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Here is a screenshot of a semi-answer which froze my computer when I hit 'post':

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(2017-11-26 edit by j.c.: earlier versions of this answer consisted of David Hansen's screenshot of the following, with the text "Here is a screenshot of a semi-answer which froze my computer when I hit 'post'":)

Things get more complicated. $\newcommand{\SU}{\mathop{\rm SU}\nolimits} \newcommand{\C}{\bf C} \newcommand{\Irr}{\mathop{\rm Irr}\nolimits} $

Let $\pi=\pi_\infty\otimes\bigotimes'_p\pi_p$ be a unitarily normalized cuspidal automorphic representation of $GL_n(\bf{A}_{\bf Q})$; suppose for simplicity that $\pi$ has trivial central character. The local factor $\pi_p$ (for $p$ outside a finite set $S$) is determined by a matrix $s(\pi_p)\in GL_n(\C)$, which is well-defined up to conjugacy. The generalized Ramanujan conjecture predicts that $s(\pi_p)$ may be chosen as an element of the unitary group $\SU_n$. Suppose this is true, and let $f:\SU_n\rightarrow\C$ be a continuous function which is invariant under conjugation. Here is a generalized Sato-Tate conjecture:

If $\pi$ is not a functorial lift from a "smaller" group, then $\frac{1}{\pi(X)}\sum_{p\leq X}f(s(\pi_p))\rightarrow \int_{\SU_n}f(g)dg$ where $dg$ is the Haar probability measure on $\SU_n$.

If $n=2$ and $\pi$ is associated with an elliptic curve $E$, this simplifies drastically: the conjugacy class $s(\pi_p)\in\SU_2$ is determined uniquely by its trace, which is $\frac{a_p(E)}{\sqrt{p}}$. Likewise, any $f$ determines a unique continuous $f':[-2,2]\rightarrow\C$, with $f=f'\circ\mathrm{tr}$. The Haar probability measure $dg$ pushes forward to the measure $\frac{1}{2\pi}\sqrt{4-x^2}$ on $[-2,2]$, and the conjecture now takes the more familiar form.

What's the meaning of the vague condition "$\pi$ is not a functorial lift from a smaller group"? Well, let's try to give a plausibility argument for the above conjecture. Let $\Irr_n$ denote the set of irreducible algebraic representations of $\SU_n$; for $\sigma\in\Irr_n$, write $\chi_\sigma$ for its character. In $L^2(\SU_n)$ we have

$$f(g)=\sum_{\sigma\in\Irr_n}\chi_\sigma(g)\langle f,\chi_\sigma\rangle.$$

Changing $g$ to $s(\pi_p)$, summing over $p\leq X$ and switching the order of summation gives

$$\frac{1}{\pi(X)}\sum_{p\leq X}f(s(\pi_p)) \approx \int_{\SU_n}f(g)dg + \sum_{\sigma\text{ nontrivial}}\frac{1}{\pi(X)}\sum_{p\leq X}\chi_\sigma(s(\pi_p)).$$

The behavior of the sum $\sum_{p\leq X}\chi_\sigma(s(\pi_p))$ is ontrolled by the Langlands L-function $L(s,\pi,\sigma)$, and in particular this sum is $o(X)$ if the L-function is nonvanishing and without poles in $\mathrm{Re}(s)\geq 1$. The truth of this latter property for all $\sigma$ is expected to be equivalent to $\pi$ not arising via functoriality from a smaller group.

Here is a screenshot of a semi-answer which froze my computer when I hit 'post'.

^{(Source: Wayback Machine)}

Here is a screenshot of a semi-answer which froze my computer when I hit 'post':

^{(Source: Wayback Machine)}

Here is a screenshot of a semi-answer which froze my computer when I hit 'post'.

alt text http://www2.bc.edu/%7Ehansendd/satotate.png

Here is a screenshot of a semi-answer which froze my computer when I hit 'post'.

^{(Source: Wayback Machine)}