Maximal abelian extension of $X_0(p)$ unramified outside cusps

Let $p$ be a prime number. Mazur proved that a geometrically maximal unramified abelian covering of $X_0(p)$ over $\mathbb{Q}$ is given by the Shimura covering $X_2(p)\to X_0(p)$, that is, a unique subcovering of $X_1(p) \to X_0(p)$ofdegree$N_p := (p-1)/\gcd(p-1, 12)$. In this short paper, we show that a geometrically maximal abelian covering $X_2^{\mathrm{\prime }}(p)\to X_0(p)$ of $X_0(p)$ over $\mathbb{Q}$ unramified outside cusps is cyclic of degree $2N_p$. The main ingredient for the construction of $X_2^{\mathrm{\prime }}(p)$ is the generalized Dedelind eta functions.