Prime values of Ramanujan's tau function

We study the prime values of Ramanujan’s tau function $\tau (n)$. Lehmer found that $n={251}^2=63001$ is the smallest $n$ such that $\tau (n)$ is prime: $$\tau ({251}^2)=-80561663527802406257321747.$$ We prove that in most arithmetic progressions (mod 23), the prime values $\tau$ belonging to the progression form a thin set. As a consequence, there exists a set of primes of Dirichlet density $\frac{9}{11}$ which are not values of $\tau$.