Properties

Label 19.8.a
Level 19
Weight 8
Character orbit a
Rep. character \(\chi_{19}(1,\cdot)\)
Character field \(\Q\)
Dimension 10
Newform subspaces 2
Sturm bound 13
Trace bound 1

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Defining parameters

Level: \( N \) = \( 19 \)
Weight: \( k \) = \( 8 \)
Character orbit: \([\chi]\) = 19.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 2 \)
Sturm bound: \(13\)
Trace bound: \(1\)
Distinguishing \(T_p\): \(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{8}(\Gamma_0(19))\).

Total New Old
Modular forms 12 10 2
Cusp forms 10 10 0
Eisenstein series 2 0 2

The following table gives the dimensions of the cuspidal new subspaces with specified eigenvalues for the Atkin-Lehner operators.

\(19\)Dim.
\(+\)\(6\)
\(-\)\(4\)

Trace form

\( 10q + 6q^{2} + 26q^{3} + 394q^{4} - 3q^{5} + 322q^{6} + 859q^{7} + 2280q^{8} + 1018q^{9} + O(q^{10}) \) \( 10q + 6q^{2} + 26q^{3} + 394q^{4} - 3q^{5} + 322q^{6} + 859q^{7} + 2280q^{8} + 1018q^{9} + 1768q^{10} - 1461q^{11} + 744q^{12} + 2370q^{13} + 6924q^{14} + 890q^{15} + 10234q^{16} + 975q^{17} - 6258q^{18} - 13718q^{19} + 13608q^{20} - 49166q^{21} - 108688q^{22} - 31248q^{23} - 60978q^{24} - 77473q^{25} - 78006q^{26} + 124508q^{27} + 129054q^{28} + 127380q^{29} - 126436q^{30} - 39380q^{31} + 210096q^{32} + 134066q^{33} + 537772q^{34} + 175905q^{35} - 300692q^{36} + 811840q^{37} - 164616q^{38} + 859224q^{39} - 293364q^{40} - 343332q^{41} - 1024798q^{42} + 236159q^{43} - 1690416q^{44} - 332011q^{45} + 343556q^{46} + 88545q^{47} - 1571712q^{48} - 606421q^{49} - 1581846q^{50} + 975142q^{51} - 909908q^{52} + 3303498q^{53} - 442034q^{54} - 481635q^{55} + 2745012q^{56} - 370386q^{57} + 2661970q^{58} - 885954q^{59} + 1316620q^{60} + 932281q^{61} - 2095068q^{62} - 3233781q^{63} + 462754q^{64} + 296448q^{65} - 111604q^{66} - 3534352q^{67} - 848814q^{68} - 3319512q^{69} + 6172716q^{70} + 4767558q^{71} + 6970200q^{72} - 4576669q^{73} + 12391656q^{74} - 5252924q^{75} - 2194880q^{76} - 1691319q^{77} - 420236q^{78} + 7345288q^{79} + 14281620q^{80} - 13798874q^{81} - 2233388q^{82} + 4739088q^{83} - 16782156q^{84} - 11785041q^{85} + 2052972q^{86} - 15657084q^{87} - 19763640q^{88} + 15372558q^{89} + 5785580q^{90} + 8899364q^{91} - 15201174q^{92} + 18645168q^{93} - 20440488q^{94} - 3024819q^{95} - 7681042q^{96} - 3884078q^{97} + 57894318q^{98} + 36282547q^{99} + O(q^{100}) \)

Decomposition of \(S_{8}^{\mathrm{new}}(\Gamma_0(19))\) into newform subspaces

Label Dim. \(A\) Field CM Traces A-L signs $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\) 19
19.8.a.a \(4\) \(5.935\) \(\mathbb{Q}[x]/(x^{4} - \cdots)\) None \(-9\) \(-14\) \(-222\) \(-1246\) \(-\) \(q+(-2+\beta _{1})q^{2}+(-4-\beta _{1}-\beta _{2})q^{3}+\cdots\)
19.8.a.b \(6\) \(5.935\) \(\mathbb{Q}[x]/(x^{6} - \cdots)\) None \(15\) \(40\) \(219\) \(2105\) \(+\) \(q+(2+\beta _{1})q^{2}+(6+\beta _{1}+\beta _{4})q^{3}+(57+\cdots)q^{4}+\cdots\)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ (\( 1 + 9 T + 278 T^{2} + 3060 T^{3} + 41640 T^{4} + 391680 T^{5} + 4554752 T^{6} + 18874368 T^{7} + 268435456 T^{8} \))(\( 1 - 15 T + 318 T^{2} - 4950 T^{3} + 79632 T^{4} - 961800 T^{5} + 12185088 T^{6} - 123110400 T^{7} + 1304690688 T^{8} - 10380902400 T^{9} + 85362475008 T^{10} - 515396075520 T^{11} + 4398046511104 T^{12} \))
$3$ (\( 1 + 14 T + 6921 T^{2} + 110754 T^{3} + 20790216 T^{4} + 242218998 T^{5} + 33102928449 T^{6} + 146444944842 T^{7} + 22876792454961 T^{8} \))(\( 1 - 40 T + 4403 T^{2} - 192110 T^{3} + 18342663 T^{4} - 626151150 T^{5} + 43652875050 T^{6} - 1369392565050 T^{7} + 87732388506447 T^{8} - 2009538453828330 T^{9} + 100726517179193283 T^{10} - 2001261803959988280 T^{11} + \)\(10\!\cdots\!09\)\( T^{12} \))
$5$ (\( 1 + 222 T + 192845 T^{2} + 55255950 T^{3} + 18908232000 T^{4} + 4316871093750 T^{5} + 1177032470703125 T^{6} + 105857849121093750 T^{7} + 37252902984619140625 T^{8} \))(\( 1 - 219 T + 285139 T^{2} - 48299412 T^{3} + 38290594185 T^{4} - 4849644909525 T^{5} + 3437556118802750 T^{6} - 378878508556640625 T^{7} + \)\(23\!\cdots\!25\)\( T^{8} - \)\(23\!\cdots\!00\)\( T^{9} + \)\(10\!\cdots\!75\)\( T^{10} - \)\(63\!\cdots\!75\)\( T^{11} + \)\(22\!\cdots\!25\)\( T^{12} \))
$7$ (\( 1 + 1246 T + 2457124 T^{2} + 2119942888 T^{3} + 2721611058013 T^{4} + 1745864125812184 T^{5} + 1666478189651026276 T^{6} + \)\(69\!\cdots\!22\)\( T^{7} + \)\(45\!\cdots\!01\)\( T^{8} \))(\( 1 - 2105 T + 4955572 T^{2} - 6241638925 T^{3} + 8601965585212 T^{4} - 7996912907814185 T^{5} + 8564543806733580254 T^{6} - \)\(65\!\cdots\!55\)\( T^{7} + \)\(58\!\cdots\!88\)\( T^{8} - \)\(34\!\cdots\!75\)\( T^{9} + \)\(22\!\cdots\!72\)\( T^{10} - \)\(79\!\cdots\!15\)\( T^{11} + \)\(31\!\cdots\!49\)\( T^{12} \))
$11$ (\( 1 + 8718 T + 83348441 T^{2} + 397129760262 T^{3} + 2253310600559136 T^{4} + 7738935547414598802 T^{5} + \)\(31\!\cdots\!81\)\( T^{6} + \)\(64\!\cdots\!98\)\( T^{7} + \)\(14\!\cdots\!81\)\( T^{8} \))(\( 1 - 7257 T + 58346045 T^{2} - 338244396294 T^{3} + 2244764293443763 T^{4} - 9991304550445286181 T^{5} + \)\(49\!\cdots\!22\)\( T^{6} - \)\(19\!\cdots\!51\)\( T^{7} + \)\(85\!\cdots\!83\)\( T^{8} - \)\(25\!\cdots\!34\)\( T^{9} + \)\(84\!\cdots\!45\)\( T^{10} - \)\(20\!\cdots\!07\)\( T^{11} + \)\(54\!\cdots\!21\)\( T^{12} \))
$13$ (\( 1 + 4480 T + 135751603 T^{2} + 1216594982680 T^{3} + 8905550799721468 T^{4} + 76339530952810685560 T^{5} + \)\(53\!\cdots\!67\)\( T^{6} + \)\(11\!\cdots\!40\)\( T^{7} + \)\(15\!\cdots\!21\)\( T^{8} \))(\( 1 - 6850 T + 91496725 T^{2} - 704167820960 T^{3} + 12383134645314715 T^{4} - 66878843073577695430 T^{5} + \)\(63\!\cdots\!42\)\( T^{6} - \)\(41\!\cdots\!10\)\( T^{7} + \)\(48\!\cdots\!35\)\( T^{8} - \)\(17\!\cdots\!80\)\( T^{9} + \)\(14\!\cdots\!25\)\( T^{10} - \)\(66\!\cdots\!50\)\( T^{11} + \)\(61\!\cdots\!69\)\( T^{12} \))
$17$ (\( 1 + 4440 T + 779869562 T^{2} - 4771177186320 T^{3} + 272572121869323963 T^{4} - \)\(19\!\cdots\!60\)\( T^{5} + \)\(13\!\cdots\!98\)\( T^{6} + \)\(30\!\cdots\!80\)\( T^{7} + \)\(28\!\cdots\!41\)\( T^{8} \))(\( 1 - 5415 T + 1463911262 T^{2} - 1596609888645 T^{3} + 880328636051219356 T^{4} + \)\(24\!\cdots\!25\)\( T^{5} + \)\(36\!\cdots\!88\)\( T^{6} + \)\(10\!\cdots\!25\)\( T^{7} + \)\(14\!\cdots\!24\)\( T^{8} - \)\(11\!\cdots\!65\)\( T^{9} + \)\(41\!\cdots\!42\)\( T^{10} - \)\(62\!\cdots\!95\)\( T^{11} + \)\(47\!\cdots\!89\)\( T^{12} \))
$19$ (\( ( 1 - 6859 T )^{4} \))(\( ( 1 + 6859 T )^{6} \))
$23$ (\( 1 + 30528 T + 6766373099 T^{2} + 281312350347456 T^{3} + 30869060512593572088 T^{4} + \)\(95\!\cdots\!32\)\( T^{5} + \)\(78\!\cdots\!91\)\( T^{6} + \)\(12\!\cdots\!44\)\( T^{7} + \)\(13\!\cdots\!81\)\( T^{8} \))(\( 1 + 720 T + 6210655833 T^{2} - 67672690840440 T^{3} + 32507715687552091011 T^{4} - \)\(58\!\cdots\!00\)\( T^{5} + \)\(10\!\cdots\!90\)\( T^{6} - \)\(19\!\cdots\!00\)\( T^{7} + \)\(37\!\cdots\!99\)\( T^{8} - \)\(26\!\cdots\!20\)\( T^{9} + \)\(83\!\cdots\!73\)\( T^{10} + \)\(32\!\cdots\!40\)\( T^{11} + \)\(15\!\cdots\!29\)\( T^{12} \))
$29$ (\( 1 + 254244 T + 50526434375 T^{2} + 4158825389649144 T^{3} + \)\(56\!\cdots\!48\)\( T^{4} + \)\(71\!\cdots\!96\)\( T^{5} + \)\(15\!\cdots\!75\)\( T^{6} + \)\(13\!\cdots\!76\)\( T^{7} + \)\(88\!\cdots\!61\)\( T^{8} \))(\( 1 - 381624 T + 120598336869 T^{2} - 25554410115932760 T^{3} + \)\(50\!\cdots\!95\)\( T^{4} - \)\(78\!\cdots\!44\)\( T^{5} + \)\(11\!\cdots\!86\)\( T^{6} - \)\(13\!\cdots\!96\)\( T^{7} + \)\(15\!\cdots\!95\)\( T^{8} - \)\(13\!\cdots\!40\)\( T^{9} + \)\(10\!\cdots\!09\)\( T^{10} - \)\(58\!\cdots\!76\)\( T^{11} + \)\(26\!\cdots\!41\)\( T^{12} \))
$31$ (\( 1 + 303460 T + 76861833628 T^{2} + 13630700228047828 T^{3} + \)\(27\!\cdots\!06\)\( T^{4} + \)\(37\!\cdots\!08\)\( T^{5} + \)\(58\!\cdots\!88\)\( T^{6} + \)\(63\!\cdots\!60\)\( T^{7} + \)\(57\!\cdots\!41\)\( T^{8} \))(\( 1 - 264080 T + 77213082234 T^{2} - 11907345848265264 T^{3} + \)\(31\!\cdots\!23\)\( T^{4} - \)\(52\!\cdots\!08\)\( T^{5} + \)\(11\!\cdots\!24\)\( T^{6} - \)\(14\!\cdots\!88\)\( T^{7} + \)\(23\!\cdots\!83\)\( T^{8} - \)\(24\!\cdots\!84\)\( T^{9} + \)\(44\!\cdots\!94\)\( T^{10} - \)\(41\!\cdots\!80\)\( T^{11} + \)\(43\!\cdots\!61\)\( T^{12} \))
$37$ (\( 1 + 270460 T + 190182623716 T^{2} + 33517908738047860 T^{3} + \)\(19\!\cdots\!78\)\( T^{4} + \)\(31\!\cdots\!80\)\( T^{5} + \)\(17\!\cdots\!24\)\( T^{6} + \)\(23\!\cdots\!20\)\( T^{7} + \)\(81\!\cdots\!21\)\( T^{8} \))(\( 1 - 1082300 T + 714019226394 T^{2} - 345140153247749340 T^{3} + \)\(13\!\cdots\!71\)\( T^{4} - \)\(47\!\cdots\!00\)\( T^{5} + \)\(15\!\cdots\!44\)\( T^{6} - \)\(45\!\cdots\!00\)\( T^{7} + \)\(12\!\cdots\!19\)\( T^{8} - \)\(29\!\cdots\!80\)\( T^{9} + \)\(57\!\cdots\!74\)\( T^{10} - \)\(83\!\cdots\!00\)\( T^{11} + \)\(73\!\cdots\!69\)\( T^{12} \))
$41$ (\( 1 + 828564 T + 615809837432 T^{2} + 263636106124383612 T^{3} + \)\(13\!\cdots\!62\)\( T^{4} + \)\(51\!\cdots\!72\)\( T^{5} + \)\(23\!\cdots\!52\)\( T^{6} + \)\(61\!\cdots\!24\)\( T^{7} + \)\(14\!\cdots\!21\)\( T^{8} \))(\( 1 - 485232 T + 691050034414 T^{2} - 303482610769790160 T^{3} + \)\(21\!\cdots\!51\)\( T^{4} - \)\(85\!\cdots\!48\)\( T^{5} + \)\(45\!\cdots\!12\)\( T^{6} - \)\(16\!\cdots\!88\)\( T^{7} + \)\(80\!\cdots\!11\)\( T^{8} - \)\(22\!\cdots\!60\)\( T^{9} + \)\(99\!\cdots\!94\)\( T^{10} - \)\(13\!\cdots\!32\)\( T^{11} + \)\(54\!\cdots\!81\)\( T^{12} \))
$43$ (\( 1 - 37454 T + 745997453341 T^{2} + 58604976088183150 T^{3} + \)\(25\!\cdots\!72\)\( T^{4} + \)\(15\!\cdots\!50\)\( T^{5} + \)\(55\!\cdots\!09\)\( T^{6} - \)\(75\!\cdots\!22\)\( T^{7} + \)\(54\!\cdots\!01\)\( T^{8} \))(\( 1 - 198705 T + 834541775061 T^{2} - 117667646359549570 T^{3} + \)\(35\!\cdots\!11\)\( T^{4} - \)\(20\!\cdots\!85\)\( T^{5} + \)\(10\!\cdots\!66\)\( T^{6} - \)\(55\!\cdots\!95\)\( T^{7} + \)\(26\!\cdots\!39\)\( T^{8} - \)\(23\!\cdots\!10\)\( T^{9} + \)\(45\!\cdots\!61\)\( T^{10} - \)\(29\!\cdots\!35\)\( T^{11} + \)\(40\!\cdots\!49\)\( T^{12} \))
$47$ (\( 1 - 335670 T + 80999396117 T^{2} + 82836172922388930 T^{3} + \)\(13\!\cdots\!48\)\( T^{4} + \)\(41\!\cdots\!90\)\( T^{5} + \)\(20\!\cdots\!73\)\( T^{6} - \)\(43\!\cdots\!90\)\( T^{7} + \)\(65\!\cdots\!61\)\( T^{8} \))(\( 1 + 247125 T + 1905455742645 T^{2} + 541718263873959150 T^{3} + \)\(18\!\cdots\!83\)\( T^{4} + \)\(51\!\cdots\!25\)\( T^{5} + \)\(11\!\cdots\!86\)\( T^{6} + \)\(25\!\cdots\!75\)\( T^{7} + \)\(47\!\cdots\!27\)\( T^{8} + \)\(70\!\cdots\!50\)\( T^{9} + \)\(12\!\cdots\!45\)\( T^{10} + \)\(82\!\cdots\!75\)\( T^{11} + \)\(16\!\cdots\!09\)\( T^{12} \))
$53$ (\( 1 - 76728 T + 3451678167515 T^{2} - 456735752197732368 T^{3} + \)\(55\!\cdots\!72\)\( T^{4} - \)\(53\!\cdots\!16\)\( T^{5} + \)\(47\!\cdots\!35\)\( T^{6} - \)\(12\!\cdots\!84\)\( T^{7} + \)\(19\!\cdots\!61\)\( T^{8} \))(\( 1 - 3226770 T + 7054947419693 T^{2} - 12112694846956504920 T^{3} + \)\(17\!\cdots\!99\)\( T^{4} - \)\(23\!\cdots\!50\)\( T^{5} + \)\(26\!\cdots\!62\)\( T^{6} - \)\(27\!\cdots\!50\)\( T^{7} + \)\(24\!\cdots\!31\)\( T^{8} - \)\(19\!\cdots\!60\)\( T^{9} + \)\(13\!\cdots\!73\)\( T^{10} - \)\(72\!\cdots\!90\)\( T^{11} + \)\(26\!\cdots\!09\)\( T^{12} \))
$59$ (\( 1 + 3191334 T + 9572074588361 T^{2} + 18383692890303807138 T^{3} + \)\(32\!\cdots\!96\)\( T^{4} + \)\(45\!\cdots\!22\)\( T^{5} + \)\(59\!\cdots\!21\)\( T^{6} + \)\(49\!\cdots\!06\)\( T^{7} + \)\(38\!\cdots\!21\)\( T^{8} \))(\( 1 - 2305380 T + 11140169488955 T^{2} - 22637394594512479326 T^{3} + \)\(61\!\cdots\!35\)\( T^{4} - \)\(97\!\cdots\!70\)\( T^{5} + \)\(19\!\cdots\!62\)\( T^{6} - \)\(24\!\cdots\!30\)\( T^{7} + \)\(37\!\cdots\!35\)\( T^{8} - \)\(34\!\cdots\!34\)\( T^{9} + \)\(42\!\cdots\!55\)\( T^{10} - \)\(22\!\cdots\!20\)\( T^{11} + \)\(23\!\cdots\!81\)\( T^{12} \))
$61$ (\( 1 - 346550 T + 10546139924665 T^{2} - 3501286532199067106 T^{3} + \)\(47\!\cdots\!20\)\( T^{4} - \)\(11\!\cdots\!26\)\( T^{5} + \)\(10\!\cdots\!65\)\( T^{6} - \)\(10\!\cdots\!50\)\( T^{7} + \)\(97\!\cdots\!81\)\( T^{8} \))(\( 1 - 585731 T + 5872558224827 T^{2} - 8225508365873418048 T^{3} + \)\(22\!\cdots\!41\)\( T^{4} - \)\(37\!\cdots\!01\)\( T^{5} + \)\(84\!\cdots\!46\)\( T^{6} - \)\(11\!\cdots\!21\)\( T^{7} + \)\(22\!\cdots\!81\)\( T^{8} - \)\(25\!\cdots\!28\)\( T^{9} + \)\(57\!\cdots\!87\)\( T^{10} - \)\(17\!\cdots\!31\)\( T^{11} + \)\(96\!\cdots\!21\)\( T^{12} \))
$67$ (\( 1 + 270322 T + 9026732328853 T^{2} - 10148100690291020534 T^{3} + \)\(32\!\cdots\!48\)\( T^{4} - \)\(61\!\cdots\!82\)\( T^{5} + \)\(33\!\cdots\!37\)\( T^{6} + \)\(60\!\cdots\!74\)\( T^{7} + \)\(13\!\cdots\!41\)\( T^{8} \))(\( 1 + 3264030 T + 18405648119883 T^{2} + 48646827165388850090 T^{3} + \)\(19\!\cdots\!91\)\( T^{4} + \)\(46\!\cdots\!60\)\( T^{5} + \)\(14\!\cdots\!02\)\( T^{6} + \)\(28\!\cdots\!80\)\( T^{7} + \)\(72\!\cdots\!39\)\( T^{8} + \)\(10\!\cdots\!30\)\( T^{9} + \)\(24\!\cdots\!03\)\( T^{10} + \)\(26\!\cdots\!90\)\( T^{11} + \)\(49\!\cdots\!89\)\( T^{12} \))
$71$ (\( 1 + 2066124 T + 18123756199196 T^{2} + 31467662292373515468 T^{3} + \)\(15\!\cdots\!42\)\( T^{4} + \)\(28\!\cdots\!88\)\( T^{5} + \)\(14\!\cdots\!76\)\( T^{6} + \)\(15\!\cdots\!04\)\( T^{7} + \)\(68\!\cdots\!61\)\( T^{8} \))(\( 1 - 6833682 T + 63784648336002 T^{2} - \)\(29\!\cdots\!74\)\( T^{3} + \)\(15\!\cdots\!23\)\( T^{4} - \)\(53\!\cdots\!92\)\( T^{5} + \)\(19\!\cdots\!28\)\( T^{6} - \)\(48\!\cdots\!72\)\( T^{7} + \)\(12\!\cdots\!63\)\( T^{8} - \)\(22\!\cdots\!54\)\( T^{9} + \)\(43\!\cdots\!22\)\( T^{10} - \)\(42\!\cdots\!82\)\( T^{11} + \)\(56\!\cdots\!41\)\( T^{12} \))
$73$ (\( 1 + 416044 T + 27420631550362 T^{2} + 7627031467546596208 T^{3} + \)\(42\!\cdots\!63\)\( T^{4} + \)\(84\!\cdots\!76\)\( T^{5} + \)\(33\!\cdots\!58\)\( T^{6} + \)\(56\!\cdots\!12\)\( T^{7} + \)\(14\!\cdots\!81\)\( T^{8} \))(\( 1 + 4160625 T + 47308389751458 T^{2} + \)\(15\!\cdots\!75\)\( T^{3} + \)\(93\!\cdots\!44\)\( T^{4} + \)\(25\!\cdots\!65\)\( T^{5} + \)\(11\!\cdots\!12\)\( T^{6} + \)\(28\!\cdots\!05\)\( T^{7} + \)\(11\!\cdots\!96\)\( T^{8} + \)\(20\!\cdots\!75\)\( T^{9} + \)\(70\!\cdots\!98\)\( T^{10} + \)\(68\!\cdots\!25\)\( T^{11} + \)\(18\!\cdots\!29\)\( T^{12} \))
$79$ (\( 1 - 16025864 T + 165936409446616 T^{2} - \)\(11\!\cdots\!40\)\( T^{3} + \)\(57\!\cdots\!46\)\( T^{4} - \)\(21\!\cdots\!60\)\( T^{5} + \)\(61\!\cdots\!96\)\( T^{6} - \)\(11\!\cdots\!56\)\( T^{7} + \)\(13\!\cdots\!61\)\( T^{8} \))(\( 1 + 8680576 T + 104444400096262 T^{2} + \)\(52\!\cdots\!56\)\( T^{3} + \)\(35\!\cdots\!79\)\( T^{4} + \)\(12\!\cdots\!60\)\( T^{5} + \)\(74\!\cdots\!40\)\( T^{6} + \)\(24\!\cdots\!40\)\( T^{7} + \)\(13\!\cdots\!99\)\( T^{8} + \)\(36\!\cdots\!24\)\( T^{9} + \)\(14\!\cdots\!82\)\( T^{10} + \)\(22\!\cdots\!24\)\( T^{11} + \)\(50\!\cdots\!41\)\( T^{12} \))
$83$ (\( 1 - 8524128 T + 119138452816904 T^{2} - \)\(65\!\cdots\!20\)\( T^{3} + \)\(49\!\cdots\!66\)\( T^{4} - \)\(17\!\cdots\!40\)\( T^{5} + \)\(87\!\cdots\!16\)\( T^{6} - \)\(17\!\cdots\!24\)\( T^{7} + \)\(54\!\cdots\!41\)\( T^{8} \))(\( 1 + 3785040 T + 103158618010030 T^{2} + \)\(42\!\cdots\!80\)\( T^{3} + \)\(54\!\cdots\!87\)\( T^{4} + \)\(19\!\cdots\!20\)\( T^{5} + \)\(18\!\cdots\!40\)\( T^{6} + \)\(51\!\cdots\!40\)\( T^{7} + \)\(40\!\cdots\!23\)\( T^{8} + \)\(84\!\cdots\!40\)\( T^{9} + \)\(55\!\cdots\!30\)\( T^{10} + \)\(55\!\cdots\!80\)\( T^{11} + \)\(39\!\cdots\!89\)\( T^{12} \))
$89$ (\( 1 - 2899092 T + 139779480449732 T^{2} - \)\(34\!\cdots\!16\)\( T^{3} + \)\(86\!\cdots\!54\)\( T^{4} - \)\(15\!\cdots\!64\)\( T^{5} + \)\(27\!\cdots\!12\)\( T^{6} - \)\(25\!\cdots\!88\)\( T^{7} + \)\(38\!\cdots\!81\)\( T^{8} \))(\( 1 - 12473466 T + 279390358891030 T^{2} - \)\(25\!\cdots\!78\)\( T^{3} + \)\(31\!\cdots\!03\)\( T^{4} - \)\(21\!\cdots\!68\)\( T^{5} + \)\(18\!\cdots\!88\)\( T^{6} - \)\(97\!\cdots\!72\)\( T^{7} + \)\(62\!\cdots\!23\)\( T^{8} - \)\(22\!\cdots\!42\)\( T^{9} + \)\(10\!\cdots\!30\)\( T^{10} - \)\(21\!\cdots\!34\)\( T^{11} + \)\(74\!\cdots\!21\)\( T^{12} \))
$97$ (\( 1 + 4766908 T + 110311755135064 T^{2} + \)\(75\!\cdots\!60\)\( T^{3} + \)\(86\!\cdots\!82\)\( T^{4} + \)\(61\!\cdots\!80\)\( T^{5} + \)\(72\!\cdots\!16\)\( T^{6} + \)\(25\!\cdots\!76\)\( T^{7} + \)\(42\!\cdots\!61\)\( T^{8} \))(\( 1 - 882830 T + 208938548614706 T^{2} - \)\(68\!\cdots\!50\)\( T^{3} + \)\(28\!\cdots\!47\)\( T^{4} - \)\(11\!\cdots\!80\)\( T^{5} + \)\(25\!\cdots\!44\)\( T^{6} - \)\(94\!\cdots\!40\)\( T^{7} + \)\(18\!\cdots\!43\)\( T^{8} - \)\(36\!\cdots\!50\)\( T^{9} + \)\(89\!\cdots\!66\)\( T^{10} - \)\(30\!\cdots\!90\)\( T^{11} + \)\(27\!\cdots\!09\)\( T^{12} \))
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