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We use the notation $\chi_{q}(n,\cdot)$ to identify Dirichlet characters $\Z\to \C$, where $q$ is the modulus, and $n$ is the index, a positive integer coprime to $q$ that identifies a Dirichlet character of modulus $q$ as described below. The LMFDB label $\texttt{q.n}$, with $1\le n < \max(q,2)$ uniquely identifies $\chi_{q}(n,\cdot)$.

Introduced by Brian Conrey, this labeling system is based on an explicit isomorphism between the multiplicative group $(\Z/q\Z)^\times$ and the group of Dirichlet characters of modulus $q$ that makes it easy to recover the order, the conductor, and the parity of a Dirichlet character from its label, or to induce characters.

As an example, $\chi_q(1, \cdot)$ is always trivial, $\chi_q(m,\cdot)$ is real if $m^2=1\bmod q$, and for all $m,n$ coprime to $q$ we have $\chi_q(m,n)=\chi_q(n,m)$.

For prime powers $q=p^e$ we define $\chi_q(n,\cdot)$ as follows:

  • For each odd prime $p$ we choose the least positive integer $g_p$ which is a primitive root for all $p^e$, and then for $n \equiv g_p^a $ mod $p^{e}$ and $m \equiv g_p^{b} $ mod $p^{e}$ coprime to $p$ we define $$ \chi_{p^e}(n, m) = \exp\left(2\pi i \frac{a b}{\phi(p^{e})} \right). $$

  • $\chi_2(1, \cdot)$ is trivial, $\chi_4(3, \cdot)$ is the unique nontrivial character of modulus $4$, and for larger powers of $2$ we choose $-1$ and $5$ as generators of the multiplicative group. For $e > 2$, if $$ n \equiv \epsilon_a 5^a \pmod{2^e} $$ and $$ m \equiv \epsilon_b 5^b \pmod{2^e} $$ with $\epsilon_a, \epsilon_b \in \{\pm 1\}$, then \[ \chi_{2^e}(n, m) = \exp\left(2 \pi i \left(\frac{(1 - \epsilon_a)(1 - \epsilon_b)}{8} + \frac{ab}{2^{e-2}}\right)\right). \]

For general $q$, the function $\chi_q(n, m)$ is defined multiplicatively: $\chi_{q_1 q_2}(n, m) := \chi_{q_1}(n, m)\chi_{q_2}(n, m)$ for all coprime positive integers $q_1$ and $q_2$. The Chinese remainder theorem implies that this definition is well founded and that every Dirichlet character can be defined in this way. In particular, every Dirichlet character $\chi$ of modulus $q$ can be written uniquely as a product of Dirichlet characters of prime power modulus.

Knowl status:
  • Review status: reviewed
  • Last edited by John Cremona on 2018-11-28 10:41:05
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