Properties

Label 224.3.r
Level 224
Weight 3
Character orbit r
Rep. character \(\chi_{224}(95,\cdot)\)
Character field \(\Q(\zeta_{6})\)
Dimension 32
Newform subspaces 4
Sturm bound 96
Trace bound 5

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Defining parameters

Level: \( N \) \(=\) \( 224 = 2^{5} \cdot 7 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 224.r (of order \(6\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 28 \)
Character field: \(\Q(\zeta_{6})\)
Newform subspaces: \( 4 \)
Sturm bound: \(96\)
Trace bound: \(5\)
Distinguishing \(T_p\): \(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{3}(224, [\chi])\).

Total New Old
Modular forms 144 32 112
Cusp forms 112 32 80
Eisenstein series 32 0 32

Trace form

\( 32q + 48q^{9} + O(q^{10}) \) \( 32q + 48q^{9} + 32q^{13} + 80q^{21} - 96q^{25} + 96q^{29} + 48q^{33} - 80q^{37} + 32q^{41} + 80q^{45} + 160q^{49} + 16q^{53} + 32q^{57} - 272q^{61} - 176q^{65} - 512q^{69} - 144q^{73} - 224q^{77} + 64q^{81} - 224q^{85} - 368q^{89} + 112q^{93} - 544q^{97} + O(q^{100}) \)

Decomposition of \(S_{3}^{\mathrm{new}}(224, [\chi])\) into newform subspaces

Label Dim. \(A\) Field CM Traces $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\)
224.3.r.a \(4\) \(6.104\) \(\Q(\zeta_{12})\) None \(0\) \(0\) \(-18\) \(0\) \(q+5\zeta_{12}q^{3}-9\zeta_{12}^{2}q^{5}+(8\zeta_{12}-3\zeta_{12}^{3})q^{7}+\cdots\)
224.3.r.b \(4\) \(6.104\) \(\Q(\zeta_{12})\) None \(0\) \(0\) \(-2\) \(0\) \(q+\zeta_{12}q^{3}-\zeta_{12}^{2}q^{5}-7\zeta_{12}^{3}q^{7}+\cdots\)
224.3.r.c \(12\) \(6.104\) \(\mathbb{Q}[x]/(x^{12} - \cdots)\) None \(0\) \(0\) \(2\) \(0\) \(q-\beta _{1}q^{3}+(\beta _{5}+\beta _{11})q^{5}+(\beta _{3}-\beta _{6}+\cdots)q^{7}+\cdots\)
224.3.r.d \(12\) \(6.104\) \(\mathbb{Q}[x]/(x^{12} - \cdots)\) None \(0\) \(0\) \(18\) \(0\) \(q+(\beta _{7}+\beta _{9}-\beta _{10})q^{3}+(-3\beta _{1}+\beta _{4}+\cdots)q^{5}+\cdots\)

Decomposition of \(S_{3}^{\mathrm{old}}(224, [\chi])\) into lower level spaces

\( S_{3}^{\mathrm{old}}(224, [\chi]) \cong \) \(S_{3}^{\mathrm{new}}(28, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(112, [\chi])\)\(^{\oplus 2}\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ (\( 1 - 7 T^{2} - 32 T^{4} - 567 T^{6} + 6561 T^{8} \))(\( 1 + 17 T^{2} + 208 T^{4} + 1377 T^{6} + 6561 T^{8} \))(\( 1 + 11 T^{2} - 117 T^{4} - 884 T^{6} + 18105 T^{8} + 64793 T^{10} - 1253162 T^{12} + 5248233 T^{14} + 118786905 T^{16} - 469793844 T^{18} - 5036466357 T^{20} + 38354628411 T^{22} + 282429536481 T^{24} \))(\( 1 + 27 T^{2} + 411 T^{4} + 2972 T^{6} - 423 T^{8} - 321159 T^{10} - 4031882 T^{12} - 26013879 T^{14} - 2775303 T^{16} + 1579442652 T^{18} + 17692202331 T^{20} + 94143178827 T^{22} + 282429536481 T^{24} \))
$5$ (\( ( 1 + 9 T + 56 T^{2} + 225 T^{3} + 625 T^{4} )^{2} \))(\( ( 1 + T - 24 T^{2} + 25 T^{3} + 625 T^{4} )^{2} \))(\( ( 1 - T - 29 T^{2} - 132 T^{3} + 201 T^{4} + 2405 T^{5} + 10726 T^{6} + 60125 T^{7} + 125625 T^{8} - 2062500 T^{9} - 11328125 T^{10} - 9765625 T^{11} + 244140625 T^{12} )^{2} \))(\( ( 1 - 9 T + 3 T^{2} + 140 T^{3} + 345 T^{4} - 3363 T^{5} + 5766 T^{6} - 84075 T^{7} + 215625 T^{8} + 2187500 T^{9} + 1171875 T^{10} - 87890625 T^{11} + 244140625 T^{12} )^{2} \))
$7$ (\( 1 - 94 T^{2} + 2401 T^{4} \))(\( ( 1 + 49 T^{2} )^{2} \))(\( 1 + 6 T^{2} + 4335 T^{4} + 71444 T^{6} + 10408335 T^{8} + 34588806 T^{10} + 13841287201 T^{12} \))(\( 1 - 90 T^{2} + 7791 T^{4} - 412972 T^{6} + 18706191 T^{8} - 518832090 T^{10} + 13841287201 T^{12} \))
$11$ (\( 1 + 233 T^{2} + 39648 T^{4} + 3411353 T^{6} + 214358881 T^{8} \))(\( 1 - 47 T^{2} - 12432 T^{4} - 688127 T^{6} + 214358881 T^{8} \))(\( 1 + 283 T^{2} + 46555 T^{4} + 3528284 T^{6} - 210042215 T^{8} - 117264051335 T^{10} - 18469284955402 T^{12} - 1716862975595735 T^{14} - 45024414170161415 T^{16} + 11073266626730676764 T^{18} + \)\(21\!\cdots\!55\)\( T^{20} + \)\(19\!\cdots\!83\)\( T^{22} + \)\(98\!\cdots\!41\)\( T^{24} \))(\( 1 + 363 T^{2} + 55851 T^{4} + 5892268 T^{6} + 719654457 T^{8} + 93270469209 T^{10} + 11000493019158 T^{12} + 1365572939688969 T^{14} + 154264324109182617 T^{16} + 18492461094445093228 T^{18} + \)\(25\!\cdots\!11\)\( T^{20} + \)\(24\!\cdots\!63\)\( T^{22} + \)\(98\!\cdots\!41\)\( T^{24} \))
$13$ (\( ( 1 - 16 T + 169 T^{2} )^{4} \))(\( ( 1 - 24 T + 169 T^{2} )^{4} \))(\( ( 1 + 32 T + 803 T^{2} + 11632 T^{3} + 135707 T^{4} + 913952 T^{5} + 4826809 T^{6} )^{4} \))(\( ( 1 + 339 T^{2} - 784 T^{3} + 57291 T^{4} + 4826809 T^{6} )^{4} \))
$17$ (\( ( 1 - 7 T - 240 T^{2} - 2023 T^{3} + 83521 T^{4} )^{2} \))(\( ( 1 + T - 288 T^{2} + 289 T^{3} + 83521 T^{4} )^{2} \))(\( ( 1 + 3 T - 181 T^{2} + 11996 T^{3} - 1671 T^{4} - 1164367 T^{5} + 69728918 T^{6} - 336502063 T^{7} - 139563591 T^{8} + 289554277724 T^{9} - 1262612096821 T^{10} + 6047981701347 T^{11} + 582622237229761 T^{12} )^{2} \))(\( ( 1 + 3 T - 261 T^{2} - 5460 T^{3} - 14535 T^{4} + 628833 T^{5} + 36283574 T^{6} + 181732737 T^{7} - 1213977735 T^{8} - 131791126740 T^{9} - 1820672692101 T^{10} + 6047981701347 T^{11} + 582622237229761 T^{12} )^{2} \))
$19$ (\( ( 1 - 46 T^{2} + 130321 T^{4} )( 1 + 647 T^{2} + 130321 T^{4} ) \))(\( 1 + 673 T^{2} + 322608 T^{4} + 87706033 T^{6} + 16983563041 T^{8} \))(\( 1 + 843 T^{2} + 169755 T^{4} + 10759612 T^{6} + 34418758617 T^{8} + 15878060416809 T^{10} + 3783625584664182 T^{12} + 2069244711578965689 T^{14} + \)\(58\!\cdots\!97\)\( T^{16} + \)\(23\!\cdots\!32\)\( T^{18} + \)\(48\!\cdots\!55\)\( T^{20} + \)\(31\!\cdots\!43\)\( T^{22} + \)\(48\!\cdots\!21\)\( T^{24} \))(\( 1 + 411 T^{2} - 128661 T^{4} - 89218804 T^{6} + 6305474169 T^{8} + 6575128475913 T^{10} + 1202874182390166 T^{12} + 856877318109458073 T^{14} + \)\(10\!\cdots\!29\)\( T^{16} - \)\(19\!\cdots\!44\)\( T^{18} - \)\(37\!\cdots\!41\)\( T^{20} + \)\(15\!\cdots\!11\)\( T^{22} + \)\(48\!\cdots\!21\)\( T^{24} \))
$23$ (\( 1 + 697 T^{2} + 205968 T^{4} + 195049177 T^{6} + 78310985281 T^{8} \))(\( 1 + 1009 T^{2} + 738240 T^{4} + 282359569 T^{6} + 78310985281 T^{8} \))(\( 1 + 2659 T^{2} + 3876571 T^{4} + 4127666252 T^{6} + 3503571279769 T^{8} + 2424695206910689 T^{10} + 1397846635041452342 T^{12} + \)\(67\!\cdots\!49\)\( T^{14} + \)\(27\!\cdots\!89\)\( T^{16} + \)\(90\!\cdots\!92\)\( T^{18} + \)\(23\!\cdots\!31\)\( T^{20} + \)\(45\!\cdots\!59\)\( T^{22} + \)\(48\!\cdots\!41\)\( T^{24} \))(\( 1 + 579 T^{2} + 62859 T^{4} - 35156516 T^{6} - 57866315751 T^{8} - 16658355952143 T^{10} + 6113539249932246 T^{12} - 4661690988003649263 T^{14} - \)\(45\!\cdots\!31\)\( T^{16} - \)\(77\!\cdots\!36\)\( T^{18} + \)\(38\!\cdots\!99\)\( T^{20} + \)\(99\!\cdots\!79\)\( T^{22} + \)\(48\!\cdots\!41\)\( T^{24} \))
$29$ (\( ( 1 + 32 T + 841 T^{2} )^{4} \))(\( ( 1 - 24 T + 841 T^{2} )^{4} \))(\( ( 1 - 32 T - 109 T^{2} + 37136 T^{3} - 91669 T^{4} - 22632992 T^{5} + 594823321 T^{6} )^{4} \))(\( ( 1 + 1347 T^{2} - 5488 T^{3} + 1132827 T^{4} + 594823321 T^{6} )^{4} \))
$31$ (\( 1 + 1801 T^{2} + 2320080 T^{4} + 1663261321 T^{6} + 852891037441 T^{8} \))(\( 1 + 241 T^{2} - 865440 T^{4} + 222568561 T^{6} + 852891037441 T^{8} \))(\( 1 + 2963 T^{2} + 4291147 T^{4} + 3752683836 T^{6} + 1856892008697 T^{8} - 613860419386879 T^{10} - 1734915478854125546 T^{12} - \)\(56\!\cdots\!59\)\( T^{14} + \)\(15\!\cdots\!77\)\( T^{16} + \)\(29\!\cdots\!96\)\( T^{18} + \)\(31\!\cdots\!07\)\( T^{20} + \)\(19\!\cdots\!63\)\( T^{22} + \)\(62\!\cdots\!21\)\( T^{24} \))(\( 1 + 5235 T^{2} + 15533691 T^{4} + 32305589068 T^{6} + 51858557764857 T^{8} + 66810392242083729 T^{10} + 70681395526523023542 T^{12} + \)\(61\!\cdots\!09\)\( T^{14} + \)\(44\!\cdots\!37\)\( T^{16} + \)\(25\!\cdots\!48\)\( T^{18} + \)\(11\!\cdots\!71\)\( T^{20} + \)\(35\!\cdots\!35\)\( T^{22} + \)\(62\!\cdots\!21\)\( T^{24} \))
$37$ (\( ( 1 - T - 1368 T^{2} - 1369 T^{3} + 1874161 T^{4} )^{2} \))(\( ( 1 - 49 T + 1032 T^{2} - 67081 T^{3} + 1874161 T^{4} )^{2} \))(\( ( 1 + 33 T - 1333 T^{2} - 122132 T^{3} - 783591 T^{4} + 87702163 T^{5} + 5266681334 T^{6} + 120064261147 T^{7} - 1468575692151 T^{8} - 313357297783988 T^{9} - 4682135112076693 T^{10} + 158683284289789017 T^{11} + 6582952005840035281 T^{12} )^{2} \))(\( ( 1 + 57 T - 1269 T^{2} - 55332 T^{3} + 5310729 T^{4} + 108467211 T^{5} - 4432135786 T^{6} + 148491611859 T^{7} + 9953161173369 T^{8} - 141966773662788 T^{9} - 4457336427025749 T^{10} + 274089309227817393 T^{11} + 6582952005840035281 T^{12} )^{2} \))
$41$ (\( ( 1 + 40 T + 1681 T^{2} )^{4} \))(\( ( 1 + 48 T + 1681 T^{2} )^{4} \))(\( ( 1 - 96 T + 7339 T^{2} - 328752 T^{3} + 12336859 T^{4} - 271273056 T^{5} + 4750104241 T^{6} )^{4} \))(\( ( 1 + 3195 T^{2} + 22736 T^{3} + 5370795 T^{4} + 4750104241 T^{6} )^{4} \))
$43$ (\( ( 1 - 2098 T^{2} + 3418801 T^{4} )^{2} \))(\( ( 1 - 3122 T^{2} + 3418801 T^{4} )^{2} \))(\( ( 1 - 5206 T^{2} + 17138783 T^{4} - 37018566196 T^{6} + 58594088459183 T^{8} - 60848770645190806 T^{10} + 39959630797262576401 T^{12} )^{2} \))(\( ( 1 - 5718 T^{2} + 18746463 T^{4} - 40352107060 T^{6} + 64090426450863 T^{8} - 66833129187322518 T^{10} + 39959630797262576401 T^{12} )^{2} \))
$47$ (\( 1 - 2807 T^{2} + 2999568 T^{4} - 13697264567 T^{6} + 23811286661761 T^{8} \))(\( 1 + 1393 T^{2} - 2939232 T^{4} + 6797395633 T^{6} + 23811286661761 T^{8} \))(\( 1 + 6995 T^{2} + 22181707 T^{4} + 50014251324 T^{6} + 105476166484281 T^{8} + 174006129657696449 T^{10} + \)\(27\!\cdots\!62\)\( T^{12} + \)\(84\!\cdots\!69\)\( T^{14} + \)\(25\!\cdots\!41\)\( T^{16} + \)\(58\!\cdots\!84\)\( T^{18} + \)\(12\!\cdots\!47\)\( T^{20} + \)\(19\!\cdots\!95\)\( T^{22} + \)\(13\!\cdots\!81\)\( T^{24} \))(\( 1 + 8307 T^{2} + 34424571 T^{4} + 100893075532 T^{6} + 253028343980217 T^{8} + 616841824418715921 T^{10} + \)\(14\!\cdots\!38\)\( T^{12} + \)\(30\!\cdots\!01\)\( T^{14} + \)\(60\!\cdots\!37\)\( T^{16} + \)\(11\!\cdots\!12\)\( T^{18} + \)\(19\!\cdots\!91\)\( T^{20} + \)\(22\!\cdots\!07\)\( T^{22} + \)\(13\!\cdots\!81\)\( T^{24} \))
$53$ (\( ( 1 + 7 T - 2760 T^{2} + 19663 T^{3} + 7890481 T^{4} )^{2} \))(\( ( 1 - 25 T - 2184 T^{2} - 70225 T^{3} + 7890481 T^{4} )^{2} \))(\( ( 1 + T - 2837 T^{2} - 53748 T^{3} + 54009 T^{4} + 72242131 T^{5} + 22496954230 T^{6} + 202928145979 T^{7} + 426156988329 T^{8} - 1191290081961492 T^{9} - 176630741697031157 T^{10} + 174887470365513049 T^{11} + \)\(49\!\cdots\!41\)\( T^{12} )^{2} \))(\( ( 1 + 9 T - 2997 T^{2} - 116484 T^{3} + 179145 T^{4} + 137286747 T^{5} + 22274291990 T^{6} + 385638472323 T^{7} + 1413540218745 T^{8} - 2581793441750436 T^{9} - 186592292162848917 T^{10} + 1573987233289617441 T^{11} + \)\(49\!\cdots\!41\)\( T^{12} )^{2} \))
$59$ (\( 1 + 4153 T^{2} + 5130048 T^{4} + 50323400233 T^{6} + 146830437604321 T^{8} \))(\( 1 + 6673 T^{2} + 32411568 T^{4} + 80859149953 T^{6} + 146830437604321 T^{8} \))(\( 1 + 18811 T^{2} + 200120491 T^{4} + 1494527327372 T^{6} + 8628303999799801 T^{8} + 40087667805525876841 T^{10} + \)\(15\!\cdots\!62\)\( T^{12} + \)\(48\!\cdots\!01\)\( T^{14} + \)\(12\!\cdots\!21\)\( T^{16} + \)\(26\!\cdots\!32\)\( T^{18} + \)\(43\!\cdots\!31\)\( T^{20} + \)\(49\!\cdots\!11\)\( T^{22} + \)\(31\!\cdots\!61\)\( T^{24} \))(\( 1 + 1707 T^{2} - 9128709 T^{4} - 112178055524 T^{6} - 114189887648679 T^{8} + 551687617569890985 T^{10} + \)\(57\!\cdots\!54\)\( T^{12} + \)\(66\!\cdots\!85\)\( T^{14} - \)\(16\!\cdots\!59\)\( T^{16} - \)\(19\!\cdots\!44\)\( T^{18} - \)\(19\!\cdots\!69\)\( T^{20} + \)\(44\!\cdots\!07\)\( T^{22} + \)\(31\!\cdots\!61\)\( T^{24} \))
$61$ (\( ( 1 + 79 T + 2520 T^{2} + 293959 T^{3} + 13845841 T^{4} )^{2} \))(\( ( 1 - T - 3720 T^{2} - 3721 T^{3} + 13845841 T^{4} )^{2} \))(\( ( 1 + 217 T + 20699 T^{2} + 1814604 T^{3} + 173780089 T^{4} + 12006150107 T^{5} + 685616953942 T^{6} + 44674884548147 T^{7} + 2406131481259849 T^{8} + 93489077396968044 T^{9} + 3968149671730719419 T^{10} + \)\(15\!\cdots\!17\)\( T^{11} + \)\(26\!\cdots\!21\)\( T^{12} )^{2} \))(\( ( 1 - 159 T + 7899 T^{2} - 401828 T^{3} + 49683657 T^{4} - 2161110333 T^{5} + 23418423222 T^{6} - 8041491549093 T^{7} + 687912015120537 T^{8} - 20702328988731908 T^{9} + 1514296065365522619 T^{10} - \)\(11\!\cdots\!59\)\( T^{11} + \)\(26\!\cdots\!21\)\( T^{12} )^{2} \))
$67$ (\( 1 + 8857 T^{2} + 58295328 T^{4} + 178478478697 T^{6} + 406067677556641 T^{8} \))(\( 1 + 4753 T^{2} + 2439888 T^{4} + 95778278113 T^{6} + 406067677556641 T^{8} \))(\( 1 + 9899 T^{2} + 24620203 T^{4} - 79278218964 T^{6} - 82489071404871 T^{8} + 4300819315073397209 T^{10} + \)\(30\!\cdots\!86\)\( T^{12} + \)\(86\!\cdots\!89\)\( T^{14} - \)\(33\!\cdots\!11\)\( T^{16} - \)\(64\!\cdots\!04\)\( T^{18} + \)\(40\!\cdots\!43\)\( T^{20} + \)\(32\!\cdots\!99\)\( T^{22} + \)\(66\!\cdots\!21\)\( T^{24} \))(\( 1 + 23547 T^{2} + 312061755 T^{4} + 2891423739964 T^{6} + 20714504811186393 T^{8} + \)\(12\!\cdots\!29\)\( T^{10} + \)\(59\!\cdots\!86\)\( T^{12} + \)\(24\!\cdots\!09\)\( T^{14} + \)\(84\!\cdots\!13\)\( T^{16} + \)\(23\!\cdots\!04\)\( T^{18} + \)\(51\!\cdots\!55\)\( T^{20} + \)\(78\!\cdots\!47\)\( T^{22} + \)\(66\!\cdots\!21\)\( T^{24} \))
$71$ (\( ( 1 - 7778 T^{2} + 25411681 T^{4} )^{2} \))(\( ( 1 - 866 T^{2} + 25411681 T^{4} )^{2} \))(\( ( 1 - 12262 T^{2} + 100517039 T^{4} - 635265930196 T^{6} + 2554306930132559 T^{8} - 7918229800135521382 T^{10} + \)\(16\!\cdots\!41\)\( T^{12} )^{2} \))(\( ( 1 - 10086 T^{2} + 56255151 T^{4} - 272956768468 T^{6} + 1429537951818831 T^{8} - 6513070116144745446 T^{10} + \)\(16\!\cdots\!41\)\( T^{12} )^{2} \))
$73$ (\( ( 1 + 46 T + 5329 T^{2} )^{2}( 1 + 97 T + 5329 T^{2} )^{2} \))(\( ( 1 + 95 T + 3696 T^{2} + 506255 T^{3} + 28398241 T^{4} )^{2} \))(\( ( 1 + 5 T - 12813 T^{2} - 167716 T^{3} + 95820777 T^{4} + 946458799 T^{5} - 562891511514 T^{6} + 5043678939871 T^{7} + 2721141518053257 T^{8} - 25381171096285924 T^{9} - 10333173157438859853 T^{10} + 21488129148517788245 T^{11} + \)\(22\!\cdots\!21\)\( T^{12} )^{2} \))(\( ( 1 - 171 T + 4275 T^{2} - 355908 T^{3} + 155672073 T^{4} - 8204564097 T^{5} - 20860196506 T^{6} - 43722122072913 T^{7} + 4420813046023593 T^{8} - 53861061810065412 T^{9} + 3447616892847196275 T^{10} - \)\(73\!\cdots\!79\)\( T^{11} + \)\(22\!\cdots\!21\)\( T^{12} )^{2} \))
$79$ (\( 1 + 11257 T^{2} + 87769968 T^{4} + 438461061817 T^{6} + 1517108809906561 T^{8} \))(\( 1 + 10801 T^{2} + 77711520 T^{4} + 420699824881 T^{6} + 1517108809906561 T^{8} \))(\( 1 + 28083 T^{2} + 421723515 T^{4} + 4556484807052 T^{6} + 39635396888713497 T^{8} + \)\(29\!\cdots\!89\)\( T^{10} + \)\(19\!\cdots\!22\)\( T^{12} + \)\(11\!\cdots\!09\)\( T^{14} + \)\(60\!\cdots\!17\)\( T^{16} + \)\(26\!\cdots\!32\)\( T^{18} + \)\(97\!\cdots\!15\)\( T^{20} + \)\(25\!\cdots\!83\)\( T^{22} + \)\(34\!\cdots\!81\)\( T^{24} \))(\( 1 + 16371 T^{2} + 121567851 T^{4} + 486780511900 T^{6} + 101914373627865 T^{8} - 22023984316104222399 T^{10} - \)\(21\!\cdots\!94\)\( T^{12} - \)\(85\!\cdots\!19\)\( T^{14} + \)\(15\!\cdots\!65\)\( T^{16} + \)\(28\!\cdots\!00\)\( T^{18} + \)\(27\!\cdots\!71\)\( T^{20} + \)\(14\!\cdots\!71\)\( T^{22} + \)\(34\!\cdots\!81\)\( T^{24} \))
$83$ (\( ( 1 - 13714 T^{2} + 47458321 T^{4} )^{2} \))(\( ( 1 - 8594 T^{2} + 47458321 T^{4} )^{2} \))(\( ( 1 - 29238 T^{2} + 396631039 T^{4} - 3342660983284 T^{6} + 18823443167425519 T^{8} - 65852520283281280758 T^{10} + \)\(10\!\cdots\!61\)\( T^{12} )^{2} \))(\( ( 1 - 29238 T^{2} + 403545087 T^{4} - 3424551164404 T^{6} + 19151572276818927 T^{8} - 65852520283281280758 T^{10} + \)\(10\!\cdots\!61\)\( T^{12} )^{2} \))
$89$ (\( ( 1 - 97 T + 1488 T^{2} - 768337 T^{3} + 62742241 T^{4} )^{2} \))(\( ( 1 + 95 T + 1104 T^{2} + 752495 T^{3} + 62742241 T^{4} )^{2} \))(\( ( 1 + 261 T + 22931 T^{2} + 2459452 T^{3} + 465950601 T^{4} + 40453273231 T^{5} + 2330338704422 T^{6} + 320430377262751 T^{7} + 29234784902036841 T^{8} + 1222301630016613372 T^{9} + 90269917903554419411 T^{10} + \)\(81\!\cdots\!61\)\( T^{11} + \)\(24\!\cdots\!21\)\( T^{12} )^{2} \))(\( ( 1 - 75 T - 7725 T^{2} + 660828 T^{3} + 10149225 T^{4} + 61763775 T^{5} - 80037126682 T^{6} + 489230861775 T^{7} + 636785120913225 T^{8} + 328419152543175708 T^{9} - 30410148524048575725 T^{10} - \)\(23\!\cdots\!75\)\( T^{11} + \)\(24\!\cdots\!21\)\( T^{12} )^{2} \))
$97$ (\( ( 1 + 88 T + 9409 T^{2} )^{4} \))(\( ( 1 - 144 T + 9409 T^{2} )^{4} \))(\( ( 1 + 192 T + 28555 T^{2} + 2621616 T^{3} + 268673995 T^{4} + 16997621952 T^{5} + 832972004929 T^{6} )^{4} \))(\( ( 1 + 28059 T^{2} - 784 T^{3} + 264007131 T^{4} + 832972004929 T^{6} )^{4} \))
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