Properties

Label 210.3.o
Level 210
Weight 3
Character orbit o
Rep. character \(\chi_{210}(31,\cdot)\)
Character field \(\Q(\zeta_{6})\)
Dimension 24
Newform subspaces 2
Sturm bound 144
Trace bound 1

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Defining parameters

Level: \( N \) \(=\) \( 210 = 2 \cdot 3 \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 210.o (of order \(6\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 7 \)
Character field: \(\Q(\zeta_{6})\)
Newform subspaces: \( 2 \)
Sturm bound: \(144\)
Trace bound: \(1\)
Distinguishing \(T_p\): \(11\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{3}(210, [\chi])\).

Total New Old
Modular forms 208 24 184
Cusp forms 176 24 152
Eisenstein series 32 0 32

Trace form

\( 24q - 12q^{3} - 24q^{4} + 4q^{7} + 36q^{9} + O(q^{10}) \) \( 24q - 12q^{3} - 24q^{4} + 4q^{7} + 36q^{9} - 8q^{11} + 24q^{12} - 32q^{14} - 48q^{16} + 96q^{17} + 36q^{19} - 24q^{21} - 96q^{22} + 60q^{25} - 96q^{26} + 32q^{28} + 144q^{29} - 12q^{31} + 80q^{35} - 144q^{36} - 52q^{37} - 240q^{38} + 60q^{39} - 96q^{42} - 168q^{43} - 16q^{44} + 16q^{46} - 48q^{47} + 116q^{49} + 72q^{51} - 24q^{52} + 64q^{53} + 32q^{56} + 360q^{57} + 16q^{58} + 264q^{59} + 192q^{61} + 60q^{63} + 192q^{64} + 40q^{65} - 284q^{67} - 192q^{68} - 128q^{71} - 324q^{73} - 128q^{74} - 60q^{75} - 120q^{77} - 192q^{78} + 292q^{79} - 108q^{81} + 288q^{82} - 24q^{84} + 240q^{85} - 128q^{86} + 96q^{88} + 192q^{89} - 580q^{91} - 252q^{93} + 480q^{94} + 160q^{95} + 384q^{98} - 48q^{99} + O(q^{100}) \)

Decomposition of \(S_{3}^{\mathrm{new}}(210, [\chi])\) into newform subspaces

Label Dim. \(A\) Field CM Traces $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\)
210.3.o.a \(8\) \(5.722\) 8.0.3317760000.3 None \(0\) \(12\) \(0\) \(0\) \(q+(\beta _{2}+\beta _{4})q^{2}+(1+\beta _{3})q^{3}+(-2+2\beta _{3}+\cdots)q^{4}+\cdots\)
210.3.o.b \(16\) \(5.722\) \(\mathbb{Q}[x]/(x^{16} + \cdots)\) None \(0\) \(-24\) \(0\) \(4\) \(q+\beta _{9}q^{2}+(-1+\beta _{5})q^{3}+(-2-2\beta _{5}+\cdots)q^{4}+\cdots\)

Decomposition of \(S_{3}^{\mathrm{old}}(210, [\chi])\) into lower level spaces

\( S_{3}^{\mathrm{old}}(210, [\chi]) \cong \) \(S_{3}^{\mathrm{new}}(14, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(21, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(35, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(42, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(70, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(105, [\chi])\)\(^{\oplus 2}\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ (\( ( 1 + 2 T^{2} + 4 T^{4} )^{2} \))(\( ( 1 + 2 T^{2} + 4 T^{4} )^{4} \))
$3$ (\( ( 1 - 3 T + 3 T^{2} )^{4} \))(\( ( 1 + 3 T + 3 T^{2} )^{8} \))
$5$ (\( ( 1 - 5 T^{2} + 25 T^{4} )^{2} \))(\( ( 1 - 5 T^{2} + 25 T^{4} )^{4} \))
$7$ (\( 1 - 78 T^{2} + 5243 T^{4} - 187278 T^{6} + 5764801 T^{8} \))(\( 1 - 4 T + 28 T^{2} - 352 T^{3} + 2898 T^{4} - 16836 T^{5} + 70272 T^{6} - 1158948 T^{7} + 7794283 T^{8} - 56788452 T^{9} + 168723072 T^{10} - 1980738564 T^{11} + 16706393298 T^{12} - 99431287648 T^{13} + 387556041628 T^{14} - 2712892291396 T^{15} + 33232930569601 T^{16} \))
$11$ (\( 1 + 4 T - 168 T^{2} + 1928 T^{3} + 16250 T^{4} - 341700 T^{5} + 2353408 T^{6} + 32486668 T^{7} - 296735661 T^{8} + 3930886828 T^{9} + 34456246528 T^{10} - 605342393700 T^{11} + 3483331816250 T^{12} + 50007354630728 T^{13} - 527255967289128 T^{14} + 1518999334332964 T^{15} + 45949729863572161 T^{16} \))(\( 1 + 4 T - 364 T^{2} - 2552 T^{3} + 54108 T^{4} + 655724 T^{5} - 3422504 T^{6} - 97911028 T^{7} - 146055574 T^{8} + 11147160032 T^{9} + 52178728220 T^{10} - 1271915177612 T^{11} - 12776124769680 T^{12} + 140823145942564 T^{13} + 3084355376442156 T^{14} - 7698059096283072 T^{15} - 472018303966569005 T^{16} - 931465150650251712 T^{17} + 45158047066489605996 T^{18} + \)\(24\!\cdots\!04\)\( T^{19} - \)\(27\!\cdots\!80\)\( T^{20} - \)\(32\!\cdots\!12\)\( T^{21} + \)\(16\!\cdots\!20\)\( T^{22} + \)\(42\!\cdots\!12\)\( T^{23} - \)\(67\!\cdots\!14\)\( T^{24} - \)\(54\!\cdots\!68\)\( T^{25} - \)\(23\!\cdots\!04\)\( T^{26} + \)\(53\!\cdots\!04\)\( T^{27} + \)\(53\!\cdots\!28\)\( T^{28} - \)\(30\!\cdots\!72\)\( T^{29} - \)\(52\!\cdots\!84\)\( T^{30} + \)\(69\!\cdots\!04\)\( T^{31} + \)\(21\!\cdots\!21\)\( T^{32} \))
$13$ (\( 1 - 860 T^{2} + 357498 T^{4} - 95438800 T^{6} + 18541152803 T^{8} - 2725827566800 T^{10} + 291622101296058 T^{12} - 20036353205333660 T^{14} + 665416609183179841 T^{16} \))(\( 1 - 1336 T^{2} + 822548 T^{4} - 307870064 T^{6} + 77337463770 T^{8} - 13349067637096 T^{10} + 1484717408181296 T^{12} - 75519988778385192 T^{14} - 843289248891793565 T^{16} - \)\(21\!\cdots\!12\)\( T^{18} + \)\(12\!\cdots\!16\)\( T^{20} - \)\(31\!\cdots\!76\)\( T^{22} + \)\(51\!\cdots\!70\)\( T^{24} - \)\(58\!\cdots\!64\)\( T^{26} + \)\(44\!\cdots\!28\)\( T^{28} - \)\(20\!\cdots\!56\)\( T^{30} + \)\(44\!\cdots\!81\)\( T^{32} \))
$17$ (\( 1 - 84 T + 4208 T^{2} - 155904 T^{3} + 4746250 T^{4} - 123749124 T^{5} + 2818194464 T^{6} - 56792172564 T^{7} + 1020121281523 T^{8} - 16412937870996 T^{9} + 235378419827744 T^{10} - 2987003019239556 T^{11} + 33108688754346250 T^{12} - 314301513055600896 T^{13} + 2451674374262834288 T^{14} - 14143737430989678036 T^{15} + 48661191875666868481 T^{16} \))(\( 1 - 12 T + 420 T^{2} - 4464 T^{3} + 92044 T^{4} - 785292 T^{5} + 23989560 T^{6} + 227912772 T^{7} - 3236850998 T^{8} + 177940653384 T^{9} - 2004860299572 T^{10} + 43692613137348 T^{11} - 466147839356624 T^{12} + 8985915182672820 T^{13} - 29797959923559588 T^{14} - 769960022203743096 T^{15} + 34516652247342574675 T^{16} - \)\(22\!\cdots\!44\)\( T^{17} - \)\(24\!\cdots\!48\)\( T^{18} + \)\(21\!\cdots\!80\)\( T^{19} - \)\(32\!\cdots\!84\)\( T^{20} + \)\(88\!\cdots\!52\)\( T^{21} - \)\(11\!\cdots\!92\)\( T^{22} + \)\(29\!\cdots\!36\)\( T^{23} - \)\(15\!\cdots\!38\)\( T^{24} + \)\(32\!\cdots\!48\)\( T^{25} + \)\(97\!\cdots\!60\)\( T^{26} - \)\(92\!\cdots\!88\)\( T^{27} + \)\(31\!\cdots\!24\)\( T^{28} - \)\(43\!\cdots\!16\)\( T^{29} + \)\(11\!\cdots\!20\)\( T^{30} - \)\(98\!\cdots\!88\)\( T^{31} + \)\(23\!\cdots\!61\)\( T^{32} \))
$19$ (\( 1 - 108 T + 6142 T^{2} - 243432 T^{3} + 7548345 T^{4} - 198492408 T^{5} + 4660679966 T^{6} - 100240172052 T^{7} + 1986343374068 T^{8} - 36186702110772 T^{9} + 607384473849086 T^{10} - 9338250206171448 T^{11} + 128197793162717145 T^{12} - 1492497721269013032 T^{13} + 13594180232904360862 T^{14} - 86292722064551485068 T^{15} + \)\(28\!\cdots\!81\)\( T^{16} \))(\( 1 + 72 T + 3812 T^{2} + 150048 T^{3} + 4883342 T^{4} + 136887912 T^{5} + 3384543112 T^{6} + 75691203624 T^{7} + 1567163213049 T^{8} + 30524546336184 T^{9} + 578634766673960 T^{10} + 10979211701479464 T^{11} + 213751725988675694 T^{12} + 4327693265152902096 T^{13} + 88720456587214840044 T^{14} + \)\(17\!\cdots\!80\)\( T^{15} + \)\(34\!\cdots\!48\)\( T^{16} + \)\(64\!\cdots\!80\)\( T^{17} + \)\(11\!\cdots\!24\)\( T^{18} + \)\(20\!\cdots\!76\)\( T^{19} + \)\(36\!\cdots\!54\)\( T^{20} + \)\(67\!\cdots\!64\)\( T^{21} + \)\(12\!\cdots\!60\)\( T^{22} + \)\(24\!\cdots\!64\)\( T^{23} + \)\(45\!\cdots\!69\)\( T^{24} + \)\(78\!\cdots\!84\)\( T^{25} + \)\(12\!\cdots\!12\)\( T^{26} + \)\(18\!\cdots\!32\)\( T^{27} + \)\(23\!\cdots\!82\)\( T^{28} + \)\(26\!\cdots\!88\)\( T^{29} + \)\(24\!\cdots\!92\)\( T^{30} + \)\(16\!\cdots\!72\)\( T^{31} + \)\(83\!\cdots\!61\)\( T^{32} \))
$23$ (\( 1 - 12 T - 1512 T^{2} + 15144 T^{3} + 1340890 T^{4} - 9662580 T^{5} - 850652928 T^{6} + 2479269564 T^{7} + 454065720819 T^{8} + 1311533599356 T^{9} - 238047566024448 T^{10} - 1430408620333620 T^{11} + 105006417053440090 T^{12} + 627363085819500456 T^{13} - 33134912141214725352 T^{14} - \)\(13\!\cdots\!08\)\( T^{15} + \)\(61\!\cdots\!61\)\( T^{16} \))(\( 1 + 12 T - 1804 T^{2} - 41256 T^{3} + 1078172 T^{4} + 48109188 T^{5} - 226519400 T^{6} - 29579538300 T^{7} + 68149425194 T^{8} + 16885870838304 T^{9} + 1241401844668 T^{10} - 10296857200481412 T^{11} - 100210062266615184 T^{12} + 3750005340988049388 T^{13} + 80231845600155444876 T^{14} - \)\(51\!\cdots\!40\)\( T^{15} - \)\(38\!\cdots\!85\)\( T^{16} - \)\(27\!\cdots\!60\)\( T^{17} + \)\(22\!\cdots\!16\)\( T^{18} + \)\(55\!\cdots\!32\)\( T^{19} - \)\(78\!\cdots\!04\)\( T^{20} - \)\(42\!\cdots\!88\)\( T^{21} + \)\(27\!\cdots\!28\)\( T^{22} + \)\(19\!\cdots\!36\)\( T^{23} + \)\(41\!\cdots\!34\)\( T^{24} - \)\(95\!\cdots\!00\)\( T^{25} - \)\(38\!\cdots\!00\)\( T^{26} + \)\(43\!\cdots\!52\)\( T^{27} + \)\(51\!\cdots\!52\)\( T^{28} - \)\(10\!\cdots\!84\)\( T^{29} - \)\(24\!\cdots\!24\)\( T^{30} + \)\(85\!\cdots\!88\)\( T^{31} + \)\(37\!\cdots\!21\)\( T^{32} \))
$29$ (\( ( 1 - 36 T + 2904 T^{2} - 82956 T^{3} + 3490070 T^{4} - 69765996 T^{5} + 2053944024 T^{6} - 21413639556 T^{7} + 500246412961 T^{8} )^{2} \))(\( ( 1 - 36 T + 1444 T^{2} + 19116 T^{3} - 1274860 T^{4} + 68800452 T^{5} + 84102156 T^{6} - 30035473452 T^{7} + 2426711187974 T^{8} - 25259833173132 T^{9} + 59483856997836 T^{10} + 40924113344941092 T^{11} - 637744142027460460 T^{12} + 8042239471766642316 T^{13} + \)\(51\!\cdots\!04\)\( T^{14} - \)\(10\!\cdots\!16\)\( T^{15} + \)\(25\!\cdots\!21\)\( T^{16} )^{2} \))
$31$ (\( 1 + 132 T + 11278 T^{2} + 722040 T^{3} + 38602473 T^{4} + 1778529960 T^{5} + 72524765390 T^{6} + 2637318955548 T^{7} + 86278260959732 T^{8} + 2534463516281628 T^{9} + 66978143857738190 T^{10} + 1578451886268782760 T^{11} + 32923703244758191593 T^{12} + \)\(59\!\cdots\!40\)\( T^{13} + \)\(88\!\cdots\!58\)\( T^{14} + \)\(99\!\cdots\!72\)\( T^{15} + \)\(72\!\cdots\!81\)\( T^{16} \))(\( 1 - 120 T + 11588 T^{2} - 814560 T^{3} + 49123694 T^{4} - 2488742424 T^{5} + 113293093576 T^{6} - 4558808688792 T^{7} + 170270490089625 T^{8} - 5869377562140936 T^{9} + 196357797692622056 T^{10} - 6377368910762118360 T^{11} + \)\(21\!\cdots\!82\)\( T^{12} - \)\(69\!\cdots\!28\)\( T^{13} + \)\(23\!\cdots\!88\)\( T^{14} - \)\(74\!\cdots\!40\)\( T^{15} + \)\(23\!\cdots\!16\)\( T^{16} - \)\(71\!\cdots\!40\)\( T^{17} + \)\(21\!\cdots\!48\)\( T^{18} - \)\(61\!\cdots\!68\)\( T^{19} + \)\(17\!\cdots\!62\)\( T^{20} - \)\(52\!\cdots\!60\)\( T^{21} + \)\(15\!\cdots\!16\)\( T^{22} - \)\(44\!\cdots\!56\)\( T^{23} + \)\(12\!\cdots\!25\)\( T^{24} - \)\(31\!\cdots\!72\)\( T^{25} + \)\(76\!\cdots\!76\)\( T^{26} - \)\(16\!\cdots\!64\)\( T^{27} + \)\(30\!\cdots\!74\)\( T^{28} - \)\(48\!\cdots\!60\)\( T^{29} + \)\(66\!\cdots\!08\)\( T^{30} - \)\(66\!\cdots\!20\)\( T^{31} + \)\(52\!\cdots\!61\)\( T^{32} \))
$37$ (\( 1 + 96 T + 1110 T^{2} - 46464 T^{3} + 7652401 T^{4} + 360085920 T^{5} - 3532027338 T^{6} + 170035213440 T^{7} + 29111429055396 T^{8} + 232778207199360 T^{9} - 6619587887813418 T^{10} + 923881954453061280 T^{11} + 26878901285664514321 T^{12} - \)\(22\!\cdots\!36\)\( T^{13} + \)\(73\!\cdots\!10\)\( T^{14} + \)\(86\!\cdots\!44\)\( T^{15} + \)\(12\!\cdots\!41\)\( T^{16} \))(\( 1 - 44 T - 1956 T^{2} - 11728 T^{3} + 3652806 T^{4} + 242885524 T^{5} - 1534355576 T^{6} - 317275895964 T^{7} - 9605253544351 T^{8} + 75954637404076 T^{9} + 6072970847630504 T^{10} + 248299820626414612 T^{11} + 4168440621293518230 T^{12} + \)\(29\!\cdots\!60\)\( T^{13} + \)\(10\!\cdots\!28\)\( T^{14} - \)\(69\!\cdots\!04\)\( T^{15} - \)\(27\!\cdots\!16\)\( T^{16} - \)\(95\!\cdots\!76\)\( T^{17} + \)\(20\!\cdots\!08\)\( T^{18} + \)\(75\!\cdots\!40\)\( T^{19} + \)\(14\!\cdots\!30\)\( T^{20} + \)\(11\!\cdots\!88\)\( T^{21} + \)\(39\!\cdots\!24\)\( T^{22} + \)\(68\!\cdots\!64\)\( T^{23} - \)\(11\!\cdots\!91\)\( T^{24} - \)\(53\!\cdots\!56\)\( T^{25} - \)\(35\!\cdots\!76\)\( T^{26} + \)\(76\!\cdots\!56\)\( T^{27} + \)\(15\!\cdots\!66\)\( T^{28} - \)\(69\!\cdots\!52\)\( T^{29} - \)\(15\!\cdots\!76\)\( T^{30} - \)\(48\!\cdots\!56\)\( T^{31} + \)\(15\!\cdots\!81\)\( T^{32} \))
$41$ (\( 1 - 5456 T^{2} + 19963884 T^{4} - 49821420976 T^{6} + 96897341176550 T^{8} - 140783428358562736 T^{10} + \)\(15\!\cdots\!64\)\( T^{12} - \)\(12\!\cdots\!36\)\( T^{14} + \)\(63\!\cdots\!41\)\( T^{16} \))(\( 1 - 11416 T^{2} + 66218264 T^{4} - 265318066664 T^{6} + 835818107018940 T^{8} - 2208514761298495288 T^{10} + \)\(50\!\cdots\!56\)\( T^{12} - \)\(10\!\cdots\!32\)\( T^{14} + \)\(18\!\cdots\!14\)\( T^{16} - \)\(28\!\cdots\!52\)\( T^{18} + \)\(40\!\cdots\!76\)\( T^{20} - \)\(49\!\cdots\!28\)\( T^{22} + \)\(53\!\cdots\!40\)\( T^{24} - \)\(47\!\cdots\!64\)\( T^{26} + \)\(33\!\cdots\!04\)\( T^{28} - \)\(16\!\cdots\!36\)\( T^{30} + \)\(40\!\cdots\!81\)\( T^{32} \))
$43$ (\( ( 1 + 56 T + 5082 T^{2} + 180688 T^{3} + 11078435 T^{4} + 334092112 T^{5} + 17374346682 T^{6} + 353996330744 T^{7} + 11688200277601 T^{8} )^{2} \))(\( ( 1 + 28 T + 5732 T^{2} + 194288 T^{3} + 21755610 T^{4} + 646615948 T^{5} + 59916131504 T^{6} + 1611056221164 T^{7} + 122468758030675 T^{8} + 2978842952932236 T^{9} + 204841330302006704 T^{10} + 4087494160581305452 T^{11} + \)\(25\!\cdots\!10\)\( T^{12} + \)\(41\!\cdots\!12\)\( T^{13} + \)\(22\!\cdots\!32\)\( T^{14} + \)\(20\!\cdots\!72\)\( T^{15} + \)\(13\!\cdots\!01\)\( T^{16} )^{2} \))
$47$ (\( 1 + 24 T + 3580 T^{2} + 81312 T^{3} + 3430986 T^{4} + 200307384 T^{5} + 5424417392 T^{6} + 942687331128 T^{7} + 25778360177363 T^{8} + 2082396314461752 T^{9} + 26469426483811952 T^{10} + 2159156424124689336 T^{11} + 81696191178488726346 T^{12} + \)\(42\!\cdots\!88\)\( T^{13} + \)\(41\!\cdots\!80\)\( T^{14} + \)\(61\!\cdots\!56\)\( T^{15} + \)\(56\!\cdots\!21\)\( T^{16} \))(\( 1 + 24 T + 10040 T^{2} + 236352 T^{3} + 48191588 T^{4} + 529473000 T^{5} + 142018937488 T^{6} - 1533668366952 T^{7} + 289429105887690 T^{8} - 11114511768111792 T^{9} + 616929826339364168 T^{10} - 28449623988679606104 T^{11} + \)\(19\!\cdots\!48\)\( T^{12} - \)\(39\!\cdots\!52\)\( T^{13} + \)\(60\!\cdots\!84\)\( T^{14} - \)\(28\!\cdots\!36\)\( T^{15} + \)\(14\!\cdots\!27\)\( T^{16} - \)\(62\!\cdots\!24\)\( T^{17} + \)\(29\!\cdots\!04\)\( T^{18} - \)\(42\!\cdots\!08\)\( T^{19} + \)\(46\!\cdots\!28\)\( T^{20} - \)\(14\!\cdots\!96\)\( T^{21} + \)\(71\!\cdots\!88\)\( T^{22} - \)\(28\!\cdots\!48\)\( T^{23} + \)\(16\!\cdots\!90\)\( T^{24} - \)\(19\!\cdots\!28\)\( T^{25} + \)\(39\!\cdots\!88\)\( T^{26} + \)\(32\!\cdots\!00\)\( T^{27} + \)\(65\!\cdots\!28\)\( T^{28} + \)\(70\!\cdots\!08\)\( T^{29} + \)\(66\!\cdots\!40\)\( T^{30} + \)\(34\!\cdots\!76\)\( T^{31} + \)\(32\!\cdots\!41\)\( T^{32} \))
$53$ (\( 1 - 32 T - 3572 T^{2} + 570944 T^{3} - 366070 T^{4} - 2018388320 T^{5} + 123180800432 T^{6} + 3809526158624 T^{7} - 445561676340941 T^{8} + 10700958979574816 T^{9} + 971955765373487792 T^{10} - 44736287623035613280 T^{11} - 22791404868886921270 T^{12} + \)\(99\!\cdots\!56\)\( T^{13} - \)\(17\!\cdots\!52\)\( T^{14} - \)\(44\!\cdots\!08\)\( T^{15} + \)\(38\!\cdots\!21\)\( T^{16} \))(\( 1 - 32 T - 12072 T^{2} + 345280 T^{3} + 71529764 T^{4} - 1574568864 T^{5} - 290206976304 T^{6} + 2770742930720 T^{7} + 1006589576083018 T^{8} + 3889510628997504 T^{9} - 3360408915215254232 T^{10} - 25194864185334307040 T^{11} + \)\(10\!\cdots\!32\)\( T^{12} + \)\(34\!\cdots\!88\)\( T^{13} - \)\(28\!\cdots\!00\)\( T^{14} - \)\(63\!\cdots\!04\)\( T^{15} + \)\(75\!\cdots\!55\)\( T^{16} - \)\(17\!\cdots\!36\)\( T^{17} - \)\(22\!\cdots\!00\)\( T^{18} + \)\(75\!\cdots\!52\)\( T^{19} + \)\(64\!\cdots\!52\)\( T^{20} - \)\(44\!\cdots\!60\)\( T^{21} - \)\(16\!\cdots\!12\)\( T^{22} + \)\(53\!\cdots\!76\)\( T^{23} + \)\(39\!\cdots\!78\)\( T^{24} + \)\(30\!\cdots\!80\)\( T^{25} - \)\(88\!\cdots\!04\)\( T^{26} - \)\(13\!\cdots\!76\)\( T^{27} + \)\(17\!\cdots\!84\)\( T^{28} + \)\(23\!\cdots\!20\)\( T^{29} - \)\(22\!\cdots\!92\)\( T^{30} - \)\(17\!\cdots\!68\)\( T^{31} + \)\(15\!\cdots\!41\)\( T^{32} \))
$59$ (\( 1 - 132 T + 15632 T^{2} - 1296768 T^{3} + 88851370 T^{4} - 4629146772 T^{5} + 218386477856 T^{6} - 7743785317668 T^{7} + 393482793133843 T^{8} - 26956116690802308 T^{9} + 2646267789699658016 T^{10} - \)\(19\!\cdots\!52\)\( T^{11} + \)\(13\!\cdots\!70\)\( T^{12} - \)\(66\!\cdots\!68\)\( T^{13} + \)\(27\!\cdots\!92\)\( T^{14} - \)\(81\!\cdots\!52\)\( T^{15} + \)\(21\!\cdots\!41\)\( T^{16} \))(\( 1 - 132 T + 9660 T^{2} - 508464 T^{3} + 17070268 T^{4} - 1368116580 T^{5} + 85370423112 T^{6} - 5735724829716 T^{7} + 274189008275050 T^{8} - 2776853544451464 T^{9} + 245103035882958612 T^{10} + 4667418467579246988 T^{11} + \)\(23\!\cdots\!68\)\( T^{12} - \)\(15\!\cdots\!44\)\( T^{13} + \)\(45\!\cdots\!16\)\( T^{14} - \)\(18\!\cdots\!68\)\( T^{15} - \)\(23\!\cdots\!33\)\( T^{16} - \)\(62\!\cdots\!08\)\( T^{17} + \)\(55\!\cdots\!76\)\( T^{18} - \)\(65\!\cdots\!04\)\( T^{19} + \)\(34\!\cdots\!28\)\( T^{20} + \)\(23\!\cdots\!88\)\( T^{21} + \)\(43\!\cdots\!72\)\( T^{22} - \)\(17\!\cdots\!04\)\( T^{23} + \)\(59\!\cdots\!50\)\( T^{24} - \)\(43\!\cdots\!36\)\( T^{25} + \)\(22\!\cdots\!12\)\( T^{26} - \)\(12\!\cdots\!80\)\( T^{27} + \)\(54\!\cdots\!48\)\( T^{28} - \)\(56\!\cdots\!24\)\( T^{29} + \)\(37\!\cdots\!60\)\( T^{30} - \)\(17\!\cdots\!32\)\( T^{31} + \)\(46\!\cdots\!81\)\( T^{32} \))
$61$ (\( 1 - 96 T + 11380 T^{2} - 797568 T^{3} + 60840138 T^{4} - 3862327392 T^{5} + 197844384848 T^{6} - 13588740778464 T^{7} + 649379857320947 T^{8} - 50563704436664544 T^{9} + 2739321895348217168 T^{10} - \)\(19\!\cdots\!12\)\( T^{11} + \)\(11\!\cdots\!78\)\( T^{12} - \)\(56\!\cdots\!68\)\( T^{13} + \)\(30\!\cdots\!80\)\( T^{14} - \)\(94\!\cdots\!36\)\( T^{15} + \)\(36\!\cdots\!61\)\( T^{16} \))(\( 1 - 96 T + 17624 T^{2} - 1396992 T^{3} + 159354980 T^{4} - 12864972960 T^{5} + 1094820209872 T^{6} - 90213896412000 T^{7} + 6336184490820426 T^{8} - 516522392725304640 T^{9} + 33635567216910203432 T^{10} - \)\(25\!\cdots\!96\)\( T^{11} + \)\(16\!\cdots\!12\)\( T^{12} - \)\(10\!\cdots\!72\)\( T^{13} + \)\(72\!\cdots\!12\)\( T^{14} - \)\(43\!\cdots\!24\)\( T^{15} + \)\(28\!\cdots\!43\)\( T^{16} - \)\(16\!\cdots\!04\)\( T^{17} + \)\(10\!\cdots\!92\)\( T^{18} - \)\(56\!\cdots\!92\)\( T^{19} + \)\(31\!\cdots\!72\)\( T^{20} - \)\(18\!\cdots\!96\)\( T^{21} + \)\(89\!\cdots\!72\)\( T^{22} - \)\(51\!\cdots\!40\)\( T^{23} + \)\(23\!\cdots\!86\)\( T^{24} - \)\(12\!\cdots\!00\)\( T^{25} + \)\(55\!\cdots\!72\)\( T^{26} - \)\(24\!\cdots\!60\)\( T^{27} + \)\(11\!\cdots\!80\)\( T^{28} - \)\(36\!\cdots\!12\)\( T^{29} + \)\(17\!\cdots\!44\)\( T^{30} - \)\(34\!\cdots\!96\)\( T^{31} + \)\(13\!\cdots\!21\)\( T^{32} \))
$67$ (\( 1 + 120 T - 4522 T^{2} - 749040 T^{3} + 46360561 T^{4} + 4026690720 T^{5} - 240439443082 T^{6} - 2848051223400 T^{7} + 1781423144538724 T^{8} - 12784901941842600 T^{9} - 4845124310717994922 T^{10} + \)\(36\!\cdots\!80\)\( T^{11} + \)\(18\!\cdots\!01\)\( T^{12} - \)\(13\!\cdots\!60\)\( T^{13} - \)\(37\!\cdots\!42\)\( T^{14} + \)\(44\!\cdots\!80\)\( T^{15} + \)\(16\!\cdots\!81\)\( T^{16} \))(\( 1 + 164 T - 6340 T^{2} - 2597968 T^{3} - 31915482 T^{4} + 22697073892 T^{5} + 967634972296 T^{6} - 123392891205548 T^{7} - 9455087268413407 T^{8} + 333882071018564764 T^{9} + 51639925158822741896 T^{10} + \)\(26\!\cdots\!32\)\( T^{11} - \)\(16\!\cdots\!86\)\( T^{12} - \)\(60\!\cdots\!36\)\( T^{13} + \)\(18\!\cdots\!60\)\( T^{14} + \)\(15\!\cdots\!28\)\( T^{15} + \)\(54\!\cdots\!88\)\( T^{16} + \)\(70\!\cdots\!92\)\( T^{17} + \)\(36\!\cdots\!60\)\( T^{18} - \)\(55\!\cdots\!84\)\( T^{19} - \)\(67\!\cdots\!26\)\( T^{20} + \)\(47\!\cdots\!68\)\( T^{21} + \)\(42\!\cdots\!56\)\( T^{22} + \)\(12\!\cdots\!56\)\( T^{23} - \)\(15\!\cdots\!67\)\( T^{24} - \)\(91\!\cdots\!32\)\( T^{25} + \)\(32\!\cdots\!96\)\( T^{26} + \)\(33\!\cdots\!88\)\( T^{27} - \)\(21\!\cdots\!22\)\( T^{28} - \)\(78\!\cdots\!92\)\( T^{29} - \)\(85\!\cdots\!40\)\( T^{30} + \)\(99\!\cdots\!36\)\( T^{31} + \)\(27\!\cdots\!61\)\( T^{32} \))
$71$ (\( ( 1 - 4 T + 13176 T^{2} - 338828 T^{3} + 79144886 T^{4} - 1708031948 T^{5} + 334824308856 T^{6} - 512401135684 T^{7} + 645753531245761 T^{8} )^{2} \))(\( ( 1 + 68 T + 26092 T^{2} + 2195028 T^{3} + 351820772 T^{4} + 28773766364 T^{5} + 3188424927972 T^{6} + 217589962697708 T^{7} + 19655929349959526 T^{8} + 1096871001959146028 T^{9} + 81023237162072440932 T^{10} + \)\(36\!\cdots\!44\)\( T^{11} + \)\(22\!\cdots\!92\)\( T^{12} + \)\(71\!\cdots\!28\)\( T^{13} + \)\(42\!\cdots\!72\)\( T^{14} + \)\(56\!\cdots\!08\)\( T^{15} + \)\(41\!\cdots\!21\)\( T^{16} )^{2} \))
$73$ (\( 1 - 24 T + 9630 T^{2} - 226512 T^{3} + 47274361 T^{4} + 2709573216 T^{5} - 18252715698 T^{6} + 34150744827912 T^{7} - 1287593588654412 T^{8} + 181989319187943048 T^{9} - 518345019296287218 T^{10} + \)\(41\!\cdots\!24\)\( T^{11} + \)\(38\!\cdots\!41\)\( T^{12} - \)\(97\!\cdots\!88\)\( T^{13} + \)\(22\!\cdots\!30\)\( T^{14} - \)\(29\!\cdots\!16\)\( T^{15} + \)\(65\!\cdots\!61\)\( T^{16} \))(\( 1 + 348 T + 82708 T^{2} + 14734320 T^{3} + 2214323678 T^{4} + 290251205388 T^{5} + 34312832193656 T^{6} + 3733330194438684 T^{7} + 379285162573239545 T^{8} + 36467064302261366820 T^{9} + \)\(33\!\cdots\!56\)\( T^{10} + \)\(29\!\cdots\!16\)\( T^{11} + \)\(24\!\cdots\!70\)\( T^{12} + \)\(20\!\cdots\!16\)\( T^{13} + \)\(16\!\cdots\!72\)\( T^{14} + \)\(12\!\cdots\!12\)\( T^{15} + \)\(91\!\cdots\!12\)\( T^{16} + \)\(65\!\cdots\!48\)\( T^{17} + \)\(45\!\cdots\!52\)\( T^{18} + \)\(30\!\cdots\!24\)\( T^{19} + \)\(20\!\cdots\!70\)\( T^{20} + \)\(12\!\cdots\!84\)\( T^{21} + \)\(76\!\cdots\!76\)\( T^{22} + \)\(44\!\cdots\!80\)\( T^{23} + \)\(24\!\cdots\!45\)\( T^{24} + \)\(12\!\cdots\!96\)\( T^{25} + \)\(63\!\cdots\!56\)\( T^{26} + \)\(28\!\cdots\!52\)\( T^{27} + \)\(11\!\cdots\!98\)\( T^{28} + \)\(41\!\cdots\!80\)\( T^{29} + \)\(12\!\cdots\!48\)\( T^{30} + \)\(27\!\cdots\!52\)\( T^{31} + \)\(42\!\cdots\!21\)\( T^{32} \))
$79$ (\( 1 - 12 T - 2418 T^{2} + 386472 T^{3} - 58917335 T^{4} - 324907032 T^{5} + 66125106126 T^{6} - 10581169939908 T^{7} + 1991860414252788 T^{8} - 66037081594965828 T^{9} + 2575578239741296206 T^{10} - 78980823689760123672 T^{11} - \)\(89\!\cdots\!35\)\( T^{12} + \)\(36\!\cdots\!72\)\( T^{13} - \)\(14\!\cdots\!38\)\( T^{14} - \)\(44\!\cdots\!72\)\( T^{15} + \)\(23\!\cdots\!21\)\( T^{16} \))(\( 1 - 280 T + 14532 T^{2} + 1622944 T^{3} + 35326638 T^{4} - 30655524856 T^{5} + 55399514824 T^{6} + 161497193433480 T^{7} + 10355496367899353 T^{8} - 1123575678148193512 T^{9} - 61265347082641448728 T^{10} + \)\(14\!\cdots\!40\)\( T^{11} + \)\(44\!\cdots\!62\)\( T^{12} - \)\(73\!\cdots\!60\)\( T^{13} + \)\(53\!\cdots\!92\)\( T^{14} - \)\(10\!\cdots\!56\)\( T^{15} + \)\(35\!\cdots\!48\)\( T^{16} - \)\(66\!\cdots\!96\)\( T^{17} + \)\(20\!\cdots\!52\)\( T^{18} - \)\(17\!\cdots\!60\)\( T^{19} + \)\(67\!\cdots\!82\)\( T^{20} + \)\(13\!\cdots\!40\)\( T^{21} - \)\(36\!\cdots\!48\)\( T^{22} - \)\(41\!\cdots\!72\)\( T^{23} + \)\(23\!\cdots\!13\)\( T^{24} + \)\(23\!\cdots\!80\)\( T^{25} + \)\(49\!\cdots\!24\)\( T^{26} - \)\(17\!\cdots\!96\)\( T^{27} + \)\(12\!\cdots\!78\)\( T^{28} + \)\(35\!\cdots\!24\)\( T^{29} + \)\(19\!\cdots\!52\)\( T^{30} - \)\(23\!\cdots\!80\)\( T^{31} + \)\(52\!\cdots\!41\)\( T^{32} \))
$83$ (\( 1 - 2384 T^{2} + 54393900 T^{4} - 170213772976 T^{6} + 3204332319622694 T^{8} - 8078059876516133296 T^{10} + \)\(12\!\cdots\!00\)\( T^{12} - \)\(25\!\cdots\!24\)\( T^{14} + \)\(50\!\cdots\!81\)\( T^{16} \))(\( 1 - 66184 T^{2} + 2113671608 T^{4} - 43986302935160 T^{6} + 678322851950602236 T^{8} - \)\(83\!\cdots\!72\)\( T^{10} + \)\(84\!\cdots\!60\)\( T^{12} - \)\(72\!\cdots\!04\)\( T^{14} + \)\(53\!\cdots\!06\)\( T^{16} - \)\(34\!\cdots\!84\)\( T^{18} + \)\(19\!\cdots\!60\)\( T^{20} - \)\(88\!\cdots\!92\)\( T^{22} + \)\(34\!\cdots\!16\)\( T^{24} - \)\(10\!\cdots\!60\)\( T^{26} + \)\(24\!\cdots\!68\)\( T^{28} - \)\(35\!\cdots\!44\)\( T^{30} + \)\(25\!\cdots\!61\)\( T^{32} \))
$89$ (\( 1 - 492 T + 137248 T^{2} - 27827520 T^{3} + 4512312618 T^{4} - 615183153660 T^{5} + 72837035721440 T^{6} - 7634393359602348 T^{7} + 716618506839255347 T^{8} - 60472029801410198508 T^{9} + \)\(45\!\cdots\!40\)\( T^{10} - \)\(30\!\cdots\!60\)\( T^{11} + \)\(17\!\cdots\!58\)\( T^{12} - \)\(86\!\cdots\!20\)\( T^{13} + \)\(33\!\cdots\!08\)\( T^{14} - \)\(96\!\cdots\!72\)\( T^{15} + \)\(15\!\cdots\!61\)\( T^{16} \))(\( 1 + 300 T + 54668 T^{2} + 7400400 T^{3} + 757620572 T^{4} + 53632718700 T^{5} + 870152589352 T^{6} - 480156239482500 T^{7} - 98736006237054870 T^{8} - 12632355432189921000 T^{9} - \)\(12\!\cdots\!92\)\( T^{10} - \)\(89\!\cdots\!00\)\( T^{11} - \)\(37\!\cdots\!44\)\( T^{12} + \)\(17\!\cdots\!40\)\( T^{13} + \)\(60\!\cdots\!12\)\( T^{14} + \)\(83\!\cdots\!20\)\( T^{15} + \)\(83\!\cdots\!23\)\( T^{16} + \)\(66\!\cdots\!20\)\( T^{17} + \)\(37\!\cdots\!92\)\( T^{18} + \)\(84\!\cdots\!40\)\( T^{19} - \)\(14\!\cdots\!64\)\( T^{20} - \)\(27\!\cdots\!00\)\( T^{21} - \)\(30\!\cdots\!32\)\( T^{22} - \)\(24\!\cdots\!00\)\( T^{23} - \)\(15\!\cdots\!70\)\( T^{24} - \)\(58\!\cdots\!00\)\( T^{25} + \)\(84\!\cdots\!52\)\( T^{26} + \)\(41\!\cdots\!00\)\( T^{27} + \)\(46\!\cdots\!52\)\( T^{28} + \)\(35\!\cdots\!00\)\( T^{29} + \)\(20\!\cdots\!08\)\( T^{30} + \)\(90\!\cdots\!00\)\( T^{31} + \)\(24\!\cdots\!21\)\( T^{32} \))
$97$ (\( 1 - 35880 T^{2} + 724155484 T^{4} - 9811041724440 T^{6} + 103976225413471686 T^{8} - \)\(86\!\cdots\!40\)\( T^{10} + \)\(56\!\cdots\!24\)\( T^{12} - \)\(24\!\cdots\!80\)\( T^{14} + \)\(61\!\cdots\!21\)\( T^{16} \))(\( 1 - 70448 T^{2} + 2604307832 T^{4} - 66719407643536 T^{6} + 1323480529449253148 T^{8} - \)\(21\!\cdots\!72\)\( T^{10} + \)\(29\!\cdots\!68\)\( T^{12} - \)\(34\!\cdots\!84\)\( T^{14} + \)\(34\!\cdots\!38\)\( T^{16} - \)\(30\!\cdots\!04\)\( T^{18} + \)\(23\!\cdots\!48\)\( T^{20} - \)\(14\!\cdots\!52\)\( T^{22} + \)\(81\!\cdots\!08\)\( T^{24} - \)\(36\!\cdots\!36\)\( T^{26} + \)\(12\!\cdots\!92\)\( T^{28} - \)\(30\!\cdots\!28\)\( T^{30} + \)\(37\!\cdots\!41\)\( T^{32} \))
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