In 2019, Hartman, Mazáč, and Rastelli [3] discovered a remarkable equivalence between the spinless modular bootstrap bound for the spectral gap of conformal field theories with current algebra U(1)c×U(1)c (i.e., theories of c free bosons) and the linear programming bound for sphere packing density in dimension 2c. This bound is sharp for the E8 root lattice and the Leech lattice [4,2].
Under this equivalence, the scaling dimensions Δn of the conformal field theory are given by half the squared distances between points in the sphere packing and the degeneracies dn are the corresponding theta series coefficients.
Other than in dimensions 1, 8, and 24, this bound has not been exactly optimized. It is conjectured to match the densest sphere packing in dimension 2, while other cases are a mystery. The paper [1] computes the optimum numerically to high precision, and this web page displays part of the full data set. The tables below show the scaling dimensions and degeneracies for hypothetical conformal field theories achieving the bound. (To obtain odd sphere packing dimensions, we allow c to be a half-integer, in which case the CFT must be chiral.)
I believe that all the digits presented here are accurate, aside from rounding the last digit, except in the case of dimension 3, which is much more challenging numerically for unclear reasons. In that case Δ1 is an upper bound for the spectral gap, but it is unclear how close to the optimum we have come.
The scaling dimensions and degeneracies show numerous patterns that have yet to be explained (see [1, Section 4]). Can they somehow be characterized exactly, perhaps in terms of automorphic forms?
References
- N. Afkhami-Jeddi, H. Cohn, T. Hartman, D. de Laat, and A. Tajdini, High-dimensional sphere packing and the modular bootstrap, J. High Energy Phys. (2020), Paper No. 66, 44 pp., doi:10.1007/jhep12(2020)066, arXiv:2006.02560.
- H. Cohn, A. Kumar, S. D. Miller, D. Radchenko, and M. Viazovska, The sphere packing problem in dimension 24, Ann. of Math. (2) 185 (2017), 1017-1033, doi:10.4007/annals.2017.185.3.8, arXiv:1603.06518.
- T. Hartman, D. Mazáč, and L. Rastelli, Sphere packing and quantum gravity, J. High Energy Phys. (2019), Paper No. 48, 66 pp., doi:10.1007/JHEP12(2019)048, arXiv:1905.01319.
- M. S. Viazovska, The sphere packing problem in dimension 8, Ann. of Math. (2) 185 (2017), 991-1015, doi:10.4007/annals.2017.185.3.7, arXiv:1603.04246.
Dimension 1
| n | Scaling dimension Δn | Degeneracy dn |
|---|---|---|
| 1 | 0.500000000000 | 2.0000 |
| 2 | 2.000000000000 | 2.0000 |
| 3 | 4.500000000000 | 2.0000 |
| 4 | 8.000000000000 | 2.0000 |
| 5 | 12.500000000000 | 2.0000 |
| 6 | 18.000000000000 | 2.0000 |
| 7 | 24.500000000000 | 2.0000 |
| 8 | 32.000000000000 | 2.0000 |
| 9 | 40.500000000000 | 2.0000 |
| 10 | 50.000000000000 | 2.0000 |
| 11 | 60.500000000000 | 2.0000 |
| 12 | 72.000000000000 | 2.0000 |
| 13 | 84.500000000000 | 2.0000 |
| 14 | 98.000000000000 | 2.0000 |
| 15 | 112.500000000000 | 2.0000 |
Dimension 2
| n | Scaling dimension Δn | Degeneracy dn |
|---|---|---|
| 1 | 0.577350269190 | 6.0000 |
| 2 | 1.732050807569 | 6.0000 |
| 3 | 2.309401076759 | 6.0000 |
| 4 | 4.041451884327 | 12.0000 |
| 5 | 5.196152422707 | 6.0000 |
| 6 | 6.928203230276 | 6.0000 |
| 7 | 7.505553499465 | 12.0000 |
| 8 | 9.237604307034 | 6.0000 |
| 9 | 10.969655114603 | 12.0000 |
| 10 | 12.124355652982 | 12.0000 |
| 11 | 14.433756729741 | 6.0000 |
| 12 | 15.588457268120 | 6.0000 |
| 13 | 16.165807537310 | 12.0000 |
| 14 | 17.897858344878 | 12.0000 |
Dimension 3
| n | Scaling dimension Δn | Degeneracy dn |
|---|---|---|
| 1 | 0.652038604084 | 13.4018 |
| 2 | 1.749909380418 | 24.8614 |
| 3 | 2.514640815146 | 23.7455 |
| 4 | 3.865125850442 | 47.6248 |
| 5 | 4.456191053793 | 23.0409 |
| 6 | 5.570309782339 | 41.0548 |
| 7 | 6.827051035245 | 64.4510 |
| 8 | 7.585504096334 | 38.5651 |
| 9 | 8.337857048838 | 32.8679 |
| 10 | 9.786857841408 | 79.9850 |
| 11 | 10.566078989360 | 43.5815 |
| 12 | 11.168623110808 | 32.7852 |
| 13 | 11.774456230127 | 39.1912 |
| 14 | 12.496435590785 | 45.7352 |
| 15 | 13.382419122342 | 51.1772 |
Dimension 4
| n | Scaling dimension Δn | Degeneracy dn |
|---|---|---|
| 1 | 0.724575648447 | 26.4302 |
| 2 | 1.786743532669 | 75.7139 |
| 3 | 2.660501420369 | 89.1337 |
| 4 | 3.839787768752 | 194.2921 |
| 5 | 4.629238766583 | 124.3223 |
| 6 | 5.739214724958 | 253.3545 |
| 7 | 6.795587623716 | 311.1023 |
| 8 | 7.732454260043 | 252.8666 |
| 9 | 8.609054459161 | 270.4661 |
| 10 | 9.856544494859 | 613.6197 |
| 11 | 10.673227076942 | 252.7442 |
| 12 | 11.753833785031 | 474.6704 |
| 13 | 12.693697947788 | 466.2060 |
| 14 | 13.701460922165 | 555.1642 |
| 15 | 14.812721681468 | 823.2113 |
Dimension 5
| n | Scaling dimension Δn | Degeneracy dn |
|---|---|---|
| 1 | 0.795367272698 | 48.5767 |
| 2 | 1.833339375556 | 197.5092 |
| 3 | 2.769101852829 | 302.0124 |
| 4 | 3.856506720124 | 676.3753 |
| 5 | 4.756184134560 | 586.7003 |
| 6 | 5.823494473118 | 1217.1148 |
| 7 | 6.816239227355 | 1313.5583 |
| 8 | 7.821561716476 | 1536.6110 |
| 9 | 8.760175645322 | 1737.1171 |
| 10 | 9.862016364118 | 3085.8336 |
| 11 | 10.746012000123 | 1697.4011 |
| 12 | 11.834737748848 | 3495.6842 |
| 13 | 12.798350324566 | 3285.3816 |
| 14 | 13.813014793307 | 3976.7058 |
| 15 | 14.788237497193 | 4182.7772 |
Dimension 6
| n | Scaling dimension Δn | Degeneracy dn |
|---|---|---|
| 1 | 0.864721288837 | 85.2621 |
| 2 | 1.885680671483 | 467.0991 |
| 3 | 2.856119698789 | 923.2615 |
| 4 | 3.894252020927 | 2132.5634 |
| 5 | 4.852182538423 | 2434.1515 |
| 6 | 5.884484699003 | 4925.4351 |
| 7 | 6.867763392120 | 5403.1184 |
| 8 | 7.885155853240 | 7868.9727 |
| 9 | 8.856041493877 | 9171.9763 |
| 10 | 9.893583771293 | 14206.2662 |
| 11 | 10.847442835673 | 11148.3548 |
| 12 | 11.889592640213 | 20596.0809 |
| 13 | 12.866547856577 | 19233.0958 |
| 14 | 13.880814833853 | 24579.0526 |
| 15 | 14.855310483034 | 25390.9126 |
Dimension 7
| n | Scaling dimension Δn | Degeneracy dn |
|---|---|---|
| 1 | 0.932871948832 | 144.8267 |
| 2 | 1.941647755354 | 1030.8824 |
| 3 | 2.931262731499 | 2583.3087 |
| 4 | 3.943776319695 | 6277.3480 |
| 5 | 4.930610078168 | 8998.1218 |
| 6 | 5.941684052623 | 17901.4294 |
| 7 | 6.932759933982 | 21618.2718 |
| 8 | 7.942476586752 | 35000.4892 |
| 9 | 8.931524540325 | 42577.9628 |
| 10 | 9.942764890238 | 63290.4925 |
| 11 | 10.929962747812 | 63765.8805 |
| 12 | 11.943226716667 | 105022.9315 |
| 13 | 12.932843708428 | 103350.9777 |
| 14 | 13.940909966279 | 140543.4581 |
| 15 | 14.930990496605 | 152871.3624 |
Dimension 8
| n | Scaling dimension Δn | Degeneracy dn |
|---|---|---|
| 1 | 1.000000000000 | 240.0000 |
| 2 | 2.000000000000 | 2160.0000 |
| 3 | 3.000000000000 | 6720.0000 |
| 4 | 4.000000000000 | 17520.0000 |
| 5 | 5.000000000000 | 30240.0000 |
| 6 | 6.000000000000 | 60480.0000 |
| 7 | 7.000000000000 | 82560.0000 |
| 8 | 8.000000000000 | 140400.0000 |
| 9 | 9.000000000000 | 181680.0000 |
| 10 | 10.000000000000 | 272160.0000 |
| 11 | 11.000000000000 | 319680.0000 |
| 12 | 12.000000000000 | 490560.0000 |
| 13 | 13.000000000000 | 527520.0000 |
| 14 | 14.000000000000 | 743040.0000 |
| 15 | 15.000000000000 | 846720.0000 |
Dimension 9
| n | Scaling dimension Δn | Degeneracy dn |
|---|---|---|
| 1 | 1.066247148837 | 390.0761 |
| 2 | 2.059961093056 | 4345.1572 |
| 3 | 3.065304683606 | 16467.2642 |
| 4 | 4.059910320630 | 46760.2894 |
| 5 | 5.065030210528 | 94113.9547 |
| 6 | 6.060155499784 | 193425.4888 |
| 7 | 7.065659175129 | 297978.5313 |
| 8 | 8.059312629843 | 522181.5488 |
| 9 | 9.065498356323 | 727601.3807 |
| 10 | 10.060306031363 | 1118129.3324 |
| 11 | 11.064956220708 | 1450025.8259 |
| 12 | 12.059794412001 | 2161957.1255 |
| 13 | 13.065665774074 | 2552777.3314 |
| 14 | 14.059750223809 | 3642586.1934 |
| 15 | 15.065360648423 | 4340318.4045 |
Dimension 10
| n | Scaling dimension Δn | Degeneracy dn |
|---|---|---|
| 1 | 1.131726299211 | 624.1171 |
| 2 | 2.121022129771 | 8457.3899 |
| 3 | 3.128789125093 | 38409.2555 |
| 4 | 4.121724249192 | 119984.0666 |
| 5 | 5.128087140569 | 275219.6400 |
| 6 | 6.121746969216 | 591383.2623 |
| 7 | 7.129456854615 | 1016157.9508 |
| 8 | 8.120375264698 | 1833500.5528 |
| 9 | 9.129277299081 | 2759768.6430 |
| 10 | 10.121858615823 | 4374625.6819 |
| 11 | 11.128123541221 | 6104038.1020 |
| 12 | 12.121476161280 | 9100850.4785 |
| 13 | 13.129384120631 | 11661964.5504 |
| 14 | 14.120729825648 | 16746731.8190 |
| 15 | 15.129094373888 | 20924851.6314 |
Dimension 11
| n | Scaling dimension Δn | Degeneracy dn |
|---|---|---|
| 1 | 1.196528819622 | 985.6577 |
| 2 | 2.182838026955 | 16017.4995 |
| 3 | 3.191336461111 | 85963.3207 |
| 4 | 4.184425039584 | 297140.9683 |
| 5 | 5.190337442751 | 764546.6278 |
| 6 | 6.184205905953 | 1738421.1646 |
| 7 | 7.192026332083 | 3289006.0683 |
| 8 | 8.182679631702 | 6146017.3350 |
| 9 | 9.191971654335 | 9964559.7364 |
| 10 | 10.184082357000 | 16336520.0463 |
| 11 | 11.190651326266 | 24237613.8132 |
| 12 | 12.184063281875 | 36757999.7099 |
| 13 | 13.191843338791 | 50381503.2164 |
| 14 | 14.182866335545 | 73054698.3078 |
| 15 | 15.191879034486 | 95815488.5160 |
Dimension 12
| n | Scaling dimension Δn | Degeneracy dn |
|---|---|---|
| 1 | 1.260729735848 | 1539.5887 |
| 2 | 2.245168717551 | 29643.0183 |
| 3 | 3.253434963410 | 185775.5197 |
| 4 | 4.247463316686 | 712498.6599 |
| 5 | 5.252328081628 | 2034113.8823 |
| 6 | 6.247089093589 | 4931819.4840 |
| 7 | 7.253978427978 | 10163185.7318 |
| 8 | 8.245685879532 | 19802776.4511 |
| 9 | 9.254050667428 | 34395888.0636 |
| 10 | 10.246737846156 | 58476733.3293 |
| 11 | 11.252899931934 | 91684712.6544 |
| 12 | 12.246993986448 | 142769910.3496 |
| 13 | 13.253742083856 | 206912243.3764 |
| 14 | 14.245767212922 | 304963162.8014 |
| 15 | 15.254053839255 | 418948593.5413 |
Dimension 13
| n | Scaling dimension Δn | Degeneracy dn |
|---|---|---|
| 1 | 1.324391476210 | 2382.1984 |
| 2 | 2.307843800283 | 53782.9347 |
| 3 | 3.315354699712 | 389603.4159 |
| 4 | 4.310563009643 | 1658868.7270 |
| 5 | 5.314298799277 | 5215213.7118 |
| 6 | 6.310120091190 | 13542014.5533 |
| 7 | 7.315713149795 | 30149498.0094 |
| 8 | 8.308982269872 | 61598886.7157 |
| 9 | 9.315857483703 | 113976895.6750 |
| 10 | 10.309652611691 | 201486964.2941 |
| 11 | 11.315003959437 | 332507226.8842 |
| 12 | 12.310014982515 | 534378877.0270 |
| 13 | 13.315494074155 | 812542838.5923 |
| 14 | 14.309018229786 | 1224833642.8499 |
| 15 | 15.315907206291 | 1755831782.6128 |
Dimension 14
| n | Scaling dimension Δn | Degeneracy dn |
|---|---|---|
| 1 | 1.387566606866 | 3655.7809 |
| 2 | 2.370740354929 | 95915.5463 |
| 3 | 3.377242676607 | 796059.2590 |
| 4 | 4.373600506765 | 3759700.1953 |
| 5 | 5.376340858297 | 12946519.9750 |
| 6 | 6.373151755476 | 36079811.0559 |
| 7 | 7.377448607789 | 86279308.0622 |
| 8 | 8.372312777926 | 185551965.6174 |
| 9 | 9.377611741906 | 363995121.1903 |
| 10 | 10.372682270420 | 670676061.0395 |
| 11 | 11.377045838861 | 1161363935.0925 |
| 12 | 12.373026123354 | 1931933184.6574 |
| 13 | 13.377287511975 | 3066755355.7930 |
| 14 | 14.372330337982 | 4749609749.6948 |
| 15 | 15.377666856235 | 7077866477.0113 |
Dimension 15
| n | Scaling dimension Δn | Degeneracy dn |
|---|---|---|
| 1 | 1.450299859156 | 5569.8306 |
| 2 | 2.433768550885 | 168486.8584 |
| 3 | 3.439175414989 | 1589858.5719 |
| 4 | 4.436532838943 | 8314079.1619 |
| 5 | 5.438476935802 | 31234425.5709 |
| 6 | 6.436118560746 | 93482410.7493 |
| 7 | 7.439284099158 | 239135897.3719 |
| 8 | 8.435547427980 | 542566152.7292 |
| 9 | 9.439432645388 | 1124311599.7227 |
| 10 | 10.435719937225 | 2163000420.9000 |
| 11 | 11.439088794290 | 3920516807.7848 |
| 12 | 12.435987967044 | 6762258088.9933 |
| 13 | 13.439188281155 | 11171224401.6387 |
| 14 | 14.435553941970 | 17827847196.0744 |
| 15 | 15.439474750525 | 27532613756.0868 |
Dimension 16
| n | Scaling dimension Δn | Degeneracy dn |
|---|---|---|
| 1 | 1.512629655841 | 8431.7163 |
| 2 | 2.496862059014 | 292026.0935 |
| 3 | 3.501188743278 | 3111809.1445 |
| 4 | 4.499356769355 | 17976615.3339 |
| 5 | 5.500700822965 | 73453050.3608 |
| 6 | 6.499000100254 | 236038222.7312 |
| 7 | 7.501251027920 | 644048112.9629 |
| 8 | 8.498639002092 | 1543252274.4660 |
| 9 | 9.501370436096 | 3369293779.5773 |
| 10 | 10.498700631625 | 6775380546.1015 |
| 11 | 11.501176403749 | 12829695127.5041 |
| 12 | 12.498883782080 | 22968181925.2368 |
| 13 | 13.501207762340 | 39406853585.8334 |
| 14 | 14.498636709209 | 64911931898.9957 |
| 15 | 15.501396842832 | 103663759947.2294 |
Dimension 17
| n | Scaling dimension Δn | Degeneracy dn |
|---|---|---|
| 1 | 1.574589277869 | 12690.9757 |
| 2 | 2.559971518684 | 500122.0004 |
| 3 | 3.563295044487 | 5982249.7723 |
| 4 | 4.562086410525 | 38076692.1294 |
| 5 | 5.562996579735 | 168789400.4403 |
| 6 | 6.561798022476 | 581904296.0420 |
| 7 | 7.563346222437 | 1690050404.5968 |
| 8 | 8.561586232562 | 4277931694.7046 |
| 9 | 9.563434224840 | 9822186036.3116 |
| 10 | 10.561594993516 | 20655929040.6427 |
| 11 | 11.563331809042 | 40803260414.4221 |
| 12 | 12.561707699299 | 75856512918.6674 |
| 13 | 13.563336552887 | 134984465919.6214 |
| 14 | 14.561578009461 | 229703526353.6277 |
| 15 | 15.563448625360 | 378786969928.1335 |
Dimension 18
| n | Scaling dimension Δn | Degeneracy dn |
|---|---|---|
| 1 | 1.636207769259 | 19003.1357 |
| 2 | 2.623059998669 | 847325.8476 |
| 3 | 3.625493403343 | 11316588.0758 |
| 4 | 4.624741523534 | 79142535.7983 |
| 5 | 5.625347219938 | 379777604.7477 |
| 6 | 6.624523061416 | 1403127537.6560 |
| 7 | 7.625551594571 | 4330766775.0393 |
| 8 | 8.624408793954 | 11576940827.3652 |
| 9 | 9.625612090964 | 27917694048.1091 |
| 10 | 10.624398181608 | 61401908817.8177 |
| 11 | 11.625561323947 | 126397510786.5781 |
| 12 | 12.624461214944 | 244061248082.4853 |
| 13 | 13.625558713336 | 450019466793.5992 |
| 14 | 14.624398299893 | 791402947736.3826 |
| 15 | 15.625619032594 | 1346321500636.5312 |
Dimension 19
| n | Scaling dimension Δn | Degeneracy dn |
|---|---|---|
| 1 | 1.697510648238 | 28321.4824 |
| 2 | 2.686099788120 | 1421635.8605 |
| 3 | 3.687775639022 | 21098222.8746 |
| 4 | 4.687341780752 | 161668680.1142 |
| 5 | 5.687738154875 | 838147625.2893 |
| 6 | 6.687188415743 | 3314442655.5520 |
| 7 | 7.687844815406 | 10857703147.0477 |
| 8 | 8.687132710767 | 30634366563.2380 |
| 9 | 9.687883800763 | 77516629559.9564 |
| 10 | 10.687119180620 | 178260994580.4329 |
| 11 | 11.687860106170 | 382104376778.8927 |
| 12 | 12.687151097406 | 766253017000.8087 |
| 13 | 13.687857200494 | 1463058223074.6471 |
| 14 | 14.687123062066 | 2659046938706.7805 |
| 15 | 15.687886679711 | 4663864132807.5660 |
Dimension 20
| n | Scaling dimension Δn | Degeneracy dn |
|---|---|---|
| 1 | 1.758520472847 | 42028.7657 |
| 2 | 2.749070092205 | 2364121.9825 |
| 3 | 3.750129906657 | 38818365.9614 |
| 4 | 4.749904254907 | 325015073.9176 |
| 5 | 5.750158119357 | 1817081851.2401 |
| 6 | 6.749806725062 | 7681017131.3202 |
| 7 | 7.750204560441 | 26676313725.7074 |
| 8 | 8.749782997289 | 79380789105.9521 |
| 9 | 9.750227953848 | 210610329159.7587 |
| 10 | 10.749772846229 | 506190236568.6063 |
| 11 | 11.750217524643 | 1129156959541.0439 |
| 12 | 12.749787205690 | 2351096055442.9899 |
| 13 | 13.750215911968 | 4646315624273.7588 |
| 14 | 14.749775939520 | 8725839697489.9737 |
| 15 | 15.750228837691 | 15773507180775.2201 |
Dimension 21
| n | Scaling dimension Δn | Degeneracy dn |
|---|---|---|
| 1 | 1.819257295700 | 62125.8754 |
| 2 | 2.811955353901 | 3899633.3513 |
| 3 | 3.812542840164 | 70565290.1847 |
| 4 | 4.812442588734 | 643838751.2216 |
| 5 | 5.812598985903 | 3874946663.0661 |
| 6 | 6.812388925084 | 17485961132.8520 |
| 7 | 7.812612680460 | 64319532404.8449 |
| 8 | 8.812380648973 | 201695374703.9243 |
| 9 | 9.812625440869 | 560745703093.1158 |
| 10 | 10.812374958936 | 1407807029326.2817 |
| 11 | 11.812621128888 | 3266598357766.8267 |
| 12 | 12.812380449340 | 7059757207322.0347 |
| 13 | 13.812620540876 | 14434989934812.7323 |
| 14 | 14.812376534436 | 28005340615449.4547 |
| 15 | 15.812625465662 | 52160803841896.3879 |
Dimension 22
| n | Scaling dimension Δn | Degeneracy dn |
|---|---|---|
| 1 | 1.879739033245 | 91501.6671 |
| 2 | 2.874744016162 | 6384641.3351 |
| 3 | 3.875000810170 | 126865395.2329 |
| 4 | 4.874966992080 | 1258130292.4893 |
| 5 | 5.875055172641 | 8137717142.9445 |
| 6 | 6.874944004754 | 39150435207.0381 |
| 7 | 7.875054750664 | 152380818261.5732 |
| 8 | 8.874941959117 | 503132371340.9894 |
| 9 | 9.875060700971 | 1464908986817.9242 |
| 10 | 10.874939654698 | 3839568264756.7685 |
| 11 | 11.875059077669 | 9263451571177.6343 |
| 12 | 12.874941257897 | 20771762661616.1219 |
| 13 | 13.875058959748 | 43929423611018.0942 |
| 14 | 14.874940195902 | 88018659717256.9435 |
| 15 | 15.875060466053 | 168875597254049.0979 |
Dimension 23
| n | Scaling dimension Δn | Degeneracy dn |
|---|---|---|
| 1 | 1.939981768308 | 134318.2129 |
| 2 | 2.937427598354 | 10381480.7601 |
| 3 | 3.937490641514 | 225774253.4669 |
| 4 | 4.937484606854 | 2427609261.5744 |
| 5 | 5.937522980106 | 16847623451.4586 |
| 6 | 6.937479147476 | 86302890718.3005 |
| 7 | 7.937519844671 | 355115567962.3011 |
| 8 | 8.937478917289 | 1233556232187.9055 |
| 9 | 9.937521804435 | 3759322585599.2702 |
| 10 | 10.937478423123 | 10280710157740.7658 |
| 11 | 11.937521328762 | 25780476002595.3542 |
| 12 | 12.937478676514 | 59953795666202.8985 |
| 13 | 13.937521329105 | 131109961005020.9847 |
| 14 | 14.937478515898 | 271212274348265.9619 |
| 15 | 15.937521617452 | 535929617386133.8710 |
Dimension 24
| n | Scaling dimension Δn | Degeneracy dn |
|---|---|---|
| 1 | 2.000000000000 | 196560.0000 |
| 2 | 3.000000000000 | 16773120.0000 |
| 3 | 4.000000000000 | 398034000.0000 |
| 4 | 5.000000000000 | 4629381120.0000 |
| 5 | 6.000000000000 | 34417656000.0000 |
| 6 | 7.000000000000 | 187489935360.0000 |
| 7 | 8.000000000000 | 814879774800.0000 |
| 8 | 9.000000000000 | 2975551488000.0000 |
| 9 | 10.000000000000 | 9486551299680.0000 |
| 10 | 11.000000000000 | 27052945920000.0000 |
| 11 | 12.000000000000 | 70486236999360.0000 |
| 12 | 13.000000000000 | 169931095326720.0000 |
| 13 | 14.000000000000 | 384163586352000.0000 |
| 14 | 15.000000000000 | 820166620815360.0000 |
| 15 | 16.000000000000 | 1668890090322000.0000 |
Dimension 25
| n | Scaling dimension Δn | Degeneracy dn |
|---|---|---|
| 1 | 2.059806851746 | 286815.6936 |
| 2 | 3.062456970954 | 26939953.8347 |
| 3 | 4.062517577419 | 695628275.1846 |
| 4 | 5.062515665363 | 8731845733.2943 |
| 5 | 6.062484642906 | 69438192458.9121 |
| 6 | 7.062510952696 | 401770764471.3965 |
| 7 | 8.062489615212 | 1842862074141.0864 |
| 8 | 9.062511008562 | 7068224659759.6681 |
| 9 | 10.062489089839 | 23561901889060.1643 |
| 10 | 11.062510691206 | 70027344472844.0957 |
| 11 | 12.062489259264 | 189506625614947.8761 |
| 12 | 13.062510776741 | 473431132947112.1316 |
| 13 | 14.062489271793 | 1106148049420333.4404 |
| 14 | 15.062510718976 | 2436530220343840.2428 |
| 15 | 16.062489308602 | 5104404041067761.0718 |
Dimension 26
| n | Scaling dimension Δn | Degeneracy dn |
|---|---|---|
| 1 | 2.119414245636 | 417389.3795 |
| 2 | 3.124795704833 | 43031222.0707 |
| 3 | 4.125033154887 | 1205895353.3507 |
| 4 | 5.125032480368 | 16302052765.1657 |
| 5 | 6.124975787088 | 138459801976.7024 |
| 6 | 7.125015385861 | 849916694827.2334 |
| 7 | 8.124984890466 | 4110771141062.7998 |
| 8 | 9.125015804850 | 16548370663894.7941 |
| 9 | 10.124984816250 | 57647842393112.9029 |
| 10 | 11.125014848610 | 178467329462141.5303 |
| 11 | 12.124984998849 | 501448128589984.8223 |
| 12 | 13.125015018432 | 1297624960202082.3655 |
| 13 | 14.124985071203 | 3132610510239279.5568 |
| 14 | 15.125014880400 | 7117092615989578.7759 |
| 15 | 16.124985218482 | 15347635386548681.4548 |
Dimension 27
| n | Scaling dimension Δn | Degeneracy dn |
|---|---|---|
| 1 | 2.178833049525 | 605878.0311 |
| 2 | 3.187014524658 | 68380376.8319 |
| 3 | 4.187537594352 | 2074690879.5982 |
| 4 | 5.187550097574 | 30145045572.2734 |
| 5 | 6.187472532140 | 273061844709.0562 |
| 6 | 7.187515870335 | 1776199690041.1412 |
| 7 | 8.187483354037 | 9051281169439.0931 |
| 8 | 9.187516898359 | 38215427350348.0300 |
| 9 | 10.187484338014 | 139047443831568.7066 |
| 10 | 11.187515349076 | 448159953469579.5757 |
| 11 | 12.187484451902 | 1306930751135141.5222 |
| 12 | 13.187515492322 | 3501868297689205.3247 |
| 13 | 14.187484611322 | 8732597918285572.9841 |
| 14 | 15.187515306130 | 20457268695769650.1949 |
| 15 | 16.187484856325 | 45401197838132475.9507 |
Dimension 28
| n | Scaling dimension Δn | Degeneracy dn |
|---|---|---|
| 1 | 2.238073201878 | 877407.9284 |
| 2 | 3.249112638147 | 108138827.8410 |
| 3 | 4.250022789386 | 3544218607.7996 |
| 4 | 5.250067268284 | 55243972012.1743 |
| 5 | 6.249974036015 | 532951726149.1517 |
| 6 | 7.250014324967 | 3669590164575.0201 |
| 7 | 8.249983482639 | 19685726900605.1145 |
| 8 | 9.250015886706 | 87110332070036.7539 |
| 9 | 10.249985816960 | 330871678254674.1289 |
| 10 | 11.250014009511 | 1109706445627116.7412 |
| 11 | 12.249985834768 | 3357531159390103.4611 |
| 12 | 13.250014018586 | 9311740065995990.5142 |
| 13 | 14.249986072730 | 23979749197314636.4485 |
| 14 | 15.250013817370 | 57907157861036116.5667 |
| 15 | 16.249986382686 | 132234656811776814.8897 |
Dimension 29
| n | Scaling dimension Δn | Degeneracy dn |
|---|---|---|
| 1 | 2.297143818336 | 1267801.4017 |
| 2 | 3.311089946073 | 170240111.1781 |
| 3 | 4.312481594618 | 6014534570.8230 |
| 4 | 5.312582103352 | 100388432525.5856 |
| 5 | 6.312479414725 | 1030050396545.8900 |
| 6 | 7.312512138393 | 7499334853027.7191 |
| 7 | 8.312484407684 | 42317715956650.8051 |
| 8 | 9.312513771341 | 196124714003914.3188 |
| 9 | 10.312488107806 | 777245583719659.1165 |
| 10 | 11.312511917131 | 2711293611832295.4843 |
| 11 | 12.312488050728 | 8507858011004073.8066 |
| 12 | 13.312511732487 | 24414042285442244.8521 |
| 13 | 14.312488330528 | 64909124966894200.7519 |
| 14 | 15.312511535203 | 161531386423043022.4927 |
| 15 | 16.312488673796 | 379466837896950889.8458 |
Dimension 30
| n | Scaling dimension Δn | Degeneracy dn |
|---|---|---|
| 1 | 2.356053283145 | 1828055.6208 |
| 2 | 3.372946891381 | 266862766.9796 |
| 3 | 4.374907745733 | 10143104904.1159 |
| 4 | 5.375092279402 | 180978119370.6788 |
| 5 | 6.374987687103 | 1972451145152.8634 |
| 6 | 7.375010263207 | 15168934470343.1337 |
| 7 | 8.374985693951 | 89965510309400.8808 |
| 8 | 9.375011175178 | 436404223145097.5399 |
| 9 | 10.374990533705 | 1803529103080315.3103 |
| 10 | 11.375009677472 | 6540467503399862.6293 |
| 11 | 12.374990462712 | 21277638438557274.8823 |
| 12 | 13.375009296098 | 63154096824308413.6438 |
| 13 | 14.374990736042 | 173300983189215924.1309 |
| 14 | 15.375009107519 | 444324129982113802.2045 |
| 15 | 16.374991086013 | 1073568328057041406.8888 |
Dimension 31
| n | Scaling dimension Δn | Degeneracy dn |
|---|---|---|
| 1 | 2.414809328000 | 2630669.8857 |
| 2 | 3.434684339818 | 416644125.6710 |
| 3 | 4.437295777285 | 17005298296.9932 |
| 4 | 5.437595199352 | 323822829273.2948 |
| 5 | 6.437497750675 | 3744088142291.8216 |
| 6 | 7.437509289553 | 30383876267187.0208 |
| 7 | 8.437487176385 | 189255205606844.3745 |
| 8 | 9.437508487111 | 960245223794961.6163 |
| 9 | 10.437492728066 | 4136178953197849.0967 |
| 10 | 11.437507592413 | 15586742266730702.6842 |
| 11 | 12.437492728650 | 52550964595318020.8494 |
| 12 | 13.437507058884 | 161276684736875173.8262 |
| 13 | 14.437492951798 | 456651949976311051.6535 |
| 14 | 15.437506874615 | 1205922624538000093.4519 |
| 15 | 16.437493286192 | 2996196370483018112.6978 |
Dimension 32
| n | Scaling dimension Δn | Degeneracy dn |
|---|---|---|
| 1 | 2.473419100338 | 3778574.6399 |
| 2 | 3.496303485126 | 648022165.0090 |
| 3 | 4.499640942569 | 28352096172.5803 |
| 4 | 5.500088115554 | 575316472691.1290 |
| 5 | 6.500008378053 | 7048164890529.0025 |
| 6 | 7.500009503681 | 60297389364059.5488 |
| 7 | 8.499988842139 | 394144574759065.5852 |
| 8 | 9.500005954204 | 2090436469436921.5868 |
| 9 | 10.499994525224 | 9380254667209781.7785 |
| 10 | 11.500005783060 | 36715453857476778.7677 |
| 11 | 12.499994687761 | 128240670800707207.1353 |
| 12 | 13.500005173636 | 406805836361623140.3947 |
| 13 | 14.499994832837 | 1188217877975336955.3199 |
| 14 | 15.500004985872 | 3231140501694212926.5242 |
| 15 | 16.499995132983 | 8253450010963973061.3425 |
Dimension 33
| n | Scaling dimension Δn | Degeneracy dn |
|---|---|---|
| 1 | 2.531889222761 | 5417718.8673 |
| 2 | 3.557805773473 | 1004268344.0385 |
| 3 | 4.561939137943 | 47022369187.6584 |
| 4 | 5.562568222887 | 1015286034860.2670 |
| 5 | 6.562518226134 | 13163741588689.6943 |
| 6 | 7.562510935563 | 118608834317198.5789 |
| 7 | 8.562490747096 | 813021255868124.4599 |
| 8 | 9.562503737733 | 4504665857709947.0880 |
| 9 | 10.562495885913 | 21046574580579684.4536 |
| 10 | 11.562504271345 | 85527549124963291.5248 |
| 11 | 12.562496285338 | 309368987095202086.7502 |
| 12 | 13.562503676643 | 1014073519105866282.7445 |
| 13 | 14.562496345304 | 3054604923758787655.4987 |
| 14 | 15.562503478982 | 8551320676603088744.5174 |
| 15 | 16.562496596388 | 22451719905092192377.9423 |
Dimension 34
| n | Scaling dimension Δn | Degeneracy dn |
|---|---|---|
| 1 | 2.590225844945 | 7754796.9733 |
| 2 | 3.619192843042 | 1551049913.4617 |
| 3 | 4.624186832797 | 77600083373.4565 |
| 4 | 5.625032727991 | 1780342079042.1187 |
| 5 | 6.625025852632 | 24402003085097.8454 |
| 6 | 7.625013398492 | 231355984658305.8895 |
| 7 | 8.624992958236 | 1661788378549029.1822 |
| 8 | 9.625001945104 | 9612887887254966.4694 |
| 9 | 10.624996846868 | 46740995566373592.6996 |
| 10 | 11.625003031938 | 197119151514230755.0679 |
| 11 | 12.624997525099 | 738137965672334646.1489 |
| 12 | 13.625002540956 | 2499337696257065043.5965 |
| 13 | 14.624997513274 | 7761928755773234321.3041 |
| 14 | 15.625002331645 | 22364590168984997855.4151 |
| 15 | 16.624997705198 | 60342373842533177823.7481 |
Dimension 35
| n | Scaling dimension Δn | Degeneracy dn |
|---|---|---|
| 1 | 2.648434689160 | 11082189.4814 |
| 2 | 3.680466475632 | 2387768401.8286 |
| 3 | 4.686381005631 | 127458397501.7055 |
| 4 | 5.687478899733 | 3103068427229.7554 |
| 5 | 6.687529736155 | 44912960209709.3823 |
| 6 | 7.687516522712 | 447670932020159.5928 |
| 7 | 8.687495515165 | 3367078561419166.2222 |
| 8 | 9.687500646261 | 20323234048265222.0033 |
| 9 | 10.687497486855 | 102789720327291089.9180 |
| 10 | 11.687502023478 | 449682759851024452.0994 |
| 11 | 12.687498440320 | 1742605229178501778.0013 |
| 12 | 13.687501710316 | 6093221817259043480.0970 |
| 13 | 14.687498385897 | 19504478014252119564.8696 |
| 14 | 15.687501493794 | 57827614013316454527.2445 |
| 15 | 16.687498514159 | 160306323251437736987.4184 |
Dimension 36
| n | Scaling dimension Δn | Degeneracy dn |
|---|---|---|
| 1 | 2.706521090344 | 15813020.4496 |
| 2 | 3.741628557816 | 3664524584.2439 |
| 3 | 4.748519086264 | 208413040443.8186 |
| 4 | 5.749904105061 | 5377506236535.4722 |
| 5 | 6.750028297354 | 82103945849354.4664 |
| 6 | 7.750019784449 | 859625443625152.2228 |
| 7 | 8.749998405666 | 6765489908816954.4416 |
| 8 | 9.749999880609 | 42584309749044362.9249 |
| 9 | 10.749997903712 | 223929011859171157.5936 |
| 10 | 11.750001205877 | 1015816588633700539.7371 |
| 11 | 12.749999077236 | 4072305009211161490.4212 |
| 12 | 13.750001119728 | 14699984721249482813.7254 |
| 13 | 14.749999018489 | 48487706999631315385.0831 |
| 14 | 15.750000906590 | 147890502952482182259.1750 |
| 15 | 16.749999084633 | 421133015211927679457.9063 |
Dimension 37
| n | Scaling dimension Δn | Degeneracy dn |
|---|---|---|
| 1 | 2.764490031484 | 22530390.9417 |
| 2 | 3.802681049745 | 5607451366.9995 |
| 3 | 4.810598903942 | 339332399639.7643 |
| 4 | 5.812305833538 | 9268115636177.8153 |
| 5 | 6.812519919515 | 149120875903496.9615 |
| 6 | 7.812522531300 | 1638623109504139.0497 |
| 7 | 8.812501551411 | 13485528428057228.5042 |
| 8 | 9.812499658497 | 88467499675493312.0687 |
| 9 | 10.812498198711 | 483440550058158836.7540 |
| 10 | 11.812500548498 | 2273118954027238614.5996 |
| 11 | 12.812499486057 | 9423879143894923262.6385 |
| 12 | 13.812500707148 | 35107908235010355187.7237 |
| 13 | 14.812499462746 | 119298033358084306172.5543 |
| 14 | 15.812500512809 | 374239027465757182449.9002 |
| 15 | 16.812499474174 | 1094466013549162377218.9852 |
Dimension 38
| n | Scaling dimension Δn | Degeneracy dn |
|---|---|---|
| 1 | 2.822346174983 | 32056460.8375 |
| 2 | 3.863625960074 | 8556469589.1343 |
| 3 | 4.872618640965 | 550245679995.8393 |
| 4 | 5.874681713216 | 15890353750999.0674 |
| 5 | 6.875002967660 | 269166866822682.0876 |
| 6 | 7.875024004607 | 3101744198589921.0235 |
| 7 | 8.875004800939 | 26674975188733170.8699 |
| 8 | 9.874999959943 | 182282487727042855.4366 |
| 9 | 10.874998465787 | 1034668212377447720.0608 |
| 10 | 11.875000032625 | 5040609390678117379.9653 |
| 11 | 12.874999716455 | 21603575678482120310.9555 |
| 12 | 13.875000419426 | 83036972705412575199.8018 |
| 13 | 14.874999762612 | 290603579728144773896.7770 |
| 14 | 15.875000261856 | 937399391438979247186.8969 |
| 15 | 16.874999731726 | 2814904037094715046240.1617 |
Dimension 39
| n | Scaling dimension Δn | Degeneracy dn |
|---|---|---|
| 1 | 2.880093890529 | 45549299.4418 |
| 2 | 3.924465325832 | 13021455118.5186 |
| 3 | 4.934576791385 | 888790543356.8149 |
| 4 | 5.937029519901 | 27108789498531.9987 |
| 5 | 6.937475805636 | 482982853087715.9737 |
| 6 | 7.937523359268 | 5832017591371909.3336 |
| 7 | 8.937507927752 | 52377310349458800.5149 |
| 8 | 9.937500732255 | 372626661914042323.1613 |
| 9 | 10.937498784029 | 2195981869958519740.6547 |
| 10 | 11.937499650511 | 11080135166871969626.3603 |
| 11 | 12.937499815498 | 49076929396900054950.0853 |
| 12 | 13.937500214706 | 194566316845946962259.6154 |
| 13 | 14.937499953407 | 701111706078123441566.0065 |
| 14 | 15.937500111623 | 2324988703992428598527.2826 |
| 15 | 16.937499896193 | 7167339838516203232556.5720 |
Dimension 40
| n | Scaling dimension Δn | Degeneracy dn |
|---|---|---|
| 1 | 2.937737279944 | 64638556.9132 |
| 2 | 3.985201196280 | 19765647157.6728 |
| 3 | 4.996472124315 | 1430304826002.0725 |
| 4 | 5.999347181456 | 46027445028465.9983 |
| 5 | 6.999936810975 | 861746018232733.7294 |
| 6 | 8.000019681310 | 10895280377517562.7986 |
| 7 | 9.000010631928 | 102120542718046080.7950 |
| 8 | 10.000001887580 | 755967589000189683.3853 |
| 9 | 10.999999212393 | 4623405865475409694.6780 |
| 10 | 11.999999402543 | 24151658315129730195.5935 |
| 11 | 12.999999826759 | 110516175411467535833.5392 |
| 12 | 14.000000062685 | 451789436463209224392.5430 |
| 13 | 15.000000062653 | 1675853919188206683681.9930 |
| 14 | 16.000000028625 | 5711930433075498502917.2215 |
| 15 | 16.999999996881 | 18073063452955291671784.1194 |