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).