Let \(k \ge 2\) be an even integer, \( \ell \ge \max \{5, k-1\} \) be a prime, and N be a squarefree positive integer. It is known that if the \(\,{\text { mod}}\,\ell \) Galois representation \(\overline{\rho }_f\) associated with a newform f of weight k, level N, and trivial nebentypus is reducible, then \(\overline{\rho }_f \simeq 1 \oplus \overline{\chi }_\ell ^{k-1}\) , up to semisimplification, where \(\overline{\chi }_\ell \) is the \(\mathrm {mod\,\ell }\) cyclotomic character. In this paper, we determine the necessary and sufficient conditions under which the \(\mathrm {mod\,\ell }\) representation \(1 \oplus \overline{\chi }_\ell ^{k-1}\) arises from a newform of weight k, level N with exactly two prime factors with specified Atkin–Lehner eigenvalues. Specifically, this proves a conjecture of Billerey and Menares when N is a product of two primes under certain assumptions. As an application, we show that for any \(\ell \ge 5\) and \(k=2\) or \(\ell +1\) , there exists a large class of distinct primes p and q such that the \(\mathrm {mod\,\ell }\) representation \(1 \oplus \overline{\chi }_\ell ^{k-1}\) arises from a newform of weight k and level pq with explicit Atkin–Lehner eigenvalues. Additionally, we extend the notion of admissible tuples, previously defined by Ribet for weight 2 newforms, to arbitrary weights.
