Results from empirical tests of the accuracy of our eigenvalues and
coefficients of Maass waveforms on PSL(2,Z)\H.
===================================================================
In each case we made four distinct runs, using different Y-parameters;
the resulting files are referred to as (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, with D=1050 and appropriate Y (as indicated).
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=9.53
======
Files ((1) to (3) start with SAME R-values):
(1) Old, Princeton, only phase 1, Y=0.86, M=455.
(2) phase 1 (Y=0.859, M=455) and phase 2 (Y=0.11, Q=3540).
(3) phase 1 (Y=0.858, M=456) and phase 2 (Y=0.115, Q=3400)
(4) Finding R (again) starting with 415 decimals, Y=0.859, M=456.
Tests:
* R is SAME in (1) and (2) (copied).
* R in (4) agrees with R in (1),(2),(3) with error < 7.0E-1047.
* Coefficients between (1) and [(2) phase 1] agree almost as predicted
by (Y=0.859)-heuristics (8 below to 0 above for n<=445).
* Coefficients between [(2) phase 1] and [(2) phase 2] agree almost as
predicted by (Y=0.859)-heuristics (8 below to 0 above for n<=445).
* Coefficients between (1) and [(2) phase 2] agree as predicted by
(Y=0.859)-heuristics (1 below to 8 ABOVE for n<=445).
* Coefficients between [(3) phase 1] and [(3) phase 2] agree almost as
predicted by (Y=0.858)-heuristics (13 below to 2 below for n<=445).
* Coefficients between [(2) phase 1] and [(3) phase 1] agree almost as
predicted by (Y=0.858)-heuristics (13 below to 0 above for n<=445).
* Coefficients between [(2) phase 2] and [(3) phase 2] agree almost as
predicted by (Y=0.115)-heuristics (7 below to 3 above for n<=445).
* Coefficients in [(3) phase 2] satisfy Hecke multiplicativity almost as
predicted by (Y=0.115)-heuristics (7 below to 3 below for n<=455).
R=12.17
=======
Files ((1) to (3) start with SAME R-values):
(1) Old, Princeton, only phase 1, Y=0.86, M=455.
(2) phase 1 (Y=0.859, M=455) and phase 2 (Y=0.11, Q=3540).
(3) phase 1 (Y=0.858, M=456) and phase 2 (Y=0.115, Q=3400)
(4) Finding R (again) starting with 415 decimals, Y=0.859, M=456.
Tests:
* R is SAME in (1) and (2) (copied).
* R in (4) agrees with R in (1),(2),(3) with error < 1.1E-1047.
* Coefficients between (1) and [(2) phase 1] agree almost as predicted
by (Y=0.859)-heuristics (5 below to 0 above for n<=445).
* Coefficients between [(2) phase 1] and [(2) phase 2] agree almost as
predicted by (Y=0.859)-heuristics (5 below to 0 above for n<=445).
* Coefficients between (1) and [(2) phase 2] agree as predicted by
(Y=0.859)-heuristics (2 below to 8 ABOVE for n<=445).
* Coefficients between [(3) phase 1] and [(3) phase 2] agree almost as
predicted by (Y=0.858)-heuristics (10 below to 2 below for n<=445).
* Coefficients between [(2) phase 1] and [(3) phase 1] agree almost as
predicted by (Y=0.858)-heuristics (10 below to 0 above for n<=445).
* Coefficients between [(2) phase 2] and [(3) phase 2] agree almost as
predicted by (Y=0.115)-heuristics (5 below to 0 above for n<=445).
* Coefficients in [(3) phase 2] satisfy Hecke multiplicativity almost as
predicted by (Y=0.115)-heuristics (5 below to 0 below for n<=455).
R=13.77.
=======
Files ((1) to (3) start with SAME R-values):
(1) Old, Princeton, only phase 1, Y=0.86, M=450.
(2) phase 1 (Y=0.859, M=455) and phase 2 (Y=0.11, Q=3540).
(3) phase 1 (Y=0.858, M=456) and phase 2 (Y=0.115, Q=3400)
(4) Finding R (again) starting with 415 decimals, Y=0.859, M=456.
Tests:
* R agree up to 9E-1048 between (1) and (2).
* R in (4) agrees with R in (1),(2),(3) with error < 1.0E-1047.
* Coefficients between (1) and [(2) phase 1] agree as predicted by
(Y=0.859)-heuristics (3 below to 0 above for n<=455).
* Coefficients between [(2) phase 1] and [(2) phase 2] agree as
predicted by (Y=0.859)-heuristics (6 below to 2 above for n<=455).
* Coefficients between [(3) phase 1] and [(3) phase 2] agree almost as
predicted by (Y=0.858)-heuristics (7 below to 0 below for n<=445).
* Coefficients between [(2) phase 1] and [(3) phase 1] agree almost as
predicted by (Y=0.858)-heuristics (7 below to 0 above for n<=445).
* Coefficients between [(2) phase 2] and [(3) phase 2] agree almost as
predicted by (Y=0.115)-heuristics (5 below to 4 above for n<=445).
* Coefficients in [(3) phase 2] satisfy Hecke multiplicativity almost as
predicted by (Y=0.115)-heuristics (3 below to 4 above for n<=455).
R=14.35
=======
Files ((1) to (3) start with SAME R-values):
(1) Old, Princeton, only phase 1, Y=0.86, M=455.
(2) phase 1 (Y=0.859, M=455) and phase 2 (Y=0.11, Q=3540).
(3) phase 1 (Y=0.858, M=456) and phase 2 (Y=0.115, Q=3400)
(4) Finding R (again) starting with 415 decimals, Y=0.859, M=456.
Tests:
* R is SAME in (1) and (2) (copied).
* R in (4) agrees with R in (1),(2),(3) with error < 1.9E-1047.
* Coefficients between (1) and [(2) phase 1] agree almost as predicted
by (Y=0.859)-heuristics (5 below to 0 above for n<=445).
* Coefficients between [(2) phase 1] and [(2) phase 2] agree almost as
predicted by (Y=0.859)-heuristics (5 below to 0 above for n<=445).
* Coefficients between (1) and [(2) phase 2] agree as predicted by
(Y=0.859)-heuristics (3 below to 1 above for n<=445).
* Coefficients between [(3) phase 1] and [(3) phase 2] agree almost as
predicted by (Y=0.858)-heuristics (6 below to 0 below for n<=445).
* Coefficients between [(2) phase 1] and [(3) phase 1] agree almost as
predicted by (Y=0.858)-heuristics (6 below to 0 above for n<=445).
* Coefficients between [(2) phase 2] and [(3) phase 2] agree almost as
predicted by (Y=0.115)-heuristics (3 below to 1 above for n<=445).
* Coefficients in [(3) phase 2] satisfy Hecke multiplicativity almost as
predicted by (Y=0.115)-heuristics (3 below to 1 above for n<=455).
R=16.13
=======
Files ((1) to (3) start with SAME R-values):
(1) Old, Princeton, only phase 1, Y=0.86, M=455.
(2) phase 1 (Y=0.859, M=455) and phase 2 (Y=0.11, Q=3540).
(3) phase 1 (Y=0.858, M=456) and phase 2 (Y=0.115, Q=3400)
(4) Finding R (again) starting with 415 decimals, Y=0.859, M=456.
Tests:
* R is SAME in (1) and (2) (copied).
* R in (4) agrees with R in (1),(2),(3) with error < 1.5E-1047.
* Coefficients between (1) and [(2) phase 1] agree almost as predicted
by (Y=0.859)-heuristics (6 below to 0 above for n<=445).
* Coefficients between [(2) phase 1] and [(2) phase 2] agree almost as
predicted by (Y=0.859)-heuristics (6 below to 0 above for n<=445).
* Coefficients between (1) and [(2) phase 2] agree as predicted by
(Y=0.859)-heuristics (4 below to 8 above for n<=445).
* Coefficients between [(3) phase 1] and [(3) phase 2] agree almost as
predicted by (Y=0.858)-heuristics (7 below to 2 below for n<=445).
* Coefficients between [(2) phase 1] and [(3) phase 1] agree almost as
predicted by (Y=0.858)-heuristics (7 below to 0 above for n<=445).
* Coefficients between [(2) phase 2] and [(3) phase 2] agree almost as
predicted by (Y=0.115)-heuristics (4 below to 1 above for n<=445).
* Coefficients in [(3) phase 2] satisfy Hecke multiplicativity almost as
predicted by (Y=0.115)-heuristics (5 below to 1 above for n<=455).
R=16.64
=======
Files ((1) to (3) start with SAME R-values):
(1) Old, Princeton, only phase 1, Y=0.86, M=455.
(2) phase 1 (Y=0.859, M=455) and phase 2 (Y=0.11, Q=3540).
(3) phase 1 (Y=0.858, M=456) and phase 2 (Y=0.115, Q=3400)
(4) Finding R (again) starting with 415 decimals, Y=0.859, M=456.
Tests:
* R is SAME in (1) and (2) (copied).
* R in (4) agrees with R in (1),(2),(3) with error < 1.1E-1047.
* Coefficients between (1) and [(2) phase 1] agree as predicted
by (Y=0.859)-heuristics (5 below to 0 above for n<=445).
* Coefficients between [(2) phase 1] and [(2) phase 2] agree almost as
predicted by (Y=0.859)-heuristics (5 below to 2 below for n<=445).
* Coefficients between (1) and [(2) phase 2] agree as predicted by
(Y=0.859)-heuristics (4 below to 7 above for n<=445).
* Coefficients between [(3) phase 1] and [(3) phase 2] agree almost as
predicted by (Y=0.858)-heuristics (6 below to 0 below for n<=445).
* Coefficients between [(2) phase 1] and [(3) phase 1] agree almost as
predicted by (Y=0.858)-heuristics (6 below to 0 above for n<=445).
* Coefficients between [(2) phase 2] and [(3) phase 2] agree almost as
predicted by (Y=0.115)-heuristics (4 below to 1 above for n<=445).
* Coefficients in [(3) phase 2] satisfy Hecke multiplicativity almost as
predicted by (Y=0.115)-heuristics (3 below to 1 above for n<=455).
R=17.73
=======
Files ((1) to (3) start with SAME R-values):
(1) Old, Princeton, only phase 1, Y=0.86, M=455.
(2) phase 1 (Y=0.858, M=456) and phase 2 (Y=0.115, Q=3400)
(3) phase 1 (Y=0.859, M=455) and phase 2 (Y=0.11, Q=3540).
(4) Finding R (again) starting with 415 decimals, Y=0.859, M=456.
(note order (2) <--> (3) different from earlier!)
Tests:
* R in (1),(2) differ by ~ 1E-1047 (copied).
* R in (4) agrees with R in (1),(2),(3) with error < 9E-1047.
* Coefficients between (1) and [(2) phase 1] agree as predicted
by (Y=0.858)-heuristics (4 below to 0 above for n<=445).
* Coefficients between [(2) phase 1] and [(2) phase 2] agree almost as
predicted by (Y=0.858)-heuristics (4 below to 0 below for n<=445).
* Coefficients between (1) and [(2) phase 2] agree as predicted by
(Y=0.858)-heuristics (2 below to 8 above for n<=445).
* Coefficients between [(3) phase 1] and [(3) phase 2] agree almost as
predicted by (Y=0.859)-heuristics (5 below to 2 above for n<=445).
* Coefficients between [(2) phase 1] and [(3) phase 1] agree almost as
predicted by (Y=0.859)-heuristics (4 below to 3 above for n<=445).
* Coefficients between [(2) phase 2] and [(3) phase 2] agree almost as
predicted by (Y=0.11)-heuristics (4 below to 2 above for n<=455).
* Coefficients in [(3) phase 2] satisfy Hecke multiplicativity almost as
predicted by (Y=0.11)-heuristics (4 below to 2 above for n<=455).
R=18.18
=======
Files ((1) to (3) start with SAME R-values):
(1) Old, Princeton, only phase 1, Y=0.86, M=455.
(2) phase 1 (Y=0.858, M=456) and phase 2 (Y=0.115, Q=3400)
(3) phase 1 (Y=0.859, M=455) and phase 2 (Y=0.11, Q=3540).
(4) Finding R (again) starting with 415 decimals, Y=0.859, M=456.
Tests:
* R in (1),(2) differ by ~ 1E-1047 (copied).
* R in (4) agrees with R in (1),(2),(3) with error < 3E-1046.
* Coefficients between (1) and [(2) phase 1] agree as predicted
by (Y=0.858)-heuristics (5 below to 2 above for n<=445).
* Coefficients between [(2) phase 1] and [(2) phase 2] agree almost as
predicted by (Y=0.858)-heuristics (5 below to 2 above for n<=445).
* Coefficients between (1) and [(2) phase 2] agree as predicted by
(Y=0.858)-heuristics (4 below to 7 above for n<=445).
* Coefficients between [(3) phase 1] and [(3) phase 2] agree almost as
predicted by (Y=0.859)-heuristics (6 below to 1 below for n<=445).
* Coefficients between [(2) phase 1] and [(3) phase 1] agree almost as
predicted by (Y=0.859)-heuristics (6 below to 1 below for n<=445).
* Coefficients between [(2) phase 2] and [(3) phase 2] agree almost as
predicted by (Y=0.11)-heuristics (7 below to 1 below for n<=455).
* Coefficients in [(3) phase 2] satisfy Hecke multiplicativity almost as
predicted by (Y=0.11)-heuristics (4 below to 2 above for n<=455).
R=19.42
=======
Files ((1) to (3) start with SAME R-values):
(1) Old, Princeton, only phase 1, Y=0.86, M=455.
(2) phase 1 (Y=0.858, M=456) and phase 2 (Y=0.115, Q=3400)
(3) phase 1 (Y=0.859, M=455) and phase 2 (Y=0.11, Q=3540).
(4) Finding R (again) starting with 415 decimals, Y=0.859, M=456.
Tests:
* R in (1),(2) differ by ~ 1E-1047 (copied).
* R in (4) agrees with R in (1),(2),(3) with error < 1.5E-1046.
* Coefficients between (1) and [(2) phase 1] agree as predicted
by (Y=0.858)-heuristics (5 below to 0 above for n<=445).
* Coefficients between [(2) phase 1] and [(2) phase 2] agree almost as
predicted by (Y=0.858)-heuristics (5 below to 1 below for n<=445).
* Coefficients between (1) and [(2) phase 2] agree as predicted by
(Y=0.858)-heuristics (4 below to 8 above for n<=445).
* Coefficients between [(3) phase 1] and [(3) phase 2] agree almost as
predicted by (Y=0.859)-heuristics (4 below to 0 below for n<=445).
* Coefficients between [(2) phase 1] and [(3) phase 1] agree almost as
predicted by (Y=0.859)-heuristics (4 below to 0 below for n<=445).
* Coefficients between [(2) phase 2] and [(3) phase 2] agree almost as
predicted by (Y=0.11)-heuristics (5 below to 2 above for n<=455).
* Coefficients in [(3) phase 2] satisfy Hecke multiplicativity almost as
predicted by (Y=0.11)-heuristics (3 below to 4 above for n<=455).
R=19.48
=======
Files ((1) to (3) start with SAME R-values):
(1) Old, Princeton, only phase 1, Y=0.86, M=455.
(2) phase 1 (Y=0.858, M=456) and phase 2 (Y=0.115, Q=3400)
(3) phase 1 (Y=0.859, M=455) and phase 2 (Y=0.11, Q=3540).
(4) Finding R (again) starting with 415 decimals, Y=0.859, M=456.
Tests:
* R in (1),(2) differ by ~ 1E-1047 (copied).
* R in (4) agrees with R in (1),(2),(3) with error < 5E-1048.
* Coefficients between (1) and [(2) phase 1] agree as predicted
by (Y=0.858)-heuristics (4 below to 0 above for n<=445).
* Coefficients between [(2) phase 1] and [(2) phase 2] agree almost as
predicted by (Y=0.858)-heuristics (4 below to 0 below for n<=445).
* Coefficients between (1) and [(2) phase 2] agree as predicted by
(Y=0.858)-heuristics (4 below to 6 above for n<=445).
* Coefficients between [(3) phase 1] and [(3) phase 2] agree almost as
predicted by (Y=0.859)-heuristics (5 below to 0 below for n<=445).
* Coefficients between [(2) phase 1] and [(3) phase 1] agree almost as
predicted by (Y=0.859)-heuristics (5 below to 0 below for n<=445).
* Coefficients between [(2) phase 2] and [(3) phase 2] agree almost as
predicted by (Y=0.11)-heuristics (7 below to 1 below for n<=455).
* Coefficients in [(3) phase 2] satisfy Hecke multiplicativity almost as
predicted by (Y=0.11)-heuristics (4 below to 2 below for n<=455).