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Properties

Label 31.2
Level 31
Weight 2
Dimension 26
Nonzero newspaces 4
Newform subspaces 4
Sturm bound 160
Trace bound 2

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Defining parameters

Level: \( N \) = \( 31 \)
Weight: \( k \) = \( 2 \)
Nonzero newspaces: \( 4 \)
Newform subspaces: \( 4 \)
Sturm bound: \(160\)
Trace bound: \(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_1(31))\).

Total New Old
Modular forms 55 55 0
Cusp forms 26 26 0
Eisenstein series 29 29 0

Trace form

\( 26q - 12q^{2} - 11q^{3} - 8q^{4} - 9q^{5} - 3q^{6} - 7q^{7} - 2q^{9} + O(q^{10}) \) \( 26q - 12q^{2} - 11q^{3} - 8q^{4} - 9q^{5} - 3q^{6} - 7q^{7} - 2q^{9} + 3q^{10} - 3q^{11} + 13q^{12} - q^{13} + 9q^{14} + 9q^{15} + 16q^{16} + 3q^{17} + 24q^{18} + 5q^{19} + 27q^{20} + 12q^{21} - 9q^{22} - 6q^{23} - 15q^{24} - 19q^{25} - 3q^{26} - 20q^{27} - 39q^{28} - 15q^{29} - 18q^{30} - 14q^{31} - 57q^{32} + 3q^{33} - 21q^{34} + 3q^{35} - 4q^{36} - 22q^{37} + 15q^{38} + 6q^{39} + 15q^{40} + 12q^{41} + 51q^{42} + 24q^{43} + 69q^{44} + 63q^{45} + 57q^{46} + 33q^{47} + 64q^{48} + 12q^{49} + 3q^{50} - 33q^{51} + 8q^{52} + 9q^{53} - 60q^{54} - 3q^{55} - 30q^{56} - 25q^{57} - 30q^{58} + 15q^{59} - 102q^{60} - 58q^{61} - 12q^{62} - 46q^{63} - 8q^{64} - 21q^{65} - 96q^{66} + 23q^{67} + 6q^{68} - 9q^{69} - 6q^{70} - 3q^{71} + 15q^{72} + 29q^{73} + 24q^{74} + 4q^{75} + 6q^{77} + 18q^{78} + 30q^{79} + 21q^{80} + 46q^{81} + 51q^{82} - 36q^{83} + 29q^{84} + 33q^{85} - 3q^{86} + 45q^{87} - 15q^{88} - 6q^{90} + 32q^{91} - 42q^{92} - 11q^{93} + 54q^{94} - 15q^{95} + 12q^{96} + 18q^{97} - 24q^{98} + 66q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_1(31))\)

We only show spaces with even parity, since no modular forms exist when this condition is not satisfied. Within each space \( S_k^{\mathrm{new}}(N, \chi) \) we list the newforms together with their dimension.

Label \(\chi\) Newforms Dimension \(\chi\) degree
31.2.a \(\chi_{31}(1, \cdot)\) 31.2.a.a 2 1
31.2.c \(\chi_{31}(5, \cdot)\) 31.2.c.a 4 2
31.2.d \(\chi_{31}(2, \cdot)\) 31.2.d.a 4 4
31.2.g \(\chi_{31}(7, \cdot)\) 31.2.g.a 16 8

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ (\( 1 - T + 3 T^{2} - 2 T^{3} + 4 T^{4} \))(\( ( 1 + 2 T + 3 T^{2} + 4 T^{3} + 4 T^{4} )^{2} \))(\( 1 + 3 T + 2 T^{2} + T^{4} + 8 T^{6} + 24 T^{7} + 16 T^{8} \))(\( 1 + 6 T + 21 T^{2} + 51 T^{3} + 98 T^{4} + 159 T^{5} + 231 T^{6} + 306 T^{7} + 369 T^{8} + 399 T^{9} + 378 T^{10} + 288 T^{11} + 70 T^{12} - 408 T^{13} - 1395 T^{14} - 2958 T^{15} - 4795 T^{16} - 5916 T^{17} - 5580 T^{18} - 3264 T^{19} + 1120 T^{20} + 9216 T^{21} + 24192 T^{22} + 51072 T^{23} + 94464 T^{24} + 156672 T^{25} + 236544 T^{26} + 325632 T^{27} + 401408 T^{28} + 417792 T^{29} + 344064 T^{30} + 196608 T^{31} + 65536 T^{32} \))
$3$ (\( 1 + 2 T + 2 T^{2} + 6 T^{3} + 9 T^{4} \))(\( 1 - 2 T - T^{2} + 2 T^{3} + 4 T^{4} + 6 T^{5} - 9 T^{6} - 54 T^{7} + 81 T^{8} \))(\( 1 - T - 2 T^{2} + 5 T^{3} + T^{4} + 15 T^{5} - 18 T^{6} - 27 T^{7} + 81 T^{8} \))(\( 1 + 12 T + 80 T^{2} + 375 T^{3} + 1362 T^{4} + 4038 T^{5} + 10105 T^{6} + 21936 T^{7} + 42530 T^{8} + 76515 T^{9} + 134002 T^{10} + 238260 T^{11} + 435636 T^{12} + 804669 T^{13} + 1467485 T^{14} + 2617221 T^{15} + 4573453 T^{16} + 7851663 T^{17} + 13207365 T^{18} + 21726063 T^{19} + 35286516 T^{20} + 57897180 T^{21} + 97687458 T^{22} + 167338305 T^{23} + 279039330 T^{24} + 431766288 T^{25} + 596690145 T^{26} + 715319586 T^{27} + 723822642 T^{28} + 597871125 T^{29} + 382637520 T^{30} + 172186884 T^{31} + 43046721 T^{32} \))
$5$ (\( ( 1 - T + 5 T^{2} )^{2} \))(\( ( 1 + T - 4 T^{2} + 5 T^{3} + 25 T^{4} )^{2} \))(\( ( 1 + 3 T + 11 T^{2} + 15 T^{3} + 25 T^{4} )^{2} \))(\( 1 + 3 T - 9 T^{2} - 18 T^{3} + 61 T^{4} - 21 T^{5} - 291 T^{6} + 591 T^{7} + 172 T^{8} - 4164 T^{9} + 1989 T^{10} + 14124 T^{11} - 21959 T^{12} - 41793 T^{13} + 231951 T^{14} + 136788 T^{15} - 1318469 T^{16} + 683940 T^{17} + 5798775 T^{18} - 5224125 T^{19} - 13724375 T^{20} + 44137500 T^{21} + 31078125 T^{22} - 325312500 T^{23} + 67187500 T^{24} + 1154296875 T^{25} - 2841796875 T^{26} - 1025390625 T^{27} + 14892578125 T^{28} - 21972656250 T^{29} - 54931640625 T^{30} + 91552734375 T^{31} + 152587890625 T^{32} \))
$7$ (\( 1 + 4 T + 13 T^{2} + 28 T^{3} + 49 T^{4} \))(\( 1 + 2 T - 9 T^{2} - 2 T^{3} + 92 T^{4} - 14 T^{5} - 441 T^{6} + 686 T^{7} + 2401 T^{8} \))(\( 1 + 3 T + 2 T^{2} - 15 T^{3} - 59 T^{4} - 105 T^{5} + 98 T^{6} + 1029 T^{7} + 2401 T^{8} \))(\( 1 - 2 T + 8 T^{2} - 57 T^{3} + 180 T^{4} - 447 T^{5} + 2037 T^{6} - 5539 T^{7} + 16183 T^{8} - 49082 T^{9} + 136106 T^{10} - 412273 T^{11} + 1249403 T^{12} - 2948414 T^{13} + 10225949 T^{14} - 27209176 T^{15} + 63276497 T^{16} - 190464232 T^{17} + 501071501 T^{18} - 1011306002 T^{19} + 2999816603 T^{20} - 6929072311 T^{21} + 16012734794 T^{22} - 40421137526 T^{23} + 93291774583 T^{24} - 223518629173 T^{25} + 575402082213 T^{26} - 883865054121 T^{27} + 2491431696180 T^{28} - 5522673593199 T^{29} + 5425784582792 T^{30} - 9495123019886 T^{31} + 33232930569601 T^{32} \))
$11$ (\( ( 1 - 2 T + 11 T^{2} )^{2} \))(\( 1 - 2 T - T^{2} + 34 T^{3} - 140 T^{4} + 374 T^{5} - 121 T^{6} - 2662 T^{7} + 14641 T^{8} \))(\( 1 + 2 T + 13 T^{2} + 34 T^{3} + 225 T^{4} + 374 T^{5} + 1573 T^{6} + 2662 T^{7} + 14641 T^{8} \))(\( 1 + 7 T + 35 T^{2} + 158 T^{3} + 743 T^{4} + 2658 T^{5} + 9655 T^{6} + 30605 T^{7} + 105459 T^{8} + 298602 T^{9} + 903884 T^{10} + 2105465 T^{11} + 6983649 T^{12} + 12627823 T^{13} + 32620956 T^{14} + 51556382 T^{15} + 388064517 T^{16} + 567120202 T^{17} + 3947135676 T^{18} + 16807632413 T^{19} + 102247605009 T^{20} + 339087243715 T^{21} + 1601285642924 T^{22} + 5818908234942 T^{23} + 22606073231379 T^{24} + 72164989083055 T^{25} + 250425834522655 T^{26} + 758358420484038 T^{27} + 2331852283903703 T^{28} + 5454588518741098 T^{29} + 13291244175413435 T^{30} + 29240737185909557 T^{31} + 45949729863572161 T^{32} \))
$13$ (\( 1 + 2 T + 22 T^{2} + 26 T^{3} + 169 T^{4} \))(\( 1 - 2 T - 15 T^{2} + 14 T^{3} + 140 T^{4} + 182 T^{5} - 2535 T^{6} - 4394 T^{7} + 28561 T^{8} \))(\( 1 - 6 T + 23 T^{2} - 120 T^{3} + 601 T^{4} - 1560 T^{5} + 3887 T^{6} - 13182 T^{7} + 28561 T^{8} \))(\( 1 + 7 T + 44 T^{2} + 249 T^{3} + 975 T^{4} + 3375 T^{5} + 9585 T^{6} + 9146 T^{7} - 38519 T^{8} - 357284 T^{9} - 1668070 T^{10} - 3915103 T^{11} - 4062397 T^{12} + 34368784 T^{13} + 322154357 T^{14} + 1609383854 T^{15} + 6221461841 T^{16} + 20921990102 T^{17} + 54444086333 T^{18} + 75508218448 T^{19} - 116026120717 T^{20} - 1453650338179 T^{21} - 8051455288630 T^{22} - 22419041147828 T^{23} - 31421131642199 T^{24} + 96988751265458 T^{25} + 1321373644372665 T^{26} + 6048541329874875 T^{27} + 22715632994418975 T^{28} + 75415901541470997 T^{29} + 173244560970768716 T^{30} + 358301251098635299 T^{31} + 665416609183179841 T^{32} \))
$17$ (\( 1 - 6 T + 38 T^{2} - 102 T^{3} + 289 T^{4} \))(\( 1 - 6 T + T^{2} - 6 T^{3} + 324 T^{4} - 102 T^{5} + 289 T^{6} - 29478 T^{7} + 83521 T^{8} \))(\( 1 + 3 T + 2 T^{2} + 75 T^{3} + 511 T^{4} + 1275 T^{5} + 578 T^{6} + 14739 T^{7} + 83521 T^{8} \))(\( 1 + 6 T + 75 T^{2} + 495 T^{3} + 3677 T^{4} + 23094 T^{5} + 144360 T^{6} + 817152 T^{7} + 4567650 T^{8} + 23850960 T^{9} + 122226918 T^{10} + 594432255 T^{11} + 2843619706 T^{12} + 12936362562 T^{13} + 57908087850 T^{14} + 248770132872 T^{15} + 1043483763143 T^{16} + 4229092258824 T^{17} + 16735437388650 T^{18} + 63556349267106 T^{19} + 237501961464826 T^{20} + 844008798287535 T^{21} + 2950260666882342 T^{22} + 9786971276176080 T^{23} + 31862818475383650 T^{24} + 96904320455276544 T^{25} + 291028879468817640 T^{26} + 791475173328476502 T^{27} + 2142301966293831197 T^{28} + 4902766126288438815 T^{29} + 12628336991955069675 T^{30} + 17174538309058894758 T^{31} + 48661191875666868481 T^{32} \))
$19$ (\( 1 + 33 T^{2} + 361 T^{4} \))(\( 1 + 6 T - 9 T^{2} + 42 T^{3} + 980 T^{4} + 798 T^{5} - 3249 T^{6} + 41154 T^{7} + 130321 T^{8} \))(\( 1 + 5 T + 6 T^{2} - 65 T^{3} - 439 T^{4} - 1235 T^{5} + 2166 T^{6} + 34295 T^{7} + 130321 T^{8} \))(\( 1 - 16 T + 195 T^{2} - 1746 T^{3} + 13353 T^{4} - 89969 T^{5} + 558125 T^{6} - 3294825 T^{7} + 18744924 T^{8} - 103602551 T^{9} + 554225766 T^{10} - 2858170600 T^{11} + 14222538534 T^{12} - 68363619099 T^{13} + 318565822489 T^{14} - 1445186621304 T^{15} + 6375621526007 T^{16} - 27458545804776 T^{17} + 115002261918529 T^{18} - 468906063400041 T^{19} + 1853495444289414 T^{20} - 7077113364489400 T^{21} + 26074039434369846 T^{22} - 92607392427206189 T^{23} + 318355598452753884 T^{24} - 1063199493834693675 T^{25} + 3421901355135183125 T^{26} - 10480512102813865211 T^{27} + 29554394114290447833 T^{28} - 73424509125100825014 T^{29} + \)\(15\!\cdots\!95\)\( T^{30} - \)\(24\!\cdots\!84\)\( T^{31} + \)\(28\!\cdots\!81\)\( T^{32} \))
$23$ (\( 1 + 2 T + 2 T^{2} + 46 T^{3} + 529 T^{4} \))(\( ( 1 + 4 T + 23 T^{2} )^{4} \))(\( 1 - 11 T + 38 T^{2} - 125 T^{3} + 821 T^{4} - 2875 T^{5} + 20102 T^{6} - 133837 T^{7} + 279841 T^{8} \))(\( 1 - T - 32 T^{2} + 265 T^{3} + 1220 T^{4} - 3468 T^{5} - 10927 T^{6} + 173296 T^{7} + 1397175 T^{8} - 394035 T^{9} - 17319068 T^{10} + 116114803 T^{11} + 1091799191 T^{12} + 1263133445 T^{13} - 14131047870 T^{14} + 39184802221 T^{15} + 769047681799 T^{16} + 901250451083 T^{17} - 7475324323230 T^{18} + 15368544625315 T^{19} + 305530177408631 T^{20} + 747354699485429 T^{21} - 2563843628031452 T^{22} - 1341620395008645 T^{23} + 109414150859981175 T^{24} + 312132551620892048 T^{25} - 452667488031542623 T^{26} - 3304344240445498836 T^{27} + 26735841807064791620 T^{28} + \)\(13\!\cdots\!95\)\( T^{29} - \)\(37\!\cdots\!88\)\( T^{30} - \)\(26\!\cdots\!07\)\( T^{31} + \)\(61\!\cdots\!61\)\( T^{32} \))
$29$ (\( 1 - 10 T + 78 T^{2} - 290 T^{3} + 841 T^{4} \))(\( ( 1 + 8 T + 66 T^{2} + 232 T^{3} + 841 T^{4} )^{2} \))(\( 1 - 5 T + 31 T^{2} - 115 T^{3} + 96 T^{4} - 3335 T^{5} + 26071 T^{6} - 121945 T^{7} + 707281 T^{8} \))(\( 1 + 14 T + 65 T^{2} + 271 T^{3} + 4124 T^{4} + 28761 T^{5} + 57004 T^{6} + 203557 T^{7} + 4414002 T^{8} + 20510481 T^{9} - 1562410 T^{10} + 162325096 T^{11} + 3747178047 T^{12} + 10722269693 T^{13} - 22040075979 T^{14} + 162043565485 T^{15} + 2719402913313 T^{16} + 4699263399065 T^{17} - 18535703898339 T^{18} + 261505435542577 T^{19} + 2650307836260207 T^{20} + 3329474230495304 T^{21} - 929357904963610 T^{22} + 353803260288094629 T^{23} + 2208088667302679922 T^{24} + 2953031113409966033 T^{25} + 23981995127044657804 T^{26} + \)\(35\!\cdots\!69\)\( T^{27} + \)\(14\!\cdots\!84\)\( T^{28} + \)\(27\!\cdots\!19\)\( T^{29} + \)\(19\!\cdots\!65\)\( T^{30} + \)\(12\!\cdots\!86\)\( T^{31} + \)\(25\!\cdots\!21\)\( T^{32} \))
$31$ (\( ( 1 - T )^{2} \))(\( ( 1 + 10 T + 31 T^{2} )^{2} \))(\( 1 + 11 T + 61 T^{2} + 341 T^{3} + 961 T^{4} \))(\( 1 - 15 T + 158 T^{2} - 1635 T^{3} + 13788 T^{4} - 99390 T^{5} + 688351 T^{6} - 4312200 T^{7} + 24371915 T^{8} - 133678200 T^{9} + 661505311 T^{10} - 2960927490 T^{11} + 12733507548 T^{12} - 46808661885 T^{13} + 140225581598 T^{14} - 412689211665 T^{15} + 852891037441 T^{16} \))
$37$ (\( ( 1 + 2 T + 37 T^{2} )^{2} \))(\( ( 1 - 10 T + 37 T^{2} )^{2}( 1 + 11 T + 37 T^{2} )^{2} \))(\( ( 1 + 4 T + 73 T^{2} + 148 T^{3} + 1369 T^{4} )^{2} \))(\( 1 + 8 T - 120 T^{2} - 534 T^{3} + 11739 T^{4} + 10042 T^{5} - 710515 T^{6} + 2077152 T^{7} + 30531852 T^{8} - 207297908 T^{9} - 600747795 T^{10} + 11697189185 T^{11} - 16371022929 T^{12} - 395056153140 T^{13} + 2176783134259 T^{14} + 6190914463491 T^{15} - 102798721686295 T^{16} + 229063835149167 T^{17} + 2980016110800571 T^{18} - 20010779325000420 T^{19} - 30681932703637569 T^{20} + 811129383865505045 T^{21} - 1541354482780018155 T^{22} - 19679179532183937764 T^{23} + \)\(10\!\cdots\!92\)\( T^{24} + \)\(26\!\cdots\!04\)\( T^{25} - \)\(34\!\cdots\!35\)\( T^{26} + \)\(17\!\cdots\!46\)\( T^{27} + \)\(77\!\cdots\!59\)\( T^{28} - \)\(13\!\cdots\!98\)\( T^{29} - \)\(10\!\cdots\!80\)\( T^{30} + \)\(26\!\cdots\!44\)\( T^{31} + \)\(12\!\cdots\!41\)\( T^{32} \))
$41$ (\( ( 1 - 7 T + 41 T^{2} )^{2} \))(\( 1 + 2 T - 7 T^{2} - 142 T^{3} - 1724 T^{4} - 5822 T^{5} - 11767 T^{6} + 137842 T^{7} + 2825761 T^{8} \))(\( 1 - 8 T + 23 T^{2} - 356 T^{3} + 3905 T^{4} - 14596 T^{5} + 38663 T^{6} - 551368 T^{7} + 2825761 T^{8} \))(\( 1 + 8 T + 106 T^{2} + 976 T^{3} + 7985 T^{4} + 55050 T^{5} + 396119 T^{6} + 1966432 T^{7} + 11020956 T^{8} + 52272537 T^{9} + 100609408 T^{10} + 252719935 T^{11} - 1900853239 T^{12} - 50449760350 T^{13} - 359337416133 T^{14} - 1943490188168 T^{15} - 20983245640805 T^{16} - 79683097714888 T^{17} - 604046196519573 T^{18} - 3477047933082350 T^{19} - 5371356949489879 T^{20} + 29279171586066935 T^{21} + 477905175625299328 T^{22} + 10180299987352706097 T^{23} + 88001509613432459676 T^{24} + \)\(64\!\cdots\!52\)\( T^{25} + \)\(53\!\cdots\!19\)\( T^{26} + \)\(30\!\cdots\!50\)\( T^{27} + \)\(18\!\cdots\!85\)\( T^{28} + \)\(90\!\cdots\!96\)\( T^{29} + \)\(40\!\cdots\!66\)\( T^{30} + \)\(12\!\cdots\!08\)\( T^{31} + \)\(63\!\cdots\!41\)\( T^{32} \))
$43$ (\( 1 + 2 T + 82 T^{2} + 86 T^{3} + 1849 T^{4} \))(\( 1 - 2 T + 15 T^{2} + 194 T^{3} - 1900 T^{4} + 8342 T^{5} + 27735 T^{6} - 159014 T^{7} + 3418801 T^{8} \))(\( 1 - T - 27 T^{2} + 235 T^{3} + 1196 T^{4} + 10105 T^{5} - 49923 T^{6} - 79507 T^{7} + 3418801 T^{8} \))(\( 1 - 23 T + 289 T^{2} - 3096 T^{3} + 26895 T^{4} - 183450 T^{5} + 1103565 T^{6} - 5143929 T^{7} + 16278591 T^{8} - 38987724 T^{9} - 224772150 T^{10} + 3232536297 T^{11} - 26862162747 T^{12} + 323838946599 T^{13} - 2730755315628 T^{14} + 21633024576604 T^{15} - 167060950817939 T^{16} + 930220056793972 T^{17} - 5049166578596172 T^{18} + 25747463127246693 T^{19} - 91836388861606347 T^{20} + 475210127962955571 T^{21} - 1420866363454285350 T^{22} - 10597588987903050468 T^{23} + \)\(19\!\cdots\!91\)\( T^{24} - \)\(25\!\cdots\!47\)\( T^{25} + \)\(23\!\cdots\!85\)\( T^{26} - \)\(17\!\cdots\!50\)\( T^{27} + \)\(10\!\cdots\!95\)\( T^{28} - \)\(53\!\cdots\!28\)\( T^{29} + \)\(21\!\cdots\!61\)\( T^{30} - \)\(73\!\cdots\!61\)\( T^{31} + \)\(13\!\cdots\!01\)\( T^{32} \))
$47$ (\( 1 + 4 T + 78 T^{2} + 188 T^{3} + 2209 T^{4} \))(\( ( 1 - 8 T + 78 T^{2} - 376 T^{3} + 2209 T^{4} )^{2} \))(\( 1 - 7 T - 23 T^{2} + 385 T^{3} - 1284 T^{4} + 18095 T^{5} - 50807 T^{6} - 726761 T^{7} + 4879681 T^{8} \))(\( 1 - 14 T - 64 T^{2} + 2051 T^{3} - 6367 T^{4} - 96576 T^{5} + 874231 T^{6} - 748939 T^{7} - 39432966 T^{8} + 323481909 T^{9} + 441998288 T^{10} - 20651272567 T^{11} + 54844789185 T^{12} + 746473355167 T^{13} - 4553042500275 T^{14} - 12229830802363 T^{15} + 232672311744285 T^{16} - 574802047711061 T^{17} - 10057670883107475 T^{18} + 77501103153503441 T^{19} + 267625075735049985 T^{20} - 4736266251437522969 T^{21} + 4764394721401356752 T^{22} + \)\(16\!\cdots\!67\)\( T^{23} - \)\(93\!\cdots\!26\)\( T^{24} - \)\(83\!\cdots\!13\)\( T^{25} + \)\(45\!\cdots\!19\)\( T^{26} - \)\(23\!\cdots\!28\)\( T^{27} - \)\(73\!\cdots\!47\)\( T^{28} + \)\(11\!\cdots\!77\)\( T^{29} - \)\(16\!\cdots\!16\)\( T^{30} - \)\(16\!\cdots\!02\)\( T^{31} + \)\(56\!\cdots\!21\)\( T^{32} \))
$53$ (\( 1 + 12 T + 122 T^{2} + 636 T^{3} + 2809 T^{4} \))(\( 1 + 6 T - 71 T^{2} + 6 T^{3} + 6732 T^{4} + 318 T^{5} - 199439 T^{6} + 893262 T^{7} + 7890481 T^{8} \))(\( 1 - 21 T + 118 T^{2} + 555 T^{3} - 9989 T^{4} + 29415 T^{5} + 331462 T^{6} - 3126417 T^{7} + 7890481 T^{8} \))(\( 1 - 6 T + 37 T^{2} + 126 T^{3} + 1690 T^{4} - 1806 T^{5} - 120187 T^{6} + 3508437 T^{7} - 15460642 T^{8} + 56872014 T^{9} + 164905174 T^{10} + 7036349469 T^{11} - 9006554212 T^{12} - 185117474262 T^{13} + 4685233247216 T^{14} - 8469368678412 T^{15} + 48444353804527 T^{16} - 448876539955836 T^{17} + 13160820191429744 T^{18} - 27559734215703774 T^{19} - 71066044885255972 T^{20} + 2942569635108743217 T^{21} + 3655017828576581446 T^{22} + 66808188390765821718 T^{23} - \)\(96\!\cdots\!62\)\( T^{24} + \)\(11\!\cdots\!21\)\( T^{25} - \)\(21\!\cdots\!63\)\( T^{26} - \)\(16\!\cdots\!82\)\( T^{27} + \)\(83\!\cdots\!90\)\( T^{28} + \)\(32\!\cdots\!98\)\( T^{29} + \)\(51\!\cdots\!53\)\( T^{30} - \)\(43\!\cdots\!42\)\( T^{31} + \)\(38\!\cdots\!21\)\( T^{32} \))
$59$ (\( 1 + 113 T^{2} + 3481 T^{4} \))(\( 1 - 6 T - 41 T^{2} + 246 T^{3} + 324 T^{4} + 14514 T^{5} - 142721 T^{6} - 1232274 T^{7} + 12117361 T^{8} \))(\( 1 - 5 T + 26 T^{2} - 515 T^{3} + 6161 T^{4} - 30385 T^{5} + 90506 T^{6} - 1026895 T^{7} + 12117361 T^{8} \))(\( 1 - 4 T + 157 T^{2} - 458 T^{3} + 14849 T^{4} - 42150 T^{5} + 924686 T^{6} - 2509544 T^{7} + 45834753 T^{8} - 195555639 T^{9} + 1870878718 T^{10} - 18828370418 T^{11} + 109287552140 T^{12} - 1909212618757 T^{13} + 8459795171586 T^{14} - 146067983115188 T^{15} + 587919268827301 T^{16} - 8618011003796092 T^{17} + 29448546992290866 T^{18} - 392112178427693903 T^{19} + 1324276722086702540 T^{20} - 13460859522400986982 T^{21} + 78914662702829952238 T^{22} - \)\(48\!\cdots\!41\)\( T^{23} + \)\(67\!\cdots\!13\)\( T^{24} - \)\(21\!\cdots\!16\)\( T^{25} + \)\(47\!\cdots\!86\)\( T^{26} - \)\(12\!\cdots\!50\)\( T^{27} + \)\(26\!\cdots\!69\)\( T^{28} - \)\(48\!\cdots\!82\)\( T^{29} + \)\(97\!\cdots\!77\)\( T^{30} - \)\(14\!\cdots\!96\)\( T^{31} + \)\(21\!\cdots\!41\)\( T^{32} \))
$61$ (\( 1 + 6 T + 6 T^{2} + 366 T^{3} + 3721 T^{4} \))(\( ( 1 + 114 T^{2} + 3721 T^{4} )^{2} \))(\( ( 1 - 4 T + 46 T^{2} - 244 T^{3} + 3721 T^{4} )^{2} \))(\( ( 1 + 30 T + 776 T^{2} + 13665 T^{3} + 208005 T^{4} + 2593710 T^{5} + 28390399 T^{6} + 268109040 T^{7} + 2240276459 T^{8} + 16354651440 T^{9} + 105640674679 T^{10} + 588722889510 T^{11} + 2880004157205 T^{12} + 11541408453165 T^{13} + 39979810504136 T^{14} + 94282285080630 T^{15} + 191707312997281 T^{16} )^{2} \))
$67$ (\( ( 1 - 8 T + 67 T^{2} )^{2} \))(\( 1 - 2 T - 113 T^{2} + 34 T^{3} + 8932 T^{4} + 2278 T^{5} - 507257 T^{6} - 601526 T^{7} + 20151121 T^{8} \))(\( ( 1 + 4 T + 133 T^{2} + 268 T^{3} + 4489 T^{4} )^{2} \))(\( 1 - 13 T - 258 T^{2} + 4011 T^{3} + 35028 T^{4} - 644333 T^{5} - 3445609 T^{6} + 71925183 T^{7} + 280896507 T^{8} - 6225465362 T^{9} - 19715222826 T^{10} + 429031194656 T^{11} + 1171728043770 T^{12} - 21978520425855 T^{13} - 61484782388300 T^{14} + 555554083027185 T^{15} + 3540421883673161 T^{16} + 37222123562821395 T^{17} - 276005188141078700 T^{18} - 6610325738841427365 T^{19} + 23611633589102566170 T^{20} + \)\(57\!\cdots\!92\)\( T^{21} - \)\(17\!\cdots\!94\)\( T^{22} - \)\(37\!\cdots\!26\)\( T^{23} + \)\(11\!\cdots\!87\)\( T^{24} + \)\(19\!\cdots\!01\)\( T^{25} - \)\(62\!\cdots\!41\)\( T^{26} - \)\(78\!\cdots\!39\)\( T^{27} + \)\(28\!\cdots\!08\)\( T^{28} + \)\(21\!\cdots\!57\)\( T^{29} - \)\(94\!\cdots\!82\)\( T^{30} - \)\(31\!\cdots\!59\)\( T^{31} + \)\(16\!\cdots\!81\)\( T^{32} \))
$71$ (\( 1 - 4 T + 21 T^{2} - 284 T^{3} + 5041 T^{4} \))(\( 1 - 14 T + 55 T^{2} + 14 T^{3} + 924 T^{4} + 994 T^{5} + 277255 T^{6} - 5010754 T^{7} + 25411681 T^{8} \))(\( 1 + 7 T + 53 T^{2} + 799 T^{3} + 10580 T^{4} + 56729 T^{5} + 267173 T^{6} + 2505377 T^{7} + 25411681 T^{8} \))(\( 1 + 14 T + 196 T^{2} + 2695 T^{3} + 17090 T^{4} + 118017 T^{5} + 577928 T^{6} - 4343303 T^{7} - 27945240 T^{8} - 279502470 T^{9} - 2826368429 T^{10} + 131657464 T^{11} - 99138557149 T^{12} - 2207242010815 T^{13} - 23038847142600 T^{14} - 294198893013362 T^{15} - 2969589635047583 T^{16} - 20888121403948702 T^{17} - 116138828445846600 T^{18} - 789996195332807465 T^{19} - 2519277389070657469 T^{20} + 237540260827025864 T^{21} - \)\(36\!\cdots\!09\)\( T^{22} - \)\(25\!\cdots\!70\)\( T^{23} - \)\(18\!\cdots\!40\)\( T^{24} - \)\(19\!\cdots\!93\)\( T^{25} + \)\(18\!\cdots\!28\)\( T^{26} + \)\(27\!\cdots\!07\)\( T^{27} + \)\(28\!\cdots\!90\)\( T^{28} + \)\(31\!\cdots\!45\)\( T^{29} + \)\(16\!\cdots\!76\)\( T^{30} + \)\(82\!\cdots\!14\)\( T^{31} + \)\(41\!\cdots\!21\)\( T^{32} \))
$73$ (\( 1 - 8 T + 142 T^{2} - 584 T^{3} + 5329 T^{4} \))(\( 1 + 2 T - 135 T^{2} - 14 T^{3} + 13700 T^{4} - 1022 T^{5} - 719415 T^{6} + 778034 T^{7} + 28398241 T^{8} \))(\( 1 - 21 T + 233 T^{2} - 2595 T^{3} + 26956 T^{4} - 189435 T^{5} + 1241657 T^{6} - 8169357 T^{7} + 28398241 T^{8} \))(\( 1 - 2 T + 14 T^{2} + 1041 T^{3} - 15525 T^{4} + 39720 T^{5} + 424590 T^{6} - 12956116 T^{7} + 131189071 T^{8} - 324722351 T^{9} - 2585580235 T^{10} + 84857314523 T^{11} - 864267778837 T^{12} + 4587359474806 T^{13} + 4622090434727 T^{14} - 447090776699989 T^{15} + 5618971555814261 T^{16} - 32637626699099197 T^{17} + 24631119926660183 T^{18} + 1784560820810605702 T^{19} - 24543684671947825717 T^{20} + \)\(17\!\cdots\!39\)\( T^{21} - \)\(39\!\cdots\!15\)\( T^{22} - \)\(35\!\cdots\!47\)\( T^{23} + \)\(10\!\cdots\!51\)\( T^{24} - \)\(76\!\cdots\!08\)\( T^{25} + \)\(18\!\cdots\!10\)\( T^{26} + \)\(12\!\cdots\!40\)\( T^{27} - \)\(35\!\cdots\!25\)\( T^{28} + \)\(17\!\cdots\!53\)\( T^{29} + \)\(17\!\cdots\!26\)\( T^{30} - \)\(17\!\cdots\!14\)\( T^{31} + \)\(65\!\cdots\!61\)\( T^{32} \))
$79$ (\( 1 + 10 T + 138 T^{2} + 790 T^{3} + 6241 T^{4} \))(\( 1 - 22 T + 223 T^{2} - 2266 T^{3} + 23644 T^{4} - 179014 T^{5} + 1391743 T^{6} - 10846858 T^{7} + 38950081 T^{8} \))(\( 1 - 79 T^{2} + 6241 T^{4} - 493039 T^{6} + 38950081 T^{8} \))(\( 1 - 18 T + 226 T^{2} - 2498 T^{3} + 40839 T^{4} - 469468 T^{5} + 5768280 T^{6} - 65435894 T^{7} + 780288208 T^{8} - 7561516414 T^{9} + 85206384568 T^{10} - 852745741894 T^{11} + 8569871201298 T^{12} - 77514808977636 T^{13} + 796329583287446 T^{14} - 7002575994516930 T^{15} + 64612388642450549 T^{16} - 553203503566837470 T^{17} + 4969892929296950486 T^{18} - 38217823903524675804 T^{19} + \)\(33\!\cdots\!38\)\( T^{20} - \)\(26\!\cdots\!06\)\( T^{21} + \)\(20\!\cdots\!28\)\( T^{22} - \)\(14\!\cdots\!26\)\( T^{23} + \)\(11\!\cdots\!88\)\( T^{24} - \)\(78\!\cdots\!86\)\( T^{25} + \)\(54\!\cdots\!80\)\( T^{26} - \)\(35\!\cdots\!72\)\( T^{27} + \)\(24\!\cdots\!99\)\( T^{28} - \)\(11\!\cdots\!22\)\( T^{29} + \)\(83\!\cdots\!06\)\( T^{30} - \)\(52\!\cdots\!82\)\( T^{31} + \)\(23\!\cdots\!21\)\( T^{32} \))
$83$ (\( 1 + 12 T + 122 T^{2} + 996 T^{3} + 6889 T^{4} \))(\( 1 - 6 T - 89 T^{2} + 246 T^{3} + 5748 T^{4} + 20418 T^{5} - 613121 T^{6} - 3430722 T^{7} + 47458321 T^{8} \))(\( 1 + 14 T + 13 T^{2} + 70 T^{3} + 6651 T^{4} + 5810 T^{5} + 89557 T^{6} + 8005018 T^{7} + 47458321 T^{8} \))(\( 1 + 16 T + 185 T^{2} + 3773 T^{3} + 53729 T^{4} + 585909 T^{5} + 6686260 T^{6} + 69214526 T^{7} + 675636522 T^{8} + 5645389119 T^{9} + 32133713345 T^{10} + 172336220420 T^{11} + 356263146291 T^{12} - 19484713577420 T^{13} - 322899642624249 T^{14} - 3433895006273167 T^{15} - 33440240762794785 T^{16} - 285013285520672861 T^{17} - 2224455638038451361 T^{18} - 11141105922292249540 T^{19} + 16907650757148237411 T^{20} + \)\(67\!\cdots\!60\)\( T^{21} + \)\(10\!\cdots\!05\)\( T^{22} + \)\(15\!\cdots\!13\)\( T^{23} + \)\(15\!\cdots\!02\)\( T^{24} + \)\(12\!\cdots\!78\)\( T^{25} + \)\(10\!\cdots\!40\)\( T^{26} + \)\(75\!\cdots\!03\)\( T^{27} + \)\(57\!\cdots\!69\)\( T^{28} + \)\(33\!\cdots\!99\)\( T^{29} + \)\(13\!\cdots\!65\)\( T^{30} + \)\(97\!\cdots\!12\)\( T^{31} + \)\(50\!\cdots\!81\)\( T^{32} \))
$89$ (\( 1 - 10 T + 158 T^{2} - 890 T^{3} + 7921 T^{4} \))(\( ( 1 + 8 T + 122 T^{2} + 712 T^{3} + 7921 T^{4} )^{2} \))(\( 1 - 5 T - 29 T^{2} + 785 T^{3} - 624 T^{4} + 69865 T^{5} - 229709 T^{6} - 3524845 T^{7} + 62742241 T^{8} \))(\( 1 - T - 247 T^{2} + 1489 T^{3} + 31955 T^{4} - 480540 T^{5} - 2146982 T^{6} + 82322077 T^{7} - 115178013 T^{8} - 9374121609 T^{9} + 52814885666 T^{10} + 761747156077 T^{11} - 8384072473029 T^{12} - 44513145443047 T^{13} + 944971959403383 T^{14} + 1347434269510390 T^{15} - 88986302178103107 T^{16} + 119921649986424710 T^{17} + 7485122890434196743 T^{18} - 31380387629839400543 T^{19} - \)\(52\!\cdots\!89\)\( T^{20} + \)\(42\!\cdots\!73\)\( T^{21} + \)\(26\!\cdots\!26\)\( T^{22} - \)\(41\!\cdots\!61\)\( T^{23} - \)\(45\!\cdots\!53\)\( T^{24} + \)\(28\!\cdots\!93\)\( T^{25} - \)\(66\!\cdots\!82\)\( T^{26} - \)\(13\!\cdots\!60\)\( T^{27} + \)\(78\!\cdots\!55\)\( T^{28} + \)\(32\!\cdots\!41\)\( T^{29} - \)\(48\!\cdots\!27\)\( T^{30} - \)\(17\!\cdots\!49\)\( T^{31} + \)\(15\!\cdots\!61\)\( T^{32} \))
$97$ (\( 1 + 14 T + 163 T^{2} + 1358 T^{3} + 9409 T^{4} \))(\( ( 1 - 16 T + 250 T^{2} - 1552 T^{3} + 9409 T^{4} )^{2} \))(\( 1 + 3 T + 182 T^{2} - 345 T^{3} + 16591 T^{4} - 33465 T^{5} + 1712438 T^{6} + 2738019 T^{7} + 88529281 T^{8} \))(\( 1 - 3 T - 107 T^{2} - 512 T^{3} + 11949 T^{4} + 162824 T^{5} - 68553 T^{6} - 17532563 T^{7} - 79037522 T^{8} + 475597250 T^{9} + 13552416748 T^{10} + 101483946140 T^{11} - 861406877856 T^{12} - 8094085378701 T^{13} - 80335748639044 T^{14} + 503130707568075 T^{15} + 13011656869749317 T^{16} + 48803678634103275 T^{17} - 755879058944764996 T^{18} - 7387253184835177773 T^{19} - 76259731545046501536 T^{20} + \)\(87\!\cdots\!80\)\( T^{21} + \)\(11\!\cdots\!92\)\( T^{22} + \)\(38\!\cdots\!50\)\( T^{23} - \)\(61\!\cdots\!42\)\( T^{24} - \)\(13\!\cdots\!71\)\( T^{25} - \)\(50\!\cdots\!97\)\( T^{26} + \)\(11\!\cdots\!72\)\( T^{27} + \)\(82\!\cdots\!09\)\( T^{28} - \)\(34\!\cdots\!24\)\( T^{29} - \)\(69\!\cdots\!83\)\( T^{30} - \)\(18\!\cdots\!79\)\( T^{31} + \)\(61\!\cdots\!21\)\( T^{32} \))
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