Properties

Label 31.2.g
Level 31
Weight 2
Character orbit g
Rep. character \(\chi_{31}(7,\cdot)\)
Character field \(\Q(\zeta_{15})\)
Dimension 16
Newform subspaces 1
Sturm bound 5
Trace bound 0

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Defining parameters

Level: \( N \) = \( 31 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 31.g (of order \(15\) and degree \(8\))
Character conductor: \(\operatorname{cond}(\chi)\) = \( 31 \)
Character field: \(\Q(\zeta_{15})\)
Newform subspaces: \( 1 \)
Sturm bound: \(5\)
Trace bound: \(0\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(31, [\chi])\).

Total New Old
Modular forms 32 32 0
Cusp forms 16 16 0
Eisenstein series 16 16 0

Trace form

\( 16q - 6q^{2} - 12q^{3} - 14q^{4} - 3q^{5} + 11q^{6} + 2q^{7} + 17q^{8} - 10q^{9} + O(q^{10}) \) \( 16q - 6q^{2} - 12q^{3} - 14q^{4} - 3q^{5} + 11q^{6} + 2q^{7} + 17q^{8} - 10q^{9} - 2q^{10} - 7q^{11} + 5q^{12} - 7q^{13} - 6q^{14} + 14q^{15} - 2q^{16} - 6q^{17} - 3q^{18} + 16q^{19} + 37q^{20} + 9q^{21} + 9q^{22} + q^{23} - 20q^{24} - 13q^{25} + 9q^{26} + 9q^{27} - 30q^{28} - 14q^{29} - 22q^{30} + 15q^{31} - 42q^{32} - 13q^{33} - 32q^{34} - 9q^{35} + q^{36} - 8q^{37} + 8q^{38} - 3q^{39} - q^{40} - 8q^{41} + 69q^{42} + 23q^{43} + 39q^{44} + 65q^{45} + 34q^{46} + 14q^{47} + 34q^{48} + 2q^{49} + 3q^{50} - 42q^{51} + 29q^{52} + 6q^{53} - 46q^{54} - 7q^{55} - 30q^{56} - 17q^{57} - 15q^{58} + 4q^{59} - 75q^{60} - 60q^{61} - 25q^{62} - 46q^{63} + 23q^{64} - 12q^{65} - 30q^{66} + 13q^{67} + 30q^{68} + 38q^{69} + 12q^{70} - 14q^{71} + 37q^{72} + 2q^{73} + 13q^{74} + 13q^{75} - 12q^{76} + 18q^{77} - 15q^{78} + 18q^{79} + 36q^{80} + 23q^{81} + 14q^{82} - 16q^{83} + 8q^{84} + 37q^{85} - 26q^{86} + 15q^{87} - 17q^{88} + q^{89} - 23q^{90} + 8q^{91} - 64q^{92} + 17q^{93} + 44q^{94} - 22q^{95} + 8q^{96} + 3q^{97} - 10q^{98} + 6q^{99} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(31, [\chi])\) into newform subspaces

Label Dim. \(A\) Field CM Traces $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\)
31.2.g.a \(16\) \(0.248\) \(\mathbb{Q}[x]/(x^{16} + \cdots)\) None \(-6\) \(-12\) \(-3\) \(2\) \(q+(-1-\beta _{1}+\beta _{3}-\beta _{4}+\beta _{5}+\beta _{6}+\cdots)q^{2}+\cdots\)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( 1 + 6 T + 21 T^{2} + 51 T^{3} + 98 T^{4} + 159 T^{5} + 231 T^{6} + 306 T^{7} + 369 T^{8} + 399 T^{9} + 378 T^{10} + 288 T^{11} + 70 T^{12} - 408 T^{13} - 1395 T^{14} - 2958 T^{15} - 4795 T^{16} - 5916 T^{17} - 5580 T^{18} - 3264 T^{19} + 1120 T^{20} + 9216 T^{21} + 24192 T^{22} + 51072 T^{23} + 94464 T^{24} + 156672 T^{25} + 236544 T^{26} + 325632 T^{27} + 401408 T^{28} + 417792 T^{29} + 344064 T^{30} + 196608 T^{31} + 65536 T^{32} \)
$3$ \( 1 + 12 T + 80 T^{2} + 375 T^{3} + 1362 T^{4} + 4038 T^{5} + 10105 T^{6} + 21936 T^{7} + 42530 T^{8} + 76515 T^{9} + 134002 T^{10} + 238260 T^{11} + 435636 T^{12} + 804669 T^{13} + 1467485 T^{14} + 2617221 T^{15} + 4573453 T^{16} + 7851663 T^{17} + 13207365 T^{18} + 21726063 T^{19} + 35286516 T^{20} + 57897180 T^{21} + 97687458 T^{22} + 167338305 T^{23} + 279039330 T^{24} + 431766288 T^{25} + 596690145 T^{26} + 715319586 T^{27} + 723822642 T^{28} + 597871125 T^{29} + 382637520 T^{30} + 172186884 T^{31} + 43046721 T^{32} \)
$5$ \( 1 + 3 T - 9 T^{2} - 18 T^{3} + 61 T^{4} - 21 T^{5} - 291 T^{6} + 591 T^{7} + 172 T^{8} - 4164 T^{9} + 1989 T^{10} + 14124 T^{11} - 21959 T^{12} - 41793 T^{13} + 231951 T^{14} + 136788 T^{15} - 1318469 T^{16} + 683940 T^{17} + 5798775 T^{18} - 5224125 T^{19} - 13724375 T^{20} + 44137500 T^{21} + 31078125 T^{22} - 325312500 T^{23} + 67187500 T^{24} + 1154296875 T^{25} - 2841796875 T^{26} - 1025390625 T^{27} + 14892578125 T^{28} - 21972656250 T^{29} - 54931640625 T^{30} + 91552734375 T^{31} + 152587890625 T^{32} \)
$7$ \( 1 - 2 T + 8 T^{2} - 57 T^{3} + 180 T^{4} - 447 T^{5} + 2037 T^{6} - 5539 T^{7} + 16183 T^{8} - 49082 T^{9} + 136106 T^{10} - 412273 T^{11} + 1249403 T^{12} - 2948414 T^{13} + 10225949 T^{14} - 27209176 T^{15} + 63276497 T^{16} - 190464232 T^{17} + 501071501 T^{18} - 1011306002 T^{19} + 2999816603 T^{20} - 6929072311 T^{21} + 16012734794 T^{22} - 40421137526 T^{23} + 93291774583 T^{24} - 223518629173 T^{25} + 575402082213 T^{26} - 883865054121 T^{27} + 2491431696180 T^{28} - 5522673593199 T^{29} + 5425784582792 T^{30} - 9495123019886 T^{31} + 33232930569601 T^{32} \)
$11$ \( 1 + 7 T + 35 T^{2} + 158 T^{3} + 743 T^{4} + 2658 T^{5} + 9655 T^{6} + 30605 T^{7} + 105459 T^{8} + 298602 T^{9} + 903884 T^{10} + 2105465 T^{11} + 6983649 T^{12} + 12627823 T^{13} + 32620956 T^{14} + 51556382 T^{15} + 388064517 T^{16} + 567120202 T^{17} + 3947135676 T^{18} + 16807632413 T^{19} + 102247605009 T^{20} + 339087243715 T^{21} + 1601285642924 T^{22} + 5818908234942 T^{23} + 22606073231379 T^{24} + 72164989083055 T^{25} + 250425834522655 T^{26} + 758358420484038 T^{27} + 2331852283903703 T^{28} + 5454588518741098 T^{29} + 13291244175413435 T^{30} + 29240737185909557 T^{31} + 45949729863572161 T^{32} \)
$13$ \( 1 + 7 T + 44 T^{2} + 249 T^{3} + 975 T^{4} + 3375 T^{5} + 9585 T^{6} + 9146 T^{7} - 38519 T^{8} - 357284 T^{9} - 1668070 T^{10} - 3915103 T^{11} - 4062397 T^{12} + 34368784 T^{13} + 322154357 T^{14} + 1609383854 T^{15} + 6221461841 T^{16} + 20921990102 T^{17} + 54444086333 T^{18} + 75508218448 T^{19} - 116026120717 T^{20} - 1453650338179 T^{21} - 8051455288630 T^{22} - 22419041147828 T^{23} - 31421131642199 T^{24} + 96988751265458 T^{25} + 1321373644372665 T^{26} + 6048541329874875 T^{27} + 22715632994418975 T^{28} + 75415901541470997 T^{29} + 173244560970768716 T^{30} + 358301251098635299 T^{31} + 665416609183179841 T^{32} \)
$17$ \( 1 + 6 T + 75 T^{2} + 495 T^{3} + 3677 T^{4} + 23094 T^{5} + 144360 T^{6} + 817152 T^{7} + 4567650 T^{8} + 23850960 T^{9} + 122226918 T^{10} + 594432255 T^{11} + 2843619706 T^{12} + 12936362562 T^{13} + 57908087850 T^{14} + 248770132872 T^{15} + 1043483763143 T^{16} + 4229092258824 T^{17} + 16735437388650 T^{18} + 63556349267106 T^{19} + 237501961464826 T^{20} + 844008798287535 T^{21} + 2950260666882342 T^{22} + 9786971276176080 T^{23} + 31862818475383650 T^{24} + 96904320455276544 T^{25} + 291028879468817640 T^{26} + 791475173328476502 T^{27} + 2142301966293831197 T^{28} + 4902766126288438815 T^{29} + 12628336991955069675 T^{30} + 17174538309058894758 T^{31} + 48661191875666868481 T^{32} \)
$19$ \( 1 - 16 T + 195 T^{2} - 1746 T^{3} + 13353 T^{4} - 89969 T^{5} + 558125 T^{6} - 3294825 T^{7} + 18744924 T^{8} - 103602551 T^{9} + 554225766 T^{10} - 2858170600 T^{11} + 14222538534 T^{12} - 68363619099 T^{13} + 318565822489 T^{14} - 1445186621304 T^{15} + 6375621526007 T^{16} - 27458545804776 T^{17} + 115002261918529 T^{18} - 468906063400041 T^{19} + 1853495444289414 T^{20} - 7077113364489400 T^{21} + 26074039434369846 T^{22} - 92607392427206189 T^{23} + 318355598452753884 T^{24} - 1063199493834693675 T^{25} + 3421901355135183125 T^{26} - 10480512102813865211 T^{27} + 29554394114290447833 T^{28} - 73424509125100825014 T^{29} + \)\(15\!\cdots\!95\)\( T^{30} - \)\(24\!\cdots\!84\)\( T^{31} + \)\(28\!\cdots\!81\)\( T^{32} \)
$23$ \( 1 - T - 32 T^{2} + 265 T^{3} + 1220 T^{4} - 3468 T^{5} - 10927 T^{6} + 173296 T^{7} + 1397175 T^{8} - 394035 T^{9} - 17319068 T^{10} + 116114803 T^{11} + 1091799191 T^{12} + 1263133445 T^{13} - 14131047870 T^{14} + 39184802221 T^{15} + 769047681799 T^{16} + 901250451083 T^{17} - 7475324323230 T^{18} + 15368544625315 T^{19} + 305530177408631 T^{20} + 747354699485429 T^{21} - 2563843628031452 T^{22} - 1341620395008645 T^{23} + 109414150859981175 T^{24} + 312132551620892048 T^{25} - 452667488031542623 T^{26} - 3304344240445498836 T^{27} + 26735841807064791620 T^{28} + \)\(13\!\cdots\!95\)\( T^{29} - \)\(37\!\cdots\!88\)\( T^{30} - \)\(26\!\cdots\!07\)\( T^{31} + \)\(61\!\cdots\!61\)\( T^{32} \)
$29$ \( 1 + 14 T + 65 T^{2} + 271 T^{3} + 4124 T^{4} + 28761 T^{5} + 57004 T^{6} + 203557 T^{7} + 4414002 T^{8} + 20510481 T^{9} - 1562410 T^{10} + 162325096 T^{11} + 3747178047 T^{12} + 10722269693 T^{13} - 22040075979 T^{14} + 162043565485 T^{15} + 2719402913313 T^{16} + 4699263399065 T^{17} - 18535703898339 T^{18} + 261505435542577 T^{19} + 2650307836260207 T^{20} + 3329474230495304 T^{21} - 929357904963610 T^{22} + 353803260288094629 T^{23} + 2208088667302679922 T^{24} + 2953031113409966033 T^{25} + 23981995127044657804 T^{26} + \)\(35\!\cdots\!69\)\( T^{27} + \)\(14\!\cdots\!84\)\( T^{28} + \)\(27\!\cdots\!19\)\( T^{29} + \)\(19\!\cdots\!65\)\( T^{30} + \)\(12\!\cdots\!86\)\( T^{31} + \)\(25\!\cdots\!21\)\( T^{32} \)
$31$ \( 1 - 15 T + 158 T^{2} - 1635 T^{3} + 13788 T^{4} - 99390 T^{5} + 688351 T^{6} - 4312200 T^{7} + 24371915 T^{8} - 133678200 T^{9} + 661505311 T^{10} - 2960927490 T^{11} + 12733507548 T^{12} - 46808661885 T^{13} + 140225581598 T^{14} - 412689211665 T^{15} + 852891037441 T^{16} \)
$37$ \( 1 + 8 T - 120 T^{2} - 534 T^{3} + 11739 T^{4} + 10042 T^{5} - 710515 T^{6} + 2077152 T^{7} + 30531852 T^{8} - 207297908 T^{9} - 600747795 T^{10} + 11697189185 T^{11} - 16371022929 T^{12} - 395056153140 T^{13} + 2176783134259 T^{14} + 6190914463491 T^{15} - 102798721686295 T^{16} + 229063835149167 T^{17} + 2980016110800571 T^{18} - 20010779325000420 T^{19} - 30681932703637569 T^{20} + 811129383865505045 T^{21} - 1541354482780018155 T^{22} - 19679179532183937764 T^{23} + \)\(10\!\cdots\!92\)\( T^{24} + \)\(26\!\cdots\!04\)\( T^{25} - \)\(34\!\cdots\!35\)\( T^{26} + \)\(17\!\cdots\!46\)\( T^{27} + \)\(77\!\cdots\!59\)\( T^{28} - \)\(13\!\cdots\!98\)\( T^{29} - \)\(10\!\cdots\!80\)\( T^{30} + \)\(26\!\cdots\!44\)\( T^{31} + \)\(12\!\cdots\!41\)\( T^{32} \)
$41$ \( 1 + 8 T + 106 T^{2} + 976 T^{3} + 7985 T^{4} + 55050 T^{5} + 396119 T^{6} + 1966432 T^{7} + 11020956 T^{8} + 52272537 T^{9} + 100609408 T^{10} + 252719935 T^{11} - 1900853239 T^{12} - 50449760350 T^{13} - 359337416133 T^{14} - 1943490188168 T^{15} - 20983245640805 T^{16} - 79683097714888 T^{17} - 604046196519573 T^{18} - 3477047933082350 T^{19} - 5371356949489879 T^{20} + 29279171586066935 T^{21} + 477905175625299328 T^{22} + 10180299987352706097 T^{23} + 88001509613432459676 T^{24} + \)\(64\!\cdots\!52\)\( T^{25} + \)\(53\!\cdots\!19\)\( T^{26} + \)\(30\!\cdots\!50\)\( T^{27} + \)\(18\!\cdots\!85\)\( T^{28} + \)\(90\!\cdots\!96\)\( T^{29} + \)\(40\!\cdots\!66\)\( T^{30} + \)\(12\!\cdots\!08\)\( T^{31} + \)\(63\!\cdots\!41\)\( T^{32} \)
$43$ \( 1 - 23 T + 289 T^{2} - 3096 T^{3} + 26895 T^{4} - 183450 T^{5} + 1103565 T^{6} - 5143929 T^{7} + 16278591 T^{8} - 38987724 T^{9} - 224772150 T^{10} + 3232536297 T^{11} - 26862162747 T^{12} + 323838946599 T^{13} - 2730755315628 T^{14} + 21633024576604 T^{15} - 167060950817939 T^{16} + 930220056793972 T^{17} - 5049166578596172 T^{18} + 25747463127246693 T^{19} - 91836388861606347 T^{20} + 475210127962955571 T^{21} - 1420866363454285350 T^{22} - 10597588987903050468 T^{23} + \)\(19\!\cdots\!91\)\( T^{24} - \)\(25\!\cdots\!47\)\( T^{25} + \)\(23\!\cdots\!85\)\( T^{26} - \)\(17\!\cdots\!50\)\( T^{27} + \)\(10\!\cdots\!95\)\( T^{28} - \)\(53\!\cdots\!28\)\( T^{29} + \)\(21\!\cdots\!61\)\( T^{30} - \)\(73\!\cdots\!61\)\( T^{31} + \)\(13\!\cdots\!01\)\( T^{32} \)
$47$ \( 1 - 14 T - 64 T^{2} + 2051 T^{3} - 6367 T^{4} - 96576 T^{5} + 874231 T^{6} - 748939 T^{7} - 39432966 T^{8} + 323481909 T^{9} + 441998288 T^{10} - 20651272567 T^{11} + 54844789185 T^{12} + 746473355167 T^{13} - 4553042500275 T^{14} - 12229830802363 T^{15} + 232672311744285 T^{16} - 574802047711061 T^{17} - 10057670883107475 T^{18} + 77501103153503441 T^{19} + 267625075735049985 T^{20} - 4736266251437522969 T^{21} + 4764394721401356752 T^{22} + \)\(16\!\cdots\!67\)\( T^{23} - \)\(93\!\cdots\!26\)\( T^{24} - \)\(83\!\cdots\!13\)\( T^{25} + \)\(45\!\cdots\!19\)\( T^{26} - \)\(23\!\cdots\!28\)\( T^{27} - \)\(73\!\cdots\!47\)\( T^{28} + \)\(11\!\cdots\!77\)\( T^{29} - \)\(16\!\cdots\!16\)\( T^{30} - \)\(16\!\cdots\!02\)\( T^{31} + \)\(56\!\cdots\!21\)\( T^{32} \)
$53$ \( 1 - 6 T + 37 T^{2} + 126 T^{3} + 1690 T^{4} - 1806 T^{5} - 120187 T^{6} + 3508437 T^{7} - 15460642 T^{8} + 56872014 T^{9} + 164905174 T^{10} + 7036349469 T^{11} - 9006554212 T^{12} - 185117474262 T^{13} + 4685233247216 T^{14} - 8469368678412 T^{15} + 48444353804527 T^{16} - 448876539955836 T^{17} + 13160820191429744 T^{18} - 27559734215703774 T^{19} - 71066044885255972 T^{20} + 2942569635108743217 T^{21} + 3655017828576581446 T^{22} + 66808188390765821718 T^{23} - \)\(96\!\cdots\!62\)\( T^{24} + \)\(11\!\cdots\!21\)\( T^{25} - \)\(21\!\cdots\!63\)\( T^{26} - \)\(16\!\cdots\!82\)\( T^{27} + \)\(83\!\cdots\!90\)\( T^{28} + \)\(32\!\cdots\!98\)\( T^{29} + \)\(51\!\cdots\!53\)\( T^{30} - \)\(43\!\cdots\!42\)\( T^{31} + \)\(38\!\cdots\!21\)\( T^{32} \)
$59$ \( 1 - 4 T + 157 T^{2} - 458 T^{3} + 14849 T^{4} - 42150 T^{5} + 924686 T^{6} - 2509544 T^{7} + 45834753 T^{8} - 195555639 T^{9} + 1870878718 T^{10} - 18828370418 T^{11} + 109287552140 T^{12} - 1909212618757 T^{13} + 8459795171586 T^{14} - 146067983115188 T^{15} + 587919268827301 T^{16} - 8618011003796092 T^{17} + 29448546992290866 T^{18} - 392112178427693903 T^{19} + 1324276722086702540 T^{20} - 13460859522400986982 T^{21} + 78914662702829952238 T^{22} - \)\(48\!\cdots\!41\)\( T^{23} + \)\(67\!\cdots\!13\)\( T^{24} - \)\(21\!\cdots\!16\)\( T^{25} + \)\(47\!\cdots\!86\)\( T^{26} - \)\(12\!\cdots\!50\)\( T^{27} + \)\(26\!\cdots\!69\)\( T^{28} - \)\(48\!\cdots\!82\)\( T^{29} + \)\(97\!\cdots\!77\)\( T^{30} - \)\(14\!\cdots\!96\)\( T^{31} + \)\(21\!\cdots\!41\)\( T^{32} \)
$61$ \( ( 1 + 30 T + 776 T^{2} + 13665 T^{3} + 208005 T^{4} + 2593710 T^{5} + 28390399 T^{6} + 268109040 T^{7} + 2240276459 T^{8} + 16354651440 T^{9} + 105640674679 T^{10} + 588722889510 T^{11} + 2880004157205 T^{12} + 11541408453165 T^{13} + 39979810504136 T^{14} + 94282285080630 T^{15} + 191707312997281 T^{16} )^{2} \)
$67$ \( 1 - 13 T - 258 T^{2} + 4011 T^{3} + 35028 T^{4} - 644333 T^{5} - 3445609 T^{6} + 71925183 T^{7} + 280896507 T^{8} - 6225465362 T^{9} - 19715222826 T^{10} + 429031194656 T^{11} + 1171728043770 T^{12} - 21978520425855 T^{13} - 61484782388300 T^{14} + 555554083027185 T^{15} + 3540421883673161 T^{16} + 37222123562821395 T^{17} - 276005188141078700 T^{18} - 6610325738841427365 T^{19} + 23611633589102566170 T^{20} + \)\(57\!\cdots\!92\)\( T^{21} - \)\(17\!\cdots\!94\)\( T^{22} - \)\(37\!\cdots\!26\)\( T^{23} + \)\(11\!\cdots\!87\)\( T^{24} + \)\(19\!\cdots\!01\)\( T^{25} - \)\(62\!\cdots\!41\)\( T^{26} - \)\(78\!\cdots\!39\)\( T^{27} + \)\(28\!\cdots\!08\)\( T^{28} + \)\(21\!\cdots\!57\)\( T^{29} - \)\(94\!\cdots\!82\)\( T^{30} - \)\(31\!\cdots\!59\)\( T^{31} + \)\(16\!\cdots\!81\)\( T^{32} \)
$71$ \( 1 + 14 T + 196 T^{2} + 2695 T^{3} + 17090 T^{4} + 118017 T^{5} + 577928 T^{6} - 4343303 T^{7} - 27945240 T^{8} - 279502470 T^{9} - 2826368429 T^{10} + 131657464 T^{11} - 99138557149 T^{12} - 2207242010815 T^{13} - 23038847142600 T^{14} - 294198893013362 T^{15} - 2969589635047583 T^{16} - 20888121403948702 T^{17} - 116138828445846600 T^{18} - 789996195332807465 T^{19} - 2519277389070657469 T^{20} + 237540260827025864 T^{21} - \)\(36\!\cdots\!09\)\( T^{22} - \)\(25\!\cdots\!70\)\( T^{23} - \)\(18\!\cdots\!40\)\( T^{24} - \)\(19\!\cdots\!93\)\( T^{25} + \)\(18\!\cdots\!28\)\( T^{26} + \)\(27\!\cdots\!07\)\( T^{27} + \)\(28\!\cdots\!90\)\( T^{28} + \)\(31\!\cdots\!45\)\( T^{29} + \)\(16\!\cdots\!76\)\( T^{30} + \)\(82\!\cdots\!14\)\( T^{31} + \)\(41\!\cdots\!21\)\( T^{32} \)
$73$ \( 1 - 2 T + 14 T^{2} + 1041 T^{3} - 15525 T^{4} + 39720 T^{5} + 424590 T^{6} - 12956116 T^{7} + 131189071 T^{8} - 324722351 T^{9} - 2585580235 T^{10} + 84857314523 T^{11} - 864267778837 T^{12} + 4587359474806 T^{13} + 4622090434727 T^{14} - 447090776699989 T^{15} + 5618971555814261 T^{16} - 32637626699099197 T^{17} + 24631119926660183 T^{18} + 1784560820810605702 T^{19} - 24543684671947825717 T^{20} + \)\(17\!\cdots\!39\)\( T^{21} - \)\(39\!\cdots\!15\)\( T^{22} - \)\(35\!\cdots\!47\)\( T^{23} + \)\(10\!\cdots\!51\)\( T^{24} - \)\(76\!\cdots\!08\)\( T^{25} + \)\(18\!\cdots\!10\)\( T^{26} + \)\(12\!\cdots\!40\)\( T^{27} - \)\(35\!\cdots\!25\)\( T^{28} + \)\(17\!\cdots\!53\)\( T^{29} + \)\(17\!\cdots\!26\)\( T^{30} - \)\(17\!\cdots\!14\)\( T^{31} + \)\(65\!\cdots\!61\)\( T^{32} \)
$79$ \( 1 - 18 T + 226 T^{2} - 2498 T^{3} + 40839 T^{4} - 469468 T^{5} + 5768280 T^{6} - 65435894 T^{7} + 780288208 T^{8} - 7561516414 T^{9} + 85206384568 T^{10} - 852745741894 T^{11} + 8569871201298 T^{12} - 77514808977636 T^{13} + 796329583287446 T^{14} - 7002575994516930 T^{15} + 64612388642450549 T^{16} - 553203503566837470 T^{17} + 4969892929296950486 T^{18} - 38217823903524675804 T^{19} + \)\(33\!\cdots\!38\)\( T^{20} - \)\(26\!\cdots\!06\)\( T^{21} + \)\(20\!\cdots\!28\)\( T^{22} - \)\(14\!\cdots\!26\)\( T^{23} + \)\(11\!\cdots\!88\)\( T^{24} - \)\(78\!\cdots\!86\)\( T^{25} + \)\(54\!\cdots\!80\)\( T^{26} - \)\(35\!\cdots\!72\)\( T^{27} + \)\(24\!\cdots\!99\)\( T^{28} - \)\(11\!\cdots\!22\)\( T^{29} + \)\(83\!\cdots\!06\)\( T^{30} - \)\(52\!\cdots\!82\)\( T^{31} + \)\(23\!\cdots\!21\)\( T^{32} \)
$83$ \( 1 + 16 T + 185 T^{2} + 3773 T^{3} + 53729 T^{4} + 585909 T^{5} + 6686260 T^{6} + 69214526 T^{7} + 675636522 T^{8} + 5645389119 T^{9} + 32133713345 T^{10} + 172336220420 T^{11} + 356263146291 T^{12} - 19484713577420 T^{13} - 322899642624249 T^{14} - 3433895006273167 T^{15} - 33440240762794785 T^{16} - 285013285520672861 T^{17} - 2224455638038451361 T^{18} - 11141105922292249540 T^{19} + 16907650757148237411 T^{20} + \)\(67\!\cdots\!60\)\( T^{21} + \)\(10\!\cdots\!05\)\( T^{22} + \)\(15\!\cdots\!13\)\( T^{23} + \)\(15\!\cdots\!02\)\( T^{24} + \)\(12\!\cdots\!78\)\( T^{25} + \)\(10\!\cdots\!40\)\( T^{26} + \)\(75\!\cdots\!03\)\( T^{27} + \)\(57\!\cdots\!69\)\( T^{28} + \)\(33\!\cdots\!99\)\( T^{29} + \)\(13\!\cdots\!65\)\( T^{30} + \)\(97\!\cdots\!12\)\( T^{31} + \)\(50\!\cdots\!81\)\( T^{32} \)
$89$ \( 1 - T - 247 T^{2} + 1489 T^{3} + 31955 T^{4} - 480540 T^{5} - 2146982 T^{6} + 82322077 T^{7} - 115178013 T^{8} - 9374121609 T^{9} + 52814885666 T^{10} + 761747156077 T^{11} - 8384072473029 T^{12} - 44513145443047 T^{13} + 944971959403383 T^{14} + 1347434269510390 T^{15} - 88986302178103107 T^{16} + 119921649986424710 T^{17} + 7485122890434196743 T^{18} - 31380387629839400543 T^{19} - \)\(52\!\cdots\!89\)\( T^{20} + \)\(42\!\cdots\!73\)\( T^{21} + \)\(26\!\cdots\!26\)\( T^{22} - \)\(41\!\cdots\!61\)\( T^{23} - \)\(45\!\cdots\!53\)\( T^{24} + \)\(28\!\cdots\!93\)\( T^{25} - \)\(66\!\cdots\!82\)\( T^{26} - \)\(13\!\cdots\!60\)\( T^{27} + \)\(78\!\cdots\!55\)\( T^{28} + \)\(32\!\cdots\!41\)\( T^{29} - \)\(48\!\cdots\!27\)\( T^{30} - \)\(17\!\cdots\!49\)\( T^{31} + \)\(15\!\cdots\!61\)\( T^{32} \)
$97$ \( 1 - 3 T - 107 T^{2} - 512 T^{3} + 11949 T^{4} + 162824 T^{5} - 68553 T^{6} - 17532563 T^{7} - 79037522 T^{8} + 475597250 T^{9} + 13552416748 T^{10} + 101483946140 T^{11} - 861406877856 T^{12} - 8094085378701 T^{13} - 80335748639044 T^{14} + 503130707568075 T^{15} + 13011656869749317 T^{16} + 48803678634103275 T^{17} - 755879058944764996 T^{18} - 7387253184835177773 T^{19} - 76259731545046501536 T^{20} + \)\(87\!\cdots\!80\)\( T^{21} + \)\(11\!\cdots\!92\)\( T^{22} + \)\(38\!\cdots\!50\)\( T^{23} - \)\(61\!\cdots\!42\)\( T^{24} - \)\(13\!\cdots\!71\)\( T^{25} - \)\(50\!\cdots\!97\)\( T^{26} + \)\(11\!\cdots\!72\)\( T^{27} + \)\(82\!\cdots\!09\)\( T^{28} - \)\(34\!\cdots\!24\)\( T^{29} - \)\(69\!\cdots\!83\)\( T^{30} - \)\(18\!\cdots\!79\)\( T^{31} + \)\(61\!\cdots\!21\)\( T^{32} \)
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