How can I comprehend the complex coefficient of complex Fourier series? Is there a way to understand visually the complex coefficient of Fourier series (shown as $C_n$ in many sources)? I mean something similar to  this which represents $e^{j\omega_0 t}$: .


$\omega_0 \triangleq 2 \pi f_0 .$

But when I read $C_n  e^{jn\omega_0 t}$ my mind shuts down. Here is complex Fourier series: http://www.thefouriertransform.com/series/complexFourier.jpg.
And its one element is $C_n e^{j n \omega_0 t}$. How can I visualize this element? And does $C_n$ have a meaning besides being a coefficient?
if you think of $e^{j n \omega_0 t}$ as a "vector" (complex numbers are not quite the same as 2-dimensional vectors, but they look the same, and are both describable with two real numbers) with length 1 and starting out with an angle of 0 (pointing straight to the right, along the positive real axis, and this vector is rotating counter-clockwise at a rate of $n \omega_0$ radians per unit time.  this vector is what you see looking down the spiral in the graphic referred to in the question, looking along the time axis.

then multiplying by $C_n$ changes the length of the vector to $|C_n|$ and the initial angle of the vector is the angle of $C_n$ or $\arg\{C_n\}$.

that's how i would visualize it.