Parity of a Dirichlet character from its Conrey label (reviewed)

Let $$\chi_q(n,\cdot)=\prod_{p|q}\chi_{p^e}(n,\cdot)$$ be the unique factorization of the Dirichlet character $\chi_q(n,\cdot)$ into characters of prime power modulus $p^e$ under the Conrey labeling system. The parity of $\chi_q(n,\cdot)$ is the sum of the parities of the Dirichlet characters $\chi_{p^e}(n,\cdot)$, which can be computed as follows:

• for $p>2$, the character $\chi_{p^e}(n,\cdot)$ is even if and only if $n$ is a square modulo $p$;
• for $p=2$ and $e>1$ the character $\chi_{p^e}(n,\cdot)$ is even if and only if $n$ is a square modulo $4$.
• for $p=2$ and $e=1$ the character $\chi_{p^e}(n,\cdot)=\chi_2(1,\cdot)$ is even.