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Let $f\in S_2(\Gamma_1(N))$ be a Hecke eigenform and $\ell$ a prime number does not divide $N$. Let $a_f(\ell)$ be the $\ell$-th Fourier coefficient of $f$. Then $a_f(\ell)$ is is called $\ell$-ordinary if $a_f(\ell)$ is a $\ell$-adic unit.

For any Hecke eigenforms $f$, the Dirichlet density of the following set $$ \{\ell:\text{ prime numbers }|f \text{ is }\ell\text{-ordinary }\} $$ is 1. Here are the references: Reference Link 1, Reference Link 2

Question. Is there the same result for Bianchi modular forms (automorphic forms on $\operatorname{GL}(2)$ over imaginary quadratic fields)?

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    $\begingroup$ I think this should be known (assuming of course that $f$ is not a theta lift, in which case it's false). Prop. 2.2 of the Fischman paper quotes Thm 15 this Serre paper. The only modular forms specific input of the proof is Prop 17, which requires knowing that the $l$-adic Galois representation $G_K \to \mathrm{GL}_2(\overline{\mathbb Q}_\ell)$ associated to $f$ is absolutely irreducible when restricted to any finite extension of $K$. But this is known for Bianchi modular forms. $\endgroup$
    – Ariel Weiss
    Commented Apr 13, 2021 at 11:04
  • $\begingroup$ Crossposted from MSE: math.stackexchange.com/q/4100189/11323 $\endgroup$
    – Kimball
    Commented Apr 13, 2021 at 15:32
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    $\begingroup$ I think this might be a bit more delicate than it looks. The argument of Serre is based on bounding how often we can have $a_\ell(f) = 0$. If the coefficient field is $\mathbf{Q}$ then it follows from the Hasse--Weil bound that $\ell$ can only divide $a_\ell(f)$ if it is zero, but this breaks down if the coefficient field is larger. The argument of Hida applies to the case when the form is for $GL2/K$ and the coeff field is equal to K, and shows only that there is a density 1 set of primes $\ell$ for which $f$ is ordinary at some prime above $\ell$ ("partially ordinary" in Hida's notation). $\endgroup$
    – David Loeffler
    Commented Apr 14, 2021 at 8:45

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