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Spectral parameter of Maass forms of level 1
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eigenvalue
number theory
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
Δ
that is invariant under
S
L
(
2
,
Z
)
. The Laplace eigenvalue of a Maass form has the form
λ
=
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
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]
.