{"Title": "Hecke-type Faber polynomials of monstrous McKay\u2013Thompson series", "Definition": "For a Monster conjugacy class $g$, write its normalized McKay\u2013Thompson series as $T_g(\\tau)=q^{-1}+\\sum_{m\\geq1}c_g(m)q^m$, where $q=e^{2\\pi i\\tau}$ CITE{Matsusaka}. The entry is the unique monic Hecke-type Faber polynomial $P_{n,g}(X)\\in\\mathbb Z[X]$ satisfying $P_{n,g}(T_g(\\tau))=q^{-n}+O(q)$ CITE{BKM}.", "Parameters": {"g": {"type": "Symbolic", "display": "$g$", "title": "Monster conjugacy class", "constraints": "a Monster conjugacy class in the ATLAS notation"}, "n": {"type": "Z", "display": "$n$", "title": "degree", "constraints": "$n\\geq1$"}}, "Tags": ["number theory", "modular function", "groups", "polynomial"], "Data properties": {"type": "Z[]", "rigour": "exact", "complete": "no", "complete-note": "it holds every degree $2\\leq n\\leq20$ for $g=1A,2A,2B,3A,3B,5A,5B,7A,7B,13A,13B$. The class selection includes both Fricke and non-Fricke series for every prime with genus-zero $X_0(p)$, together with the classical identity class. The degree bound is a noncanonical practical reference extent across all eleven classes; the longest polynomial has 748 characters. Degree one is stated in CITE{comment-polynomials}; other Monster classes and larger degrees remain outside this coverage", "rigour details": "Computed by exact integer arithmetic from the eta-product expansions and the Faber recurrence. The identity-class series is obtained from $j=E_4^3/\\Delta$, with the usual divisor-sum expansion of HREF{Q-expansion_of_the_Eisenstein_series_E4}[$E_4$] and product expansion of HREF{Q-expansion_of_the_modular_discriminant}[$\\Delta$].\n\nAn independent computation reconstructs the series from archived HREF{Q-expansion_of_the_j-invariant}[$j$-coefficients] and HREF{Canonical_modular_polynomials}[canonical modular equations], then determines the polynomials by triangular Laurent cancellation. All 209 polynomials agree. For every entry, the principal part is $q^{-n}$ with zero constant coefficient, and the first three positive coefficients agree with the twisted Hecke relation of CITE{BKM}. Ten explicit polynomial examples in CITE{BKM}, pp. 24\u201325, agree after the indicated shifts of variable. Known-value controls and deliberately incorrect polynomials precede these checks. The attached checker can repeat all tests on the stored entries."}, "Numbers": {"1A": {"2": "X^2 - 393768", "3": "X^3 - 590652*X - 64481280", "4": "X^4 - 787536*X^2 - 85975040*X + 74069419032", "5": "X^5 - 984420*X^3 - 107468800*X^2 + 189495047430*X + 21057657937920", "6": "X^6 - 1181304*X^4 - 128962560*X^3 + 343683985284*X^2 + 50659854188544*X - 12858803885140128", "7": "X^7 - 1378188*X^5 - 150456320*X^4 + 536636232594*X^3 + 88725605326848*X^2 - 47809252624358008*X - 5674272666930008064", "8": "X^8 - 1575072*X^6 - 171950080*X^5 + 768351789360*X^4 + 135254911352832*X^3 - 112484887649791744*X^2 - 19635086241677574144*X + 2016502627602420645912", "9": "X^9 - 1771956*X^7 - 193443840*X^6 + 1038830655582*X^5 + 190247772266496*X^4 - 214517584380376440*X^3 - 44381974047141482496*X^2 + 10178665610374591079949*X + 1373880968445473485012992", "10": "X^10 - 1968840*X^8 - 214937600*X^7 + 1348072831260*X^6 + 253704188067840*X^5 - 361539218235047200*X^4 - 82414443894080716800*X^3 + 30123485467790332998150*X^2 + 6219725247967316587315200*X - 264392503356231456801328368", "11": "X^11 - 2165724*X^9 - 236431360*X^8 + 1696078316394*X^7 + 325624158756864*X^6 - 561181664632739128*X^5 - 136232003593254260736*X^4 + 68990553191904045616035*X^3 + 17291944447558405263671296*X^2 - 1806143936956943373926224308*X - 308680121292331493369701251072", "12": "X^12 - 2362608*X^10 - 257925120*X^9 + 2082847110984*X^8 + 406007684333568*X^7 - 821076798992387328*X^6 - 208334160955421097984*X^5 + 135422053934751747165060*X^4 + 38001100973841504887898112*X^3 - 6687730416418166046559185216*X^2 - 1734101440714844019622157451264*X + 20900994943861369711633444933728", "13": "X^13 - 2559492*X^11 - 279418880*X^10 + 2508379215030*X^9 + 494854764797952*X^8 - 1148856496732926904*X^7 - 301220423791340212224*X^6 + 239562767008351074892617*X^5 + 72413908027857343157616640*X^4 - 18316979239508771782084104708*X^3 - 5745469926475188586673002027008*X^2 + 234432371170332764328351072091406*X + 65230639299981778514596025509453824", "14": "X^14 - 2756376*X^12 - 300912640*X^11 + 2972674628532*X^10 + 592165400150016*X^9 - 1552152633273292960*X^8 - 417390299911770587136*X^7 + 393060065884701285062034*X^6 + 125253229605064610094252032*X^5 - 41742905634772883940485875824*X^4 - 14634618532549137014877869875200*X^3 + 1154370493764070152996575608379144*X^2 + 439893250924568205954493672948629504*X + 2719676780106679295201358858717703872", "15": "X^15 - 2953260*X^13 - 322406400*X^12 + 3475733351490*X^11 + 697939590389760*X^10 - 2038597084032420600*X^9 - 559343297127471206400*X^8 + 609063918195783252952575*X^7 + 201898080495339958043033600*X^6 - 83951549510123698012683934452*X^5 - 31677489189941040228665316096000*X^4 + 3886014987027330446640994321129590*X^3 + 1699288783999389436750655401346457600*X^2 - 4732125471145437147519752624386752360*X - 13020431181324119642565741945090630021120", "16": "X^16 - 3150144*X^14 - 343900160*X^13 + 4017555383904*X^12 + 812177335517184*X^11 - 2615821724429244928*X^10 - 729578923249201053696*X^9 + 902226885733559472859440*X^8 + 308383626282978001672994816*X^7 - 154161812201437302698865802752*X^6 - 61295614797944060874236638789632*X^5 + 10433910539256301107480913877620992*X^4 + 4940099018195170071992141764928471040*X^3 - 103097182627097875055677073555793825792*X^2 - 103133971177736825790439338439656064679936*X - 1949245618099114279933367278713609675079656", "17": "X^17 - 3347028*X^15 - 365393920*X^14 + 4598140725774*X^13 + 934878635532288*X^12 - 3291458429882701048*X^11 - 930596686087719112704*X^10 + 1288704124449974058093765*X^9 + 451401183346691317976973312*X^8 - 264121293221146500986820632580*X^7 - 109217601215400405937265701853184*X^6 + 24067814229927577236064263062799310*X^5 + 12008819642642698763394868993363353600*X^4 - 538599007006170499989591020405998102584*X^3 - 457413185937416433380772012103443736940544*X^2 - 10639462391700261502948080127518848143403502*X + 2448360409675245677860106703465418348428705792", "18": "X^18 - 3543912*X^16 - 386887680*X^15 + 5217489377100*X^14 + 1066043490435072*X^13 - 4073139075811724064*X^12 - 1164896093453784367104*X^11 + 1786153384456952740982622*X^10 + 638298218859610446271610880*X^9 - 428402127006834631230272748912*X^8 - 182640609251961559360598757777408*X^7 + 49746908379294784527488216860491144*X^6 + 25768345199288533394205985502068998144*X^5 - 1877432916708819238388046518628668237760*X^4 - 1500558122886515340000182633119133719756800*X^3 - 27744628889562451933005227469536926545888243*X^2 + 22471139557701479015361955942257405537846558720*X + 659956389972972741907558573811866188194490278904", "19": "X^19 - 3740796*X^17 - 408381440*X^16 + 5875601337882*X^15 + 1205671900225536*X^14 - 4968495537635249080*X^13 - 1434976653158155800576*X^12 + 2413735010026402872869019*X^11 + 877078350789283888197353472*X^10 - 664696819669829388227215193268*X^9 - 290391836659348514661954270443520*X^8 + 94602258920395348096806441898107478*X^7 + 50402705890691441826552102696348770304*X^6 - 5228582669342872078825016501434126475976*X^5 - 4060967262671950082417487685654288811765760*X^4 - 30940858283878242049757379386294831844325275*X^3 + 113254298951731344549995705180609569889278279680*X^2 + 4966546426300752385957503058224443935345813239764*X - 428969687314085837556810166370806784513928652032000", "20": "X^20 - 3937680*X^18 - 429875200*X^17 + 6572476608120*X^16 + 1353763864903680*X^15 - 5985159690772211200*X^14 - 1743337873011592396800*X^13 + 3192111939590213424111900*X^12 + 1176401347897678107718451200*X^11 - 994114085743796644298243267136*X^10 - 443090000122612007551622737920000*X^9 + 168477519293467407173622963254838240*X^8 + 91759956252259578215231380069357977600*X^7 - 12546431271571442512849725967700049423360*X^6 - 9597015236441734633482106793660552674017280*X^5 + 72217861082859314962678931858332308589330950*X^4 + 415923251530204164973260847091175433664829849600*X^3 + 20314892597552926968269250778460173854877344941600*X^2 - 4535408474949149437067296323630459859968219836006400*X - 180612166589483962719321564578448241282015029854349168"}, "2A": {"2": "X^2 - 8744", "3": "X^3 - 13116*X - 288768", "4": "X^4 - 17488*X^2 - 385024*X + 33268760", "5": "X^5 - 21860*X^3 - 481280*X^2 + 89371910*X + 2050662400", "6": "X^6 - 26232*X^4 - 577536*X^3 + 164589444*X^2 + 4985782272*X - 107259357344", "7": "X^7 - 30604*X^5 - 673792*X^4 + 258921362*X^3 + 8762564608*X^2 - 444743039608*X - 11719211700224", "8": "X^8 - 34976*X^6 - 770048*X^5 + 372367664*X^4 + 13381009408*X^3 - 1096391274240*X^2 - 41502310531072*X + 234018624159768", "9": "X^9 - 39348*X^7 - 866304*X^6 + 504928350*X^5 + 18841116672*X^4 - 2145772148088*X^3 - 94871508111360*X^2 + 1592795656578573*X + 58687165263298560", "10": "X^10 - 43720*X^8 - 962560*X^7 + 656603420*X^6 + 25142886400*X^5 - 3676453748000*X^4 - 177346426880000*X^3 + 5345126136492550*X^2 + 275840211478118400*X + 213644656132535056", "11": "X^11 - 48092*X^9 - 1058816*X^8 + 827392874*X^7 + 32286318592*X^6 - 5772004160824*X^5 - 294446689275904*X^4 + 13125180226030499*X^3 + 783569541396381696*X^2 - 2263527674236166580*X - 263592855549972461568", "12": "X^12 - 52464*X^10 - 1155072*X^9 + 1017296712*X^8 + 40271413248*X^7 - 8515991473408*X^6 - 451691917737984*X^5 + 26932487763020676*X^4 + 1746161357781204992*X^3 - 15246692238252255552*X^2 - 1567069922180230250496*X - 6740820777199941676960", "13": "X^13 - 56836*X^11 - 1251328*X^10 + 1226314934*X^9 + 49098170368*X^8 - 11991983772600*X^7 - 654601734705152*X^6 + 49131938260990793*X^5 + 3360077582466269184*X^4 - 53060694476028731908*X^3 - 5378687508080176928768*X^2 - 24896242946524361638642*X + 1057501151296489775669248", "14": "X^14 - 61208*X^12 - 1347584*X^11 + 1454447540*X^10 + 58766589952*X^9 - 16283549145248*X^8 - 908695762616320*X^7 + 82453780909168018*X^6 + 5853955856355819520*X^5 - 138137402906356123760*X^4 - 14034521464279993327616*X^3 - 39196471180286526155000*X^2 + 7770794602184447261016064*X + 53938317320618104429620928", "15": "X^15 - 65580*X^13 - 1443840*X^12 + 1701694530*X^11 + 69276672000*X^10 - 21474255678200*X^9 - 1219493623910400*X^8 + 129993624572478975*X^7 + 9488609539424665600*X^6 - 302613061628860508916*X^5 - 30914307491036912793600*X^4 + 10481880004761202696310*X^3 + 31732701273338284946350080*X^2 + 313380494698654141315391640*X - 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1772906216803548102000*X^8 - 184042279134761728647168*X^7 + 2536080288400990656560520*X^6 + 526328740434733502046142464*X^5 + 4331970869406638765914953792*X^4 - 563956918438900616087130243072*X^3 - 9771692563155990130499934634995*X^2 + 115673153580431309341328373448704*X + 1701512642998472590782767090058232", "19": "X^19 - 83068*X^17 - 1828864*X^16 + 2881826330*X^15 + 119733624832*X^14 - 52899855150008*X^13 - 3140115851694080*X^12 + 537032775325073563*X^11 + 41777395730458951680*X^10 - 2826334966883262989492*X^9 - 294642838096082640582656*X^8 + 5602916068477241833792086*X^7 + 1047668315414991257960538112*X^6 + 6257445382572798782188010808*X^5 - 1598504726279691866357740875776*X^4 - 29198204211252267201824145274779*X^3 + 685648066343126356020669033005056*X^2 + 14915390025479728906664789710358484*X - 6516101044995270732825771088087040", "20": "X^20 - 87440*X^18 - 1925120*X^17 + 3224645240*X^16 + 134452019200*X^15 - 63839788787200*X^14 - 3817227216486400*X^13 + 714884345687433500*X^12 + 56154277026516172800*X^11 - 4321316859041349261888*X^10 - 452175336032874806476800*X^9 + 11067859412979961268257760*X^8 + 1934618369644067751774126080*X^7 + 6299048314517247261032944640*X^6 - 3914542371073783223833630081024*X^5 - 71216433228460730766007402937850*X^4 + 2805749757838212179994547784253440*X^3 + 71590327784579963457092015428453920*X^2 - 244168301981607069652752766335139840*X - 7920275773961092952870043713428137840"}, "2B": {"2": "X^2 - 552", "3": "X^3 - 828*X + 6144", "4": "X^4 - 1104*X^2 + 8192*X + 107544", "5": "X^5 - 1380*X^3 + 10240*X^2 + 324870*X - 2580480", "6": "X^6 - 1656*X^4 + 12288*X^3 + 618372*X^2 - 6488064*X - 12019872", "7": "X^7 - 1932*X^5 + 14336*X^4 + 988050*X^3 - 11526144*X^2 - 75815544*X + 840806400", "8": "X^8 - 2208*X^6 + 16384*X^5 + 1433904*X^4 - 17694720*X^3 - 213331712*X^2 + 3164995584*X - 2783526888", "9": "X^9 - 2484*X^7 + 18432*X^6 + 1955934*X^5 - 24993792*X^4 - 445592952*X^3 + 7444279296*X^2 + 1767958029*X - 223908151296", "10": "X^10 - 2760*X^8 + 20480*X^7 + 2554140*X^6 - 33423360*X^5 - 793623840*X^4 + 14146682880*X^3 + 30819561990*X^2 - 1166080737280*X + 2482913768208", "11": "X^11 - 3036*X^9 + 22528*X^8 + 3228522*X^7 - 42983424*X^6 - 1278448952*X^5 + 23740231680*X^4 + 107352345507*X^3 - 3500652412928*X^2 + 8954045378124*X + 46615300098048", "12": "X^12 - 3312*X^10 + 24576*X^9 + 3979080*X^8 - 53673984*X^7 - 1921092864*X^6 + 36692951040*X^5 + 260150152068*X^4 - 8074034151424*X^3 + 17411497907904*X^2 + 334267329576960*X - 1070875637673888", "13": "X^13 - 3588*X^11 + 26624*X^10 + 4805814*X^9 - 65495040*X^8 - 2742580152*X^7 + 53472866304*X^6 + 523799608137*X^5 - 15904870252544*X^4 + 20704765242876*X^3 + 1272165214414848*X^2 - 5785154015101682*X - 5566199400652800", "14": "X^14 - 3864*X^12 + 28672*X^11 + 5708724*X^10 - 78446592*X^9 - 3763935392*X^8 + 74548002816*X^7 + 938690123154*X^6 - 28184038342656*X^5 + 4933033670544*X^4 + 3543916130033664*X^3 - 17708319131756792*X^2 - 66951974741213184*X + 353246966446763712", "15": "X^15 - 4140*X^13 + 30720*X^12 + 6687810*X^11 - 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30429507000", "20": "X^20 - 180*X^18 - 200*X^17 + 14370*X^16 + 28680*X^15 - 648700*X^14 - 1771800*X^13 + 17783775*X^12 + 60326200*X^11 - 297017136*X^10 - 1211670000*X^9 + 2855525740*X^8 + 14290227600*X^7 - 13243548360*X^6 - 92583767280*X^5 + 10890346575*X^4 + 279223599600*X^3 + 74525879100*X^2 - 253861631400*X - 60342942918"}, "7A": {"2": "X^2 - 102", "3": "X^3 - 153*X - 612", "4": "X^4 - 204*X^2 - 816*X + 2478", "5": "X^5 - 255*X^3 - 1020*X^2 + 9600*X + 42240", "6": "X^6 - 306*X^4 - 1224*X^3 + 19323*X^2 + 113112*X + 37122", "7": "X^7 - 357*X^5 - 1428*X^4 + 31647*X^3 + 204792*X^2 - 186956*X - 2129988", "8": "X^8 - 408*X^6 - 1632*X^5 + 46572*X^4 + 317280*X^3 - 830560*X^2 - 9014496*X - 10701810", "9": "X^9 - 459*X^7 - 1836*X^6 + 64098*X^5 + 450576*X^4 - 2026341*X^3 - 22319496*X^2 - 28425870*X + 71302284", "10": "X^10 - 510*X^8 - 2040*X^7 + 84225*X^6 + 604680*X^5 - 3906950*X^4 - 43636800*X^3 - 37989300*X^2 + 479394400*X + 868859448", "11": "X^11 - 561*X^9 - 2244*X^8 + 106953*X^7 + 779592*X^6 - 6605038*X^5 - 74558220*X^4 - 17247186*X^3 + 1617562364*X^2 + 4028572251*X - 302240796", "12": "X^12 - 612*X^10 - 2448*X^9 + 132282*X^8 + 975312*X^7 - 10253256*X^6 - 116675568*X^5 + 62710587*X^4 + 3987819984*X^3 + 10603236132*X^2 - 13115444544*X - 48141654882", "13": "X^13 - 663*X^11 - 2652*X^10 + 160212*X^9 + 1191840*X^8 - 14984255*X^7 - 171580656*X^6 + 237559335*X^5 + 8200424284*X^4 + 21003754317*X^3 - 72286807944*X^2 - 315044744625*X - 180225789276", "14": "X^14 - 714*X^12 - 2856*X^11 + 190743*X^10 + 1429176*X^9 - 20930686*X^8 - 240865296*X^7 + 549739575*X^6 + 14973875504*X^5 + 34581679146*X^4 - 240437794632*X^3 - 1114987573952*X^2 - 559055270760*X + 1863267140388", "15": "X^15 - 765*X^13 - 3060*X^12 + 223875*X^11 + 1687320*X^10 - 28225200*X^9 - 326121300*X^8 + 1048457025*X^7 + 25134917100*X^6 + 49284186822*X^5 - 615867150600*X^4 - 2896895906535*X^3 + 55384706160*X^2 + 17179809440625*X + 17336946733068", "16": "X^16 - 816*X^14 - 3264*X^13 + 259608*X^12 + 1966272*X^11 - 37000448*X^10 - 428940480*X^9 + 1789682604*X^8 + 39618535744*X^7 + 61309052160*X^6 - 1339451409216*X^5 - 6217010149440*X^4 + 5589475392192*X^3 + 78885747867360*X^2 + 109307989667520*X - 29813688561714", "17": "X^17 - 867*X^15 - 3468*X^14 + 297942*X^13 + 2266032*X^12 - 47389081*X^11 - 550914648*X^10 + 2836152432*X^9 + 59467961324*X^8 + 64759623336*X^7 - 2601545396244*X^6 - 11693950888085*X^5 + 24652797708972*X^4 + 254235144582246*X^3 + 355239589058264*X^2 - 577012686527130*X - 1057699051732800", "18": "X^18 - 918*X^16 - 3672*X^15 + 338877*X^14 + 2586600*X^13 - 59523750*X^12 - 693635616*X^11 + 4257367830*X^10 + 85834666944*X^9 + 51299796636*X^8 - 4648882734864*X^7 - 19946188133397*X^6 + 72909145773936*X^5 + 658026148176450*X^4 + 764836393233240*X^3 - 3782760467157666*X^2 - 9133229346096456*X - 2708588894485728", "19": "X^19 - 969*X^17 - 3876*X^16 + 382413*X^15 + 2927976*X^14 - 73537106*X^13 - 858695196*X^12 + 6129595320*X^11 + 119978368924*X^10 + 9808991205*X^9 - 7791476350692*X^8 - 31511914281168*X^7 + 175869851332416*X^6 + 1465178168502975*X^5 + 1086959072460288*X^4 - 15599777819977887*X^3 - 41000090227192452*X^2 - 4542355910399712*X + 46044646873946808", "20": "X^20 - 1020*X^18 - 4080*X^17 + 428550*X^16 + 3290160*X^15 - 89561800*X^14 - 1047685200*X^13 + 8535866625*X^12 + 163267026800*X^11 - 73962876204*X^10 - 12409518976800*X^9 - 46751320783610*X^8 + 372109419096960*X^7 + 2921762648334000*X^6 + 493687988473664*X^5 - 49025686386830100*X^4 - 129141055144249920*X^3 + 68313385071947920*X^2 + 528348194646192000*X + 334212288470817288"}, "7B": {"2": "X^2 - 4", "3": "X^3 - 6*X - 24", "4": "X^4 - 8*X^2 - 32*X + 28", "5": "X^5 - 10*X^3 - 40*X^2 + 45*X + 100", "6": "X^6 - 12*X^4 - 48*X^3 + 66*X^2 + 216*X + 176", "7": "X^7 - 14*X^5 - 56*X^4 + 91*X^3 + 364*X^2 + 322*X - 644", "8": "X^8 - 16*X^6 - 64*X^5 + 120*X^4 + 544*X^3 + 480*X^2 - 1632*X - 1092", "9": "X^9 - 18*X^7 - 72*X^6 + 153*X^5 + 756*X^4 + 642*X^3 - 3132*X^2 - 3420*X + 228", "10": "X^10 - 20*X^8 - 80*X^7 + 190*X^6 + 1000*X^5 + 800*X^4 - 5240*X^3 - 7195*X^2 + 1920*X + 8696", "11": "X^11 - 22*X^9 - 88*X^8 + 231*X^7 + 1276*X^6 + 946*X^5 - 8052*X^4 - 12661*X^3 + 6116*X^2 + 31812*X + 6688", "12": "X^12 - 24*X^10 - 96*X^9 + 276*X^8 + 1584*X^7 + 1072*X^6 - 11664*X^5 - 20046*X^4 + 14176*X^3 + 76152*X^2 + 24816*X - 25808", "13": "X^13 - 26*X^11 - 104*X^10 + 325*X^9 + 1924*X^8 + 1170*X^7 - 16172*X^6 - 29562*X^5 + 27716*X^4 + 150150*X^3 + 59072*X^2 - 126919*X - 92716", "14": "X^14 - 28*X^12 - 112*X^11 + 378*X^10 + 2296*X^9 + 1232*X^8 - 21672*X^7 - 41405*X^6 + 48608*X^5 + 263424*X^4 + 112504*X^3 - 375046*X^2 - 418656*X + 36844", "15": "X^15 - 30*X^13 - 120*X^12 + 435*X^11 + 2700*X^10 + 1250*X^9 - 28260*X^8 - 55755*X^7 + 78980*X^6 + 426744*X^5 + 186240*X^4 - 871620*X^3 - 1177680*X^2 + 241380*X + 465616", "16": "X^16 - 32*X^14 - 128*X^13 + 496*X^12 + 3136*X^11 + 1216*X^10 - 36032*X^9 - 72776*X^8 + 121216*X^7 + 652000*X^6 + 278848*X^5 - 1751840*X^4 - 2638784*X^3 + 885472*X^2 + 2606464*X + 642172", "17": "X^17 - 34*X^15 - 136*X^14 + 561*X^13 + 3604*X^12 + 1122*X^11 - 45084*X^10 - 92616*X^9 + 177956*X^8 + 952170*X^7 + 385696*X^6 - 3188945*X^5 - 5143860*X^4 + 2466326*X^3 + 8668164*X^2 + 3250485*X - 1782348", "18": "X^18 - 36*X^16 - 144*X^15 + 630*X^14 + 4104*X^13 + 960*X^12 - 55512*X^11 - 115407*X^10 + 252096*X^9 + 1341288*X^8 + 498312*X^7 - 5398422*X^6 - 9110448*X^5 + 5795892*X^4 + 22310280*X^3 + 9916164*X^2 - 11524680*X - 5438596", "19": "X^19 - 38*X^17 - 152*X^16 + 703*X^15 + 4636*X^14 + 722*X^13 - 67412*X^12 - 141265*X^11 + 346788*X^10 + 1834412*X^9 + 603744*X^8 - 8642150*X^7 - 15032952*X^6 + 12104368*X^5 + 49020152*X^4 + 23461105*X^3 - 43380952*X^2 - 34504798*X + 203528", "20": "X^20 - 40*X^18 - 160*X^17 + 780*X^16 + 5200*X^15 + 400*X^14 - 80880*X^13 - 170290*X^12 + 465440*X^11 + 2447592*X^10 + 683920*X^9 - 13232480*X^8 - 23482320*X^7 + 23157080*X^6 + 96562688*X^5 + 47223285*X^4 - 124604720*X^3 - 127697960*X^2 + 8256720*X + 31008488"}, "13A": {"2": "X^2 - 24", "3": "X^3 - 36*X - 84", "4": "X^4 - 48*X^2 - 112*X + 24", "5": "X^5 - 60*X^3 - 140*X^2 + 390*X + 1020", "6": "X^6 - 72*X^4 - 168*X^3 + 900*X^2 + 3240*X + 2100", "7": "X^7 - 84*X^5 - 196*X^4 + 1554*X^3 + 6132*X^2 + 2674*X - 7476", "8": "X^8 - 96*X^6 - 224*X^5 + 2352*X^4 + 9696*X^3 - 1296*X^2 - 45600*X - 44808", "9": "X^9 - 108*X^7 - 252*X^6 + 3294*X^5 + 13932*X^4 - 11538*X^3 - 129276*X^2 - 170127*X - 19524", "10": "X^10 - 120*X^8 - 280*X^7 + 4380*X^6 + 18840*X^5 - 29780*X^4 - 270600*X^3 - 355650*X^2 + 231800*X + 503076", "11": "X^11 - 132*X^9 - 308*X^8 + 5610*X^7 + 24420*X^6 - 57750*X^5 - 481668*X^4 - 554433*X^3 + 1353044*X^2 + 3196974*X + 1492920", "12": "X^12 - 144*X^10 - 336*X^9 + 6984*X^8 + 30672*X^7 - 97176*X^6 - 774576*X^5 - 698796*X^4 + 4148240*X^3 + 10231992*X^2 + 5254992*X - 2482812", "13": "X^13 - 156*X^11 - 364*X^10 + 8502*X^9 + 37596*X^8 - 149786*X^7 - 1161420*X^6 - 700323*X^5 + 9614956*X^4 + 23669490*X^3 + 5859360*X^2 - 32384001*X - 24577488", "14": "X^14 - 168*X^12 - 392*X^11 + 10164*X^10 + 45192*X^9 - 217308*X^8 - 1654296*X^7 - 449862*X^6 + 18944296*X^5 + 45211404*X^4 - 12018216*X^3 - 155500758*X^2 - 162369480*X - 30593580", "15": "X^15 - 180*X^13 - 420*X^12 + 11970*X^11 + 53460*X^10 - 301470*X^9 - 2265300*X^8 + 182475*X^7 + 33520900*X^6 + 75951414*X^5 - 76541400*X^4 - 485866515*X^3 - 533513160*X^2 + 27337860*X + 226688196", "16": "X^16 - 192*X^14 - 448*X^13 + 13920*X^12 + 62400*X^11 - 404000*X^10 - 3006528*X^9 + 1347312*X^8 + 54922944*X^7 + 116126112*X^6 - 231616320*X^5 - 1188294224*X^4 - 1187188800*X^3 + 1018632096*X^2 + 2359194304*X + 941335992", "17": "X^17 - 204*X^15 - 476*X^14 + 16014*X^13 + 72012*X^12 - 526626*X^11 - 3890076*X^10 + 3216009*X^9 + 84922140*X^8 + 164866170*X^7 - 539794608*X^6 - 2477864869*X^5 - 1953110688*X^4 + 5338627776*X^3 + 11289280236*X^2 + 5858642034*X - 449825100", "18": "X^18 - 216*X^16 - 504*X^15 + 18252*X^14 + 82296*X^13 - 671076*X^12 - 4928040*X^11 + 5980662*X^10 + 125483736*X^9 + 219947508*X^8 - 1085176440*X^7 - 4616429994*X^6 - 2251096920*X^5 + 17894651004*X^4 + 36214683768*X^3 + 15250017609*X^2 - 17261297640*X - 12719333508", "19": "X^19 - 228*X^17 - 532*X^16 + 20634*X^15 + 93252*X^14 - 839078*X^13 - 6132516*X^12 + 9854103*X^11 + 178766516*X^10 + 277542462*X^9 - 1976313576*X^8 - 7906128247*X^7 - 858835416*X^6 + 47159268972*X^5 + 90262240788*X^4 + 10869810225*X^3 - 132177306612*X^2 - 126045733734*X - 26122558728", "20": "X^20 - 240*X^18 - 560*X^17 + 23160*X^16 + 104880*X^15 - 1032360*X^14 - 7515600*X^13 + 15069900*X^12 + 247122800*X^11 + 331970952*X^10 - 3349112400*X^9 - 12679915940*X^8 + 4362153840*X^7 + 106279281960*X^6 + 188036291568*X^5 - 75529872210*X^4 - 583804000080*X^3 - 593833297720*X^2 - 93980231280*X + 91698544884"}, "13B": {"2": "X^2 + 2", "3": "X^3 + 3*X - 6", "4": "X^4 + 4*X^2 - 8*X - 2", "5": "X^5 + 5*X^3 - 10*X^2 - 20", "6": "X^6 + 6*X^4 - 12*X^3 + 3*X^2 - 36*X + 20", "7": "X^7 + 7*X^5 - 14*X^4 + 7*X^3 - 56*X^2 + 35*X - 14", "8": "X^8 + 8*X^6 - 16*X^5 + 12*X^4 - 80*X^3 + 56*X^2 - 48*X + 94", "9": "X^9 + 9*X^7 - 18*X^6 + 18*X^5 - 108*X^4 + 84*X^3 - 108*X^2 + 225*X - 24", "10": "X^10 + 10*X^8 - 20*X^7 + 25*X^6 - 140*X^5 + 120*X^4 - 200*X^3 + 420*X^2 - 120*X + 132", "11": "X^11 + 11*X^9 - 22*X^8 + 33*X^7 - 176*X^6 + 165*X^5 - 330*X^4 + 693*X^3 - 308*X^2 + 495*X - 286", "12": "X^12 + 12*X^10 - 24*X^9 + 42*X^8 - 216*X^7 + 220*X^6 - 504*X^5 + 1059*X^4 - 632*X^3 + 1200*X^2 - 960*X + 136", "13": "X^13 + 13*X^11 - 26*X^10 + 52*X^9 - 260*X^8 + 286*X^7 - 728*X^6 + 1534*X^5 - 1144*X^4 + 2392*X^3 - 2236*X^2 + 793*X - 832", "14": "X^14 + 14*X^12 - 28*X^11 + 63*X^10 - 308*X^9 + 364*X^8 - 1008*X^7 + 2135*X^6 - 1904*X^5 + 4242*X^4 - 4396*X^3 + 2499*X^2 - 3276*X + 828", "15": "X^15 + 15*X^13 - 30*X^12 + 75*X^11 - 360*X^10 + 455*X^9 - 1350*X^8 + 2880*X^7 - 2980*X^6 + 6948*X^5 - 7800*X^4 + 6015*X^3 - 8760*X^2 + 3735*X - 1046", "16": "X^16 + 16*X^14 - 32*X^13 + 88*X^12 - 416*X^11 + 560*X^10 - 1760*X^9 + 3788*X^8 - 4448*X^7 + 10736*X^6 - 12896*X^5 + 12376*X^4 - 19232*X^3 + 10960*X^2 - 6176*X + 3262", "17": "X^17 + 17*X^15 - 34*X^14 + 102*X^13 - 476*X^12 + 680*X^11 - 2244*X^10 + 4879*X^9 - 6392*X^8 + 15861*X^7 - 20230*X^6 + 22933*X^5 - 37332*X^4 + 26027*X^3 - 20774*X^2 + 16014*X - 2584", "18": "X^18 + 18*X^16 - 36*X^15 + 117*X^14 - 540*X^13 + 816*X^12 - 2808*X^11 + 6174*X^10 - 8904*X^9 + 22608*X^8 - 30456*X^7 + 39396*X^6 - 66528*X^5 + 54180*X^4 - 53856*X^3 + 49473*X^2 - 16200*X + 6554", "19": "X^19 + 19*X^17 - 38*X^16 + 133*X^15 - 608*X^14 + 969*X^13 - 3458*X^12 + 7695*X^11 - 12084*X^10 + 31293*X^9 - 44346*X^8 + 63878*X^7 - 111264*X^6 + 102866*X^5 - 119244*X^4 + 122474*X^3 - 59280*X^2 + 39938*X - 11628", "20": "X^20 + 20*X^18 - 40*X^17 + 150*X^16 - 680*X^15 + 1140*X^14 - 4200*X^13 + 9465*X^12 - 16040*X^11 + 42264*X^10 - 62800*X^9 + 98940*X^8 - 177120*X^7 + 182280*X^6 - 237008*X^5 + 264420*X^4 - 166480*X^3 + 144040*X^2 - 68640*X + 11828"}}, "Keywords": ["McKay-Thompson", "McKay\u2013Thompson", "Monster", "moonshine", "Hauptmodul", "Faber polynomial", "replicable function", "twisted Hecke operator"], "Comments": {"comment-normalization": "The class labels use the ATLAS convention. Every $T_g$ has constant coefficient zero, and $T_{1A}=j-744$, whose positive Fourier coefficients are in HREF{Q-expansion_of_the_j-invariant}[the $j$-invariant expansion]. In particular, $T_{13A}=q^{-1}+12q+28q^2+\\cdots$ uses the zero-constant convention CITE{OEIS-13A}.", "comment-classes": "The selected primes $p=2,3,5,7,13$ are precisely the primes for which $X_0(p)$ has genus zero. The classes $pB$ and $pA$ give its non-Fricke and Fricke Hauptmoduln, respectively CITE{Matsusaka}. The identity class $1A$ supplies the classical anchor.", "comment-polynomials": "The degree-one polynomial is $P_{1,g}(X)=X$ for every class and is omitted from the entries. The normalization is monic, without division by $n$; in the twisted Hecke-operator convention of CITE{BKM}, Definitions 2.2\u20132.3, $T_g\\mid\\widetilde T_n=P_{n,g}(T_g)/n$.", "comment-translations": "Changing a series from $T_g$ to $T_g+C$ changes its Faber polynomial to $P_{n,g}(X-C)$. The examples for $1A$ and $2A$ in CITE{BKM}, pp. 24\u201325, use shifts $C=744$ and $C=104$, respectively.", "comment-cm": "The HREF{Values_of_monstrous_McKay_Thompson_series_at_rectangular_CM_points}[rectangular CM values] use the same normalized series for the ten nonidentity classes. Substituting a value $T_g(i\\sqrt r)$ into the polynomials gives $P_{n,g}(T_g(i\\sqrt r))$; those CM values are not defined as polynomial zeros."}, "Formulas": {"formula-eta": "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 $T_{pA}=R_p+b_p+C_p/R_p$ CITE{Matsusaka}. Here $\\eta(\\tau)=q^{1/24}\\prod_{m\\geq1}(1-q^m)$; its HREF{Dedekind_eta_values_at_rectangular_CM_points}[rectangular CM values] provide the factors at those arguments.", "formula-recurrence": "With $P_{0,g}(X)=1$ as an auxiliary initial value and $P_{1,g}(X)=X$, the recurrence is $P_{n,g}(X)=XP_{n-1,g}(X)-\\sum_{k=1}^{n-2}c_g(k)P_{n-1-k,g}(X)-n c_g(n-1)$ for $n\\geq2$ CITE{BKM}, \u00a72.2.", "formula-first": "$P_{2,g}(X)=X^2-2c_g(1)$ and $P_{3,g}(X)=X^3-3c_g(1)X-3c_g(2)$.", "formula-canonical": "For $m_p=C_p/R_p$, the HREF{Canonical_modular_polynomials}[canonical modular equation] is $\\Phi^c_p(m_p,j)=0$. Its branch $m_p=C_pq+O(q^2)$ determines the same eta quotient as CITE{formula-eta}."}, "Links": {"OEIS-13A": {"title": "OEIS A034319: zero-constant McKay\u2013Thompson series of class 13A", "url": "https://oeis.org/A034319"}}, "References": {"Matsusaka": {"bib": "Toshiki Matsusaka, The Fourier coefficients of the McKay\u2013Thompson series and the traces of CM values, Research in Number Theory 3 (2017), article 23, \u00a71.", "doi": "10.1007/s40993-017-0090-x"}, "BKM": {"bib": "Eiichi Bannai, Koji Kojima and Tsuyoshi Miezaki, On the zeros of Hecke-type Faber polynomials, Kyushu Journal of Mathematics 62 (2008), 15\u201361, \u00a7\u00a71\u20133.", "doi": "10.2206/kyushujm.62.15"}}, "Similar tables": [{"table": "HREF{Values_of_monstrous_McKay_Thompson_series_at_rectangular_CM_points}[McKay\u2013Thompson values at rectangular CM points]", "relation": "These supply arguments $T_g(i\\sqrt r)$ for evaluating the polynomials, with the same normalization and the same ten nonidentity classes."}, {"table": "HREF{Q-expansion_of_the_j-invariant}[$q$-expansion of the $j$-invariant]", "relation": "The positive coefficients are $c_{1A}(m)$, which determine the identity-class polynomials."}, {"table": "HREF{Canonical_modular_polynomials}[Canonical modular polynomials]", "relation": "The equations for $m_p=C_p/R_p$ and $j$ independently determine the series used for $pA$ and $pB$."}, {"table": "HREF{Hecke_polynomials_of_level_one_cusp_forms}[Hecke polynomials of level one cusp forms]", "relation": "These are characteristic polynomials of operators on cusp forms; the present polynomials are instead evaluated in a Hauptmodul."}, {"table": "HREF{Degrees_of_the_irreducible_characters_of_the_sporadic_simple_groups}[Sporadic irreducible character degrees]", "relation": "The Monster representations are the constituents of the graded module whose traces give the McKay\u2013Thompson series CITE{Matsusaka}."}], "Display properties": {"number-header": "$P_{n,g}(X)$", "group parameters": [["g"], ["n"]]}, "Programs": {"program-sage": {"language": "Sage", "code": "# Save the attached generate.py beside this script.\n# This computes one additional degree in the same normalization.\nfrom generate import faber\nfaber('3A', 21)"}}}