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

The distribution of primes raising the level of even Galois representations of tetrahedral type is studied. Data are presented on primes $v\le {10}^8$ raising the level of $3$-adic even representations of various conductors. Based on the data, a conjecture is formulated concerning the distribution of certain lines in the plane. By an application of Wiles’ formula, the conjecture is shown to imply that the density of primes raising the level of a $p$-adic even representation is $$\frac{p-1}{p},$$ in agreement with the density of $2\mathrm{/}3$ for $p=3$ observed in the data.