Squarefree-level weight-four Eisenstein congruence conjecture

Let $N_1$ and $N_2$ be coprime squarefree positive integers, with $\omega(N_1)$ odd, and let $E_4$ be the weight-$4$ Eisenstein series on $\operatorname{SL}_2(\mathbb Z)$. Squarefree-level weight-four congruence conjecture. If $p$ is an odd prime dividing

$$\frac{1}{45}\prod_{\ell\mid N_1}(\ell^2-1)\prod_{\ell\mid N_2}(\ell^2+1),$$

then there exists an eigenform $f\in S_4(\Gamma_0(N_1N_2))$ Hecke congruent to $E_4$ modulo $p$. If $p\ge7$ and $p\mid(\ell^2-1)$ for every $\ell\mid N_1$ while $p\mid(\ell^2+1)$ for every $\ell\mid N_2$, $f$ may be chosen new of level $N_1N_2$, with Atkin--Lehner sign $-1$ at primes dividing $N_1$ and $+1$ at primes dividing $N_2$. This is presented as a weight-four generalization of the cited prime-level Eisenstein congruence theorem and is intended to explain congruences arising from the SO(5) mass formula; its general validity is not established in the supplied text.