My conjecture about the gap is probably not wrong but it's far from explaining the big picture. $q \mod p \not = \pm 1$ is a much better conjecture (not mine of course).

For example, it looks like for $3<p<q<r$ where $q=p+2,r=mpq+3$ the coefficient $c_{mpq+q}$ of $\Phi_n(x)$ is $-2$. Here $n=pqr$ and $ (p,q,r)$ is a prime triple.

If my calculations are correct then $(5,7)$ is another pair like that. I mean the flat ones occur only within the margin of $\pm 1$. My wild guess is the gap $q-p=2$ is responsible for that.

Is $(3,5)$ the only pair where the flat ones are squeezed in the margin of $\pm 1$? For example the "next" pair $(3,7)$ has a "gap" for $w=10,11$. I mean the flat ones occur at margin of $\pm 1,\pm 2$ as well as $\pm 10$ using the theorem you mentioned.

Not a problem. It answers my question. Thank you very much.

yes it's because the odd ones are prime for which $\mu(k)=-1$ and the even ones are 2 times a prime in which case $\mu(k/2)=-1$ . For both cases @Ofir proved that the answer is 0. You have to go as high as 143 to see a $-2$ and as high as 286 to see a $2$. Other examples under 300 for a $-2$ are 187,209,221,247,253,299. If you double those numbers you will see a 2.