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The distribution of auxiliary primes raising the level of evenrepresentations of tetrahedral type is studied. Under an equidistributionassumption, the density of primes raising the level of an even, $p$-adicrepresentation is shown to be $$ \frac{p-1}{p}. $$ Data on auxiliary primes$v\leq 10^8$ raising the level of even $3$-adic representations of variousconductors are presented. The data support equidistribution for $p=3$. In theprocess, we prove existence of even, surjective representations $$\rho^{(\ell)}:\operatorname{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})\longrightarrow \operatorname{SL}(2,\mathbb{Z}_3) $$ ramified only at $\ell$and at $3$ for $\ell =163$ and $\ell = 277$. The prime $\ell = 277$ fallsoutside the class of Shanks primes. Measured by conductor, these are thesmallest known examples of totally real extensions of $\mathbb{Q}$ with Galoisgroup $\operatorname{SL}(2, \mathbb{Z}_3)$.

Level-raising of even representations of tetrahedral type and equidistribution of lines in the projective plane