Sato-Tate conjecture when Fourier coefficients are complex numbers

Question

Let $k\geq 1$ and let $f=\sum_{n\geq 1}a(n)q^{n}$, $a(n)\in\mathbb{R}$, be a normalised cuspidal Hecke eigenform of weight $2k$ for $\Gamma_{0}(N)$ without complex multiplication. So the result of Barnet-Lamb et al. tells us that the numbers $\frac{a(p)}{2p^{k-1/2}}$ are $\mu$-equidistributed in $[-1,1]$, where $\mu$ is the probability measure on the interval $[-1,1]$ defined by $\frac{2}{\pi}\sqrt{1-t^{2}}\,dt$, and $p$ runs through the primes not dividing $N$.

I am looking for any generalisation of Sato-Tate conjecture when Fourier coefficients $a(n)$ are complex numbers, $a(n)\in\mathbb{C}$?

Answer 1

First, note that if $f = \sum_{n \geq 1} a(n) q^{n}$ is a cuspidal Hecke eigenform lying in the new subspace of $S_{k}(\Gamma_{0}(N), \chi)$, where $\chi$ is a Dirichlet character modulo $n$, then one can determine the argument of $a(n)$ in terms of the values of $\chi$. In particular, for $\gcd(n,N) = 1$, then $$ \overline{a(n)} = \chi(n) a(n). $$ (A proof of this relation can be found in Iwaniec's "Topics in Classical Automorphic Forms" text.) This is the reason that if $\chi$ is trivial, all the Fourier coefficient of $f$ are real. In general, if $\zeta$ is a root of unity and $\chi(p) = \zeta^{2}$, then $\frac{a(p)}{\zeta}$ is real.

A generalization of the Sato-Tate conjecture is stated and proved in the paper of Barnet-Lamb et. al. (which can be found online here). In particular, Theorem B of that paper states that if $\zeta$ is a root of unity with $\zeta^{2}$ in the image of $\chi$, then the numbers $$ \left\{ \frac{a(p)}{2p^{(k-1)/2} \zeta} : p~\text{prime}, \chi(p) = \zeta^{2} \right\}$$ are equidistributed in $[-1,1]$ with respect to $\frac{2}{\pi} \sqrt{1-t^{2}}$.