Interest of some Dirichlet characters I bumped into papers interested in the following characters
$$\chi_m(n)  = \left\{
\begin{array}{cl}
\left( \frac{m}{n} \right) & \text{if } m \equiv 1 \mod 4 \\
\left( \frac{4m}{n} \right) & \text{if } m \equiv 2,3 \mod 4 \\
\end{array}
\right\}$$
where the symbols are the Kronecker symbols mod $m$ and $4m$ respectively. Why are these characters of particular interest and why do they arise naturally?
 A: Quadratic Dirichlet characters arise naturally for reciprocity in number theory, i.e., for Gauss sums and Gauss quadratic reciprocity law, but also in $L$-functions and many other topics. As an example, the proof of Dirchlet's theorem on primes in arithmetic progressions uses such $L$-functions. A detailed and explicit account is given inTom Apostol's book "Introduction to analytic number theory". 
A: Let $m$ be a square-free integer and consider the quadratic extension $K=\mathbb{Q}(\sqrt{m})$ over $\mathbb{Q}$. We may define a Dirichlet character
$$
\chi_K: 
 \Big(\mathbb{Z} \big/ |\Delta_K|\mathbb{Z} \Big)^\times \longrightarrow \big\{ \pm1 \big\}
$$ as you discribed in your question. here $\Delta_K = m$ if $m \equiv 1 \pmod{4}$ and $\Delta_K = 4m$ if $m \equiv 2,3 \pmod{4}$.
This character will play important role in many propositions and theorems about the quadratic field $K$.
For example, the decomposition of $p$ in $K$:

a prime number $p$ splits in $K$ $\Longleftrightarrow$ $\chi_K(p)=1$,
  $p$ is inert in $K$ $\Longleftrightarrow$ $\chi_K(p)=-1$ and $p$ is
  ramified in $K$ $\Longleftrightarrow$ $\chi_K(p)=0$.

And another example, the class number formula of $K$:

let $h_K$ be the class number of $K$.
if $K$ is imaginary and $K \neq \mathbb{Q}(\sqrt{-1}),\mathbb{Q}(\sqrt{-3})$, then $$ h_K = \frac1{2-\chi_K(2)} \left| \sum_{1 \leq k < \frac{1}{2}|\Delta_K|}
 \chi_K(k) \right|. $$ if $K$ is real, say $\varepsilon$ is a
  fundamental unit, then $$ h_K = \frac1{\log \varepsilon} \left| \sum_{1 \leq k < \frac12|\Delta_K|} \chi_K(k) \log\sin \frac{\pi
 k}{|\Delta_K|} \right|. $$

