On formulas for the index of the circular distributions

A circular distribution is a Galois equivariant map ψ from the roots of unity μ_∞ to an algebraic closure of ℚ such that ψ satisfies product conditions, ∏_ζ^d =ε ψ (ζ) = ψ(ε) for ε ∈ μ_∞ and d ∈ ℕ, and congruence conditions for each prime number l and s ∈ ℕ with (l, s) = 1, ψ(ε ζ) ≡ ψ(ζ) modulo primes over l for all ε ∈ μ_l, ζ ∈ μ_s, where μ_l and μ _s denote respectively the sets of lth and sth roots of unity. For such ψ, let Pψ_s be the group generated over ℤ[Gal(ℚ(μ_s)/ℚ)] by ψ(ζ), ζ ∈ μ_s and let Cψ_s be Pψ_s ∩ U_s, where U_s denotes the global units of ℚ(μ_s). We give formulas for the indices (P_s:P ^ψ_s) and (C_s : C^ψ _s) of P^ψ_s and C^ψ_s inside the circular numbers P_s and units C_s of Sinnott over ℚ(μ_s).