Dirichlet character tables modulo 8 and 12

The unit groups $(\mathbb{Z}\mathrm{/}8\mathbb{Z}{)}^{\times }=\{1,3,5,7\}$ and $(\mathbb{Z}\mathrm{/}12\mathbb{Z}{)}^{\times }=\{1,5,7,11\}$ are both isomorphic to $C_2\times C_2$. Hence each has four homomorphisms to $\{\pm 1\}$. For modulus $8$, taking signs independently on the generators $3$ and $5$ produces the four characters with values $(0,1,0,\pm 1,0,\pm 1,0,\pm 1)$ on the classes $0,\dots ,7$, subject to $\chi (7)=\chi (3)\chi (5)$.

The same construction for modulus $12$ uses the generators $5$ and $7$, with $\chi (11)=\chi (5)\chi (7)$. Thus all four characters have zeroes on the nonunits and values $(0,1,0,0,0,\pm 1,0,\pm 1,0,0,0,\pm 1)$ on $0,\dots ,11$. Because every homomorphism from $C_2\times C_2$ to $\{\pm 1\}$ is determined by the chosen signs on a basis, these are all the Dirichlet characters modulo $8$ and $12$.