  In this paper we describe how to explicitly construct infinitely many finite
simple groups as characteristic quotients of the rank 2 free group $F_2$. This
shows that a "baby" version of the Wiegold conjecture fails for $F_2$, and
provides counterexamples to two conjectures in the theory of noncongruence
subgroups of $\text{SL}_2(\mathbb{Z})$. Our main result explicitly produces,
for every prime power $q\ge 7$, the groups $\text{SL}_3(\mathbb{F}_q)$ and
$\text{SU}_3(\mathbb{F}_q)$ as characteristic quotients of $F_2$. Our strategy
is to study specializations of the Burau representation for the braid group
$B_4$, exploiting an exceptional relationship between $F_2$ and $B_4$ first
observed by Dyer, Formanek, and Grossman. Weisfeiler's strong approximation
theorem guarantees that our specializations are surjective for infinitely many
primes, but it is not effective. To make our result effective, we give another
proof of surjectivity via a careful analysis of the maximal subgroup structures
of $\text{SL}_3(\mathbb{F}_q)$ and $\text{SU}_3(\mathbb{F}_q)$. We also show
that our examples of $\text{PSL}_3(\mathbb{F}_q)$ and
$\text{PSU}_3(\mathbb{F}_q)$ are minimal in the sense that no group of the form
$\text{PSL}_2(\mathbb{F}_q)$ is a characteristic quotient of $F_2$.
