Properties

Label 1008.2.cc
Level 1008
Weight 2
Character orbit cc
Rep. character \(\chi_{1008}(209,\cdot)\)
Character field \(\Q(\zeta_{6})\)
Dimension 92
Newform subspaces 4
Sturm bound 384
Trace bound 7

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Defining parameters

Level: \( N \) = \( 1008 = 2^{4} \cdot 3^{2} \cdot 7 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 1008.cc (of order \(6\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) = \( 63 \)
Character field: \(\Q(\zeta_{6})\)
Newform subspaces: \( 4 \)
Sturm bound: \(384\)
Trace bound: \(7\)
Distinguishing \(T_p\): \(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(1008, [\chi])\).

Total New Old
Modular forms 408 100 308
Cusp forms 360 92 268
Eisenstein series 48 8 40

Trace form

\( 92q + q^{7} - 4q^{9} + O(q^{10}) \) \( 92q + q^{7} - 4q^{9} + 6q^{11} - 2q^{15} - q^{21} + 6q^{23} - 40q^{25} - 30q^{29} - 8q^{37} - 14q^{39} - 10q^{43} - q^{49} + 24q^{51} + 16q^{57} - 7q^{63} + 18q^{65} + 2q^{67} - 27q^{77} + 2q^{79} - 12q^{81} - 12q^{85} + 30q^{91} - 26q^{93} + 84q^{95} - 38q^{99} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(1008, [\chi])\) into newform subspaces

Label Dim. \(A\) Field CM Traces $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\)
1008.2.cc.a \(12\) \(8.049\) \(\mathbb{Q}[x]/(x^{12} - \cdots)\) None \(0\) \(0\) \(0\) \(2\) \(q-\beta _{8}q^{3}-\beta _{11}q^{5}+(-\beta _{2}-\beta _{3}+\beta _{4}+\cdots)q^{7}+\cdots\)
1008.2.cc.b \(16\) \(8.049\) \(\mathbb{Q}[x]/(x^{16} - \cdots)\) None \(0\) \(0\) \(0\) \(-2\) \(q-\beta _{9}q^{3}+(-\beta _{9}-\beta _{13}+\beta _{15})q^{5}+\cdots\)
1008.2.cc.c \(16\) \(8.049\) \(\mathbb{Q}[x]/(x^{16} - \cdots)\) None \(0\) \(0\) \(0\) \(1\) \(q+(-\beta _{1}-\beta _{8})q^{3}-\beta _{15}q^{5}+(\beta _{10}-\beta _{12}+\cdots)q^{7}+\cdots\)
1008.2.cc.d \(48\) \(8.049\) None \(0\) \(0\) \(0\) \(0\)

Decomposition of \(S_{2}^{\mathrm{old}}(1008, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(1008, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(63, [\chi])\)\(^{\oplus 5}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(126, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(252, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(504, [\chi])\)\(^{\oplus 2}\)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ 1
$3$ (\( 1 + 6 T^{2} + 12 T^{4} + 9 T^{6} + 108 T^{8} + 486 T^{10} + 729 T^{12} \))(\( 1 - 6 T^{2} + 9 T^{4} + 54 T^{6} - 288 T^{8} + 486 T^{10} + 729 T^{12} - 4374 T^{14} + 6561 T^{16} \))(\( 1 + 9 T^{4} - 45 T^{6} - 18 T^{8} - 405 T^{10} + 729 T^{12} + 6561 T^{16} \))
$5$ (\( 1 - 15 T^{2} + 84 T^{4} - 457 T^{6} + 3921 T^{8} - 20514 T^{10} + 80841 T^{12} - 512850 T^{14} + 2450625 T^{16} - 7140625 T^{18} + 32812500 T^{20} - 146484375 T^{22} + 244140625 T^{24} \))(\( 1 - 16 T^{2} + 123 T^{4} - 584 T^{6} + 1481 T^{8} + 1416 T^{10} - 59414 T^{12} + 576968 T^{14} - 3477114 T^{16} + 14424200 T^{18} - 37133750 T^{20} + 22125000 T^{22} + 578515625 T^{24} - 5703125000 T^{26} + 30029296875 T^{28} - 97656250000 T^{30} + 152587890625 T^{32} \))(\( 1 - 16 T^{2} + 159 T^{4} - 998 T^{6} + 4298 T^{8} - 12066 T^{10} + 21316 T^{12} - 64633 T^{14} + 271566 T^{16} - 1615825 T^{18} + 13322500 T^{20} - 188531250 T^{22} + 1678906250 T^{24} - 9746093750 T^{26} + 38818359375 T^{28} - 97656250000 T^{30} + 152587890625 T^{32} \))
$7$ (\( 1 - 2 T - T^{2} + 30 T^{3} - 50 T^{4} - 58 T^{5} + 715 T^{6} - 406 T^{7} - 2450 T^{8} + 10290 T^{9} - 2401 T^{10} - 33614 T^{11} + 117649 T^{12} \))(\( 1 + 2 T + 6 T^{2} - 8 T^{3} - 58 T^{4} - 222 T^{5} - 104 T^{6} + 662 T^{7} + 3483 T^{8} + 4634 T^{9} - 5096 T^{10} - 76146 T^{11} - 139258 T^{12} - 134456 T^{13} + 705894 T^{14} + 1647086 T^{15} + 5764801 T^{16} \))(\( 1 - T + 3 T^{2} - 20 T^{3} + 53 T^{4} - 21 T^{5} - 74 T^{6} - 355 T^{7} - 1314 T^{8} - 2485 T^{9} - 3626 T^{10} - 7203 T^{11} + 127253 T^{12} - 336140 T^{13} + 352947 T^{14} - 823543 T^{15} + 5764801 T^{16} \))
$11$ (\( ( 1 + 22 T^{2} + 242 T^{4} - 66 T^{5} + 2665 T^{6} - 726 T^{7} + 29282 T^{8} + 322102 T^{10} + 1771561 T^{12} )^{2} \))(\( ( 1 - 6 T + 35 T^{2} - 138 T^{3} + 481 T^{4} - 1512 T^{5} + 3854 T^{6} - 13116 T^{7} + 37618 T^{8} - 144276 T^{9} + 466334 T^{10} - 2012472 T^{11} + 7042321 T^{12} - 22225038 T^{13} + 62004635 T^{14} - 116923026 T^{15} + 214358881 T^{16} )^{2} \))(\( ( 1 + 3 T + 26 T^{2} + 69 T^{3} + 265 T^{4} + 756 T^{5} + 2585 T^{6} + 8259 T^{7} + 34846 T^{8} + 90849 T^{9} + 312785 T^{10} + 1006236 T^{11} + 3879865 T^{12} + 11112519 T^{13} + 46060586 T^{14} + 58461513 T^{15} + 214358881 T^{16} )^{2} \))
$13$ (\( 1 + 33 T^{2} + 351 T^{4} + 2144 T^{6} + 53457 T^{8} + 906351 T^{10} + 9826878 T^{12} + 153173319 T^{14} + 1526785377 T^{16} + 10348678496 T^{18} + 286321483071 T^{20} + 4549330231017 T^{22} + 23298085122481 T^{24} \))(\( 1 + 68 T^{2} + 2376 T^{4} + 57352 T^{6} + 1082018 T^{8} + 16951644 T^{10} + 232506496 T^{12} + 2987085740 T^{14} + 38351015667 T^{16} + 504817490060 T^{18} + 6640618032256 T^{20} + 81822347823996 T^{22} + 882635323274978 T^{24} + 7906460224523848 T^{26} + 55356250251014856 T^{28} + 267741594227551652 T^{30} + 665416609183179841 T^{32} \))(\( 1 + 56 T^{2} + 1590 T^{4} + 26848 T^{6} + 274865 T^{8} + 1615656 T^{10} + 12863830 T^{12} + 363405056 T^{14} + 6601006116 T^{16} + 61415454464 T^{18} + 367403848630 T^{20} + 7798462921704 T^{22} + 224215824627665 T^{24} + 3701224789161952 T^{26} + 37043955344744790 T^{28} + 220493077599160184 T^{30} + 665416609183179841 T^{32} \))
$17$ (\( ( 1 + 48 T^{2} + 1338 T^{4} + 26845 T^{6} + 386682 T^{8} + 4009008 T^{10} + 24137569 T^{12} )^{2} \))(\( ( 1 + 94 T^{2} + 4285 T^{4} + 124198 T^{6} + 2503180 T^{8} + 35893222 T^{10} + 357887485 T^{12} + 2268931486 T^{14} + 6975757441 T^{16} )^{2} \))(\( ( 1 + 58 T^{2} + 1603 T^{4} + 30013 T^{6} + 497674 T^{8} + 8673757 T^{10} + 133884163 T^{12} + 1399979002 T^{14} + 6975757441 T^{16} )^{2} \))
$19$ (\( ( 1 - 51 T^{2} + 1599 T^{4} - 35471 T^{6} + 577239 T^{8} - 6646371 T^{10} + 47045881 T^{12} )^{2} \))(\( ( 1 - 98 T^{2} + 4546 T^{4} - 134984 T^{6} + 2932423 T^{8} - 48729224 T^{10} + 592439266 T^{12} - 4610496338 T^{14} + 16983563041 T^{16} )^{2} \))(\( ( 1 - 77 T^{2} + 3034 T^{4} - 84368 T^{6} + 1811980 T^{8} - 30456848 T^{10} + 395393914 T^{12} - 3622532837 T^{14} + 16983563041 T^{16} )^{2} \))
$23$ (\( ( 1 + 12 T + 130 T^{2} + 984 T^{3} + 7082 T^{4} + 39654 T^{5} + 209833 T^{6} + 912042 T^{7} + 3746378 T^{8} + 11972328 T^{9} + 36379330 T^{10} + 77236116 T^{11} + 148035889 T^{12} )^{2} \))(\( ( 1 - 24 T + 317 T^{2} - 3000 T^{3} + 22111 T^{4} - 134028 T^{5} + 704756 T^{6} - 3411156 T^{7} + 16228318 T^{8} - 78456588 T^{9} + 372815924 T^{10} - 1630718676 T^{11} + 6187564351 T^{12} - 19309029000 T^{13} + 46927376813 T^{14} - 81715810728 T^{15} + 78310985281 T^{16} )^{2} \))(\( ( 1 + 3 T + 56 T^{2} + 159 T^{3} + 1321 T^{4} + 3510 T^{5} + 30863 T^{6} + 67173 T^{7} + 842170 T^{8} + 1544979 T^{9} + 16326527 T^{10} + 42706170 T^{11} + 369669961 T^{12} + 1023378537 T^{13} + 8290009784 T^{14} + 10214476341 T^{15} + 78310985281 T^{16} )^{2} \))
$29$ (\( ( 1 - 15 T + 184 T^{2} - 1635 T^{3} + 13205 T^{4} - 85158 T^{5} + 500899 T^{6} - 2469582 T^{7} + 11105405 T^{8} - 39876015 T^{9} + 130139704 T^{10} - 307667235 T^{11} + 594823321 T^{12} )^{2} \))(\( ( 1 + 6 T + 98 T^{2} + 516 T^{3} + 4846 T^{4} + 25650 T^{5} + 193448 T^{6} + 972210 T^{7} + 6347347 T^{8} + 28194090 T^{9} + 162689768 T^{10} + 625577850 T^{11} + 3427483726 T^{12} + 10583752884 T^{13} + 58292685458 T^{14} + 103499257854 T^{15} + 500246412961 T^{16} )^{2} \))(\( ( 1 + 6 T + 53 T^{2} + 246 T^{3} + 580 T^{4} + 540 T^{5} - 22066 T^{6} - 299625 T^{7} - 1268282 T^{8} - 8689125 T^{9} - 18557506 T^{10} + 13170060 T^{11} + 410222980 T^{12} + 5045742654 T^{13} + 31525636013 T^{14} + 103499257854 T^{15} + 500246412961 T^{16} )^{2} \))
$31$ (\( 1 + 57 T^{2} + 3216 T^{4} + 97787 T^{6} + 2702697 T^{8} + 47442966 T^{10} + 1072931313 T^{12} + 45592690326 T^{14} + 2495997436137 T^{16} + 86786322453947 T^{18} + 2742897576410256 T^{20} + 46718812357905657 T^{22} + 787662783788549761 T^{24} \))(\( 1 + 104 T^{2} + 3888 T^{4} + 96880 T^{6} + 4455362 T^{8} + 148421160 T^{10} + 1870813504 T^{12} + 70884338648 T^{14} + 4079738375235 T^{16} + 68119849440728 T^{18} + 1727735558027584 T^{20} + 131724325838289960 T^{22} + 3799938318355208642 T^{24} + 79405588442700000880 T^{26} + \)\(30\!\cdots\!68\)\( T^{28} + \)\(78\!\cdots\!84\)\( T^{30} + \)\(72\!\cdots\!81\)\( T^{32} \))(\( 1 + 176 T^{2} + 16407 T^{4} + 1045570 T^{6} + 50906954 T^{8} + 2036295270 T^{10} + 71233973380 T^{12} + 2306080860179 T^{14} + 72156636517806 T^{16} + 2216143706632019 T^{18} + 65786070329870980 T^{20} + 1807219547727888870 T^{22} + 43418084810021264714 T^{24} + \)\(85\!\cdots\!70\)\( T^{26} + \)\(12\!\cdots\!27\)\( T^{28} + \)\(13\!\cdots\!96\)\( T^{30} + \)\(72\!\cdots\!81\)\( T^{32} \))
$37$ (\( ( 1 + T + 107 T^{2} + 73 T^{3} + 3959 T^{4} + 1369 T^{5} + 50653 T^{6} )^{4} \))(\( ( 1 + 2 T + 46 T^{2} + 38 T^{3} + 2002 T^{4} + 1406 T^{5} + 62974 T^{6} + 101306 T^{7} + 1874161 T^{8} )^{4} \))(\( ( 1 - T + 82 T^{2} - 88 T^{3} + 3940 T^{4} - 3256 T^{5} + 112258 T^{6} - 50653 T^{7} + 1874161 T^{8} )^{4} \))
$41$ (\( 1 - 174 T^{2} + 15930 T^{4} - 1039210 T^{6} + 54933612 T^{8} - 2550462528 T^{10} + 108692576859 T^{12} - 4287327509568 T^{14} + 155229258378732 T^{16} - 4936355828289610 T^{18} + 127199858899897530 T^{20} - 2335542719966517774 T^{22} + 22563490300366186081 T^{24} \))(\( 1 - 70 T^{2} - 1569 T^{4} + 150526 T^{6} + 5171081 T^{8} - 280356900 T^{10} - 8583026738 T^{12} + 255096492224 T^{14} + 9422146980954 T^{16} + 428817203428544 T^{18} - 24253582218197618 T^{20} - 1331724499683612900 T^{22} + 41290695138728249801 T^{24} + \)\(20\!\cdots\!26\)\( T^{26} - \)\(35\!\cdots\!89\)\( T^{28} - \)\(26\!\cdots\!70\)\( T^{30} + \)\(63\!\cdots\!41\)\( T^{32} \))(\( 1 - 151 T^{2} + 9564 T^{4} - 353897 T^{6} + 12352577 T^{8} - 588962616 T^{10} + 30503470855 T^{12} - 1573469011633 T^{14} + 72313421958936 T^{16} - 2645001408555073 T^{18} + 86195518306695655 T^{20} - 2797633820052054456 T^{22} + 98634403731959794817 T^{24} - \)\(47\!\cdots\!97\)\( T^{26} + \)\(21\!\cdots\!84\)\( T^{28} - \)\(57\!\cdots\!11\)\( T^{30} + \)\(63\!\cdots\!41\)\( T^{32} \))
$43$ (\( ( 1 - 5 T - 58 T^{2} + 51 T^{3} + 2155 T^{4} + 7106 T^{5} - 129149 T^{6} + 305558 T^{7} + 3984595 T^{8} + 4054857 T^{9} - 198290458 T^{10} - 735042215 T^{11} + 6321363049 T^{12} )^{2} \))(\( ( 1 + 2 T - 129 T^{2} + 46 T^{3} + 9833 T^{4} - 11184 T^{5} - 521114 T^{6} + 232628 T^{7} + 22298490 T^{8} + 10003004 T^{9} - 963539786 T^{10} - 889206288 T^{11} + 33617070233 T^{12} + 6762388378 T^{13} - 815455833321 T^{14} + 543637222214 T^{15} + 11688200277601 T^{16} )^{2} \))(\( ( 1 + 2 T - 93 T^{2} + 64 T^{3} + 3956 T^{4} - 12192 T^{5} - 158990 T^{6} + 321485 T^{7} + 7972668 T^{8} + 13823855 T^{9} - 293972510 T^{10} - 969349344 T^{11} + 13524776756 T^{12} + 9408540352 T^{13} - 587886763557 T^{14} + 543637222214 T^{15} + 11688200277601 T^{16} )^{2} \))
$47$ (\( 1 - 189 T^{2} + 18726 T^{4} - 1271869 T^{6} + 67940493 T^{8} - 3218131254 T^{10} + 150306249081 T^{12} - 7108851940086 T^{14} + 331527932822733 T^{16} - 13709749821279901 T^{18} + 445890154028136486 T^{20} - 9941235992571879261 T^{22} + \)\(11\!\cdots\!41\)\( T^{24} \))(\( 1 - 136 T^{2} + 8016 T^{4} - 225584 T^{6} - 1533310 T^{8} + 489880632 T^{10} - 30253322048 T^{12} + 1486359308360 T^{14} - 68731587628605 T^{16} + 3283367712167240 T^{18} - 147626560784506688 T^{20} + 5280528817834607928 T^{22} - 36510083951344758910 T^{24} - \)\(11\!\cdots\!16\)\( T^{26} + \)\(93\!\cdots\!56\)\( T^{28} - \)\(34\!\cdots\!84\)\( T^{30} + \)\(56\!\cdots\!21\)\( T^{32} \))(\( 1 - 154 T^{2} + 7683 T^{4} - 245888 T^{6} + 25028264 T^{8} - 1538143278 T^{10} + 38475997042 T^{12} - 2602469435029 T^{14} + 205422570296046 T^{16} - 5748854981979061 T^{18} + 187750591721903602 T^{20} - 16579977600415908462 T^{22} + \)\(59\!\cdots\!04\)\( T^{24} - \)\(12\!\cdots\!12\)\( T^{26} + \)\(89\!\cdots\!03\)\( T^{28} - \)\(39\!\cdots\!26\)\( T^{30} + \)\(56\!\cdots\!21\)\( T^{32} \))
$53$ (\( ( 1 - 200 T^{2} + 20924 T^{4} - 1371431 T^{6} + 58775516 T^{8} - 1578096200 T^{10} + 22164361129 T^{12} )^{2} \))(\( ( 1 - 53 T^{2} )^{16} \))(\( ( 1 - 10 T^{2} + 3133 T^{4} + 54785 T^{6} + 5778574 T^{8} + 153891065 T^{10} + 24720876973 T^{12} - 221643611290 T^{14} + 62259690411361 T^{16} )^{2} \))
$59$ (\( 1 - 135 T^{2} + 4578 T^{4} + 47801 T^{6} + 11150547 T^{8} - 1706849076 T^{10} + 108381578457 T^{12} - 5941541633556 T^{14} + 135115203346467 T^{16} + 2016271688573441 T^{18} + 672189743352581538 T^{20} - 69000761695586589135 T^{22} + \)\(17\!\cdots\!81\)\( T^{24} \))(\( 1 - 178 T^{2} + 20643 T^{4} - 1496150 T^{6} + 80456981 T^{8} - 3515943660 T^{10} + 180649052698 T^{12} - 13219132050040 T^{14} + 857467356385554 T^{16} - 46015798666189240 T^{18} + 2188989785849689978 T^{20} - \)\(14\!\cdots\!60\)\( T^{22} + \)\(11\!\cdots\!01\)\( T^{24} - \)\(76\!\cdots\!50\)\( T^{26} + \)\(36\!\cdots\!83\)\( T^{28} - \)\(11\!\cdots\!58\)\( T^{30} + \)\(21\!\cdots\!41\)\( T^{32} \))(\( 1 - 376 T^{2} + 75453 T^{4} - 10695122 T^{6} + 1193881406 T^{8} - 111003832932 T^{10} + 8882998756942 T^{12} - 624280670070511 T^{14} + 38988958528434078 T^{16} - 2173121012515448791 T^{18} + \)\(10\!\cdots\!62\)\( T^{20} - \)\(46\!\cdots\!12\)\( T^{22} + \)\(17\!\cdots\!26\)\( T^{24} - \)\(54\!\cdots\!22\)\( T^{26} + \)\(13\!\cdots\!93\)\( T^{28} - \)\(23\!\cdots\!36\)\( T^{30} + \)\(21\!\cdots\!41\)\( T^{32} \))
$61$ (\( 1 + 342 T^{2} + 66840 T^{4} + 9095042 T^{6} + 949804386 T^{8} + 78844563978 T^{10} + 5324903072391 T^{12} + 293380622562138 T^{14} + 13150840509658626 T^{16} + 468579968669018162 T^{18} + 12813716800738262040 T^{20} + \)\(24\!\cdots\!42\)\( T^{22} + \)\(26\!\cdots\!21\)\( T^{24} \))(\( 1 + 248 T^{2} + 28191 T^{4} + 2052448 T^{6} + 122525357 T^{8} + 7198089144 T^{10} + 457346362462 T^{12} + 32095759051208 T^{14} + 2131627513941198 T^{16} + 119428319429544968 T^{18} + 6332345016577220542 T^{20} + \)\(37\!\cdots\!84\)\( T^{22} + \)\(23\!\cdots\!17\)\( T^{24} + \)\(14\!\cdots\!48\)\( T^{26} + \)\(74\!\cdots\!11\)\( T^{28} + \)\(24\!\cdots\!68\)\( T^{30} + \)\(36\!\cdots\!61\)\( T^{32} \))(\( 1 + 137 T^{2} + 306 T^{4} - 368483 T^{6} + 33183791 T^{8} + 2225199582 T^{10} - 143899132121 T^{12} + 1205057298539 T^{14} + 1133086682080440 T^{16} + 4484018207863619 T^{18} - 1992404503385358761 T^{20} + \)\(11\!\cdots\!02\)\( T^{22} + \)\(63\!\cdots\!71\)\( T^{24} - \)\(26\!\cdots\!83\)\( T^{26} + \)\(81\!\cdots\!26\)\( T^{28} + \)\(13\!\cdots\!17\)\( T^{30} + \)\(36\!\cdots\!61\)\( T^{32} \))
$67$ (\( ( 1 + 6 T - 150 T^{2} - 506 T^{3} + 17268 T^{4} + 28236 T^{5} - 1220289 T^{6} + 1891812 T^{7} + 77516052 T^{8} - 152186078 T^{9} - 3022668150 T^{10} + 8100750642 T^{11} + 90458382169 T^{12} )^{2} \))(\( ( 1 - 14 T + 39 T^{2} + 110 T^{3} - 2659 T^{4} + 53760 T^{5} - 92054 T^{6} - 2669060 T^{7} + 22240746 T^{8} - 178827020 T^{9} - 413230406 T^{10} + 16169018880 T^{11} - 53581830739 T^{12} + 148513761770 T^{13} + 3527876904591 T^{14} - 84849962474522 T^{15} + 406067677556641 T^{16} )^{2} \))(\( ( 1 + 7 T - 108 T^{2} + 293 T^{3} + 11753 T^{4} - 50778 T^{5} - 198827 T^{6} + 2620663 T^{7} - 10701486 T^{8} + 175584421 T^{9} - 892534403 T^{10} - 15272143614 T^{11} + 236836125113 T^{12} + 395586656351 T^{13} - 9769505274252 T^{14} + 42424981237261 T^{15} + 406067677556641 T^{16} )^{2} \))
$71$ (\( ( 1 - 263 T^{2} + 31517 T^{4} - 2539307 T^{6} + 158877197 T^{8} - 6683272103 T^{10} + 128100283921 T^{12} )^{2} \))(\( ( 1 - 478 T^{2} + 105553 T^{4} - 13986142 T^{6} + 1213269316 T^{8} - 70504141822 T^{10} + 2682279164593 T^{12} - 61231935714238 T^{14} + 645753531245761 T^{16} )^{2} \))(\( ( 1 - 361 T^{2} + 64216 T^{4} - 7488268 T^{6} + 623512600 T^{8} - 37748358988 T^{10} + 1631836507096 T^{12} - 46244202495481 T^{14} + 645753531245761 T^{16} )^{2} \))
$73$ (\( ( 1 - 159 T^{2} + 10563 T^{4} - 623423 T^{6} + 56290227 T^{8} - 4515320319 T^{10} + 151334226289 T^{12} )^{2} \))(\( ( 1 - 362 T^{2} + 64045 T^{4} - 7316714 T^{6} + 612211324 T^{8} - 38990768906 T^{10} + 1818765344845 T^{12} - 54782989916618 T^{14} + 806460091894081 T^{16} )^{2} \))(\( ( 1 - 341 T^{2} + 62398 T^{4} - 7562288 T^{6} + 647645452 T^{8} - 40299432752 T^{10} + 1771993441918 T^{12} - 51604971164549 T^{14} + 806460091894081 T^{16} )^{2} \))
$79$ (\( ( 1 - 3 T - 204 T^{2} + 151 T^{3} + 27357 T^{4} - 1896 T^{5} - 2498217 T^{6} - 149784 T^{7} + 170735037 T^{8} + 74448889 T^{9} - 7945816524 T^{10} - 9231169197 T^{11} + 243087455521 T^{12} )^{2} \))(\( ( 1 - 2 T - 183 T^{2} - 982 T^{3} + 19715 T^{4} + 144312 T^{5} - 491612 T^{6} - 7480148 T^{7} - 8945118 T^{8} - 590931692 T^{9} - 3068150492 T^{10} + 71151444168 T^{11} + 767900846915 T^{12} - 3021669383818 T^{13} - 44485004360343 T^{14} - 38407817972318 T^{15} + 1517108809906561 T^{16} )^{2} \))(\( ( 1 + 10 T - 123 T^{2} - 844 T^{3} + 11042 T^{4} - 2166 T^{5} - 1268066 T^{6} + 531391 T^{7} + 106354368 T^{8} + 41979889 T^{9} - 7913999906 T^{10} - 1067922474 T^{11} + 430086794402 T^{12} - 2597035600756 T^{13} - 29899757029083 T^{14} + 192039089861590 T^{15} + 1517108809906561 T^{16} )^{2} \))
$83$ (\( 1 - 264 T^{2} + 27306 T^{4} - 2832334 T^{6} + 401548758 T^{8} - 35501027934 T^{10} + 2433955301391 T^{12} - 244566581437326 T^{14} + 19056829854315318 T^{16} - 926004335465713246 T^{18} + 61501091690788653546 T^{20} - \)\(40\!\cdots\!36\)\( T^{22} + \)\(10\!\cdots\!61\)\( T^{24} \))(\( 1 + 44 T^{2} - 13368 T^{4} - 595880 T^{6} + 87112226 T^{8} + 3596762100 T^{10} - 160886956928 T^{12} - 11841264313180 T^{14} - 673218489607821 T^{16} - 81574469853497020 T^{18} - 7635424846602197888 T^{20} + \)\(11\!\cdots\!00\)\( T^{22} + \)\(19\!\cdots\!66\)\( T^{24} - \)\(92\!\cdots\!20\)\( T^{26} - \)\(14\!\cdots\!48\)\( T^{28} + \)\(32\!\cdots\!76\)\( T^{30} + \)\(50\!\cdots\!81\)\( T^{32} \))(\( 1 - 397 T^{2} + 87990 T^{4} - 12501089 T^{6} + 1185510017 T^{8} - 57837981066 T^{10} - 2507933243939 T^{12} + 849901181197043 T^{14} - 92913215010718536 T^{16} + 5854969237266429227 T^{18} - \)\(11\!\cdots\!19\)\( T^{20} - \)\(18\!\cdots\!54\)\( T^{22} + \)\(26\!\cdots\!97\)\( T^{24} - \)\(19\!\cdots\!61\)\( T^{26} + \)\(94\!\cdots\!90\)\( T^{28} - \)\(29\!\cdots\!13\)\( T^{30} + \)\(50\!\cdots\!81\)\( T^{32} \))
$89$ (\( ( 1 + 210 T^{2} + 31902 T^{4} + 3230233 T^{6} + 252695742 T^{8} + 13175870610 T^{10} + 496981290961 T^{12} )^{2} \))(\( ( 1 + 496 T^{2} + 119404 T^{4} + 18272464 T^{6} + 1934931814 T^{8} + 144736187344 T^{10} + 7491674544364 T^{12} + 246502720316656 T^{14} + 3936588805702081 T^{16} )^{2} \))(\( ( 1 + 64 T^{2} + 9487 T^{4} + 662335 T^{6} + 143358802 T^{8} + 5246355535 T^{10} + 595235640367 T^{12} + 31806802621504 T^{14} + 3936588805702081 T^{16} )^{2} \))
$97$ (\( 1 + 513 T^{2} + 147894 T^{4} + 30072485 T^{6} + 4748892315 T^{8} + 607745530428 T^{10} + 64510991774769 T^{12} + 5718277695797052 T^{14} + 420416022193375515 T^{16} + 25049538123647278565 T^{18} + \)\(11\!\cdots\!34\)\( T^{20} + \)\(37\!\cdots\!37\)\( T^{22} + \)\(69\!\cdots\!41\)\( T^{24} \))(\( 1 + 74 T^{2} + 11511 T^{4} - 803858 T^{6} - 150210775 T^{8} - 25062425316 T^{10} - 114467134418 T^{12} + 73367970993632 T^{14} + 27832786456667274 T^{16} + 690319239079083488 T^{18} - 10133693108155893458 T^{20} - \)\(20\!\cdots\!64\)\( T^{22} - \)\(11\!\cdots\!75\)\( T^{24} - \)\(59\!\cdots\!42\)\( T^{26} + \)\(79\!\cdots\!51\)\( T^{28} + \)\(48\!\cdots\!06\)\( T^{30} + \)\(61\!\cdots\!21\)\( T^{32} \))(\( 1 + 404 T^{2} + 81609 T^{4} + 10963666 T^{6} + 1046175986 T^{8} + 57359540784 T^{10} - 1663618077194 T^{12} - 785812116013063 T^{14} - 99406691908304130 T^{16} - 7393706199566909767 T^{18} - \)\(14\!\cdots\!14\)\( T^{20} + \)\(47\!\cdots\!36\)\( T^{22} + \)\(81\!\cdots\!46\)\( T^{24} + \)\(80\!\cdots\!34\)\( T^{26} + \)\(56\!\cdots\!69\)\( T^{28} + \)\(26\!\cdots\!76\)\( T^{30} + \)\(61\!\cdots\!21\)\( T^{32} \))
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