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The Galois orbits of Dirichlet characters of modulus $q$ are ordered as follows. Let $\chi$ be any character in the Galois orbit $[\chi]$ and define the $N$-tuple of integers \[ t([\chi]) := (n,t_1,t_2,\ldots,t_{q-1}) \in \Z^q, \] where $n$ is the order of $\chi$ and $t_i:=\mathrm{tr}_{\Q(\chi)/\Q}(\chi(i))$ is the trace of $\chi(i)$ from the field of values of $\chi$ to $\Q$. The $q$-tuple $t([\chi])$ is independent of the choice of representative $\chi$ and uniquely identifies the Galois orbit $[\chi]$.

The orbit index of $\chi$ is the index of $t([\chi])$ in the lexicographic ordering of all such tuples arising for Dirichlet characters of modulus $q$; indexing begins at $1$, which is always the index of the Galois orbit of the principal character of modulus $1$.

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  • Last edited by Andrew Sutherland on 2019-01-12 15:41:21
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