Results from empirical tests of the accuracy of our eigenvalues and
coefficients of (what appears to be) defoemations of Maass waveforms.
=====================================================================
In each case we made four final runs after having found good
a,r,R-values; these resulted in four files, (1),(2),(3),(4).
"Phase 1" refers to the system in our paper [equations (5),(6)];
"phase 2" refers to the formula [equation (8)].
Various tests were performed on this data, and the resulting numerical
errors/differences were always compared against the heuristic formula
(7) in our paper.
The results are presented as eg "B below to A above" meaning that the
numerical error/difference was at most B digits worse, and as best A
digits better, than indicated by formula (7).
R=4.13, a=0.218
===============
Files (all start with SAME finalized a,r,R-values, gave Test < 2E-214):
(1) Phase 1, Y=0.2, M=410
(2) Phase 1, Y=0.2*0.9, M=410
(3) Phase 2, Y=0.03, M=400, Q=2650.
(4) Phase 2, Y=0.03*0.99, M=400, Q=2650.
Tests:
* Coefficients between (1) and (2) agree almost as predicted
by (Y=0.2, D=215)-heuristics (2 below to 0 below for n<=380).
* Coefficients between (3) and (4) agree almost as predicted
by (Y=0.03, D=215)-heuristics (3 below to 2 below for n<=400).
* Coefficients between (1) and (3) agree almost as predicted
by (Y=0.2, D=215)-heuristics (5 below to 0 above for n<=380).
* Coefficients between (2) and (4) agree almost as predicted
by (Y=0.2*0.9, D=215)-heuristics (3 below to 1 below for n<=380).
* Coefficients in (3) satisfy relation conj(C[-n])=conj(C[-1])*C[n]
almost as predicted by (Y=0.03)-heuristics (4 below to 2 below for n<=400).
In printout: Sacrifice 10 decimals
R=4.13, a=0.374
===============
Files (all start with SAME finalized a,r,R-values, gave Test <
1E-206. That is, the search was never continued to 215 decimals; hence
we use "D=206" in the precision heuristics below!)
(1) Phase 1, Y=0.2, M=410
(2) Phase 1, Y=0.2*0.9, M=410
(3) Phase 2, Y=0.03, M=400, Q=2650.
(4) Phase 2, Y=0.03*0.99, M=400, Q=2650.
Tests:
* Coefficients between (1) and (2) agree almost as predicted
by (Y=0.2, D=206)-heuristics (1 below to 0 below for n<=380).
* Coefficients between (3) and (4) agree almost as predicted
by (Y=0.03, D=206)-heuristics (1 below to 3 above for n<=400).
* Coefficients between (1) and (3) agree almost as predicted
by (Y=0.2, D=206)-heuristics (2 below to 0 above for n<=380).
* Coefficients between (2) and (4) agree almost as predicted
by (Y=0.2*0.9, D=206)-heuristics (3 below to 0 below for n<=380).
* Coefficients in (3) satisfy relation conj(C[-n])=conj(C[-1])*C[n]
almost as predicted by (Y=0.03)-heuristics (1 below to 4 above for n<=400).
In printout: Sacrifice 10 decimals
R=4.13, a=0.413
===============
Files (all start with SAME finalized a,r,R-values, gave Test ~ 1E-213):
(1) Phase 1, Y=0.2, M=410
(2) Phase 1, Y=0.2*0.9, M=410
(3) Phase 2, Y=0.03, M=400, Q=2650.
(4) Phase 2, Y=0.03*0.99, M=400, Q=2650.
Tests:
* Coefficients between (1) and (2) agree almost as predicted
by (Y=0.2, D=215)-heuristics (3 below to 1 below for n<=380).
* Coefficients between (3) and (4) agree almost as predicted
by (Y=0.03, D=215)-heuristics (2 below to 1 above for n<=400).
* Coefficients between (1) and (3) agree almost as predicted
by (Y=0.2, D=215)-heuristics (4 below to 1 below for n<=380).
* Coefficients between (2) and (4) agree almost as predicted
by (Y=0.2*0.9, D=215)-heuristics (4 below to 1 below for n<=380).
* Coefficients in (3) satisfy relation conj(C[-n])=conj(C[-1])*C[n]
almost as predicted by (Y=0.03)-heuristics (2 below to 1 above for n<=400).
In printout: Sacrifice 10 decimal places.
R=5.37, a=0.071
===============
Files (all start with SAME finalized a,r,R-values, gave Test ~ 3E-213):
(1) Phase 1, Y=0.2, M=410
(2) Phase 1, Y=0.2*0.9, M=410
(3) Phase 2, Y=0.03, M=400, Q=2650.
(4) Phase 2, Y=0.03*0.99, M=400, Q=2650.
Tests:
* Coefficients between (1) and (2) agree almost as predicted
by (Y=0.2, D=215)-heuristics (3 below to 2 below for n<=380).
* Coefficients between (3) and (4) agree almost as predicted
by (Y=0.03, D=215)-heuristics (2 below to 1 below for n<=400).
* Coefficients between (1) and (3) agree almost as predicted
by (Y=0.2, D=215)-heuristics (3 below to 2 below for n<=380).
* Coefficients between (2) and (4) agree almost as predicted
by (Y=0.2*0.9, D=215)-heuristics (3 below to 1 below for n<=380).
* Coefficients in (3) satisfy relation conj(C[-n])=conj(C[-1])*C[n]
almost as predicted by (Y=0.03)-heuristics (2 below to 0 below for n<=400).
In printout: Sacrifice 10 decimal places.
R=5.28, a=0.143
===============
Files (all start with SAME finalized a,r,R-values, gave Test ~ 3E-214):
(1) Phase 1, Y=0.2, M=410
(2) Phase 1, Y=0.2*0.9, M=410
(3) Phase 2, Y=0.03, M=400, Q=2650.
(4) Phase 2, Y=0.03*0.99, M=400, Q=2650.
Tests:
* Coefficients between (1) and (2) agree almost as predicted
by (Y=0.2, D=215)-heuristics (3 below to 2 below for n<=380).
* Coefficients between (3) and (4) agree almost as predicted
by (Y=0.03, D=215)-heuristics (5 below to 4 below for n<=400).
* Coefficients between (1) and (3) agree almost as predicted
by (Y=0.2, D=215)-heuristics (7 below to 2 below for n<=380).
* Coefficients between (2) and (4) agree almost as predicted
by (Y=0.2*0.9, D=215)-heuristics (6 below to 1 below for n<=380).
* Coefficients in (3) satisfy relation conj(C[-n])=conj(C[-1])*C[n]
almost as predicted by (Y=0.03)-heuristics (5 below to 3 below for n<=400).
In printout: Sacrifice 10 decimal places.
R=5.33, a=0.279
===============
Files (all start with SAME finalized a,r,R-values, gave Test ~ 3E-213):
(1) Phase 1, Y=0.22, M=380
(2) Phase 1, Y=0.22*0.9, M=380
(3) Phase 2, Y=0.03, M=400, Q=2650.
(4) Phase 2, Y=0.03*0.99, M=400, Q=2650.
Tests:
* Coefficients between (1) and (2) agree almost as predicted
by (Y=0.22, D=215)-heuristics (3 below to 1 below for n<=340).
* Coefficients between (3) and (4) agree almost as predicted
by (Y=0.03, D=215)-heuristics (3 below to 2 below for n<=400).
* Coefficients between (1) and (3) agree almost as predicted
by (Y=0.22, D=215)-heuristics (4 below to 1 below for n<=340).
* Coefficients between (2) and (4) agree almost as predicted
by (Y=0.22*0.9, D=215)-heuristics (3 below to 0 below for n<=340).
* Coefficients in (3) satisfy relation conj(C[-n])=conj(C[-1])*C[n]
almost as predicted by (Y=0.03)-heuristics (3 below to 2 below for n<=400).
In printout: Sacrifice 10 decimal places.
R=6.87, a=0.007
===============
Files (all start with SAME finalized a,r,R-values, gave Test ~ 1E-211):
(1) Phase 1, Y=0.2, M=410
(2) Phase 1, Y=0.2*0.9, M=410
(3) Phase 2, Y=0.03, M=400, Q=2650.
(4) Phase 2, Y=0.03*0.99, M=400, Q=2650.
Tests:
* Coefficients between (1) and (2) agree almost as predicted
by (Y=0.2, D=215)-heuristics (5 below to 3 below for n<=380).
* Coefficients between (3) and (4) agree almost as predicted
by (Y=0.03, D=215)-heuristics (4 below to 1 below for n<=400).
* Coefficients between (1) and (3) agree almost as predicted
by (Y=0.2, D=215)-heuristics (5 below to 2 below for n<=380).
* Coefficients between (2) and (4) agree almost as predicted
by (Y=0.2*0.9, D=215)-heuristics (5 below to 2 below for n<=380).
* Coefficients in (3) satisfy relation conj(C[-n])=conj(C[-1])*C[n]
almost as predicted by (Y=0.03)-heuristics (3 below to 0 above for n<=400).
In printout: Sacrifice 10 decimal places.
R=6.87, a=0.118
===============
Files (all start with SAME finalized a,r,R-values, gave Test < 4E-214):
(1) Phase 1, Y=0.2, M=410
(2) Phase 1, Y=0.2*0.9, M=410
(3) Phase 2, Y=0.03, M=400, Q=2650.
(4) Phase 2, Y=0.03*0.99, M=400, Q=2650.
Tests:
* Coefficients between (1) and (2) agree almost as predicted
by (Y=0.2, D=215)-heuristics (2 below to 1 above for n<=380).
* Coefficients between (3) and (4) agree almost as predicted
by (Y=0.03, D=215)-heuristics (2 below to 2 above for n<=400).
* Coefficients between (1) and (3) agree almost as predicted
by (Y=0.2, D=215)-heuristics (3 below to 1 above for n<=380).
* Coefficients between (2) and (4) agree almost as predicted
by (Y=0.2*0.9, D=215)-heuristics (3 below to 1 above for n<=380).
* Coefficients in (3) satisfy relation conj(C[-n])=conj(C[-1])*C[n]
almost as predicted by (Y=0.03)-heuristics (1 below to 1 above for n<=400).
In printout: Sacrifice 10 decimal places.
R=6.87, a=0.174
===============
Files (all start with SAME finalized a,r,R-values, gave Test < 2E-213):
(1) Phase 1, Y=0.2, M=410
(2) Phase 1, Y=0.2*0.9, M=410
(3) Phase 2, Y=0.03, M=400, Q=2650.
(4) Phase 2, Y=0.03*0.99, M=400, Q=2650.
Tests:
* Coefficients between (1) and (2) agree almost as predicted
by (Y=0.2, D=215)-heuristics (3 below to 2 below for n<=380).
* Coefficients between (3) and (4) agree almost as predicted
by (Y=0.03, D=215)-heuristics (2 below to 1 above for n<=400).
* Coefficients between (1) and (3) agree almost as predicted
by (Y=0.2, D=215)-heuristics (3 below to 1 below for n<=380).
* Coefficients between (2) and (4) agree almost as predicted
by (Y=0.2*0.9, D=215)-heuristics (3 below to 1 below for n<=380).
* Coefficients in (3) satisfy relation conj(C[-n])=conj(C[-1])*C[n]
almost as predicted by (Y=0.03)-heuristics (2 below to 4 above for n<=400).
In printout: Sacrifice 10 decimal places.