Properties

Label 15.10
Level 15
Weight 10
Dimension 46
Nonzero newspaces 3
Newform subspaces 6
Sturm bound 160
Trace bound 1

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Defining parameters

Level: \( N \) = \( 15 = 3 \cdot 5 \)
Weight: \( k \) = \( 10 \)
Nonzero newspaces: \( 3 \)
Newform subspaces: \( 6 \)
Sturm bound: \(160\)
Trace bound: \(1\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{10}(\Gamma_1(15))\).

Total New Old
Modular forms 80 54 26
Cusp forms 64 46 18
Eisenstein series 16 8 8

Trace form

\( 46q + 68q^{2} - 150q^{3} + 344q^{4} - 690q^{5} - 7140q^{6} - 21188q^{7} + 67932q^{8} - 13122q^{9} + O(q^{10}) \) \( 46q + 68q^{2} - 150q^{3} + 344q^{4} - 690q^{5} - 7140q^{6} - 21188q^{7} + 67932q^{8} - 13122q^{9} + 18520q^{10} - 159064q^{11} + 233412q^{12} + 170148q^{13} + 531816q^{14} - 61290q^{15} - 1915024q^{16} - 420520q^{17} + 1440948q^{18} + 1510496q^{19} + 1173100q^{20} - 3242652q^{21} - 6848280q^{22} + 3122496q^{23} + 4436208q^{24} + 9847270q^{25} - 6770392q^{26} + 4786830q^{27} - 9172776q^{28} - 7070108q^{29} - 15876960q^{30} + 12093928q^{31} + 17859412q^{32} + 18442884q^{33} - 2414552q^{34} - 21174680q^{35} - 31019304q^{36} - 10493548q^{37} - 2385976q^{38} + 7411824q^{39} + 124068200q^{40} + 117594724q^{41} + 2551536q^{42} - 122810748q^{43} - 177425288q^{44} - 125161290q^{45} - 96583992q^{46} + 51128264q^{47} + 340587828q^{48} + 135923910q^{49} + 94360700q^{50} - 109600476q^{51} - 209781040q^{52} - 114062224q^{53} - 23383404q^{54} + 128495720q^{55} - 136664640q^{56} + 45495684q^{57} - 415626152q^{58} + 417468464q^{59} - 208559160q^{60} + 185675724q^{61} - 31830936q^{62} + 14879472q^{63} + 452073416q^{64} - 172199200q^{65} + 324822624q^{66} - 331042820q^{67} + 717440536q^{68} + 172304496q^{69} + 1164742440q^{70} - 73747472q^{71} - 1088344908q^{72} + 37028316q^{73} - 2296939904q^{74} - 390483390q^{75} - 2207705040q^{76} + 72064224q^{77} - 137852280q^{78} + 589394400q^{79} + 733243180q^{80} + 2853197046q^{81} + 5664251624q^{82} + 804884184q^{83} - 1030301856q^{84} - 2307340580q^{85} + 245612816q^{86} - 2840994132q^{87} - 2764190328q^{88} + 27621876q^{89} + 196282680q^{90} - 5025897832q^{91} - 2807106528q^{92} - 1123759992q^{93} + 2549830744q^{94} + 2062249640q^{95} + 9194942832q^{96} + 5375198660q^{97} + 5935831732q^{98} - 98992368q^{99} + O(q^{100}) \)

Decomposition of \(S_{10}^{\mathrm{new}}(\Gamma_1(15))\)

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
15.10.a \(\chi_{15}(1, \cdot)\) 15.10.a.a 1 1
15.10.a.b 1
15.10.a.c 2
15.10.a.d 2
15.10.b \(\chi_{15}(4, \cdot)\) 15.10.b.a 8 1
15.10.e \(\chi_{15}(2, \cdot)\) 15.10.e.a 32 2

Decomposition of \(S_{10}^{\mathrm{old}}(\Gamma_1(15))\) into lower level spaces

\( S_{10}^{\mathrm{old}}(\Gamma_1(15)) \cong \) \(S_{10}^{\mathrm{new}}(\Gamma_1(3))\)\(^{\oplus 2}\)\(\oplus\)\(S_{10}^{\mathrm{new}}(\Gamma_1(5))\)\(^{\oplus 2}\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ (\( 1 + 4 T + 512 T^{2} \))(\( 1 - 22 T + 512 T^{2} \))(\( 1 - 19 T - 68 T^{2} - 9728 T^{3} + 262144 T^{4} \))(\( 1 - 31 T + 722 T^{2} - 15872 T^{3} + 262144 T^{4} \))(\( 1 - 1451 T^{2} + 1581940 T^{4} - 1146579392 T^{6} + 685942325248 T^{8} - 300568908136448 T^{10} + 108710089027747840 T^{12} - 26138892237258358784 T^{14} + \)\(47\!\cdots\!96\)\( T^{16} \))
$3$ (\( 1 - 81 T \))(\( 1 + 81 T \))(\( ( 1 - 81 T )^{2} \))(\( ( 1 + 81 T )^{2} \))(\( ( 1 + 6561 T^{2} )^{4} \))
$5$ (\( 1 - 625 T \))(\( 1 + 625 T \))(\( ( 1 + 625 T )^{2} \))(\( ( 1 - 625 T )^{2} \))(\( 1 + 690 T - 609700 T^{2} - 1699931250 T^{3} - 2538785156250 T^{4} - 3320178222656250 T^{5} - 2325820922851562500 T^{6} + \)\(51\!\cdots\!50\)\( T^{7} + \)\(14\!\cdots\!25\)\( T^{8} \))
$7$ (\( 1 + 7680 T + 40353607 T^{2} \))(\( 1 + 5988 T + 40353607 T^{2} \))(\( 1 + 11872 T + 112235774 T^{2} + 479078022304 T^{3} + 1628413597910449 T^{4} \))(\( 1 - 14112 T + 103286414 T^{2} - 569470101984 T^{3} + 1628413597910449 T^{4} \))(\( 1 - 187079228 T^{2} + 15253712953314292 T^{4} - \)\(75\!\cdots\!16\)\( T^{6} + \)\(30\!\cdots\!14\)\( T^{8} - \)\(12\!\cdots\!84\)\( T^{10} + \)\(40\!\cdots\!92\)\( T^{12} - \)\(80\!\cdots\!72\)\( T^{14} + \)\(70\!\cdots\!01\)\( T^{16} \))
$11$ (\( 1 + 86404 T + 2357947691 T^{2} \))(\( 1 + 14648 T + 2357947691 T^{2} \))(\( 1 - 35488 T + 526013014 T^{2} - 83678847658208 T^{3} + 5559917313492231481 T^{4} \))(\( 1 + 21512 T + 1790961254 T^{2} + 50724170728792 T^{3} + 5559917313492231481 T^{4} \))(\( ( 1 + 35994 T + 5523704672 T^{2} + 186668393765490 T^{3} + 15626722625814770286 T^{4} + \)\(44\!\cdots\!90\)\( T^{5} + \)\(30\!\cdots\!32\)\( T^{6} + \)\(47\!\cdots\!74\)\( T^{7} + \)\(30\!\cdots\!61\)\( T^{8} )^{2} \))
$13$ (\( 1 + 149978 T + 10604499373 T^{2} \))(\( 1 - 37906 T + 10604499373 T^{2} \))(\( 1 - 143676 T + 24105149134 T^{2} - 1523612051915148 T^{3} + \)\(11\!\cdots\!29\)\( T^{4} \))(\( 1 - 24284 T + 5870669214 T^{2} - 257519662773932 T^{3} + \)\(11\!\cdots\!29\)\( T^{4} \))(\( 1 - 57204710684 T^{2} + \)\(16\!\cdots\!00\)\( T^{4} - \)\(29\!\cdots\!48\)\( T^{6} + \)\(37\!\cdots\!18\)\( T^{8} - \)\(33\!\cdots\!92\)\( T^{10} + \)\(20\!\cdots\!00\)\( T^{12} - \)\(81\!\cdots\!76\)\( T^{14} + \)\(15\!\cdots\!81\)\( T^{16} \))
$17$ (\( 1 + 207622 T + 118587876497 T^{2} \))(\( 1 + 441098 T + 118587876497 T^{2} \))(\( 1 - 385156 T + 268539949078 T^{2} - 45674832160078532 T^{3} + \)\(14\!\cdots\!09\)\( T^{4} \))(\( 1 + 156956 T + 212670095078 T^{2} + 18613078743463132 T^{3} + \)\(14\!\cdots\!09\)\( T^{4} \))(\( 1 - 118674378380 T^{2} + \)\(17\!\cdots\!36\)\( T^{4} - \)\(19\!\cdots\!60\)\( T^{6} + \)\(18\!\cdots\!86\)\( T^{8} - \)\(27\!\cdots\!40\)\( T^{10} + \)\(34\!\cdots\!16\)\( T^{12} - \)\(33\!\cdots\!20\)\( T^{14} + \)\(39\!\cdots\!61\)\( T^{16} \))
$19$ (\( 1 - 716284 T + 322687697779 T^{2} \))(\( 1 - 441820 T + 322687697779 T^{2} \))(\( 1 + 403296 T + 684929514838 T^{2} + 130138657763479584 T^{3} + \)\(10\!\cdots\!41\)\( T^{4} \))(\( 1 + 95896 T + 629883192438 T^{2} + 30944459466214984 T^{3} + \)\(10\!\cdots\!41\)\( T^{4} \))(\( ( 1 - 425792 T + 640639588492 T^{2} - 117316391220409664 T^{3} + \)\(17\!\cdots\!54\)\( T^{4} - \)\(37\!\cdots\!56\)\( T^{5} + \)\(66\!\cdots\!72\)\( T^{6} - \)\(14\!\cdots\!88\)\( T^{7} + \)\(10\!\cdots\!81\)\( T^{8} )^{2} \))
$23$ (\( 1 - 1369920 T + 1801152661463 T^{2} \))(\( 1 - 2264136 T + 1801152661463 T^{2} \))(\( 1 - 223704 T - 1375273107794 T^{2} - 402925054979918952 T^{3} + \)\(32\!\cdots\!69\)\( T^{4} \))(\( 1 + 735264 T + 3363187908526 T^{2} + 1324322710477931232 T^{3} + \)\(32\!\cdots\!69\)\( T^{4} \))(\( 1 - 10272938895992 T^{2} + \)\(48\!\cdots\!32\)\( T^{4} - \)\(14\!\cdots\!84\)\( T^{6} + \)\(30\!\cdots\!94\)\( T^{8} - \)\(46\!\cdots\!96\)\( T^{10} + \)\(51\!\cdots\!52\)\( T^{12} - \)\(35\!\cdots\!28\)\( T^{14} + \)\(11\!\cdots\!21\)\( T^{16} \))
$29$ (\( 1 + 3194402 T + 14507145975869 T^{2} \))(\( 1 + 1049350 T + 14507145975869 T^{2} \))(\( 1 + 74572 T + 28833430018078 T^{2} + 1081826889712503068 T^{3} + \)\(21\!\cdots\!61\)\( T^{4} \))(\( 1 + 2678212 T + 15397908029438 T^{2} + 38853212438324066228 T^{3} + \)\(21\!\cdots\!61\)\( T^{4} \))(\( ( 1 + 36786 T + 11698955610980 T^{2} - 38232367405598309058 T^{3} + \)\(21\!\cdots\!18\)\( T^{4} - \)\(55\!\cdots\!02\)\( T^{5} + \)\(24\!\cdots\!80\)\( T^{6} + \)\(11\!\cdots\!74\)\( T^{7} + \)\(44\!\cdots\!21\)\( T^{8} )^{2} \))
$31$ (\( 1 + 2349000 T + 26439622160671 T^{2} \))(\( 1 + 7910568 T + 26439622160671 T^{2} \))(\( 1 + 5027128 T + 52415931233342 T^{2} + \)\(13\!\cdots\!88\)\( T^{3} + \)\(69\!\cdots\!41\)\( T^{4} \))(\( 1 - 10782432 T + 69294691361342 T^{2} - \)\(28\!\cdots\!72\)\( T^{3} + \)\(69\!\cdots\!41\)\( T^{4} \))(\( ( 1 - 237044 T + 50199397014268 T^{2} - \)\(19\!\cdots\!72\)\( T^{3} + \)\(11\!\cdots\!74\)\( T^{4} - \)\(51\!\cdots\!12\)\( T^{5} + \)\(35\!\cdots\!88\)\( T^{6} - \)\(43\!\cdots\!84\)\( T^{7} + \)\(48\!\cdots\!81\)\( T^{8} )^{2} \))
$37$ (\( 1 - 18735710 T + 129961739795077 T^{2} \))(\( 1 + 20992558 T + 129961739795077 T^{2} \))(\( 1 - 5373628 T + 231061724951934 T^{2} - \)\(69\!\cdots\!56\)\( T^{3} + \)\(16\!\cdots\!29\)\( T^{4} \))(\( 1 - 21968332 T + 359210373327534 T^{2} - \)\(28\!\cdots\!64\)\( T^{3} + \)\(16\!\cdots\!29\)\( T^{4} \))(\( 1 - 624655700715068 T^{2} + \)\(20\!\cdots\!12\)\( T^{4} - \)\(45\!\cdots\!76\)\( T^{6} + \)\(70\!\cdots\!14\)\( T^{8} - \)\(77\!\cdots\!04\)\( T^{10} + \)\(59\!\cdots\!92\)\( T^{12} - \)\(30\!\cdots\!52\)\( T^{14} + \)\(81\!\cdots\!81\)\( T^{16} \))
$41$ (\( 1 + 29282630 T + 327381934393961 T^{2} \))(\( 1 - 13285562 T + 327381934393961 T^{2} \))(\( 1 - 14211332 T + 443988635955862 T^{2} - \)\(46\!\cdots\!52\)\( T^{3} + \)\(10\!\cdots\!21\)\( T^{4} \))(\( 1 - 26060372 T + 693239183881142 T^{2} - \)\(85\!\cdots\!92\)\( T^{3} + \)\(10\!\cdots\!21\)\( T^{4} \))(\( ( 1 - 46660044 T + 1629598509234068 T^{2} - \)\(39\!\cdots\!32\)\( T^{3} + \)\(78\!\cdots\!54\)\( T^{4} - \)\(12\!\cdots\!52\)\( T^{5} + \)\(17\!\cdots\!28\)\( T^{6} - \)\(16\!\cdots\!64\)\( T^{7} + \)\(11\!\cdots\!41\)\( T^{8} )^{2} \))
$43$ (\( 1 + 1516724 T + 502592611936843 T^{2} \))(\( 1 + 23130764 T + 502592611936843 T^{2} \))(\( 1 - 27748920 T + 1170232974699430 T^{2} - \)\(13\!\cdots\!60\)\( T^{3} + \)\(25\!\cdots\!49\)\( T^{4} \))(\( 1 + 7191160 T + 750634586008230 T^{2} + \)\(36\!\cdots\!80\)\( T^{3} + \)\(25\!\cdots\!49\)\( T^{4} \))(\( 1 - 1288760154444440 T^{2} + \)\(10\!\cdots\!96\)\( T^{4} - \)\(57\!\cdots\!80\)\( T^{6} + \)\(31\!\cdots\!06\)\( T^{8} - \)\(14\!\cdots\!20\)\( T^{10} + \)\(65\!\cdots\!96\)\( T^{12} - \)\(20\!\cdots\!60\)\( T^{14} + \)\(40\!\cdots\!01\)\( T^{16} \))
$47$ (\( 1 - 615752 T + 1119130473102767 T^{2} \))(\( 1 + 13873688 T + 1119130473102767 T^{2} \))(\( 1 - 95966440 T + 4320659216802910 T^{2} - \)\(10\!\cdots\!80\)\( T^{3} + \)\(12\!\cdots\!89\)\( T^{4} \))(\( 1 + 31580240 T + 382042606129310 T^{2} + \)\(35\!\cdots\!80\)\( T^{3} + \)\(12\!\cdots\!89\)\( T^{4} \))(\( 1 - 4836473602067240 T^{2} + \)\(12\!\cdots\!56\)\( T^{4} - \)\(21\!\cdots\!80\)\( T^{6} + \)\(28\!\cdots\!26\)\( T^{8} - \)\(27\!\cdots\!20\)\( T^{10} + \)\(19\!\cdots\!76\)\( T^{12} - \)\(95\!\cdots\!60\)\( T^{14} + \)\(24\!\cdots\!41\)\( T^{16} \))
$53$ (\( 1 - 4747430 T + 3299763591802133 T^{2} \))(\( 1 + 57635174 T + 3299763591802133 T^{2} \))(\( 1 + 64305596 T + 6611083028543086 T^{2} + \)\(21\!\cdots\!68\)\( T^{3} + \)\(10\!\cdots\!89\)\( T^{4} \))(\( 1 - 3131116 T + 6584849973489806 T^{2} - \)\(10\!\cdots\!28\)\( T^{3} + \)\(10\!\cdots\!89\)\( T^{4} \))(\( 1 - 4341506689340012 T^{2} + \)\(22\!\cdots\!72\)\( T^{4} - \)\(59\!\cdots\!84\)\( T^{6} + \)\(28\!\cdots\!74\)\( T^{8} - \)\(64\!\cdots\!76\)\( T^{10} + \)\(26\!\cdots\!12\)\( T^{12} - \)\(56\!\cdots\!28\)\( T^{14} + \)\(14\!\cdots\!41\)\( T^{16} \))
$59$ (\( 1 - 60616076 T + 8662995818654939 T^{2} \))(\( 1 + 32042120 T + 8662995818654939 T^{2} \))(\( 1 - 187863136 T + 23071633420288438 T^{2} - \)\(16\!\cdots\!04\)\( T^{3} + \)\(75\!\cdots\!21\)\( T^{4} \))(\( 1 + 35494664 T + 7388006896329158 T^{2} + \)\(30\!\cdots\!96\)\( T^{3} + \)\(75\!\cdots\!21\)\( T^{4} \))(\( ( 1 - 118263018 T + 24386862278408432 T^{2} - \)\(17\!\cdots\!86\)\( T^{3} + \)\(26\!\cdots\!54\)\( T^{4} - \)\(15\!\cdots\!54\)\( T^{5} + \)\(18\!\cdots\!72\)\( T^{6} - \)\(76\!\cdots\!42\)\( T^{7} + \)\(56\!\cdots\!41\)\( T^{8} )^{2} \))
$61$ (\( 1 + 126745682 T + 11694146092834141 T^{2} \))(\( 1 - 110664022 T + 11694146092834141 T^{2} \))(\( 1 - 154080060 T + 23302683905802238 T^{2} - \)\(18\!\cdots\!60\)\( T^{3} + \)\(13\!\cdots\!81\)\( T^{4} \))(\( 1 - 341497340 T + 52053805546777278 T^{2} - \)\(39\!\cdots\!40\)\( T^{3} + \)\(13\!\cdots\!81\)\( T^{4} \))(\( ( 1 + 178713880 T + 47272829372304076 T^{2} + \)\(59\!\cdots\!20\)\( T^{3} + \)\(82\!\cdots\!06\)\( T^{4} + \)\(69\!\cdots\!20\)\( T^{5} + \)\(64\!\cdots\!56\)\( T^{6} + \)\(28\!\cdots\!80\)\( T^{7} + \)\(18\!\cdots\!61\)\( T^{8} )^{2} \))
$67$ (\( 1 + 111182652 T + 27206534396294947 T^{2} \))(\( 1 + 118568268 T + 27206534396294947 T^{2} \))(\( 1 - 33592376 T - 10819815556424362 T^{2} - \)\(91\!\cdots\!72\)\( T^{3} + \)\(74\!\cdots\!09\)\( T^{4} \))(\( 1 + 288195816 T + 74605839041196758 T^{2} + \)\(78\!\cdots\!52\)\( T^{3} + \)\(74\!\cdots\!09\)\( T^{4} \))(\( 1 - 128915336297443400 T^{2} + \)\(88\!\cdots\!36\)\( T^{4} - \)\(39\!\cdots\!00\)\( T^{6} + \)\(12\!\cdots\!86\)\( T^{8} - \)\(29\!\cdots\!00\)\( T^{10} + \)\(48\!\cdots\!16\)\( T^{12} - \)\(52\!\cdots\!00\)\( T^{14} + \)\(30\!\cdots\!61\)\( T^{16} \))
$71$ (\( 1 + 175551608 T + 45848500718449031 T^{2} \))(\( 1 - 276679712 T + 45848500718449031 T^{2} \))(\( 1 + 228270976 T + 45777616900481806 T^{2} + \)\(10\!\cdots\!56\)\( T^{3} + \)\(21\!\cdots\!61\)\( T^{4} \))(\( 1 - 210286064 T + 91549406631588686 T^{2} - \)\(96\!\cdots\!84\)\( T^{3} + \)\(21\!\cdots\!61\)\( T^{4} \))(\( ( 1 + 78445332 T + 132742682733898508 T^{2} + \)\(14\!\cdots\!24\)\( T^{3} + \)\(78\!\cdots\!70\)\( T^{4} + \)\(65\!\cdots\!44\)\( T^{5} + \)\(27\!\cdots\!88\)\( T^{6} + \)\(75\!\cdots\!12\)\( T^{7} + \)\(44\!\cdots\!21\)\( T^{8} )^{2} \))
$73$ (\( 1 + 61233350 T + 58871586708267913 T^{2} \))(\( 1 + 264023294 T + 58871586708267913 T^{2} \))(\( 1 + 33122316 T + 68371107952007926 T^{2} + \)\(19\!\cdots\!08\)\( T^{3} + \)\(34\!\cdots\!69\)\( T^{4} \))(\( 1 + 232663084 T + 130536207391012086 T^{2} + \)\(13\!\cdots\!92\)\( T^{3} + \)\(34\!\cdots\!69\)\( T^{4} \))(\( 1 - 112790784848235992 T^{2} + \)\(16\!\cdots\!32\)\( T^{4} - \)\(11\!\cdots\!84\)\( T^{6} + \)\(89\!\cdots\!94\)\( T^{8} - \)\(39\!\cdots\!96\)\( T^{10} + \)\(19\!\cdots\!52\)\( T^{12} - \)\(46\!\cdots\!28\)\( T^{14} + \)\(14\!\cdots\!21\)\( T^{16} \))
$79$ (\( 1 - 234431160 T + 119851595982618319 T^{2} \))(\( 1 - 448202760 T + 119851595982618319 T^{2} \))(\( 1 + 932406760 T + 453226630902929438 T^{2} + \)\(11\!\cdots\!40\)\( T^{3} + \)\(14\!\cdots\!61\)\( T^{4} \))(\( 1 + 24755040 T + 43090694479668638 T^{2} + \)\(29\!\cdots\!60\)\( T^{3} + \)\(14\!\cdots\!61\)\( T^{4} \))(\( ( 1 - 431961140 T + 452072920533003676 T^{2} - \)\(14\!\cdots\!80\)\( T^{3} + \)\(79\!\cdots\!66\)\( T^{4} - \)\(17\!\cdots\!20\)\( T^{5} + \)\(64\!\cdots\!36\)\( T^{6} - \)\(74\!\cdots\!60\)\( T^{7} + \)\(20\!\cdots\!21\)\( T^{8} )^{2} \))
$83$ (\( 1 - 118910388 T + 186940255267540403 T^{2} \))(\( 1 - 851015796 T + 186940255267540403 T^{2} \))(\( 1 - 207040152 T + 372783310330485238 T^{2} - \)\(38\!\cdots\!56\)\( T^{3} + \)\(34\!\cdots\!09\)\( T^{4} \))(\( 1 + 372082152 T + 379908828789982198 T^{2} + \)\(69\!\cdots\!56\)\( T^{3} + \)\(34\!\cdots\!09\)\( T^{4} \))(\( 1 - 981210392397464024 T^{2} + \)\(42\!\cdots\!20\)\( T^{4} - \)\(11\!\cdots\!28\)\( T^{6} + \)\(22\!\cdots\!98\)\( T^{8} - \)\(38\!\cdots\!52\)\( T^{10} + \)\(51\!\cdots\!20\)\( T^{12} - \)\(41\!\cdots\!96\)\( T^{14} + \)\(14\!\cdots\!61\)\( T^{16} \))
$89$ (\( 1 + 316534326 T + 350356403707485209 T^{2} \))(\( 1 - 189894930 T + 350356403707485209 T^{2} \))(\( 1 - 224518164 T + 610925899926766678 T^{2} - \)\(78\!\cdots\!76\)\( T^{3} + \)\(12\!\cdots\!81\)\( T^{4} \))(\( 1 + 427639116 T + 702028302670151638 T^{2} + \)\(14\!\cdots\!44\)\( T^{3} + \)\(12\!\cdots\!81\)\( T^{4} \))(\( ( 1 - 178691112 T + 708605882924008892 T^{2} - \)\(39\!\cdots\!44\)\( T^{3} + \)\(23\!\cdots\!94\)\( T^{4} - \)\(13\!\cdots\!96\)\( T^{5} + \)\(86\!\cdots\!52\)\( T^{6} - \)\(76\!\cdots\!48\)\( T^{7} + \)\(15\!\cdots\!61\)\( T^{8} )^{2} \))
$97$ (\( 1 - 242912258 T + 760231058654565217 T^{2} \))(\( 1 + 1014149278 T + 760231058654565217 T^{2} \))(\( 1 - 387134596 T - 734969029248610362 T^{2} - \)\(29\!\cdots\!32\)\( T^{3} + \)\(57\!\cdots\!89\)\( T^{4} \))(\( 1 - 1771658884 T + 2207048700436243398 T^{2} - \)\(13\!\cdots\!28\)\( T^{3} + \)\(57\!\cdots\!89\)\( T^{4} \))(\( 1 - 1313769052010852360 T^{2} + \)\(18\!\cdots\!56\)\( T^{4} - \)\(20\!\cdots\!20\)\( T^{6} + \)\(15\!\cdots\!26\)\( T^{8} - \)\(11\!\cdots\!80\)\( T^{10} + \)\(62\!\cdots\!76\)\( T^{12} - \)\(25\!\cdots\!40\)\( T^{14} + \)\(11\!\cdots\!41\)\( T^{16} \))
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