Is there an explicit infinite set of primes, modulo which $X^5 - X - 1$ is irreducible?

Since our polynomial's Galois group over $\mathbb{Q}$ is $S_5$, Chebotarev's density theorem implies that there are infinitely many such primes, but it is seemingly unclear how to find them.

If the Galois group of a polynomial is abelian, then I think primes modulo which the polynomial is irreducible are just the primes satisfying some congruences.

If we take $X^3-X-1$ instead, then such a set of primes can be constructed using the first answer given in Galoisian sets of prime numbers

As far as I am concerned, once a representation of $S_5$ is given, the Langlands correspondence should provide some analytic object (e.g modular form) such that the set of primes for which the polynomial is irreducible (the primes which are inert) could be read off this object.

Can recent work on the Langlands correspondence help finding such a set of primes?

Is there a specific automorphic form which (at least conjecturally) gives us the required primes?