(k\pi/17)).$ The sum of the coefficients of this polynomial is obtained by plugging in $x=1.$ This is $\prod_{k=1}^8 (1-\sin^2(k\pi/17)) = \prod_{k=1}^8 \cos^2(k\pi/17) = \prod_{k=1}^{16} \cos(k\pi/17).$ It is "well-known" that the product $\prod_{k=1}^n \cos(k\pi/n)$ equals $\frac{-\sin(n\pi/2)}{2^{n-1}},$ so if we take out the $k=n$ term, we get $\frac{\sin(n\pi/2)}{2^{n-1}},$ which when $n=17$ gives $\frac1{2^{16}}.$

(For several proofs of the "well-known" identity, see this thread .)

The polynomial is

$x^{16}-\dfrac{17}{4}x^{14}+\dfrac{119}{16}x^{12}-\dfrac{221}{32}x^{10}+\dfrac{935}{256}x^8-\dfrac{561}{512}x^6+\dfrac{357}{2048}x^4-\dfrac{51}{4096}x^2+\dfrac{17}{65536}.$