Twisted adjoint L-values, dihedral congruence primes and the Bloch-Kato conjecture

We show that a dihedral congruence prime for a normalised Hecke eigenform f in Sk(Γ0(D), χD), where χD is a real quadratic character, appears in the denominator of the Bloch-Kato conjectural formula for the value at 1 of the twisted adjoint L-function of f. We then use a formula of Zagier to prove that it appears in the denominator of a suitably normalised L(1, ad0(g) ⊗ χD) for some g ∈ Sk(Γ0(D), χD).