$$\begin{array}{rrrrrrrrrrr} & 1 & g & g^2 & g^3 & g^4 & g^5\\ \phi_0 & 1 & 1 & 1 & 1 & 1 & 1 \\ \phi_1 & 1 & \zeta_6^1 & & & & \\ \phi_2 & 1 & \zeta_6^2 & & & & \\ \phi_3 & 1 & \zeta_6^3 & & & & \\ \phi_3 & 1 & \zeta_6^4 & & & & \\ \phi_5 & 1 & \zeta_6^5 & & & & \end{array}$$
I'm trying to construct a character table for $C_6$, with generator $g$. Since $\phi_0$ is the trivial character, $\phi_0(g^n)$ for $1=<n=<5$ is 1. Similarly, since 1 is the identity element, any character $\phi_n(1)$ = 1. I also know that each character of $C_6$ will send the generator $g$ to a different $n^{th}$ root of unity.
However, I'm not sure how to proceed from here. In general, how do I make the character table of a cyclic group? I'm assuming I need to use the 6th root of unity, $e^{2\pi i\over 6}$.
Thanks.
$C_{6}$ is a cyclic group then $\phi_1: 1\rightarrow e^{2\pi i\over 6}$ $\phi_i=(\phi_1)^i$ $i=1....5$\begin{array}{rrrrrrrrrrr} & 1 & g & g^2 & g^3 & g^4 & g^5\\ \phi_0 & 1 & 1 & 1 & 1 & 1 & 1 \\ \phi_1 & 1 & \zeta_6 & \zeta_6^2 & \zeta_6^3 &\zeta_6^4 &\zeta_6^5 & \\ \phi_2 & 1& \zeta_6^2 & \zeta_6^4 & 1 &\zeta_6^2 &\zeta_6^4 & \\ \phi_3 & 1 & \zeta_6^3 & 1 & \zeta_6^3 &1 &\zeta_6^3 & \\ \phi_4 & 1 & \zeta_6^4 & \zeta_6^2 & 1 &\zeta_6^4 &\zeta_6^2 & \\ \phi_5 & 1 & \zeta_6^5 & \zeta_6^4 & \zeta_6^3 &\zeta_6^2&\zeta_6 & \end{array}
Call $e^{2\pi i\over 6}=\zeta_6$