The term *representation* is used for a variety of different types of group homomorphisms.

The first type in the LMFDB are Dirichlet
characters. These are homomorphisms
$$ \chi:\modstar\Z n \to \C^\times$$
for some $n\in\Z^+$. Typically, they are thought of as functions on
$\Z$ via the projection $\Z\to\Z/n\Z$ and extension by $0$.

The LMFDB also contains Artin representations. These are continuous
homomorphisms
$$ \rho:\Gal(\overline\Q/\Q)\to\GL_n(\C)$$
for some positive integer $n$. Such homomorphisms always factor
through the Galois group of some number field, and are complex
representations of that Galois group.

A Dirichlet character can be viewed as special type of Artin
representation. If $n\in\Z^+$, then $\Gal(\Q(\zeta_n)/\Q)$
can be identified in a natural way with $\modstar\Z n$, and
$\C^\times\cong \GL_1(\C)$.

In both cases, the representations can be used to factor Dedekind
$\zeta$-functions as products of L-functions.

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