## What exactly did they do?

Bian and Booker produced an approximation to a generic
(meaning non-self-dual) Maass form on *SL(3,Z)*.
Specifically, they computed an approximation to the eigenvalues
and approximations to the first few
hundred Fourier coefficients.
It is believed that these coefficients are transcendental
numbers. Previously it was known how to directly construct
transcendental forms by transferring examples from lower
rank groups,
but this is the first indirect
construction.
In their initial run they found 4 generic Maass forms,
and one self-dual form. The self-dual form corresponds
to the smallest symmetric square lift from *SL(2,Z)*.

Their method is based on the GL(3) converse theorem, so one
has high confidence in the answer. Additional evidence is
that the L-function satisfies the Riemann Hypothesis for
the first several zeros.

The calculation required solving many (non-sparse!) system with
10,000 unknowns, which is now within the capabilities of a
personal computer.
One of the steps
was to figure out how to form such a large system that is sufficiently
well-conditioned that you don't lose all digits of accuracy
when you solve it. Their calculations are done in double
precision, and the have maybe 6 decimal place accuracy in the
answer. (Actually, a few more digits in the eigenvalues, less
in some of the coefficients).