Non-optimal levels of some reducible mod \(p\) modular representations

Let \(p \geq 5\) be a prime, \(N\) be an integer not divisible by \(p\), \(\bar\rho_0\) be a reducible, odd and semi-simple representation of \(G_{\mathbb{Q},Np}\) of dimension \(2\) and \(\{\ell_1,\cdots,\ell_r\}\) be a set of primes not dividing \(Np\). After assuming that a certain Selmer group has dimension at most \(1\), we find sufficient conditions for the existence of a cuspidal eigenform \(f\) of level \(N\prod_{i=1}^{r}\ell_i\) and appropriate weight lifting \(\bar\rho_0\) such that \(f\) is new at every \(\ell_i\). Moreover, suppose \(p \mid \ell_{i_0}+1\) for some \(1 \leq i_0 \leq r\). Then, after assuming that a certain Selmer group vanishes, we find sufficient conditions for the existence of a cuspidal eigenform of level \(N\ell_{i_0}^2 \prod_{j \neq i_0} \ell_j\) and appropriate weight which is new at every \(\ell_i\) and which lifts \(\bar\rho_0\). As a consequence, we prove a conjecture of Billerey--Menares in many cases.