3. Using expression (33). The principal character modulo $p^{\alpha}$ is $\chi\left(n ; p^{\alpha}, 0\right)=\chi(n ; p, 0)$, and the non-principal real character modulo $p^{\alpha}$ is $\chi\left(n ; p^{\alpha}, \varphi\left(p^{\alpha}\right) / 2\right)=\chi(n ; p,(p-1) / 2)=\left(\frac{n}{p}\right)$, where expression (28) is used. For a given $p$, the same primitive root is taken for all moduli $p^{\alpha}(\alpha \geqslant 1)$. The real characters modulo 2 are given by taking $\beta_{-1}=\beta_{0}=2$, and the real characters modulo 4 are given by taking $\beta_{0}=2, \beta_{-1}=1,2$ (verify directly and see the explanation after equation (9)). The real characters modulo $2^{\alpha_{0}}\left(\alpha_{0} \geqslant 3\right)$, as known from equation (32), are: $\chi\left(n ; 2^{\alpha_{0}}, l_{-1}, 0\right)=\chi\left(n ; 8, l_{-1}, 0\right)=\chi\left(n ; 4, l_{-1}\right)$, i.e.,

i.e., take $\beta_{-1}=1,2, \beta_{0}=1$.
$$\begin{array}{l} \beta_{-1}=1,2, \beta_{0}=0 \text {; } \\ \chi\left(n ; 2^{\alpha_{0}}, l_{-1}, 2^{\alpha_{0}-3}\right)=\chi\left(n ; 8, l_{-1}, 1\right)=\chi(n ; 4, l_{-1}) \cdot \chi(n ; 8,0,1) \\ =\chi\left(n ; 4, l_{-1}\right)\left(\frac{2}{n}\right)=\chi\left(n ; 4, l_{-1}\right)\left(\frac{8}{n}\right), \end{array}$$