Properties

Label 224.3.o
Level 224
Weight 3
Character orbit o
Rep. character \(\chi_{224}(79,\cdot)\)
Character field \(\Q(\zeta_{6})\)
Dimension 28
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.o (of order \(6\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 56 \)
Character field: \(\Q(\zeta_{6})\)
Newform subspaces: \( 4 \)
Sturm bound: \(96\)
Trace bound: \(5\)
Distinguishing \(T_p\): \(3\), \(5\)

Dimensions

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

Total New Old
Modular forms 144 36 108
Cusp forms 112 28 84
Eisenstein series 32 8 24

Trace form

\( 28q + 2q^{3} - 32q^{9} + O(q^{10}) \) \( 28q + 2q^{3} - 32q^{9} + 18q^{11} - 2q^{17} + 2q^{19} + 28q^{25} - 28q^{27} + 34q^{33} + 6q^{35} - 8q^{41} + 104q^{43} - 20q^{49} + 46q^{51} - 68q^{57} - 62q^{59} + 48q^{65} - 222q^{67} - 82q^{73} - 84q^{75} + 14q^{81} + 488q^{83} + 46q^{89} - 288q^{91} - 424q^{97} - 832q^{99} + O(q^{100}) \)

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.o.a \(2\) \(6.104\) \(\Q(\sqrt{-3}) \) None \(0\) \(1\) \(-9\) \(2\) \(q+\zeta_{6}q^{3}+(-3-3\zeta_{6})q^{5}+(-3+8\zeta_{6})q^{7}+\cdots\)
224.3.o.b \(2\) \(6.104\) \(\Q(\sqrt{-3}) \) None \(0\) \(1\) \(9\) \(-2\) \(q+\zeta_{6}q^{3}+(3+3\zeta_{6})q^{5}+(3-8\zeta_{6})q^{7}+\cdots\)
224.3.o.c \(12\) \(6.104\) \(\mathbb{Q}[x]/(x^{12} + \cdots)\) None \(0\) \(-6\) \(0\) \(0\) \(q+(-1+\beta _{2}-\beta _{8})q^{3}+(\beta _{1}-\beta _{4}+\beta _{5}+\cdots)q^{5}+\cdots\)
224.3.o.d \(12\) \(6.104\) \(\mathbb{Q}[x]/(x^{12} - \cdots)\) None \(0\) \(6\) \(0\) \(0\) \(q+(1-\beta _{3}+\beta _{6}+\beta _{7}-\beta _{10})q^{3}+(-\beta _{1}+\cdots)q^{5}+\cdots\)

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

\( S_{3}^{\mathrm{old}}(224, [\chi]) \cong \) \(S_{3}^{\mathrm{new}}(56, [\chi])\)\(^{\oplus 3}\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ (\( 1 - T - 8 T^{2} - 9 T^{3} + 81 T^{4} \))(\( 1 - T - 8 T^{2} - 9 T^{3} + 81 T^{4} \))(\( ( 1 + 3 T - 7 T^{2} - 22 T^{3} + 7 T^{4} - 61 T^{5} - 350 T^{6} - 549 T^{7} + 567 T^{8} - 16038 T^{9} - 45927 T^{10} + 177147 T^{11} + 531441 T^{12} )^{2} \))(\( ( 1 - 3 T + T^{2} + 14 T^{3} - 65 T^{4} + 37 T^{5} + 514 T^{6} + 333 T^{7} - 5265 T^{8} + 10206 T^{9} + 6561 T^{10} - 177147 T^{11} + 531441 T^{12} )^{2} \))
$5$ (\( 1 + 9 T + 52 T^{2} + 225 T^{3} + 625 T^{4} \))(\( 1 - 9 T + 52 T^{2} - 225 T^{3} + 625 T^{4} \))(\( 1 + 17 T^{2} - 437 T^{4} - 28164 T^{6} - 270975 T^{8} + 5916803 T^{10} + 456152134 T^{12} + 3698001875 T^{14} - 105849609375 T^{16} - 6875976562500 T^{18} - 66680908203125 T^{20} + 1621246337890625 T^{22} + 59604644775390625 T^{24} \))(\( 1 + 121 T^{2} + 7979 T^{4} + 380588 T^{6} + 14474897 T^{8} + 457025259 T^{10} + 12295779174 T^{12} + 285640786875 T^{14} + 5654256640625 T^{16} + 92916992187500 T^{18} + 1217498779296875 T^{20} + 11539459228515625 T^{22} + 59604644775390625 T^{24} \))
$7$ (\( 1 - 2 T + 49 T^{2} \))(\( 1 + 2 T + 49 T^{2} \))(\( 1 + 102 T^{2} + 8463 T^{4} + 426692 T^{6} + 20319663 T^{8} + 588009702 T^{10} + 13841287201 T^{12} \))(\( 1 - 186 T^{2} + 16527 T^{4} - 956284 T^{6} + 39681327 T^{8} - 1072252986 T^{10} + 13841287201 T^{12} \))
$11$ (\( 1 - 17 T + 168 T^{2} - 2057 T^{3} + 14641 T^{4} \))(\( 1 - 17 T + 168 T^{2} - 2057 T^{3} + 14641 T^{4} \))(\( ( 1 - 7 T - 193 T^{2} + 2268 T^{3} + 15415 T^{4} - 171941 T^{5} - 336170 T^{6} - 20804861 T^{7} + 225691015 T^{8} + 4017900348 T^{9} - 41371264033 T^{10} - 181561972207 T^{11} + 3138428376721 T^{12} )^{2} \))(\( ( 1 + 15 T - 129 T^{2} - 924 T^{3} + 35727 T^{4} + 38013 T^{5} - 5198650 T^{6} + 4599573 T^{7} + 523079007 T^{8} - 1636922364 T^{9} - 27652295649 T^{10} + 389061369015 T^{11} + 3138428376721 T^{12} )^{2} \))
$13$ (\( ( 1 - 22 T + 169 T^{2} )( 1 + 22 T + 169 T^{2} ) \))(\( ( 1 - 22 T + 169 T^{2} )( 1 + 22 T + 169 T^{2} ) \))(\( ( 1 - 950 T^{2} + 386271 T^{4} - 85941828 T^{6} + 11032286031 T^{8} - 774944184950 T^{10} + 23298085122481 T^{12} )^{2} \))(\( ( 1 - 86 T^{2} + 60895 T^{4} - 4902788 T^{6} + 1739222095 T^{8} - 70152842006 T^{10} + 23298085122481 T^{12} )^{2} \))
$17$ (\( 1 - 25 T + 336 T^{2} - 7225 T^{3} + 83521 T^{4} \))(\( 1 - 25 T + 336 T^{2} - 7225 T^{3} + 83521 T^{4} \))(\( ( 1 + 41 T + 471 T^{2} + 5456 T^{3} + 209253 T^{4} + 1462951 T^{5} - 27854634 T^{6} + 422792839 T^{7} + 17477019813 T^{8} + 131694576464 T^{9} + 3285581754711 T^{10} + 82655749918409 T^{11} + 582622237229761 T^{12} )^{2} \))(\( ( 1 - 15 T - 665 T^{2} + 3920 T^{3} + 405365 T^{4} - 1221665 T^{5} - 124870762 T^{6} - 353061185 T^{7} + 33856490165 T^{8} + 94619270480 T^{9} - 4638878698265 T^{10} - 30239908506735 T^{11} + 582622237229761 T^{12} )^{2} \))
$19$ (\( 1 + 7 T - 312 T^{2} + 2527 T^{3} + 130321 T^{4} \))(\( 1 + 7 T - 312 T^{2} + 2527 T^{3} + 130321 T^{4} \))(\( ( 1 - 47 T + 487 T^{2} - 8532 T^{3} + 642195 T^{4} - 10028861 T^{5} + 46257550 T^{6} - 3620418821 T^{7} + 83691494595 T^{8} - 401395456692 T^{9} + 8270995200967 T^{10} - 288160114116647 T^{11} + 2213314919066161 T^{12} )^{2} \))(\( ( 1 + 39 T + 151 T^{2} - 3964 T^{3} + 263147 T^{4} + 5820949 T^{5} + 41830430 T^{6} + 2101362589 T^{7} + 34293580187 T^{8} - 186489872284 T^{9} + 2564518019191 T^{10} + 239111584054239 T^{11} + 2213314919066161 T^{12} )^{2} \))
$23$ (\( 1 + 9 T + 556 T^{2} + 4761 T^{3} + 279841 T^{4} \))(\( 1 - 9 T + 556 T^{2} - 4761 T^{3} + 279841 T^{4} \))(\( 1 + 2381 T^{2} + 2947123 T^{4} + 2827993812 T^{6} + 2298643835613 T^{8} + 1531108455942719 T^{10} + 862820921151363886 T^{12} + \)\(42\!\cdots\!79\)\( T^{14} + \)\(18\!\cdots\!53\)\( T^{16} + \)\(61\!\cdots\!52\)\( T^{18} + \)\(18\!\cdots\!03\)\( T^{20} + \)\(40\!\cdots\!81\)\( T^{22} + \)\(48\!\cdots\!41\)\( T^{24} \))(\( 1 + 693 T^{2} - 15581 T^{4} - 315659500 T^{6} - 121343121715 T^{8} + 25399919636503 T^{10} + 46081428374834798 T^{12} + 7107938910998636023 T^{14} - \)\(95\!\cdots\!15\)\( T^{16} - \)\(69\!\cdots\!00\)\( T^{18} - \)\(95\!\cdots\!41\)\( T^{20} + \)\(11\!\cdots\!93\)\( T^{22} + \)\(48\!\cdots\!41\)\( T^{24} \))
$29$ (\( 1 - 1490 T^{2} + 707281 T^{4} \))(\( 1 - 1490 T^{2} + 707281 T^{4} \))(\( ( 1 - 1838 T^{2} + 2723567 T^{4} - 2423469812 T^{6} + 1926327191327 T^{8} - 919452907022318 T^{10} + 353814783205469041 T^{12} )^{2} \))(\( ( 1 - 3662 T^{2} + 6471151 T^{4} - 6858243380 T^{6} + 4576922150431 T^{8} - 1831902364263182 T^{10} + 353814783205469041 T^{12} )^{2} \))
$31$ (\( 1 + 57 T + 2044 T^{2} + 54777 T^{3} + 923521 T^{4} \))(\( 1 - 57 T + 2044 T^{2} - 54777 T^{3} + 923521 T^{4} \))(\( 1 + 3329 T^{2} + 4631887 T^{4} + 6475839496 T^{6} + 10686875408317 T^{8} + 11546553377441207 T^{10} + 9928800654054771830 T^{12} + \)\(10\!\cdots\!47\)\( T^{14} + \)\(91\!\cdots\!97\)\( T^{16} + \)\(51\!\cdots\!56\)\( T^{18} + \)\(33\!\cdots\!47\)\( T^{20} + \)\(22\!\cdots\!29\)\( T^{22} + \)\(62\!\cdots\!21\)\( T^{24} \))(\( 1 + 3561 T^{2} + 5838879 T^{4} + 8280992264 T^{6} + 11856779149293 T^{8} + 13054196823637455 T^{10} + 12193464685853753718 T^{12} + \)\(12\!\cdots\!55\)\( T^{14} + \)\(10\!\cdots\!13\)\( T^{16} + \)\(65\!\cdots\!04\)\( T^{18} + \)\(42\!\cdots\!99\)\( T^{20} + \)\(23\!\cdots\!61\)\( T^{22} + \)\(62\!\cdots\!21\)\( T^{24} \))
$37$ (\( 1 - 15 T + 1444 T^{2} - 20535 T^{3} + 1874161 T^{4} \))(\( 1 + 15 T + 1444 T^{2} + 20535 T^{3} + 1874161 T^{4} \))(\( 1 + 6401 T^{2} + 22091395 T^{4} + 55331146644 T^{6} + 111395987301489 T^{8} + 188098942339055675 T^{10} + \)\(27\!\cdots\!06\)\( T^{12} + \)\(35\!\cdots\!75\)\( T^{14} + \)\(39\!\cdots\!69\)\( T^{16} + \)\(36\!\cdots\!64\)\( T^{18} + \)\(27\!\cdots\!95\)\( T^{20} + \)\(14\!\cdots\!01\)\( T^{22} + \)\(43\!\cdots\!61\)\( T^{24} \))(\( 1 + 5785 T^{2} + 17889827 T^{4} + 38047414052 T^{6} + 62842022068961 T^{8} + 88030006321881747 T^{10} + \)\(11\!\cdots\!14\)\( T^{12} + \)\(16\!\cdots\!67\)\( T^{14} + \)\(22\!\cdots\!81\)\( T^{16} + \)\(25\!\cdots\!12\)\( T^{18} + \)\(22\!\cdots\!07\)\( T^{20} + \)\(13\!\cdots\!85\)\( T^{22} + \)\(43\!\cdots\!61\)\( T^{24} \))
$41$ (\( ( 1 - 26 T + 1681 T^{2} )^{2} \))(\( ( 1 - 26 T + 1681 T^{2} )^{2} \))(\( ( 1 - 30 T + 4755 T^{2} - 100496 T^{3} + 7993155 T^{4} - 84772830 T^{5} + 4750104241 T^{6} )^{4} \))(\( ( 1 + 58 T + 3139 T^{2} + 88960 T^{3} + 5276659 T^{4} + 163894138 T^{5} + 4750104241 T^{6} )^{4} \))
$43$ (\( ( 1 + 14 T + 1849 T^{2} )^{2} \))(\( ( 1 + 14 T + 1849 T^{2} )^{2} \))(\( ( 1 + 10 T + 3855 T^{2} + 53388 T^{3} + 7127895 T^{4} + 34188010 T^{5} + 6321363049 T^{6} )^{4} \))(\( ( 1 - 50 T + 4047 T^{2} - 107900 T^{3} + 7482903 T^{4} - 170940050 T^{5} + 6321363049 T^{6} )^{4} \))
$47$ (\( 1 - 87 T + 4732 T^{2} - 192183 T^{3} + 4879681 T^{4} \))(\( 1 + 87 T + 4732 T^{2} + 192183 T^{3} + 4879681 T^{4} \))(\( 1 + 8313 T^{2} + 34702703 T^{4} + 101154992824 T^{6} + 247422958854189 T^{8} + 586359622432093519 T^{10} + \)\(13\!\cdots\!26\)\( T^{12} + \)\(28\!\cdots\!39\)\( T^{14} + \)\(58\!\cdots\!29\)\( T^{16} + \)\(11\!\cdots\!84\)\( T^{18} + \)\(19\!\cdots\!63\)\( T^{20} + \)\(22\!\cdots\!13\)\( T^{22} + \)\(13\!\cdots\!81\)\( T^{24} \))(\( 1 + 3905 T^{2} + 6537087 T^{4} + 2079696952 T^{6} - 28977286497379 T^{8} - 99900087904130553 T^{10} - \)\(24\!\cdots\!46\)\( T^{12} - \)\(48\!\cdots\!93\)\( T^{14} - \)\(68\!\cdots\!19\)\( T^{16} + \)\(24\!\cdots\!32\)\( T^{18} + \)\(37\!\cdots\!27\)\( T^{20} + \)\(10\!\cdots\!05\)\( T^{22} + \)\(13\!\cdots\!81\)\( T^{24} \))
$53$ (\( ( 1 - 53 T )^{2}( 1 - 53 T + 2809 T^{2} ) \))(\( ( 1 + 53 T )^{2}( 1 + 53 T + 2809 T^{2} ) \))(\( 1 + 9273 T^{2} + 34839939 T^{4} + 130830291652 T^{6} + 620089921398753 T^{8} + 1876428047292173523 T^{10} + \)\(44\!\cdots\!94\)\( T^{12} + \)\(14\!\cdots\!63\)\( T^{14} + \)\(38\!\cdots\!33\)\( T^{16} + \)\(64\!\cdots\!32\)\( T^{18} + \)\(13\!\cdots\!19\)\( T^{20} + \)\(28\!\cdots\!73\)\( T^{22} + \)\(24\!\cdots\!81\)\( T^{24} \))(\( 1 + 10561 T^{2} + 59733731 T^{4} + 209884512884 T^{6} + 494098524300977 T^{8} + 757507316642438811 T^{10} + \)\(12\!\cdots\!10\)\( T^{12} + \)\(59\!\cdots\!91\)\( T^{14} + \)\(30\!\cdots\!97\)\( T^{16} + \)\(10\!\cdots\!44\)\( T^{18} + \)\(23\!\cdots\!51\)\( T^{20} + \)\(32\!\cdots\!61\)\( T^{22} + \)\(24\!\cdots\!81\)\( T^{24} \))
$59$ (\( 1 + 55 T - 456 T^{2} + 191455 T^{3} + 12117361 T^{4} \))(\( 1 + 55 T - 456 T^{2} + 191455 T^{3} + 12117361 T^{4} \))(\( ( 1 + 31 T - 7567 T^{2} - 176030 T^{3} + 36214647 T^{4} + 412965247 T^{5} - 131958052142 T^{6} + 1437532024807 T^{7} + 438825951186567 T^{8} - 7425039336825230 T^{9} - 1111065921351897007 T^{10} + 15844619352319883431 T^{11} + \)\(17\!\cdots\!81\)\( T^{12} )^{2} \))(\( ( 1 - 55 T - 7367 T^{2} + 181142 T^{3} + 51119807 T^{4} - 598293727 T^{5} - 186579818926 T^{6} - 2082660463687 T^{7} + 619437155669327 T^{8} + 7640666224798022 T^{9} - 1081699833831032807 T^{10} - 28111421431535277055 T^{11} + \)\(17\!\cdots\!81\)\( T^{12} )^{2} \))
$61$ (\( 1 - 39 T + 4228 T^{2} - 145119 T^{3} + 13845841 T^{4} \))(\( 1 + 39 T + 4228 T^{2} + 145119 T^{3} + 13845841 T^{4} \))(\( 1 - 503 T^{2} - 13520125 T^{4} - 91919186268 T^{6} + 20373908662561 T^{8} + 680714329682975843 T^{10} + \)\(44\!\cdots\!06\)\( T^{12} + \)\(94\!\cdots\!63\)\( T^{14} + \)\(39\!\cdots\!41\)\( T^{16} - \)\(24\!\cdots\!28\)\( T^{18} - \)\(49\!\cdots\!25\)\( T^{20} - \)\(25\!\cdots\!03\)\( T^{22} + \)\(70\!\cdots\!41\)\( T^{24} \))(\( 1 + 9201 T^{2} + 56941411 T^{4} + 154999649300 T^{6} - 75695642240335 T^{8} - 3911957905117164149 T^{10} - \)\(19\!\cdots\!14\)\( T^{12} - \)\(54\!\cdots\!09\)\( T^{14} - \)\(14\!\cdots\!35\)\( T^{16} + \)\(41\!\cdots\!00\)\( T^{18} + \)\(20\!\cdots\!71\)\( T^{20} + \)\(46\!\cdots\!01\)\( T^{22} + \)\(70\!\cdots\!41\)\( T^{24} \))
$67$ (\( 1 - 17 T - 4200 T^{2} - 76313 T^{3} + 20151121 T^{4} \))(\( 1 - 17 T - 4200 T^{2} - 76313 T^{3} + 20151121 T^{4} \))(\( ( 1 - 89 T - 19 T^{2} + 251662 T^{3} - 15244389 T^{4} + 746405371 T^{5} - 32107721582 T^{6} + 3350613710419 T^{7} - 307191527310069 T^{8} + 22764937373414878 T^{9} - 7715285873576179 T^{10} - \)\(16\!\cdots\!61\)\( T^{11} + \)\(81\!\cdots\!61\)\( T^{12} )^{2} \))(\( ( 1 + 217 T + 18053 T^{2} + 1666970 T^{3} + 209974835 T^{4} + 15078221277 T^{5} + 815515698066 T^{6} + 67686135312453 T^{7} + 4231228307040035 T^{8} + 150791409324257930 T^{9} + 7330739782930039973 T^{10} + \)\(39\!\cdots\!33\)\( T^{11} + \)\(81\!\cdots\!61\)\( T^{12} )^{2} \))
$71$ (\( ( 1 - 71 T )^{2}( 1 + 71 T )^{2} \))(\( ( 1 - 71 T )^{2}( 1 + 71 T )^{2} \))(\( ( 1 - 7830 T^{2} + 72414895 T^{4} - 385201987444 T^{6} + 1840184211388495 T^{8} - 5056250149654308630 T^{10} + \)\(16\!\cdots\!41\)\( T^{12} )^{2} \))(\( ( 1 - 23062 T^{2} + 245626031 T^{4} - 1559141837940 T^{6} + 6241770345068111 T^{8} - 14892367937589740182 T^{10} + \)\(16\!\cdots\!41\)\( T^{12} )^{2} \))
$73$ (\( 1 + 119 T + 8832 T^{2} + 634151 T^{3} + 28398241 T^{4} \))(\( 1 + 119 T + 8832 T^{2} + 634151 T^{3} + 28398241 T^{4} \))(\( ( 1 - 27 T - 12877 T^{2} + 216188 T^{3} + 104738937 T^{4} - 821298961 T^{5} - 619422939130 T^{6} - 4376702163169 T^{7} + 2974401575009817 T^{8} + 32716643712966332 T^{9} - 10384786603320081037 T^{10} - \)\(11\!\cdots\!23\)\( T^{11} + \)\(22\!\cdots\!21\)\( T^{12} )^{2} \))(\( ( 1 - 51 T - 9165 T^{2} + 579660 T^{3} + 45654729 T^{4} - 1993134921 T^{5} - 160997144218 T^{6} - 10621415994009 T^{7} + 1296513996931689 T^{8} + 87722397610681740 T^{9} - 7391206742209252365 T^{10} - \)\(21\!\cdots\!99\)\( T^{11} + \)\(22\!\cdots\!21\)\( T^{12} )^{2} \))
$79$ (\( 1 + 129 T + 11788 T^{2} + 805089 T^{3} + 38950081 T^{4} \))(\( 1 - 129 T + 11788 T^{2} - 805089 T^{3} + 38950081 T^{4} \))(\( 1 + 27485 T^{2} + 389982499 T^{4} + 4285373244996 T^{6} + 40173777270526125 T^{8} + \)\(31\!\cdots\!47\)\( T^{10} + \)\(20\!\cdots\!50\)\( T^{12} + \)\(12\!\cdots\!07\)\( T^{14} + \)\(60\!\cdots\!25\)\( T^{16} + \)\(25\!\cdots\!36\)\( T^{18} + \)\(89\!\cdots\!79\)\( T^{20} + \)\(24\!\cdots\!85\)\( T^{22} + \)\(34\!\cdots\!81\)\( T^{24} \))(\( 1 + 16693 T^{2} + 149251283 T^{4} + 640487711012 T^{6} - 978609215845699 T^{8} - 44625107831251066425 T^{10} - \)\(37\!\cdots\!74\)\( T^{12} - \)\(17\!\cdots\!25\)\( T^{14} - \)\(14\!\cdots\!39\)\( T^{16} + \)\(37\!\cdots\!92\)\( T^{18} + \)\(34\!\cdots\!43\)\( T^{20} + \)\(14\!\cdots\!93\)\( T^{22} + \)\(34\!\cdots\!81\)\( T^{24} \))
$83$ (\( ( 1 + 110 T + 6889 T^{2} )^{2} \))(\( ( 1 + 110 T + 6889 T^{2} )^{2} \))(\( ( 1 - 98 T + 12183 T^{2} - 526716 T^{3} + 83928687 T^{4} - 4650915458 T^{5} + 326940373369 T^{6} )^{4} \))(\( ( 1 - 134 T + 22583 T^{2} - 1661172 T^{3} + 155574287 T^{4} - 6359415014 T^{5} + 326940373369 T^{6} )^{4} \))
$89$ (\( 1 + 71 T - 2880 T^{2} + 562391 T^{3} + 62742241 T^{4} \))(\( 1 + 71 T - 2880 T^{2} + 562391 T^{3} + 62742241 T^{4} \))(\( ( 1 + 13 T - 19181 T^{2} - 141748 T^{3} + 218594713 T^{4} + 635752183 T^{5} - 1943770650106 T^{6} + 5035793041543 T^{7} + 13715122164371833 T^{8} - 70446104031139828 T^{9} - 75507709882171615661 T^{10} + \)\(40\!\cdots\!13\)\( T^{11} + \)\(24\!\cdots\!21\)\( T^{12} )^{2} \))(\( ( 1 - 107 T + 1395 T^{2} + 497084 T^{3} - 62702935 T^{4} + 2422278015 T^{5} + 93422604838 T^{6} + 19186864156815 T^{7} - 3934122659177335 T^{8} + 247041448036057724 T^{9} + 5491541383954402995 T^{10} - \)\(33\!\cdots\!07\)\( T^{11} + \)\(24\!\cdots\!21\)\( T^{12} )^{2} \))
$97$ (\( ( 1 + 22 T + 9409 T^{2} )^{2} \))(\( ( 1 + 22 T + 9409 T^{2} )^{2} \))(\( ( 1 + 46 T + 18423 T^{2} + 458352 T^{3} + 173342007 T^{4} + 4072346926 T^{5} + 832972004929 T^{6} )^{4} \))(\( ( 1 + 38 T + 25191 T^{2} + 597344 T^{3} + 237022119 T^{4} + 3364112678 T^{5} + 832972004929 T^{6} )^{4} \))
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