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Values of monstrous McKay–Thompson series at rectangular CM points

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Numbers
$g$
$r$ 
$T_g(i\sqrt r)$
2A
1:
544
comment: The square lattice has $j(i)=1728$; the level-two modular equation gives $R_2(i)=512$.
2A
2:
7228.955262170047126884390682945215004281146875980064990259367318775904444001478301011007589251868089
2A
3:
53252.37735580382259096992902503984721422492038329882360925384678561457515870464056734153665738590321
2A
4:
286751.3283844847564869827304728758726326358510695381235848480677066706607673100603641892634117958651
2A
5:
1263794.757133210670609885484221656537599763317609044122005691837930839039620876892225747014600939619
2A
6:
4831163.863502468009700882622664714481872069357478678281776182019557545301600621663430077625566362615
2A
7:
16580630.98838926632809964913103187015411188618636281461878537215197770827581292876359428878563664091
2A
8:
52249023.13403366792150756116673719398917048220898312491529781526162665812851267958940490108087295962
2A
9:
153552935.3954751661925675396731651059132461491720294729890536941998161012328956132924175152770477414
2A
10:
425670670.6183373278259870505787283402651695069891063660106869529812260859643461700806119483659227640
2A
11:
1122661623.131952164689578230681766836670100598282072337033225043870545554212417535188745847619476513
2A
12:
2835806946.422215641582734255984046783768467595637285359593061119850691150968939903545238965857494009
2A
13:
6896961562.314953718843591119133658739207900910304779375684155623397518996653559100322339617670105068
2A
14:
16220256788.05949082883678780226315518462845938365493190940606188083861577360540962901302660810145605
2A
15:
37018071743.20182025703073078408067413543579177777567180068185470447765541587727603224800354329969680
2A
16:
82226315585.59499532813723189934163975848218167519031389763105782610208565269919226253432660463005077
2A
17:
178211464838.5723620926151368179514928610951875239946012533733253457616526319922412024154689376432218
2A
18:
377674151959.6866869917868954150632833090550884897386156579498925587128229844895487386338966471329316
2A
19:
784074436680.4955197767191114700309917953564014790998283294019649341767728751927277134981612904037917
2A
20:
1597177179413.238601153828255792972883807914490552030148157576096242279864622822721115519143026464939
2A
21:
3196802717869.913292743069829617845074765398451477990423419435361002848194514781165876090981943188697
2A
22:
6294840129559.923266525041630031974805063945582892450366357556364561091178112562875049567949611965655
2A
23:
12207823870552.28622910972009712491018641958316514198860578555166539617547577284770974421023001368722
2A
24:
23340144267268.05350193336542924719248529360294941933981734249992706732615656709082568136662749378879
2A
25:
44031505860632.02901140064385907320290622141447232787674991766990697867610327734096507508930640705878
2A
26:
82028223157979.22122617333857337153317072288624844841892376797480363505597846598519842061524130356896
2A
27:
151013228703771.4978534073284116136551238660697672859934109865963531775086990194175808443511194781791
2A
28:
274917323964390.4072179651911875090465867391035019791461671972865413350598721025021634633332070524506
2A
29:
495202751155208.5242151842789598799662039335016251948842976804660820601452267703276295123383699979432
2A
30:
883067941287422.3893653018738734262972136733315237680073354388462926591649914648655321834855881378893
2B
1:
536
comment: The square lattice has $j(i)=1728$; the level-two modular equation gives $R_2(i)=512$.
2B
2:
7228.386719675123324932323133976827081138360000964987067332300129256275144862994019456992087878762426
2B
3:
53252.30040425795522858157580553019955624148869852744644693719520438169113971916856956640542859133812
2B
4:
286751.3140991348836673199961054823524130790560382134878663590851185484957365745631704968866799989921
2B
5:
1263794.753892116518537380314606282467112449272691763482405998921369499485271263658544270484238968597
2B
6:
4831163.862654634968959786635803928560278282684257720901197513651837311685937138975674804299623243392
2B
7:
16580630.98814223073990138286708340418833755453463787269317335392783899709346269993056952548619479393
2B
8:
52249023.13395527407534988203393452962614279677954355236988671978993834003019833438322897851909885112
2B
9:
153552935.3954484913482889348014341156307568649744961312069400876942684383179378852891346884310817288
2B
10:
425670670.6183277053630981010677443616306412717717448313938269610533865206312459894192657917067286131
2B
11:
1122661623.131948516217052612042998569743620525038204140971027033149110865830507848030686222617567475
2B
12:
2835806946.422214197196669882677711583079187514033614527550807045764328857496294571932503851630834698
2B
13:
6896961562.314953124958881563290090272422387433791969163873128898473262398366303597662829099681820724
2B
14:
16220256788.05949057631304174937501134719188681446296700957577701588804915262459946360424496138964869
2B
15:
37018071743.20182014638207163538319220829430407791681021097458016102472011879433116010366703057715813
2B
16:
82226315585.59499527832349560306161067174802204021324098694560039361708765012321059536730051448183727
2B
17:
178211464838.5723620696312058780321334386025442980278690435879224883404478923194943148285432767075968
2B
18:
377674151959.6866869809415675603323115026614476858950792446382470754320371071095326781064087030917326
2B
19:
784074436680.4955197714951176647152022731787646568334001125807720082938134507203850030087408027207737
2B
20:
1597177179413.238601151263731297596414324310935692004046100624359679885182589199389573771552615367238
2B
21:
3196802717869.913292741788549425878775190369356369015594459698973990107296200403784988905989674582961
2B
22:
6294840129559.923266524390938367736785589090486316054659069954405272492818910889847592041806597899811
2B
23:
12207823870552.28622910938457459088277273068322588934129278840010662667909443959287445846598288970570
2B
24:
23340144267268.05350193318993761555983962497195511487334450278397447132247106017909863962201490576305
2B
25:
44031505860632.02901140055083477344430063356139437855139621013705575147587428708573211080548111272992
2B
26:
82028223157979.22122617328863933855878020396917872485187179116765614920227393165172457134363430932066
2B
27:
151013228703771.4978534073012881620531758645213128555452084588888265295207264949681493094151697669024
2B
28:
274917323964390.4072179651762884843473251320794827617900474230584689848850414872108063204814739221696
2B
29:
495202751155208.5242151842706885204262073291508288853500870343332228116259571052959291960925073856399
2B
30:
883067941287422.3893653018692350515940978928198499052470899459833608259654396446571439369231505919341
3A
1:
536.9845356715713853126109853729865576463912826401439455133755725479247468406784062296325593752177607
3A
2:
7228.457065492307310975245612771777866702419531836111706998000377917398309577850678642035334600356182
3A
3:
53252.30992875493446456055937262844428617686193388159872120843276943898583969741907075033978328578760
3A
4:
286751.3158673478324504873361506420162543044231208658411217781089616156700332706007909730996452916562
3A
5:
1263794.754293295967138590572219616995120777618257863439180375492591689112349405812027357476442478051
3A
6:
4831163.862759579084570210359884700692778883511949804793311792181772778913902414357576361146741293067
3A
7:
16580630.98817280862604460679989679147475750601872087429495895043030085072964632227077860357970621232
3A
8:
52249023.13396497760997619517008897390118720449361723499222804432926165209651099981451366878391151817
3A
9:
153552935.3954517931416965787328830838672939686262455331071530899515776334621845578054130696196930874
3A
10:
425670670.6183288964247807096505868642316366724204791777804666176122006578161915032320309545096888976
3A
11:
1122661623.131948967822415352003675468001392073491512654814643292203176946897183412585342857430474200
3A
12:
2835806946.422214375981761166794885477586488591481397570597923914224976372300701603984980525124084807
3A
13:
6896961562.314953198469513306485418490375809581792416772340719841611906142745833639212192750100356883
3A
14:
16220256788.05949060757025347453888313898372012500847232503644483109975217027510289411626051230954226
3A
15:
37018071743.20182016007808486391382870139326428805730071622023238279450500869023039575218443186813031
3A
16:
82226315585.59499528448940485631765513764109890432799922027987654099110291007130802415931520001675528
3A
17:
178211464838.5723620724761406891533167102848868054706365285328823301544670150340101605341946341442714
3A
18:
377674151959.6866869822839946165596117459073901915149795927586482933265257975314450405737560817117957
3A
19:
784074436680.4955197721417399448141909955906461088913506547831201571638910242536113367142435623155025
3A
20:
1597177179413.238601151581166336452260279294701149385561228191138864940292612550195190832282302444521
3A
21:
3196802717869.913292741947145387140087086215705102428046547365088333470207460548892875132232417031832
3A
22:
6294840129559.923266524471480524418554679755159000714646788731094639793494284056870077699785428863744
3A
23:
12207823870552.28622910942610533423039454296830571481607213415798994209436110601890616596921077741529
3A
24:
23340144267268.05350193321165984633077235609125218627148356886038963972292199823506541392709019804274
3A
25:
44031505860632.02901140056234925586070931859236535057295103982192483959826867378772416558002245266561
3A
26:
82028223157979.22122617329482013805048311208255541007704638856079802900353553860617833069856608124158
3A
27:
151013228703771.4978534073046454838212880170384310918362008839396167538416410868348161920455686136003
3A
28:
274917323964390.4072179651781326751487229729699730121241951054938589719749998575375974004535904389771
3A
29:
495202751155208.5242151842717123434942684263724951095403382934263312966466728678147585346389170532256
3A
30:
883067941287422.3893653018698091863534923491464367470846733334851736228812710940125523460281888782322
3B
1:
535.5922308261581215514463073239513234447344511172816617450407306712131487392256060071456822546742693
3B
2:
7228.356044977873667171640180921189649486351426797425047254865763051900230332765308695115053466565462
3B
3:
53252.29623611887611300363193476521210759323656276761041491237547639906374065526688253379236682168515
3B
4:
286751.3133249688626933616445559691084057560777808143343586361655220589608100270918386277165968205979
3B
5:
1263794.753716456318190857552374868075174037541027009314353128958690176792251351466591404640401643190
3B
6:
4831163.862608683393287518841255393434219276777936109328963620203628624399685547270192309532766565763
3B
7:
16580630.98812884163057103978974507879005813318647258494367590890528911233439546862728539069144027850
3B
8:
52249023.13395102519280735996636967675150060527903674116025196017530371638516664845704891761490454555
3B
9:
153552935.3954470455928783605616404064286056299728278684910077589385863934497960489546346045562243129
3B
10:
425670670.6183271838331673571087946615324799128073083664312274706872641749195075401737981125658915267
3B
11:
1122661623.131948318472697884627482554375298875296876158927295980312713964452202251210065973342135704
3B
12:
2835806946.422214118912074528014155013090030672980436710582920904216197717537427037148459838304333646
3B
13:
6896961562.314953092770794431256969291853591661273705427480726309765370900336525740732496624225253705
3B
14:
16220256788.05949056262645202523152675060732224089898483069711091374384567102167629763338924354485198
3B
15:
37018071743.20182014038500075911590288850927348278664417098632593299053242075278374746154677327355741
3B
16:
82226315585.59499527562363001386440983377106742802585065801235375360506399976351495986553474860682831
3B
17:
178211464838.5723620683854947775144047128432386377777015251452766217854852901596771967016314724769099
3B
18:
377674151959.6866869803537592635694579314558222408537377708884882138422829904041295816635879878384739
3B
19:
784074436680.4955197712119812817241211909735030733279585428408292886576973303571075115643746714410299
3B
20:
1597177179413.238601151124736073484633342168379138225279186604071789362144733835496219755099720805485
3B
21:
3196802717869.913292741719105040475066044407497487928714938220561994456124483365629851494436573801076
3B
22:
6294840129559.923266524355671387888131846578407859216902185238614078497204626014499721017913901991564
3B
23:
12207823870552.28622910936638953166551560077121599494658659583275166698479504357659125463914742442585
3B
24:
23340144267268.05350193318042610622820084307566531693040371398720550700672981515226757737402085632639
3B
25:
44031505860632.02901140054579292907262216237741872083465762135440450399744063342579394678237208149670
3B
26:
82028223157979.22122617328593295298253258400281003921551361576569494366471092828859414794019631168527
3B
27:
151013228703771.4978534072998180921665471669645696787854839070161017464351994969441242336512735921760
3B
28:
274917323964390.4072179651754809688484882958639041496057160561766162706861446012234616959256129787024
3B
29:
495202751155208.5242151842702402192011391835858499149036051919078676483261800542701532365619634043038
3B
30:
883067941287422.3893653018689836553089192636865875086719203229772784346892860892018596406589713747506
5A
1:
535.7445651649735017475558822601408056655179133110522412581330529217256366303634173262162154557265666
5A
2:
7228.367128522558780266812001257939998999054682439077848491650779381832085030482420050152669854771921
5A
3:
53252.29773869636663138028498529523171741813579273854518460039457211077159600999285251837856728543127
5A
4:
286751.3136039664184036644915429508500366598811523951995338817811428928259192847574875674069101657681
5A
5:
1263794.753779758259641010530350646451126333062673113279783181959899604837622691636912482233409949263
5A
6:
4831163.862625242585920726244486151140113017650161350012581545703301063867002059462775487030886157718
5A
7:
16580630.98813366654046002909279448327344360835806361757929766539570232569950053456783614687270491499
5A
8:
52249023.13395255632221805163966830865199001142200442492162039496749506567641461330308421095215297576
5A
9:
153552935.3954475665858847029785796878945831878911129481870058873838791797120831036498455327614446889
5A
10:
425670670.6183273717718896744157933829643197910016298289696416161700460996242117344694086632249217508
5A
11:
1122661623.131948389731926040551004526814977674392273903592864755148011489913118400348387331876881873
5A
12:
2835806946.422214147122739713009618359137324260387201859764790153837441324902546704396551798774087512
5A
13:
6896961562.314953104370105141981147477045113667571606256090629002408833797888384632465949887955868363
5A
14:
16220256788.05949056755855643620684067762890791797745195603345182996498816017175943353300407959579779
5A
15:
37018071743.20182014254610738232285465882330530015891919667879157221576012938054589209134806687765993
5A
16:
82226315585.59499527659655455074080179946437061257187713688116725630029446073398507194297562647175943
5A
17:
178211464838.5723620688343996786505729950757171125433529200688324194205790835330653852624652498577166
5A
18:
377674151959.6866869805655820732245725541563392760744497804929038570666894984372062540475190946349982
5A
19:
784074436680.4955197713140124107324319518293530969979375184667538533831715176002998572129428918963055
5A
20:
1597177179413.238601151174824442534530076196564547887405901846433752168495355979417240333684201939932
5A
21:
3196802717869.913292741744130044224301761750892101748663620994681285301213843802744300827052882602648
5A
22:
6294840129559.923266524368380209455253307988256250429971686573261844016718234552362406120455236481108
5A
23:
12207823870552.28622910937294270615823125072629445576218996993979335268895380523675805887341824495009
5A
24:
23340144267268.05350193318385367715852396393254396877581896258032953367058948783343023925404346253322
5A
25:
44031505860632.02901140054760980992728186864880736964941598945929879619381280305306891296523960601322
5A
26:
82028223157979.22122617328690822706406348772276296806308275641533763885885859774453628384815170784290
5A
27:
151013228703771.4978534073003478470806476663363673651976744258805454933770428261568165289211855037919
5A
28:
274917323964390.4072179651757719654246457347835851575195404893198408704852371254077371123439466982019
5A
29:
495202751155208.5242151842704017691921547378441599735610547889397070198266336327992949241873296112872
5A
30:
883067941287422.3893653018690742485648394923846221369665833106331908623462765579234454902196374460028
5B
1:
535.5084971874737120511467085914095294300772949514407155338621556763151157047256124268018010473477844
5B
2:
7228.349821136030064839876398378178425331594175244868276779872599523244877504719507608221365714999719
5B
3:
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Definition
For a Monster conjugacy class $g$, let $T_g(\tau)=q^{-1}+\sum_{n\geq1}c_g(n)q^n$ be its normalized McKay–Thompson series, where $q=e^{2\pi i\tau}$ [1]. The entry is $T_g(i\sqrt r)$, with $r$ a positive integer and the positive square root.
Parameters
$g$
—   Monster conjugacy class (a Monster conjugacy class in the ATLAS notation)
$r$
—   rectangular CM index ($r>0$)
Formulas
(1)
For $p\in\{2,3,5,7,13\}$, set $e_p=b_p=24/(p-1)$, $C_p=p^{12/(p-1)}$ and $R_p(\tau)=(\eta(\tau)/\eta(p\tau))^{e_p}$. Then $T_{pB}=R_p+b_p$ and $T_{pA}=R_p+b_p+C_p/R_p$ [1]. The eta normalization is $\eta(\tau)=e^{\pi i\tau/12}\prod_{n\geq1}(1-q^n)$ [3]; its rectangular values are in the eta table. All exponents $e_p$ are integers.
(2)
$T_{pA}(\tau)-T_{pB}(\tau)=C_p/(T_{pB}(\tau)-b_p)$, with $b_p$ and $C_p$ as in (1).
(3)
Let $m_p(\tau)=C_p/(T_{pB}(\tau)-b_p)$. The canonical modular polynomial satisfies $\Phi^c_p(m_p(\tau),j(\tau))=0$, and at $\tau=i\sqrt r$ the Hilbert class polynomial satisfies $H_{-4r}(j(\tau))=0$.
(4)
At the square-lattice value $j(i)=1728$, the level-two equation becomes $(m_2-8)^2(m_2+64)=0$. Since $m_2(i)>0$, it follows that $R_2(i)=512$, $T_{2B}(i)=536$ and $T_{2A}(i)=544$.
Comments
(5)
The class labels use the ATLAS convention. The identity series is $T_{1A}=j-744$, whose positive Fourier coefficients are in the $j$-invariant expansion. Every series here has constant coefficient zero, including $T_{13A}$ [1].
(6)
The lattice $L_r=\mathbb Z+\mathbb Z i\sqrt r$ has endomorphism order $\mathbb Z[\sqrt{-r}]$, of discriminant $-4r$. Square factors of $r$ are retained because removing them generally changes the lattice and its order. The level-$p$ structure is fixed by the sublattice $\mathbb Z+\mathbb Z p i\sqrt r\subset L_r$.
(7)
The eta factors in (1) are positive real on the positive imaginary axis, so the listed values are real. They are algebraic integers by [2], Corollary 3.11. The table covers the principal rectangular CM points, rather than all ideal classes or all cyclic level structures.
(8)
The selected primes $p=2,3,5,7,13$ are precisely those for which $X_0(p)$ has genus zero. The classes $pB$ and $pA$ give its non-Fricke and Fricke Hauptmoduln, respectively [1]. Here $q=e^{2\pi i\tau}$; the theta nome $e^{\pi i\tau}$ used in [3] is its square root.
Programs
(P1)
Sage
import numberdb.sage as numberdb
from sage.rings.complex_arb import ComplexBallField
from sage.rings.integer_ring import ZZ

def mckay_cm(g, r, digits=100):
    p = int(g[:-1])
    assert p in (2, 3, 5, 7, 13) and g[-1] in ('A', 'B') and r > 0
    if p == 2 and r == 1:
        return ZZ(544 if g[-1] == 'A' else 536)
    F = ComplexBallField(numberdb.bits(digits, losing=96))
    tau = F(0, 1) * F(r).sqrt()
    e = 24 // (p-1)
    R = (tau.modular_eta() / (p*tau).modular_eta())**e
    value = R + e
    if g[-1] == 'A':
        value += ZZ(p)**(12 // (p-1)) / R
    assert value.real().is_finite() and value.imag().is_finite()
    assert value.imag().contains_zero()
    return value.real()

mckay_cm('3A', 1)
References
[1]
Toshiki Matsusaka, The Fourier coefficients of the McKay–Thompson series and the traces of CM values, Research in Number Theory 3 (2017), article 23, §1. (doi)
[2]
Kazuki Tomiyama, Generalized modular equations and singular values of Hauptmoduln, version 2 (2025), Corollary 3.11. (arXiv)
Links
Similar tables
Dedekind eta values at rectangular CM points —   The factors $\eta(i\sqrt r)$ and $\eta(i\sqrt{p^2r})$ determine $R_p(i\sqrt r)$ in (1).
Canonical modular polynomials —   These give algebraic equations for $m_p=C_p/R_p$ and $j$ in (3).
Hilbert class polynomials —   These give the CM equations for $j(i\sqrt r)$ used in (3).
Rational singular moduli —   These include $j(i\sqrt r)$ for $r=1,2,3,4,7$ and supply the square-lattice value in (4).
$q$-expansion of the $j$-invariant —   The positive coefficients are those of the identity-class series $T_{1A}=j-744$.
Ramanujan and Weber class invariants —   These are other eta-based invariants evaluated on the same lattices $\mathbb Z+\mathbb Z i\sqrt r$.
Sporadic irreducible character degrees —   The Monster character degrees give the dimensions of the irreducible constituents of the graded module whose traces define $T_g$ [1].
Hecke-type Faber polynomials of monstrous McKay–Thompson series —   These monic polynomials use the same zero-constant series. For each of the ten shared classes, a CM entry supplies $X=T_g(i\sqrt r)$ and yields the specialization $P_{n,g}(T_g(i\sqrt r))$.
Data properties
Entries are of type: real number
How well the digits are known: proven
Table is complete: no (it holds $g=2A,2B,3A,3B,5A,5B,7A,7B,13A,13B$ for every $1\leq r\leq30$. The class selection includes both Fricke and non-Fricke cases for every prime with genus-zero $X_0(p)$. The bound $|-4r|\leq120$ is a noncanonical practical extent covering small principal rectangular CM orders across fields and conductors; all square factors are retained. Other Monster classes, nonrectangular CM points, nonprincipal ideal classes and other cyclic level structures are outside this coverage)
How they were obtained:

The generator evaluates Dedekind eta and the positive square root using Arb complex balls at 429 bits for 100 significant decimal digits. It requires finite components and an imaginary part containing zero. The 298 approximate entries retain at least 418 bits of relative accuracy, and their full enclosures lie inside the intervals represented by the stored decimals. The two square-lattice entries at level two are exact integers by (4).

more

Independent real-ball checks at 560 bits use the eta product with 128 factors. For $0<q<1$, its omitted product lies between $1-q^{129}/(1-q)$ and $1$; this bound is propagated through the quotient and powers. These independent enclosures retain at least 545 bits and lie inside every approximate entry's represented interval. Theta sums with explicit geometric tail bounds [4] give $\lambda=(\theta_2/\theta_3)^4$ and $j=256(1-\lambda+\lambda^2)^3/(\lambda^2(1-\lambda)^2)$ [5] [6]. All thirty $j$-values satisfy the archived Hilbert class polynomials, and all 150 level-order pairs satisfy the archived canonical modular equations. The A/B relation is checked for all 150 pairs. Known-value controls and deliberately disjoint, wide and nonfinite enclosures precede these comparisons. The attached checks can repeat these tests on values read from the database.



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