Let $\omega,a_1,\dots,a_n\in\Bbb R$ arbitrary real numbers. Consider the sum $$ z = \langle a, t\rangle = a_1e^{-i\omega}+a_2e^{-i2\omega}+\cdots+a_ne^{-i n\omega} $$ which is the hermitian product of $a=(a_1,\dots,a_n)\in\Bbb R^n\subset\Bbb C^n$ and $t=(e^{i\omega},\dots,e^{in\omega})\in\Bbb C^n$
I was wondering whether there exists a known way to write it in polar form. Namely, I ask if there exists a formula to get $r=|z|$ and $\theta=\arg(z)$ in function of $a$ and $t$ such that $$ \langle a, t\rangle = re^{i\theta}\;. $$ Clearly \begin{align*} r&=|z|=\sqrt{[\Re(z)]^2+[\Im(z)]^2}\\ \theta&=\arg(z)=\arctan(\Im(z)/\Re(z)) \end{align*} (this last one only if $\Re(z)>0$, see Wikipedia for the all the cases), where \begin{align*} \Re(z)&=\sum_{k=1}^{n}a_k\cos(\omega k)\\ \Im(z)&=\sum_{k=1}^{n}a_k\sin(\omega k) \end{align*} but the problem is that these expressions "unpack" and "split" both $a$ and $t$ with projections (e.g., $a_k=\pi_k(a)$ and $\cos(\omega k) = \pi_1(\pi_k(t))$).
What I am searching for is a similar expression but depending only on the vectors $a$ and $t$ "as a whole", without passing through projections.