Spectral parameter of Maass forms of level 1

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Numbers

$R'$

$R$

9.533:

9.533695261353557554344235235928770323821256395107251982375790464135348991298347781769255509975435366

12.17:

12.17300832467967784952795117639554812398247167309994790041359894085944536082660887402607610119914083

13.77:

13.77975135189073894424367328151771259715513256879348706925238822161445033353997009415783160955742757

14.35:

14.35850951825981277986694256903716549561438589919676624781520226663201120679288581901319549358192409

16.13:

16.13807317152103058019829428598600394563144288541378695827382712175947030542755279355556642723837034

16.64:

16.64425920189981994352627455936865570143168145997231928907651455001829017618970424409102246827670179

17.73:

17.73856338105737789321732636154654617200548005325129188079624689810214157759980377279197640233860653

18.18:

18.18091783453070386031830826819331393824992456541781787106792508774526862910670490089247820557750868

19.42:

19.42348147082825519163378035720852444158552560076333197577947593786262611347059612969474807916047941

19.48:

19.48471385474101336412852642787287621877406238534520661308580751557226039659320657991642152653178001

20.10:

20.1066946826

21.31:

21.3157959402

21.47:

21.4790575447

22.19:

22.1946739776

22.78:

22.7859084942

23.20:

23.2013961812

23.26:

23.2637115379

24.11:

24.1123527298

24.41:

24.4197154423

25.05:

25.0508548508

25.82:

25.8262437127

26.05:

26.0569177607

26.15:

26.1520854492

26.44:

26.446996418

27.28:

27.2843840117

27.33:

27.3327080831

27.77:

27.7759207018

28.51:

28.5102777031

28.53:

28.5307476929

28.86:

28.8633943539

29.13:

29.1375875578

29.54:

29.5463881241

31.52:

31.5265821968

31.56:

31.5662754118

32.50:

32.5081177599

32.89:

32.8911702135

34.02:

34.0278842001

34.45:

34.456271533

35.50:

35.5023497714

35.84:

35.8416764326

36.67:

36.6775529931

36.85:

36.8563494959

37.82:

37.8250722906

38.30:

38.3032761525

39.16:

39.1680849679

39.40:

39.4075318615

39.77:

39.773622619

40.54:

40.5433512105

40.68:

40.6886664449

41.55:

41.5555776736

41.88:

41.8830032854

42.64:

42.6434884147

42.92:

42.9222277836

43.26:

43.2671820388

44.07:

44.0774047617

44.42:

44.4263481186

45.28:

45.2874384425

45.36:

45.3616136021

45.39:

45.3984695313

46.10:

46.1014563216

46.48:

46.4814024123

46.65:

46.65331836

47.42:

47.4228958985

47.92:

47.9265583306

48.03:

48.0393309051

48.74:

48.7416663476

48.99:

48.9983076541

49.68:

49.6835200753

49.96:

49.9616962905

Definition

A Maass form (of weight 0) of level 1 is a smooth, square-integrable, automorphic eigenform of the Laplace-Beltrami operator $\Delta$ that is invariant under $SL(2,\mathbb{Z})$. The Laplace eigenvalue of a Maass form has the form $\lambda = 1/4 + R^2$, where $R > 0$ is called the spectral parameter.

Parameters

$R'$

— spectral parameter (truncated)

References

[1]

A. Booker, A. Strömbergsson, A. Venkatesh, "Effective Computation of Maass Cusp Forms". IMRN 2006, Article ID 71281, 34 pages. (data files) http://www2.math.uu.se/~astrombe/papers/bsv27april06.pdf

[2]

Strömberg, Fredrik, "Maass waveforms on (Γ0(N),χ) (computational aspects)". Hyperbolic geometry and applications in quantum chaos and cosmology, 187–228, London Math. Soc. Lecture Note Ser., 397, Cambridge Univ. Press, 2012 (MR)

Links

[3]

A. Booker, A. Strömbergsson, A. Venkatesh: data files

[4]

LMFDB: Maass forms of level 1

[5]

LMFDB: Maass forms: Reliability

[6]

Wikipedia: Birch and Swinnerton-Dyer conjecture

Data properties

Entries are of type: real number

Table is complete: no

Sources of data: [3] (higher precision available, from Booker, Strömbergsson, Venkatesh [1]), [4] (computed by Fredrik Stroemberg [2] and Holger Then)

Reliability: For heuristics see [3] and [5].

How well the digits are known: heuristic

How they were obtained:

Not computed: the digits are decimal strings from the LMFDB, pasted into the script and truncated to a hundred characters. Truncation is sound under this database's convention, in which a written digit string denotes an interval of one unit in its last place, but it is the only thing here that is sound by construction.

What bounds the original computation's error is not recorded, and spectral parameters of Maass forms are hard to certify.