  The distribution of auxiliary primes raising the level of even
representations of tetrahedral type is studied. Under an equidistribution
assumption, the density of primes raising the level of an even, $p$-adic
representation 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 various
conductors are presented. The data support equidistribution for $p=3$. In the
process, 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 $\ell = 277$ falls
outside the class of Shanks primes. Measured by conductor, these are the
smallest known examples of totally real extensions of $\mathbb{Q}$ with Galois
group $\operatorname{SL}(2, \mathbb{Z}_3)$.
