I am looking for a reference that covers the following result from page 360 of the text. For context, we have a Galois representation $\overline{\rho}: G_{\mathbb Q} \to \operatorname{GL}_2(k)$, where $k$ is a finite field of characteristic $\ell$. We assume $\ell$ is an odd prime, that this representation is irreducible, that its restriction to a decomposition group at $\ell$ is finite flat or ordinary, that it has cyclotomic determinant, and that it has squarefree conductor. We also assume that $\overline{\rho}$ is equivalent over $\overline{\mathbb F}_\ell$ to $\overline{\rho}_f$ for some newform of weight 2 and level $N_f$ with trivial character. We know that if this holds, then there are infinitely many choices for $f$. Given a finite set of primes $\Sigma$, we might ask which of these $f$ give representations of so-called type $\Sigma$, meaning those $f$ such that $\rho_{f, \lambda}$ (the attached $\ell$-adic representation to $f$) is semistable at $\ell$ (meaning flat or ordinary at $\ell$) and such that $\Sigma$ contains the set of primes dividing $N(\rho_{f, \lambda})/N(\overline{\rho})$ (the quotient of the conductors).
The authors claim without reference that a sufficient condition is that $N_f \mid N_\Sigma$, where
$$N_\Sigma = N(\overline{\rho})\prod_{p \in \Sigma} p^{m_p}$$
where $m_p = 2$ if $p \nmid \ell N(\overline{\rho})$, $m_p = 1$ if $p \neq \ell$ and $p \mid N(\overline{\rho})$, $m_\ell = 1$ if $\overline{\rho}$ is flat and ordinary at $\ell$, and $m_\ell = 0$ otherwise.
Can someone provide either a detailed proof of this fact, or a reference(s) containing such a proof?
There is a paper by Gouvea, Deforming Galois representations: controlling the conductor (link), which carefully works out the relation between the conductor of $\bar\rho$ and the levels of forms lifting $\rho$. I'm not sure if it contains the exact statement you're looking for, but you can certainly put together a proof using the results in the article.