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For integers $N>0$, $\\mu\\geq 0$ and an odd prime $p$ with $\\gcd(N,p)=1$, let $K_{(Np^\\mu)}$ and $(K_i)_{(Np^\\mu)}$ for $i=1,2$ be the ray class fields of $K$ and $K_i$, respectively, modulo $Np^\\mu$. 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