Modular forms of CM type mod $\ell$

We say that a normalized modular form is of CM type modulo $\ell$ by an imaginary quadratic field $K$ if its Fourier coefficients $a_p$ are congruent to $0$ modulo a prime $\mathcal L\mid \ell$ for every prime $p$ that is inert in $K$. In this paper, we address the following question. Let $f$ be a weight~$2$ cuspidal Hecke eigenform without complex multiplication which is of CM type modulo $\ell$ by an imaginary quadratic field $K$. Does there exist a congruence modulo $\ell$ between $f$ and a genuine CM modular form of weight~$2$? We conjecture that such a congruence always exists. We prove this conjecture for $\ell>2$ and $\ell\neq 3$ when $K=\mathbb{Q}(\sqrt{-3})$. In this setting, we discuss three situations: (i) modular forms attached to abelian surfaces with quaternionic multiplication, (ii) $\mathbb{Q}$-curves completely defined over an imaginary quadratic field, and (iii) elliptic curves over $\mathbb{Q}$ whose $5$-torsion Galois representation has image the maximal cyclic of order $16$ inside $\operatorname{GL}_2({\mathbb F}_5)$. In all these cases, the modular forms under consideration are of CM type modulo suitable primes~$\ell$, and we show that the associated residual Galois representations are monomial with respect to an imaginary quadratic field $K$ (in some instances, more than one such field). Finally, we present numerical evidence that motivated the conjecture and provides further support for its validity beyond the cases treated in this paper.