Squarefree-level conjecture for 1⊕χ_l^{k−1}

Let $k$ be an even integer with $k\geq4$, let $l>k+1$, and let $N$ be squarefree. Consider the representation $\mathbf{1}\oplus\chi_l^{k-1}$, where $\chi_l$ is the mod $l$ cyclotomic character. Squarefree-level conjecture. The representation arises from a weight-$k$ newform of squarefree level $N$ and trivial Nebentypus if and only if either

$$(p^k-1)(p^{k-2}-1)\equiv0\pmod l$$

for every prime $p\mid N$, with some prime divisor $p_0\mid N$ satisfying $p_0^k\equiv1\pmod l$, or $p^{k-2}\equiv1\pmod l$ for every prime $p\mid N$ and $l$ divides the numerator of $B_k/k$, where $B_k$ is the $k$-th Bernoulli number. This is proposed as a conjectural description of the squarefree non-optimal levels, and the paper explains its equivalence with sufficiency of the Steinberg level-raising condition.