Iwasawa invariants of Λ-adic representation

It is a classical result that the number of primes ℓ for which τ(ℓ) vanishes has Dirichlet density 0, where τ(ℓ) is the Ramanujan τ function. We study an analogous question that arises in studying the Λ-adic representation whose image is full. In particular, we show that the set of primes ℓ for which the trace of the Frobenius at ℓ has positive μ-invariant has Dirichlet density 0. We also discuss the analogous Dirichlet densities related to λ-invariants.