Properties

Label 144.3.m
Level $144$
Weight $3$
Character orbit 144.m
Rep. character $\chi_{144}(19,\cdot)$
Character field $\Q(\zeta_{4})$
Dimension $38$
Newform subspaces $3$
Sturm bound $72$
Trace bound $4$

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Defining parameters

Level: \( N \) \(=\) \( 144 = 2^{4} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 144.m (of order \(4\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 16 \)
Character field: \(\Q(i)\)
Newform subspaces: \( 3 \)
Sturm bound: \(72\)
Trace bound: \(4\)
Distinguishing \(T_p\): \(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{3}(144, [\chi])\).

Total New Old
Modular forms 104 42 62
Cusp forms 88 38 50
Eisenstein series 16 4 12

Trace form

\( 38q + 2q^{2} - 8q^{4} + 2q^{5} - 4q^{7} + 8q^{8} + O(q^{10}) \) \( 38q + 2q^{2} - 8q^{4} + 2q^{5} - 4q^{7} + 8q^{8} - 4q^{10} - 14q^{11} - 2q^{13} + 32q^{14} + 24q^{16} + 4q^{17} - 34q^{19} + 4q^{20} + 84q^{22} + 68q^{23} + 4q^{26} - 64q^{28} - 14q^{29} - 168q^{32} - 4q^{34} + 4q^{35} + 46q^{37} - 208q^{38} - 224q^{40} + 14q^{43} - 108q^{44} - 60q^{46} + 178q^{49} + 190q^{50} - 212q^{52} + 82q^{53} - 260q^{55} + 392q^{56} + 280q^{58} - 78q^{59} - 34q^{61} + 420q^{62} + 112q^{64} + 20q^{65} - 162q^{67} + 336q^{68} + 344q^{70} - 252q^{71} - 256q^{74} + 148q^{76} - 12q^{77} - 784q^{80} - 16q^{82} - 158q^{83} + 108q^{85} - 796q^{86} - 440q^{88} - 4q^{91} - 328q^{92} - 744q^{94} - 4q^{97} + 430q^{98} + O(q^{100}) \)

Decomposition of \(S_{3}^{\mathrm{new}}(144, [\chi])\) into newform subspaces

Label Dim. \(A\) Field CM Traces $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\)
144.3.m.a \(6\) \(3.924\) 6.0.399424.1 None \(2\) \(0\) \(2\) \(-4\) \(q+\beta _{3}q^{2}+(-1-\beta _{1}+\beta _{4}-\beta _{5})q^{4}+\cdots\)
144.3.m.b \(16\) \(3.924\) \(\mathbb{Q}[x]/(x^{16} - \cdots)\) None \(0\) \(0\) \(0\) \(0\) \(q-\beta _{4}q^{2}+(-1-\beta _{2})q^{4}+\beta _{8}q^{5}+(\beta _{11}+\cdots)q^{7}+\cdots\)
144.3.m.c \(16\) \(3.924\) \(\mathbb{Q}[x]/(x^{16} - \cdots)\) None \(0\) \(0\) \(0\) \(0\) \(q-\beta _{2}q^{2}+(1-\beta _{1})q^{4}+(-\beta _{1}+\beta _{3}+\cdots)q^{5}+\cdots\)

Decomposition of \(S_{3}^{\mathrm{old}}(144, [\chi])\) into lower level spaces

\( S_{3}^{\mathrm{old}}(144, [\chi]) \cong \) \(S_{3}^{\mathrm{new}}(16, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(48, [\chi])\)\(^{\oplus 2}\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ (\( 1 - 2 T + 6 T^{2} - 8 T^{3} + 24 T^{4} - 32 T^{5} + 64 T^{6} \))(\( 1 + 6 T^{2} + 10 T^{4} - 88 T^{6} - 752 T^{8} - 1408 T^{10} + 2560 T^{12} + 24576 T^{14} + 65536 T^{16} \))(\( 1 - 6 T^{2} - 4 T^{3} + 10 T^{4} + 56 T^{5} + 88 T^{6} - 128 T^{7} - 496 T^{8} - 512 T^{9} + 1408 T^{10} + 3584 T^{11} + 2560 T^{12} - 4096 T^{13} - 24576 T^{14} + 65536 T^{16} \))
$3$ 1
$5$ (\( 1 - 2 T + 2 T^{2} + 14 T^{3} - 369 T^{4} - 636 T^{5} + 2108 T^{6} - 15900 T^{7} - 230625 T^{8} + 218750 T^{9} + 781250 T^{10} - 19531250 T^{11} + 244140625 T^{12} \))(\( 1 - 344 T^{4} - 8484 T^{8} + 258572952 T^{12} - 94198183354 T^{16} + 101005059375000 T^{20} - 1294555664062500 T^{24} - 20503997802734375000 T^{28} + \)\(23\!\cdots\!25\)\( T^{32} \))(\( 1 - 32 T^{3} - 344 T^{4} - 5664 T^{5} + 512 T^{6} - 145600 T^{7} + 223452 T^{8} + 2255168 T^{9} + 20875776 T^{10} + 67282720 T^{11} + 753060504 T^{12} - 1881828576 T^{13} + 2740220928 T^{14} - 75471830656 T^{15} - 399298967994 T^{16} - 1886795766400 T^{17} + 1712638080000 T^{18} - 29403571500000 T^{19} + 294164259375000 T^{20} + 657057812500000 T^{21} + 5096625000000000 T^{22} + 13764453125000000 T^{23} + 34096069335937500 T^{24} - 555419921875000000 T^{25} + 48828125000000000 T^{26} - 13504028320312500000 T^{27} - 20503997802734375000 T^{28} - 47683715820312500000 T^{29} + \)\(23\!\cdots\!25\)\( T^{32} \))
$7$ (\( ( 1 + 2 T + 87 T^{2} + 332 T^{3} + 4263 T^{4} + 4802 T^{5} + 117649 T^{6} )^{2} \))(\( ( 1 + 84 T^{2} + 352 T^{3} + 3530 T^{4} + 17248 T^{5} + 201684 T^{6} + 5764801 T^{8} )^{4} \))(\( ( 1 + 168 T^{2} - 448 T^{3} + 15076 T^{4} - 56512 T^{5} + 1070392 T^{6} - 3649664 T^{7} + 60103046 T^{8} - 178833536 T^{9} + 2570011192 T^{10} - 6648580288 T^{11} + 86910139876 T^{12} - 126548911552 T^{13} + 2325336249768 T^{14} + 33232930569601 T^{16} )^{2} \))
$11$ (\( 1 - 18 T + 162 T^{2} - 2146 T^{3} + 17759 T^{4} - 65756 T^{5} + 609308 T^{6} - 7956476 T^{7} + 260009519 T^{8} - 3801769906 T^{9} + 34726138722 T^{10} - 466873642818 T^{11} + 3138428376721 T^{12} \))(\( 1 - 27832 T^{4} + 345241884 T^{8} - 1221720802824 T^{12} - 7510669490883642 T^{16} - \)\(26\!\cdots\!44\)\( T^{20} + \)\(15\!\cdots\!24\)\( T^{24} - \)\(27\!\cdots\!12\)\( T^{28} + \)\(21\!\cdots\!21\)\( T^{32} \))(\( 1 + 32 T + 512 T^{2} + 8480 T^{3} + 137032 T^{4} + 1636576 T^{5} + 18165248 T^{6} + 219655136 T^{7} + 2263228700 T^{8} + 21867108000 T^{9} + 253620152832 T^{10} + 2916953293728 T^{11} + 33797606438392 T^{12} + 431768458252384 T^{13} + 5293166227138048 T^{14} + 61910274521995104 T^{15} + 703855220885889990 T^{16} + 7491143217161407584 T^{17} + 77497246731528160768 T^{18} + \)\(76\!\cdots\!24\)\( T^{19} + \)\(72\!\cdots\!52\)\( T^{20} + \)\(75\!\cdots\!28\)\( T^{21} + \)\(79\!\cdots\!72\)\( T^{22} + \)\(83\!\cdots\!00\)\( T^{23} + \)\(10\!\cdots\!00\)\( T^{24} + \)\(12\!\cdots\!16\)\( T^{25} + \)\(12\!\cdots\!48\)\( T^{26} + \)\(13\!\cdots\!96\)\( T^{27} + \)\(13\!\cdots\!12\)\( T^{28} + \)\(10\!\cdots\!80\)\( T^{29} + \)\(73\!\cdots\!72\)\( T^{30} + \)\(55\!\cdots\!32\)\( T^{31} + \)\(21\!\cdots\!21\)\( T^{32} \))
$13$ (\( 1 + 2 T + 2 T^{2} + 1554 T^{3} - 7825 T^{4} - 453380 T^{5} + 316348 T^{6} - 76621220 T^{7} - 223489825 T^{8} + 7500861186 T^{9} + 1631461442 T^{10} + 275716983698 T^{11} + 23298085122481 T^{12} \))(\( ( 1 - 4352 T^{3} + 12636 T^{4} + 622336 T^{5} + 9469952 T^{6} - 45147648 T^{7} - 2135161210 T^{8} - 7629952512 T^{9} + 270471299072 T^{10} + 3003897005824 T^{11} + 10307573390556 T^{12} - 599960156526848 T^{13} + 665416609183179841 T^{16} )^{2} \))(\( 1 + 3200 T^{3} + 7608 T^{4} + 95360 T^{5} + 5120000 T^{6} + 68335872 T^{7} + 2004669468 T^{8} + 7270355200 T^{9} + 184268595200 T^{10} + 4889456013184 T^{11} + 5354592144136 T^{12} + 669839496880000 T^{13} + 7008632866619392 T^{14} + 70586941744778752 T^{15} + 2398056097119178950 T^{16} + 11929193154867609088 T^{17} + \)\(20\!\cdots\!12\)\( T^{18} + \)\(32\!\cdots\!00\)\( T^{19} + \)\(43\!\cdots\!56\)\( T^{20} + \)\(67\!\cdots\!16\)\( T^{21} + \)\(42\!\cdots\!00\)\( T^{22} + \)\(28\!\cdots\!00\)\( T^{23} + \)\(13\!\cdots\!88\)\( T^{24} + \)\(76\!\cdots\!88\)\( T^{25} + \)\(97\!\cdots\!00\)\( T^{26} + \)\(30\!\cdots\!40\)\( T^{27} + \)\(41\!\cdots\!88\)\( T^{28} + \)\(29\!\cdots\!00\)\( T^{29} + \)\(44\!\cdots\!81\)\( T^{32} \))
$17$ (\( ( 1 - 2 T + 607 T^{2} + 388 T^{3} + 175423 T^{4} - 167042 T^{5} + 24137569 T^{6} )^{2} \))(\( ( 1 + 1208 T^{2} + 806844 T^{4} + 368107656 T^{6} + 122826347526 T^{8} + 30744719536776 T^{10} + 5628348036726204 T^{12} + 703807662573551288 T^{14} + 48661191875666868481 T^{16} )^{2} \))(\( ( 1 + 968 T^{2} + 2944 T^{3} + 516540 T^{4} + 3209600 T^{5} + 201700088 T^{6} + 1543904000 T^{7} + 63894476806 T^{8} + 446188256000 T^{9} + 16846193049848 T^{10} + 77471941462400 T^{11} + 3603257748574140 T^{12} + 5935086042921856 T^{13} + 563978325638408648 T^{14} + 48661191875666868481 T^{16} )^{2} \))
$19$ (\( 1 - 30 T + 450 T^{2} - 12014 T^{3} + 441215 T^{4} - 8004292 T^{5} + 113750108 T^{6} - 2889549412 T^{7} + 57499580015 T^{8} - 565209214334 T^{9} + 7642603368450 T^{10} - 183931987734030 T^{11} + 2213314919066161 T^{12} \))(\( ( 1 + 16 T + 128 T^{2} - 7984 T^{3} - 156764 T^{4} + 603504 T^{5} + 61593984 T^{6} + 384035120 T^{7} + 7522568070 T^{8} + 138636678320 T^{9} + 8026989588864 T^{10} + 28392377367024 T^{11} - 2662411276559324 T^{12} - 48950433002283184 T^{13} + 283304309640468608 T^{14} + 12784106972526145936 T^{15} + \)\(28\!\cdots\!81\)\( T^{16} )^{2} \))(\( 1 + 32 T + 512 T^{2} - 2656 T^{3} - 523448 T^{4} - 8424608 T^{5} + 1945088 T^{6} + 4454446304 T^{7} + 107916937244 T^{8} - 703649376 T^{9} - 30601835632128 T^{10} - 698985761087712 T^{11} + 998616856187896 T^{12} + 253693358084547040 T^{13} + 3161998119961945600 T^{14} - 30474951661580761248 T^{15} - \)\(19\!\cdots\!42\)\( T^{16} - \)\(11\!\cdots\!28\)\( T^{17} + \)\(41\!\cdots\!00\)\( T^{18} + \)\(11\!\cdots\!40\)\( T^{19} + \)\(16\!\cdots\!36\)\( T^{20} - \)\(42\!\cdots\!12\)\( T^{21} - \)\(67\!\cdots\!08\)\( T^{22} - \)\(56\!\cdots\!96\)\( T^{23} + \)\(31\!\cdots\!64\)\( T^{24} + \)\(46\!\cdots\!64\)\( T^{25} + \)\(73\!\cdots\!88\)\( T^{26} - \)\(11\!\cdots\!88\)\( T^{27} - \)\(25\!\cdots\!08\)\( T^{28} - \)\(46\!\cdots\!36\)\( T^{29} + \)\(32\!\cdots\!92\)\( T^{30} + \)\(73\!\cdots\!32\)\( T^{31} + \)\(83\!\cdots\!61\)\( T^{32} \))
$23$ (\( ( 1 + 30 T + 1751 T^{2} + 30772 T^{3} + 926279 T^{4} + 8395230 T^{5} + 148035889 T^{6} )^{2} \))(\( ( 1 + 2600 T^{2} + 3157852 T^{4} + 2464614168 T^{6} + 1453801462342 T^{8} + 689700093387288 T^{10} + 247294501491576412 T^{12} + 56978023523252834600 T^{14} + \)\(61\!\cdots\!61\)\( T^{16} )^{2} \))(\( ( 1 - 64 T + 3496 T^{2} - 127936 T^{3} + 4410332 T^{4} - 130001728 T^{5} + 3673719192 T^{6} - 94049622208 T^{7} + 2261818535238 T^{8} - 49752250148032 T^{9} + 1028057252408472 T^{10} - 19244921376016192 T^{11} + 345377444336323292 T^{12} - 5299942138629398464 T^{13} + 76613527014343042216 T^{14} - \)\(74\!\cdots\!76\)\( T^{15} + \)\(61\!\cdots\!61\)\( T^{16} )^{2} \))
$29$ (\( 1 - 18 T + 162 T^{2} + 4894 T^{3} + 124463 T^{4} - 24625372 T^{5} + 435069308 T^{6} - 20709937852 T^{7} + 88030315103 T^{8} + 2911065332974 T^{9} + 81039918899682 T^{10} - 7572730199403618 T^{11} + 353814783205469041 T^{12} \))(\( 1 - 1812568 T^{4} + 855482597340 T^{8} + 905028609054258072 T^{12} - \)\(12\!\cdots\!34\)\( T^{16} + \)\(45\!\cdots\!92\)\( T^{20} + \)\(21\!\cdots\!40\)\( T^{24} - \)\(22\!\cdots\!08\)\( T^{28} + \)\(62\!\cdots\!41\)\( T^{32} \))(\( 1 + 32 T + 512 T^{2} - 18368 T^{3} - 1552984 T^{4} - 20596992 T^{5} + 304715776 T^{6} + 25469097376 T^{7} + 491466517980 T^{8} - 9791032230816 T^{9} - 347423504794624 T^{10} - 2649303176415616 T^{11} + 694517140133881240 T^{12} + 20658732330776531008 T^{13} + \)\(19\!\cdots\!28\)\( T^{14} - \)\(11\!\cdots\!40\)\( T^{15} - \)\(82\!\cdots\!10\)\( T^{16} - \)\(96\!\cdots\!40\)\( T^{17} + \)\(13\!\cdots\!68\)\( T^{18} + \)\(12\!\cdots\!68\)\( T^{19} + \)\(34\!\cdots\!40\)\( T^{20} - \)\(11\!\cdots\!16\)\( T^{21} - \)\(12\!\cdots\!84\)\( T^{22} - \)\(29\!\cdots\!96\)\( T^{23} + \)\(12\!\cdots\!80\)\( T^{24} + \)\(53\!\cdots\!36\)\( T^{25} + \)\(53\!\cdots\!76\)\( T^{26} - \)\(30\!\cdots\!72\)\( T^{27} - \)\(19\!\cdots\!04\)\( T^{28} - \)\(19\!\cdots\!28\)\( T^{29} + \)\(45\!\cdots\!32\)\( T^{30} + \)\(23\!\cdots\!32\)\( T^{31} + \)\(62\!\cdots\!41\)\( T^{32} \))
$31$ (\( 1 - 3846 T^{2} + 7131791 T^{4} - 8361808916 T^{6} + 6586358756111 T^{8} - 3280218929998086 T^{10} + 787662783788549761 T^{12} \))(\( ( 1 - 3656 T^{2} + 7444644 T^{4} - 10749322008 T^{6} + 11704860550022 T^{8} - 9927224610150168 T^{10} + 6349470144538916004 T^{12} - \)\(28\!\cdots\!16\)\( T^{14} + \)\(72\!\cdots\!81\)\( T^{16} )^{2} \))(\( 1 - 7312 T^{2} + 29025544 T^{4} - 80335806576 T^{6} + 171125889681052 T^{8} - 295006946315669072 T^{10} + \)\(42\!\cdots\!64\)\( T^{12} - \)\(51\!\cdots\!00\)\( T^{14} + \)\(53\!\cdots\!38\)\( T^{16} - \)\(47\!\cdots\!00\)\( T^{18} + \)\(36\!\cdots\!24\)\( T^{20} - \)\(23\!\cdots\!92\)\( T^{22} + \)\(12\!\cdots\!12\)\( T^{24} - \)\(53\!\cdots\!76\)\( T^{26} + \)\(18\!\cdots\!24\)\( T^{28} - \)\(41\!\cdots\!92\)\( T^{30} + \)\(52\!\cdots\!61\)\( T^{32} \))
$37$ (\( 1 - 46 T + 1058 T^{2} + 6594 T^{3} - 356337 T^{4} - 78343460 T^{5} + 4002544124 T^{6} - 107252196740 T^{7} - 667832908257 T^{8} + 16918399940946 T^{9} + 3716203262248418 T^{10} - 221194881131221054 T^{11} + 6582952005840035281 T^{12} \))(\( ( 1 - 48 T + 1152 T^{2} - 50960 T^{3} - 719780 T^{4} + 84286160 T^{5} - 1918088320 T^{6} + 86111407472 T^{7} - 3715717913082 T^{8} + 117886516829168 T^{9} - 3594806323899520 T^{10} + 216255226625199440 T^{11} - 2528212461343257380 T^{12} - \)\(24\!\cdots\!40\)\( T^{13} + \)\(75\!\cdots\!12\)\( T^{14} - \)\(43\!\cdots\!72\)\( T^{15} + \)\(12\!\cdots\!41\)\( T^{16} )^{2} \))(\( 1 + 96 T + 4608 T^{2} + 145952 T^{3} + 4040888 T^{4} + 217733344 T^{5} + 12932982272 T^{6} + 602883756192 T^{7} + 21839639792924 T^{8} + 655265530977504 T^{9} + 21703692469355008 T^{10} + 815191556064282016 T^{11} + 35433653736114978312 T^{12} + \)\(14\!\cdots\!40\)\( T^{13} + \)\(50\!\cdots\!52\)\( T^{14} + \)\(14\!\cdots\!84\)\( T^{15} + \)\(43\!\cdots\!90\)\( T^{16} + \)\(20\!\cdots\!96\)\( T^{17} + \)\(95\!\cdots\!72\)\( T^{18} + \)\(37\!\cdots\!60\)\( T^{19} + \)\(12\!\cdots\!52\)\( T^{20} + \)\(39\!\cdots\!84\)\( T^{21} + \)\(14\!\cdots\!48\)\( T^{22} + \)\(59\!\cdots\!56\)\( T^{23} + \)\(26\!\cdots\!84\)\( T^{24} + \)\(10\!\cdots\!68\)\( T^{25} + \)\(29\!\cdots\!72\)\( T^{26} + \)\(68\!\cdots\!36\)\( T^{27} + \)\(17\!\cdots\!68\)\( T^{28} + \)\(86\!\cdots\!68\)\( T^{29} + \)\(37\!\cdots\!68\)\( T^{30} + \)\(10\!\cdots\!04\)\( T^{31} + \)\(15\!\cdots\!81\)\( T^{32} \))
$41$ (\( 1 - 5094 T^{2} + 15050223 T^{4} - 31243096276 T^{6} + 42528333194703 T^{8} - 40675209117142374 T^{10} + 22563490300366186081 T^{12} \))(\( ( 1 - 6200 T^{2} + 18860092 T^{4} - 40121913352 T^{6} + 71431934605958 T^{8} - 113374937995460872 T^{10} + \)\(15\!\cdots\!32\)\( T^{12} - \)\(13\!\cdots\!00\)\( T^{14} + \)\(63\!\cdots\!41\)\( T^{16} )^{2} \))(\( 1 - 13840 T^{2} + 102706104 T^{4} - 524939980080 T^{6} + 2044068651261084 T^{8} - 6376104819902485008 T^{10} + \)\(16\!\cdots\!68\)\( T^{12} - \)\(35\!\cdots\!72\)\( T^{14} + \)\(64\!\cdots\!06\)\( T^{16} - \)\(99\!\cdots\!92\)\( T^{18} + \)\(13\!\cdots\!28\)\( T^{20} - \)\(14\!\cdots\!48\)\( T^{22} + \)\(13\!\cdots\!44\)\( T^{24} - \)\(94\!\cdots\!80\)\( T^{26} + \)\(52\!\cdots\!44\)\( T^{28} - \)\(19\!\cdots\!40\)\( T^{30} + \)\(40\!\cdots\!81\)\( T^{32} \))
$43$ (\( 1 + 114 T + 6498 T^{2} + 241730 T^{3} + 12357983 T^{4} + 838941724 T^{5} + 44553879452 T^{6} + 1551203247676 T^{7} + 42249484638383 T^{8} + 1528063089834770 T^{9} + 75949925403851298 T^{10} + 2463708983714404386 T^{11} + 39959630797262576401 T^{12} \))(\( ( 1 + 16 T + 128 T^{2} + 120528 T^{3} - 217180 T^{4} - 199455632 T^{5} + 4100008320 T^{6} - 63109389264 T^{7} - 21163084456442 T^{8} - 116689260749136 T^{9} + 14017112544424320 T^{10} - 1260831462039741968 T^{11} - 2538443336289385180 T^{12} + \)\(26\!\cdots\!72\)\( T^{13} + \)\(51\!\cdots\!28\)\( T^{14} + \)\(11\!\cdots\!84\)\( T^{15} + \)\(13\!\cdots\!01\)\( T^{16} )^{2} \))(\( 1 - 160 T + 12800 T^{2} - 978464 T^{3} + 71106632 T^{4} - 3813053664 T^{5} + 178619596288 T^{6} - 8719368905312 T^{7} + 336417491247900 T^{8} - 9339737479444512 T^{9} + 288453906337733120 T^{10} - 7137460469658328480 T^{11} - \)\(12\!\cdots\!76\)\( T^{12} + \)\(13\!\cdots\!84\)\( T^{13} - \)\(44\!\cdots\!20\)\( T^{14} + \)\(28\!\cdots\!76\)\( T^{15} - \)\(17\!\cdots\!30\)\( T^{16} + \)\(53\!\cdots\!24\)\( T^{17} - \)\(15\!\cdots\!20\)\( T^{18} + \)\(83\!\cdots\!16\)\( T^{19} - \)\(15\!\cdots\!76\)\( T^{20} - \)\(15\!\cdots\!20\)\( T^{21} + \)\(11\!\cdots\!20\)\( T^{22} - \)\(69\!\cdots\!88\)\( T^{23} + \)\(45\!\cdots\!00\)\( T^{24} - \)\(22\!\cdots\!88\)\( T^{25} + \)\(83\!\cdots\!88\)\( T^{26} - \)\(32\!\cdots\!36\)\( T^{27} + \)\(11\!\cdots\!32\)\( T^{28} - \)\(28\!\cdots\!36\)\( T^{29} + \)\(69\!\cdots\!00\)\( T^{30} - \)\(16\!\cdots\!40\)\( T^{31} + \)\(18\!\cdots\!01\)\( T^{32} \))
$47$ (\( 1 - 4678 T^{2} + 12462287 T^{4} - 24905944212 T^{6} + 60811985090447 T^{8} - 111389199003717958 T^{10} + \)\(11\!\cdots\!41\)\( T^{12} \))(\( ( 1 - 6312 T^{2} + 26526428 T^{4} - 83137920920 T^{6} + 212859158930374 T^{8} - 405686533092826520 T^{10} + \)\(63\!\cdots\!08\)\( T^{12} - \)\(73\!\cdots\!92\)\( T^{14} + \)\(56\!\cdots\!21\)\( T^{16} )^{2} \))(\( 1 - 24144 T^{2} + 280869112 T^{4} - 2097883923184 T^{6} + 11327375509374492 T^{8} - 47271044690493269328 T^{10} + \)\(15\!\cdots\!16\)\( T^{12} - \)\(44\!\cdots\!04\)\( T^{14} + \)\(10\!\cdots\!58\)\( T^{16} - \)\(21\!\cdots\!24\)\( T^{18} + \)\(37\!\cdots\!76\)\( T^{20} - \)\(54\!\cdots\!48\)\( T^{22} + \)\(64\!\cdots\!32\)\( T^{24} - \)\(58\!\cdots\!84\)\( T^{26} + \)\(37\!\cdots\!72\)\( T^{28} - \)\(15\!\cdots\!84\)\( T^{30} + \)\(32\!\cdots\!41\)\( T^{32} \))
$53$ (\( 1 + 78 T + 3042 T^{2} + 270110 T^{3} + 31648463 T^{4} + 1389102820 T^{5} + 48555101564 T^{6} + 3901989821380 T^{7} + 249721595980703 T^{8} + 5986815584554190 T^{9} + 189393978231360162 T^{10} + 13641222688510017822 T^{11} + \)\(49\!\cdots\!41\)\( T^{12} \))(\( 1 - 41116504 T^{4} + 844846897443036 T^{8} - \)\(11\!\cdots\!20\)\( T^{12} + \)\(10\!\cdots\!98\)\( T^{16} - \)\(69\!\cdots\!20\)\( T^{20} + \)\(32\!\cdots\!56\)\( T^{24} - \)\(99\!\cdots\!24\)\( T^{28} + \)\(15\!\cdots\!41\)\( T^{32} \))(\( 1 - 160 T + 12800 T^{2} - 602944 T^{3} + 3948712 T^{4} + 1481707264 T^{5} - 105845942272 T^{6} + 3791430241760 T^{7} + 34861972067036 T^{8} - 14471440004155872 T^{9} + 1098860393015073792 T^{10} - 54880211634179791488 T^{11} + \)\(13\!\cdots\!12\)\( T^{12} + \)\(17\!\cdots\!00\)\( T^{13} - \)\(18\!\cdots\!48\)\( T^{14} - \)\(16\!\cdots\!08\)\( T^{15} + \)\(51\!\cdots\!54\)\( T^{16} - \)\(46\!\cdots\!72\)\( T^{17} - \)\(14\!\cdots\!88\)\( T^{18} + \)\(38\!\cdots\!00\)\( T^{19} + \)\(84\!\cdots\!32\)\( T^{20} - \)\(95\!\cdots\!12\)\( T^{21} + \)\(53\!\cdots\!72\)\( T^{22} - \)\(19\!\cdots\!68\)\( T^{23} + \)\(13\!\cdots\!56\)\( T^{24} + \)\(41\!\cdots\!40\)\( T^{25} - \)\(32\!\cdots\!72\)\( T^{26} + \)\(12\!\cdots\!76\)\( T^{27} + \)\(95\!\cdots\!72\)\( T^{28} - \)\(40\!\cdots\!76\)\( T^{29} + \)\(24\!\cdots\!00\)\( T^{30} - \)\(85\!\cdots\!40\)\( T^{31} + \)\(15\!\cdots\!41\)\( T^{32} \))
$59$ (\( 1 + 206 T + 21218 T^{2} + 1942462 T^{3} + 171214239 T^{4} + 11916831972 T^{5} + 708622973852 T^{6} + 41482492094532 T^{7} + 2074664742303279 T^{8} + 81934083737364142 T^{9} + 3115448225088482978 T^{10} + \)\(10\!\cdots\!06\)\( T^{11} + \)\(17\!\cdots\!81\)\( T^{12} \))(\( 1 - 11621240 T^{4} - 131007530650980 T^{8} + \)\(57\!\cdots\!44\)\( T^{12} + \)\(25\!\cdots\!66\)\( T^{16} + \)\(84\!\cdots\!24\)\( T^{20} - \)\(28\!\cdots\!80\)\( T^{24} - \)\(36\!\cdots\!40\)\( T^{28} + \)\(46\!\cdots\!81\)\( T^{32} \))(\( 1 - 128 T + 8192 T^{2} - 1121408 T^{3} + 136226184 T^{4} - 9279937408 T^{5} + 700645040128 T^{6} - 71627082366848 T^{7} + 5234572115355804 T^{8} - 316007889653226112 T^{9} + 25502997282495045632 T^{10} - \)\(19\!\cdots\!80\)\( T^{11} + \)\(10\!\cdots\!40\)\( T^{12} - \)\(69\!\cdots\!16\)\( T^{13} + \)\(51\!\cdots\!56\)\( T^{14} - \)\(29\!\cdots\!24\)\( T^{15} + \)\(15\!\cdots\!38\)\( T^{16} - \)\(10\!\cdots\!44\)\( T^{17} + \)\(62\!\cdots\!16\)\( T^{18} - \)\(29\!\cdots\!56\)\( T^{19} + \)\(16\!\cdots\!40\)\( T^{20} - \)\(98\!\cdots\!80\)\( T^{21} + \)\(45\!\cdots\!92\)\( T^{22} - \)\(19\!\cdots\!32\)\( T^{23} + \)\(11\!\cdots\!64\)\( T^{24} - \)\(53\!\cdots\!08\)\( T^{25} + \)\(18\!\cdots\!28\)\( T^{26} - \)\(84\!\cdots\!48\)\( T^{27} + \)\(43\!\cdots\!24\)\( T^{28} - \)\(12\!\cdots\!28\)\( T^{29} + \)\(31\!\cdots\!32\)\( T^{30} - \)\(17\!\cdots\!28\)\( T^{31} + \)\(46\!\cdots\!81\)\( T^{32} \))
$61$ (\( 1 - 30 T + 450 T^{2} - 111694 T^{3} + 33268655 T^{4} - 582006980 T^{5} + 8727089468 T^{6} - 2165647972580 T^{7} + 460632507413855 T^{8} - 5754516693877534 T^{9} + 86268290848776450 T^{10} - 21400287349886478030 T^{11} + \)\(26\!\cdots\!21\)\( T^{12} \))(\( ( 1 + 16 T + 128 T^{2} + 6576 T^{3} - 8849956 T^{4} - 175483376 T^{5} - 1653317760 T^{6} - 420993646800 T^{7} - 32622960575354 T^{8} - 1566517359742800 T^{9} - 22891574827436160 T^{10} - 9040969225652122736 T^{11} - \)\(16\!\cdots\!36\)\( T^{12} + \)\(46\!\cdots\!76\)\( T^{13} + \)\(33\!\cdots\!88\)\( T^{14} + \)\(15\!\cdots\!56\)\( T^{15} + \)\(36\!\cdots\!61\)\( T^{16} )^{2} \))(\( 1 + 32 T + 512 T^{2} - 38048 T^{3} - 60439624 T^{4} - 1520787552 T^{5} - 16996289024 T^{6} + 2981900088544 T^{7} + 2018049968078364 T^{8} + 40394929489472928 T^{9} + 275088896591278592 T^{10} - \)\(10\!\cdots\!56\)\( T^{11} - \)\(45\!\cdots\!20\)\( T^{12} - \)\(71\!\cdots\!16\)\( T^{13} - \)\(18\!\cdots\!28\)\( T^{14} + \)\(23\!\cdots\!64\)\( T^{15} + \)\(72\!\cdots\!42\)\( T^{16} + \)\(88\!\cdots\!44\)\( T^{17} - \)\(26\!\cdots\!48\)\( T^{18} - \)\(36\!\cdots\!76\)\( T^{19} - \)\(86\!\cdots\!20\)\( T^{20} - \)\(75\!\cdots\!56\)\( T^{21} + \)\(73\!\cdots\!32\)\( T^{22} + \)\(39\!\cdots\!48\)\( T^{23} + \)\(74\!\cdots\!04\)\( T^{24} + \)\(40\!\cdots\!64\)\( T^{25} - \)\(86\!\cdots\!24\)\( T^{26} - \)\(28\!\cdots\!92\)\( T^{27} - \)\(42\!\cdots\!84\)\( T^{28} - \)\(99\!\cdots\!28\)\( T^{29} + \)\(49\!\cdots\!72\)\( T^{30} + \)\(11\!\cdots\!32\)\( T^{31} + \)\(13\!\cdots\!21\)\( T^{32} \))
$67$ (\( 1 + 226 T + 25538 T^{2} + 2083538 T^{3} + 120508479 T^{4} + 5203289532 T^{5} + 268963196252 T^{6} + 23357566709148 T^{7} + 2428380941854959 T^{8} + 188473476667633922 T^{9} + 10370156349441497858 T^{10} + \)\(41\!\cdots\!74\)\( T^{11} + \)\(81\!\cdots\!61\)\( T^{12} \))(\( ( 1 + 128 T + 8192 T^{2} + 693376 T^{3} + 50920004 T^{4} + 2251590016 T^{5} + 111451987968 T^{6} + 7277319415168 T^{7} + 468898776356806 T^{8} + 32667886854689152 T^{9} + 2245882495233712128 T^{10} + \)\(20\!\cdots\!04\)\( T^{11} + \)\(20\!\cdots\!64\)\( T^{12} + \)\(12\!\cdots\!24\)\( T^{13} + \)\(67\!\cdots\!12\)\( T^{14} + \)\(47\!\cdots\!12\)\( T^{15} + \)\(16\!\cdots\!81\)\( T^{16} )^{2} \))(\( 1 - 320 T + 51200 T^{2} - 6047552 T^{3} + 641735304 T^{4} - 64228593856 T^{5} + 5982745065472 T^{6} - 525110406070976 T^{7} + 43992629224199580 T^{8} - 3502096836597496384 T^{9} + \)\(26\!\cdots\!12\)\( T^{10} - \)\(19\!\cdots\!80\)\( T^{11} + \)\(14\!\cdots\!20\)\( T^{12} - \)\(10\!\cdots\!88\)\( T^{13} + \)\(71\!\cdots\!04\)\( T^{14} - \)\(48\!\cdots\!52\)\( T^{15} + \)\(32\!\cdots\!74\)\( T^{16} - \)\(21\!\cdots\!28\)\( T^{17} + \)\(14\!\cdots\!84\)\( T^{18} - \)\(93\!\cdots\!72\)\( T^{19} + \)\(58\!\cdots\!20\)\( T^{20} - \)\(36\!\cdots\!20\)\( T^{21} + \)\(21\!\cdots\!32\)\( T^{22} - \)\(12\!\cdots\!36\)\( T^{23} + \)\(72\!\cdots\!80\)\( T^{24} - \)\(38\!\cdots\!84\)\( T^{25} + \)\(19\!\cdots\!72\)\( T^{26} - \)\(95\!\cdots\!84\)\( T^{27} + \)\(42\!\cdots\!84\)\( T^{28} - \)\(18\!\cdots\!88\)\( T^{29} + \)\(69\!\cdots\!00\)\( T^{30} - \)\(19\!\cdots\!80\)\( T^{31} + \)\(27\!\cdots\!61\)\( T^{32} \))
$71$ (\( ( 1 - 130 T + 11575 T^{2} - 918796 T^{3} + 58349575 T^{4} - 3303518530 T^{5} + 128100283921 T^{6} )^{2} \))(\( ( 1 + 776 T^{2} + 40385436 T^{4} - 127559387592 T^{6} + 690614216465094 T^{8} - 3241498466043262152 T^{10} + \)\(26\!\cdots\!96\)\( T^{12} + \)\(12\!\cdots\!16\)\( T^{14} + \)\(41\!\cdots\!21\)\( T^{16} )^{2} \))(\( ( 1 + 256 T + 68104 T^{2} + 10692864 T^{3} + 1610923548 T^{4} + 179723087616 T^{5} + 18972832358712 T^{6} + 1588998739085056 T^{7} + 125568612540426694 T^{8} + 8010142643727767296 T^{9} + \)\(48\!\cdots\!72\)\( T^{10} + \)\(23\!\cdots\!36\)\( T^{11} + \)\(10\!\cdots\!28\)\( T^{12} + \)\(34\!\cdots\!64\)\( T^{13} + \)\(11\!\cdots\!64\)\( T^{14} + \)\(21\!\cdots\!36\)\( T^{15} + \)\(41\!\cdots\!21\)\( T^{16} )^{2} \))
$73$ (\( 1 - 13126 T^{2} + 103571951 T^{4} - 653716749588 T^{6} + 2941261225338191 T^{8} - 10585595166201707206 T^{10} + \)\(22\!\cdots\!21\)\( T^{12} \))(\( ( 1 - 22536 T^{2} + 300646172 T^{4} - 2600139782456 T^{6} + 16379358145749190 T^{8} - 73839396175873059896 T^{10} + \)\(24\!\cdots\!32\)\( T^{12} - \)\(51\!\cdots\!56\)\( T^{14} + \)\(65\!\cdots\!61\)\( T^{16} )^{2} \))(\( 1 - 42768 T^{2} + 946714744 T^{4} - 14391245893936 T^{6} + 167549428359087132 T^{8} - \)\(15\!\cdots\!24\)\( T^{10} + \)\(12\!\cdots\!76\)\( T^{12} - \)\(83\!\cdots\!96\)\( T^{14} + \)\(47\!\cdots\!22\)\( T^{16} - \)\(23\!\cdots\!36\)\( T^{18} + \)\(10\!\cdots\!56\)\( T^{20} - \)\(36\!\cdots\!04\)\( T^{22} + \)\(10\!\cdots\!52\)\( T^{24} - \)\(26\!\cdots\!36\)\( T^{26} + \)\(49\!\cdots\!04\)\( T^{28} - \)\(63\!\cdots\!08\)\( T^{30} + \)\(42\!\cdots\!21\)\( T^{32} \))
$79$ (\( 1 - 70 T^{2} + 84324175 T^{4} + 17226941804 T^{6} + 3284433446508175 T^{8} - 106197616693459270 T^{10} + \)\(59\!\cdots\!41\)\( T^{12} \))(\( ( 1 - 35304 T^{2} + 608675620 T^{4} - 6614894483704 T^{6} + 49280941921260934 T^{8} - \)\(25\!\cdots\!24\)\( T^{10} + \)\(92\!\cdots\!20\)\( T^{12} - \)\(20\!\cdots\!64\)\( T^{14} + \)\(23\!\cdots\!21\)\( T^{16} )^{2} \))(\( 1 - 62928 T^{2} + 1905826568 T^{4} - 37296559235888 T^{6} + 534425714020543644 T^{8} - \)\(60\!\cdots\!08\)\( T^{10} + \)\(55\!\cdots\!96\)\( T^{12} - \)\(43\!\cdots\!36\)\( T^{14} + \)\(29\!\cdots\!62\)\( T^{16} - \)\(16\!\cdots\!16\)\( T^{18} + \)\(84\!\cdots\!56\)\( T^{20} - \)\(35\!\cdots\!28\)\( T^{22} + \)\(12\!\cdots\!24\)\( T^{24} - \)\(33\!\cdots\!88\)\( T^{26} + \)\(66\!\cdots\!08\)\( T^{28} - \)\(85\!\cdots\!08\)\( T^{30} + \)\(52\!\cdots\!41\)\( T^{32} \))
$83$ (\( 1 + 318 T + 50562 T^{2} + 6712846 T^{3} + 819490815 T^{4} + 81203275140 T^{5} + 6918697616348 T^{6} + 559409362439460 T^{7} + 38891658154821615 T^{8} + 2194700377608598174 T^{9} + \)\(11\!\cdots\!42\)\( T^{10} + \)\(49\!\cdots\!82\)\( T^{11} + \)\(10\!\cdots\!61\)\( T^{12} \))(\( 1 - 47654840 T^{4} - 1328477750370788 T^{8} + \)\(67\!\cdots\!96\)\( T^{12} + \)\(24\!\cdots\!06\)\( T^{16} + \)\(15\!\cdots\!36\)\( T^{20} - \)\(67\!\cdots\!28\)\( T^{24} - \)\(54\!\cdots\!40\)\( T^{28} + \)\(25\!\cdots\!61\)\( T^{32} \))(\( 1 - 160 T + 12800 T^{2} - 895904 T^{3} + 107479624 T^{4} - 16432771168 T^{5} + 1654826188288 T^{6} - 174484645067104 T^{7} + 18280323695716892 T^{8} - 1483531366054758688 T^{9} + \)\(11\!\cdots\!96\)\( T^{10} - \)\(11\!\cdots\!36\)\( T^{11} + \)\(13\!\cdots\!00\)\( T^{12} - \)\(12\!\cdots\!44\)\( T^{13} + \)\(89\!\cdots\!76\)\( T^{14} - \)\(74\!\cdots\!76\)\( T^{15} + \)\(61\!\cdots\!66\)\( T^{16} - \)\(51\!\cdots\!64\)\( T^{17} + \)\(42\!\cdots\!96\)\( T^{18} - \)\(39\!\cdots\!36\)\( T^{19} + \)\(30\!\cdots\!00\)\( T^{20} - \)\(17\!\cdots\!64\)\( T^{21} + \)\(12\!\cdots\!56\)\( T^{22} - \)\(10\!\cdots\!52\)\( T^{23} + \)\(92\!\cdots\!52\)\( T^{24} - \)\(60\!\cdots\!36\)\( T^{25} + \)\(39\!\cdots\!88\)\( T^{26} - \)\(27\!\cdots\!52\)\( T^{27} + \)\(12\!\cdots\!04\)\( T^{28} - \)\(70\!\cdots\!76\)\( T^{29} + \)\(69\!\cdots\!00\)\( T^{30} - \)\(59\!\cdots\!40\)\( T^{31} + \)\(25\!\cdots\!61\)\( T^{32} \))
$89$ (\( 1 - 31238 T^{2} + 466178479 T^{4} - 4433595811988 T^{6} + 29249082478431439 T^{8} - \)\(12\!\cdots\!78\)\( T^{10} + \)\(24\!\cdots\!21\)\( T^{12} \))(\( ( 1 - 17608 T^{2} + 140263452 T^{4} - 1652317947000 T^{6} + 18439815418282950 T^{8} - \)\(10\!\cdots\!00\)\( T^{10} + \)\(55\!\cdots\!12\)\( T^{12} - \)\(43\!\cdots\!68\)\( T^{14} + \)\(15\!\cdots\!61\)\( T^{16} )^{2} \))(\( 1 - 81008 T^{2} + 3201135736 T^{4} - 82544801381712 T^{6} + 1567286911309649436 T^{8} - \)\(23\!\cdots\!04\)\( T^{10} + \)\(28\!\cdots\!72\)\( T^{12} - \)\(29\!\cdots\!36\)\( T^{14} + \)\(25\!\cdots\!10\)\( T^{16} - \)\(18\!\cdots\!76\)\( T^{18} + \)\(11\!\cdots\!32\)\( T^{20} - \)\(57\!\cdots\!84\)\( T^{22} + \)\(24\!\cdots\!96\)\( T^{24} - \)\(80\!\cdots\!12\)\( T^{26} + \)\(19\!\cdots\!76\)\( T^{28} - \)\(31\!\cdots\!48\)\( T^{30} + \)\(24\!\cdots\!21\)\( T^{32} \))
$97$ (\( ( 1 + 2 T + 10687 T^{2} + 557564 T^{3} + 100553983 T^{4} + 177058562 T^{5} + 832972004929 T^{6} )^{2} \))(\( ( 1 + 25252 T^{2} - 77824 T^{3} + 327999814 T^{4} - 732246016 T^{5} + 2235541403812 T^{6} + 7837433594376961 T^{8} )^{4} \))(\( ( 1 + 38216 T^{2} + 116224 T^{3} + 770481564 T^{4} + 3485408768 T^{5} + 10857255215864 T^{6} + 49274039499776 T^{7} + 116292098553803590 T^{8} + 463619437653392384 T^{9} + \)\(96\!\cdots\!84\)\( T^{10} + \)\(29\!\cdots\!72\)\( T^{11} + \)\(60\!\cdots\!04\)\( T^{12} + \)\(85\!\cdots\!76\)\( T^{13} + \)\(26\!\cdots\!56\)\( T^{14} + \)\(61\!\cdots\!21\)\( T^{16} )^{2} \))
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