Let $\ell\ge 3$, and let $f$ be a weight $2$ newform of level $N$ with nebentypus equal to $1$. Let $K_f$ be the coefficient field of $f$, let $\lambda$ be a prime of $K_f$ above $\ell$, and let $p\mid N$ with $p\ne \ell$. Assume that $p$ divides $N$ exactly once. If $\bar{\rho}_{f,\lambda}$ is irreducible and is unramified at $p$, then there exist a divisor $M$ of $N/p$, a weight $2$ newform $g$ of level $M$, and a prime $\mu$ of the coefficient field of $g$ above $\ell$ such that, after identifying the residue fields inside a common finite extension of $\mathbb{F}_\ell$,

\begin{align*} \bar{\rho}_{g,\mu}^{\mathrm{ss}}\cong \bar{\rho}_{f,\lambda}. \end{align*}