I am trying to understand the LMFDB labeling of Dirichlet characters (see here), but I am not entirely sure what they mean by the "orbit index".
To get things started, fix some Dirichlet character $\chi:(\mathbb{Z}/N\mathbb{Z})^\times\rightarrow \mathbb{C}^\times$ having order $n$. Then $\chi$ takes values in the cyclotomic field $\mathbb{Q}(\chi):=\mathbb{Q}(\zeta_n)$. As stated on LMFDB, the Galois orbit of $\chi$ is the set $[\chi]= \{\sigma(\chi) : \sigma\in \operatorname{Gal}(\mathbb{Q}(\zeta_n)/\mathbb{Q})\}$, where $\sigma(\chi):a\mapsto \sigma(\chi(a))$.
They go on to define the tuple $$ t([\chi])=(n,tr_{\mathbb{Q}(\chi)/\mathbb{Q}}(1),\dots,tr_{\mathbb{Q}(\chi)/\mathbb{Q}}(N-1))\in \mathbb{Z}^N $$ and then they state that 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 $N$". Can someone clarify this for me? What is the index of a tuple in the "lexicographic ordering of all such tuples"? Presumably this should be something simple-ish, but I can't seem to find a concrete definition.
For example, take the unique odd character $\chi$ of conductor 3. LMFDB says that this character has label 3.b. In this case we should have $t[\chi]=(2,1,-1)$. In the words of LMFDB, "all such tuples arising for Dirichlet characters of modulus 3" would just be the two tuples $(1,1,1)$ and $(2,1,-1)$, corresponding to the trivial character mod 3 and the quadratic character $\chi$ above. In this case, it seems somewhat obvious that the index should be 2 (hence the "b" in the label), but for examples with more tuples to consider it would be nice to know exactly what they mean by this and how it is computed...
