Dieulefait–González–Lario congruence conjecture for modular forms of CM type mod \ell

Let $f(q)=\sum_{n=1}^\infty a_nq^n\in S_2(\Gamma_0(N))^{\operatorname{new}}$ be a normalized newform of CM type mod $\ell$ by an imaginary quadratic field $K$. A newform has complex multiplication by $K$ when its coefficients vanish at primes inert in $K$ (away from its level and the relevant prime). Let $\mathfrak{L}$ be a prime ideal above $\ell$ in the composite of the Fourier number fields of $f$ and a second newform. Dieulefait–González–Lario conjecture. There exists a normalized newform $g(q)=\sum_{n=1}^\infty b_nq^n\in S_2(\Gamma_0(M))^{\operatorname{new}}$ with CM by $K$ such that

$$a_p\equiv b_p\pmod{\mathfrak{L}}$$

for every prime $p\nmid NM\ell$. The conjecture asks whether every weight-$2$ newform of CM type modulo $\ell$ arises modulo $\mathfrak{L}$ from a genuine CM newform; the paper proves the assertion for $\ell>2$, $\ell\neq3$, when $K=\mathbb Q(\sqrt{-3})$, while the general case remains open.