8-ranks of class groups of quadratic number fields and their densities

For (F = mathbb{Q}left( {sqrt {varepsilon pq} } right)), ? ∈ {±1, ±2}, primes ?p ≡ q ≡ 1 mod 4, we give the necessary and sufficient conditions for 8-ranks of narrow class groups of F equal to 1 or 2 such that we can calculate their densities. All results are stated in terms of congruence relations of p, q modulo 2 ⁿ , the quartic residue symbol (left( {frac{p}{q}} right)_4) and binary quadratic forms such as q ^h(?2p)/4 = x ² + 2py ², where h(?2p) is the class number of (mathbb{Q}left( {sqrt { - 2p} } right)). The results are very useful for numerical computations.