true when $n=1$, and Ramakrishnan also showed it when $n=2$ [@ramakrishnan:SMO].

Recently, in [@walji] the second author showed that when $n=2$ one can in fact obtain stronger bounds under various assumptions, e.g., the density bound $\frac 18$ from [@ramakrishnan:SMO] may be replaced by $\frac 14$ if one restricts to non-dihedral representations (i.e., not induced from quadratic extensions) or by $\frac 29$ if the representations are not twist-equivalent.

Our motivation for this project was to try to understand an analogue of [@walji] for larger $n$. However the analytic tools known for $\operatorname{GL}(2)$ that are used in [@walji] are not known for larger $n$. Moreover, the classification of $\operatorname{GL}(2)$ cuspidal representations into dihedral, tetrahedral, octahedral and icosahedral types has no known nice generalization to $\operatorname{GL}(n)$. So, as a proxy, we consider the case of Galois (specifically Artin) representations. The strong Artin conjecture says that all Artin representations all automorphic, and Langlands’ principle of functoriality says that whatever is true for Galois representations should be true (roughly) for automorphic representations as well.

Consider $\rho, \rho'$ be irreducible $n$-dimensional Artin representations for $F$, i.e., irreducible $n$-dimensional continuous complex representations of the absolute Galois group $\operatorname{Gal}(\bar F/F)$ of $F$. For almost all places $v$ of 