Results from empirical tests of the accuracy of our eigenvalues and
coefficients of Maass waveforms on congruence subgroups
===================================================================
In each case we made four distinct runs, using different Y-parameters;
the resulting files are referred to as (1),(2),(3),.... 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 appropriate D and 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).
Note that for these congruence subgroup examples, we only ran
"Phase 1", ie solving the system in our paper [equations (5),(6)]; we
never did "phase 2" (ie [equation (8)]).
R=5.4361... on Gamma_0(5)
=========================
Files:
(1) phase 1 (Y=0.17, M=1145; the R-value resulted in
Test=-1.7E-520.)
(2) phase 1 (Y=0.169, M=1155; started with R correct to 215
decimals, the final R-value resulted in test ~ 1E-513.)
Tests:
* Difference between R-value in (1) and R-value in (2): < 4.3E-509.
* Coefficients between (1) and (2) agree almost as predicted
by (Y=0.169, D=525)-heuristics (19 below to 5 below for n<=1135).
* Coefficients in (1) satisfy Hecke multiplicativity almost as
predicted by (Y=0.17, D=525)-heuristics (13 below to 1 below for
n<=1135).
* Coefficients in (2) satisfy Hecke multiplicativity almost as
predicted by (Y=0.169, D=525)-heuristics (19 below to 5 below for
n<=1135).
The "19 below" seems to be due to the fact that already the R-value is
off by 16 digits (comparing with 525 digits).
In printout: R printed to 503 digits, coefficients n<=1050 printed to
24 digits below heuristics.
R=3.2642... (a CM-form) on Gamma_0(5) (non-trivial character (5/.)
===================================================================
Files:
(1) phase 1 (Y=0.171, M=1135).
Note that since this is a CM-form, we can compare against the known
EXACT formulas for the answer.
Tests:
* R is correct to 523 decimal places.
* The coefficients are correct as predicted by
(Y=0.171,D=525)-heuristics (2 below to 2 above) for n<=1100.
R=4.8937... on Gamma_0(5) (non-trivial character (5/.)
========================================================
Files:
(1) phase 1, W_Q-odd (Y=0.171, M=1200)
(2) phase 1, W_Q-odd (Y=0.995*0.171, M=1200, same R-value)
(3) phase 1, W_Q-even (Y=0.171, M=1200)
(4) phase 1, W_Q-even (Y=0.995*0.171, M=1200, same R-value)
(5) Hecke multiplicative coefficients, created from (1),(3).
(1),(2),(3),(4) were NOT Hecke normalized, but fixed to have REAL
coefficients.
We refer to F. Strömberg, "Maass waveforms on (Gamma_0(N),chi)", to
appear in proceedings, Reisensburg, regarding the precise
computational method. Cf. in particular his Example 1.3.1.
The "Test" value between (1),(2): 3.8E-512.
between (3),(4): -4.8E-514.
(That is, the search was never continued to 550 decimals; hence we use
"D=512" in the precision heuristics below!)
Tests:
* Coefficients between (1) and (2) agree almost as predicted
by (Y=0.171,D=512)-heuristics (2 below to 8 above for n<=1100).
* Coefficients between (3) and (4) agree almost as predicted
by (Y=0.171,D=512)-heuristics (2 below to 9 above for n<=1100).
* Coefficients in (5) satisfy Hecke multiplicativity almost as
predicted by (Y=0.171,D=512)-heuristics (2 below to 10 above for n<=1100).
* Coefficients in (5) have real part = 0 (for (n/5)=-1) or imaginary
part =0 (for (n/5)=1) almost to the precision as predicted by
(Y=0.171,D=512)-heuristics (2 below to 10 above for n<=1100).
In printout: R printed to 502 digits, coefficients n<=1050 printed to
8 digits below heuristics.
R=2.5923... on Gamma_0(6)
=========================
Files:
(1) phase 1 (Y=0.142, M=1430; the R-value resulted in
Test=4.4E-550.)
(2) phase 1 (Y=0.143, M=1430; started over from R correct to
220 decimals; the final R-value resulted in Test=-1.6E-531.)
Tests:
* Difference between R-value in (1) and R-value in (2): < 4.7E-525.
* Coefficients between (1) and (2) agree almost as predicted
by (Y=0.143, D=550)-heuristics (28 below to 16 below for n<=1300).
* Coefficients in (1) satisfy Hecke multiplicativity almost as
predicted by (Y=0.142, D=550)-heuristics (9 below to 0 below for
n<=1300).
* Coefficients in (2) satisfy Hecke multiplicativity almost as
predicted by (Y=0.143, D=550)-heuristics (28 below to 16 below for
n<=1300).
The "28 below" seems to be due to the fact that the R-value in (2) was
never finalized to ~ 550 digits, but only ~ 525 digits.
In printout: R printed to 518 digits, coefficients n<=1300 printed to
33 digits below heuristics.