Properties

Label 3234.2.e
Level 3234
Weight 2
Character orbit e
Rep. character \(\chi_{3234}(2155,\cdot)\)
Character field \(\Q\)
Dimension 80
Newform subspaces 4
Sturm bound 1344
Trace bound 6

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Defining parameters

Level: \( N \) \(=\) \( 3234 = 2 \cdot 3 \cdot 7^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3234.e (of order \(2\) and degree \(1\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 77 \)
Character field: \(\Q\)
Newform subspaces: \( 4 \)
Sturm bound: \(1344\)
Trace bound: \(6\)
Distinguishing \(T_p\): \(5\), \(13\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(3234, [\chi])\).

Total New Old
Modular forms 704 80 624
Cusp forms 640 80 560
Eisenstein series 64 0 64

Trace form

\( 80q - 80q^{4} - 80q^{9} + O(q^{10}) \) \( 80q - 80q^{4} - 80q^{9} + 16q^{11} - 8q^{15} + 80q^{16} - 12q^{22} + 16q^{23} - 56q^{25} + 80q^{36} - 24q^{37} - 16q^{44} + 8q^{60} - 80q^{64} - 64q^{67} + 16q^{71} + 80q^{81} + 48q^{86} + 12q^{88} - 16q^{92} - 48q^{93} - 16q^{99} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(3234, [\chi])\) into newform subspaces

Label Dim. \(A\) Field CM Traces $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\)
3234.2.e.a \(16\) \(25.824\) \(\mathbb{Q}[x]/(x^{16} - \cdots)\) None \(0\) \(0\) \(0\) \(0\) \(q-\beta _{12}q^{2}-\beta _{12}q^{3}-q^{4}+(-\beta _{11}+\cdots)q^{5}+\cdots\)
3234.2.e.b \(16\) \(25.824\) \(\mathbb{Q}[x]/(x^{16} - \cdots)\) None \(0\) \(0\) \(0\) \(0\) \(q+\beta _{12}q^{2}-\beta _{12}q^{3}-q^{4}+(-\beta _{11}+\cdots)q^{5}+\cdots\)
3234.2.e.c \(24\) \(25.824\) None \(0\) \(0\) \(0\) \(0\)
3234.2.e.d \(24\) \(25.824\) None \(0\) \(0\) \(0\) \(0\)

Decomposition of \(S_{2}^{\mathrm{old}}(3234, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(3234, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(77, [\chi])\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(154, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(231, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(462, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(539, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(1078, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(1617, [\chi])\)\(^{\oplus 2}\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ (\( ( 1 + T^{2} )^{8} \))(\( ( 1 + T^{2} )^{8} \))
$3$ (\( ( 1 + T^{2} )^{8} \))(\( ( 1 + T^{2} )^{8} \))
$5$ (\( 1 - 30 T^{2} + 471 T^{4} - 4930 T^{6} + 37781 T^{8} - 222460 T^{10} + 1052266 T^{12} - 4383184 T^{14} + 19642634 T^{16} - 109579600 T^{18} + 657666250 T^{20} - 3475937500 T^{22} + 14758203125 T^{24} - 48144531250 T^{26} + 114990234375 T^{28} - 183105468750 T^{30} + 152587890625 T^{32} \))(\( 1 - 30 T^{2} + 471 T^{4} - 4930 T^{6} + 37781 T^{8} - 222460 T^{10} + 1052266 T^{12} - 4383184 T^{14} + 19642634 T^{16} - 109579600 T^{18} + 657666250 T^{20} - 3475937500 T^{22} + 14758203125 T^{24} - 48144531250 T^{26} + 114990234375 T^{28} - 183105468750 T^{30} + 152587890625 T^{32} \))
$7$ 1
$11$ (\( 1 - 8 T + 33 T^{2} - 56 T^{3} - 206 T^{4} + 1320 T^{5} - 1177 T^{6} - 15528 T^{7} + 86618 T^{8} - 170808 T^{9} - 142417 T^{10} + 1756920 T^{11} - 3016046 T^{12} - 9018856 T^{13} + 58461513 T^{14} - 155897368 T^{15} + 214358881 T^{16} \))(\( 1 - 8 T + 33 T^{2} - 56 T^{3} - 206 T^{4} + 1320 T^{5} - 1177 T^{6} - 15528 T^{7} + 86618 T^{8} - 170808 T^{9} - 142417 T^{10} + 1756920 T^{11} - 3016046 T^{12} - 9018856 T^{13} + 58461513 T^{14} - 155897368 T^{15} + 214358881 T^{16} \))
$13$ (\( ( 1 + 40 T^{2} - 8 T^{3} + 576 T^{4} + 648 T^{5} + 3672 T^{6} + 30816 T^{7} + 19294 T^{8} + 400608 T^{9} + 620568 T^{10} + 1423656 T^{11} + 16451136 T^{12} - 2970344 T^{13} + 193072360 T^{14} + 815730721 T^{16} )^{2} \))(\( ( 1 + 40 T^{2} + 8 T^{3} + 576 T^{4} - 648 T^{5} + 3672 T^{6} - 30816 T^{7} + 19294 T^{8} - 400608 T^{9} + 620568 T^{10} - 1423656 T^{11} + 16451136 T^{12} + 2970344 T^{13} + 193072360 T^{14} + 815730721 T^{16} )^{2} \))
$17$ (\( ( 1 + 83 T^{2} + 68 T^{3} + 3413 T^{4} + 4044 T^{5} + 95086 T^{6} + 108260 T^{7} + 1908750 T^{8} + 1840420 T^{9} + 27479854 T^{10} + 19868172 T^{11} + 285057173 T^{12} + 96550276 T^{13} + 2003418227 T^{14} + 6975757441 T^{16} )^{2} \))(\( ( 1 + 83 T^{2} - 68 T^{3} + 3413 T^{4} - 4044 T^{5} + 95086 T^{6} - 108260 T^{7} + 1908750 T^{8} - 1840420 T^{9} + 27479854 T^{10} - 19868172 T^{11} + 285057173 T^{12} - 96550276 T^{13} + 2003418227 T^{14} + 6975757441 T^{16} )^{2} \))
$19$ (\( ( 1 - 10 T + 145 T^{2} - 1030 T^{3} + 8870 T^{4} - 49582 T^{5} + 317127 T^{6} - 1444786 T^{7} + 7373490 T^{8} - 27450934 T^{9} + 114482847 T^{10} - 340082938 T^{11} + 1155947270 T^{12} - 2550381970 T^{13} + 6821652745 T^{14} - 8938717390 T^{15} + 16983563041 T^{16} )^{2} \))(\( ( 1 + 10 T + 145 T^{2} + 1030 T^{3} + 8870 T^{4} + 49582 T^{5} + 317127 T^{6} + 1444786 T^{7} + 7373490 T^{8} + 27450934 T^{9} + 114482847 T^{10} + 340082938 T^{11} + 1155947270 T^{12} + 2550381970 T^{13} + 6821652745 T^{14} + 8938717390 T^{15} + 16983563041 T^{16} )^{2} \))
$23$ (\( ( 1 - 4 T + 57 T^{2} - 124 T^{3} + 2103 T^{4} - 4828 T^{5} + 65020 T^{6} - 100872 T^{7} + 1438254 T^{8} - 2320056 T^{9} + 34395580 T^{10} - 58742276 T^{11} + 588505623 T^{12} - 798106532 T^{13} + 8438045673 T^{14} - 13619301788 T^{15} + 78310985281 T^{16} )^{2} \))(\( ( 1 - 4 T + 57 T^{2} - 124 T^{3} + 2103 T^{4} - 4828 T^{5} + 65020 T^{6} - 100872 T^{7} + 1438254 T^{8} - 2320056 T^{9} + 34395580 T^{10} - 58742276 T^{11} + 588505623 T^{12} - 798106532 T^{13} + 8438045673 T^{14} - 13619301788 T^{15} + 78310985281 T^{16} )^{2} \))
$29$ (\( 1 - 220 T^{2} + 25766 T^{4} - 2101752 T^{6} + 132425201 T^{8} - 6784185928 T^{10} + 290786240918 T^{12} - 10599475438788 T^{14} + 331431007100868 T^{16} - 8914158844020708 T^{18} + 205667583262723958 T^{20} - 4035392003974426888 T^{22} + 66245231785889430161 T^{24} - \)\(88\!\cdots\!52\)\( T^{26} + \)\(91\!\cdots\!06\)\( T^{28} - \)\(65\!\cdots\!20\)\( T^{30} + \)\(25\!\cdots\!21\)\( T^{32} \))(\( 1 - 220 T^{2} + 25766 T^{4} - 2101752 T^{6} + 132425201 T^{8} - 6784185928 T^{10} + 290786240918 T^{12} - 10599475438788 T^{14} + 331431007100868 T^{16} - 8914158844020708 T^{18} + 205667583262723958 T^{20} - 4035392003974426888 T^{22} + 66245231785889430161 T^{24} - \)\(88\!\cdots\!52\)\( T^{26} + \)\(91\!\cdots\!06\)\( T^{28} - \)\(65\!\cdots\!20\)\( T^{30} + \)\(25\!\cdots\!21\)\( T^{32} \))
$31$ (\( 1 - 386 T^{2} + 71901 T^{4} - 8595466 T^{6} + 739336166 T^{8} - 48582533718 T^{10} + 2525237875987 T^{12} - 105975133331486 T^{14} + 3630081582174546 T^{16} - 101842103131558046 T^{18} + 2332110208469390227 T^{20} - 43117177507031615958 T^{22} + \)\(63\!\cdots\!06\)\( T^{24} - \)\(70\!\cdots\!66\)\( T^{26} + \)\(56\!\cdots\!61\)\( T^{28} - \)\(29\!\cdots\!06\)\( T^{30} + \)\(72\!\cdots\!81\)\( T^{32} \))(\( 1 - 386 T^{2} + 71901 T^{4} - 8595466 T^{6} + 739336166 T^{8} - 48582533718 T^{10} + 2525237875987 T^{12} - 105975133331486 T^{14} + 3630081582174546 T^{16} - 101842103131558046 T^{18} + 2332110208469390227 T^{20} - 43117177507031615958 T^{22} + \)\(63\!\cdots\!06\)\( T^{24} - \)\(70\!\cdots\!66\)\( T^{26} + \)\(56\!\cdots\!61\)\( T^{28} - \)\(29\!\cdots\!06\)\( T^{30} + \)\(72\!\cdots\!81\)\( T^{32} \))
$37$ (\( ( 1 + 14 T + 229 T^{2} + 2126 T^{3} + 20986 T^{4} + 154618 T^{5} + 1182051 T^{6} + 7476394 T^{7} + 49108426 T^{8} + 276626578 T^{9} + 1618227819 T^{10} + 7831865554 T^{11} + 39331142746 T^{12} + 147425252582 T^{13} + 587551347661 T^{14} + 1329046279862 T^{15} + 3512479453921 T^{16} )^{2} \))(\( ( 1 + 14 T + 229 T^{2} + 2126 T^{3} + 20986 T^{4} + 154618 T^{5} + 1182051 T^{6} + 7476394 T^{7} + 49108426 T^{8} + 276626578 T^{9} + 1618227819 T^{10} + 7831865554 T^{11} + 39331142746 T^{12} + 147425252582 T^{13} + 587551347661 T^{14} + 1329046279862 T^{15} + 3512479453921 T^{16} )^{2} \))
$41$ (\( ( 1 - 16 T + 330 T^{2} - 3640 T^{3} + 42745 T^{4} - 362832 T^{5} + 3123194 T^{6} - 21761352 T^{7} + 152229812 T^{8} - 892215432 T^{9} + 5250089114 T^{10} - 25006744272 T^{11} + 120787153945 T^{12} - 421716571640 T^{13} + 1567534399530 T^{14} - 3116068382096 T^{15} + 7984925229121 T^{16} )^{2} \))(\( ( 1 + 16 T + 330 T^{2} + 3640 T^{3} + 42745 T^{4} + 362832 T^{5} + 3123194 T^{6} + 21761352 T^{7} + 152229812 T^{8} + 892215432 T^{9} + 5250089114 T^{10} + 25006744272 T^{11} + 120787153945 T^{12} + 421716571640 T^{13} + 1567534399530 T^{14} + 3116068382096 T^{15} + 7984925229121 T^{16} )^{2} \))
$43$ (\( 1 - 370 T^{2} + 69205 T^{4} - 8714850 T^{6} + 829915302 T^{8} - 63533346462 T^{10} + 4048663635627 T^{12} - 219017168933710 T^{14} + 10156745084830162 T^{16} - 404962745358429790 T^{18} + 13841575286145223227 T^{20} - \)\(40\!\cdots\!38\)\( T^{22} + \)\(97\!\cdots\!02\)\( T^{24} - \)\(18\!\cdots\!50\)\( T^{26} + \)\(27\!\cdots\!05\)\( T^{28} - \)\(27\!\cdots\!30\)\( T^{30} + \)\(13\!\cdots\!01\)\( T^{32} \))(\( 1 - 370 T^{2} + 69205 T^{4} - 8714850 T^{6} + 829915302 T^{8} - 63533346462 T^{10} + 4048663635627 T^{12} - 219017168933710 T^{14} + 10156745084830162 T^{16} - 404962745358429790 T^{18} + 13841575286145223227 T^{20} - \)\(40\!\cdots\!38\)\( T^{22} + \)\(97\!\cdots\!02\)\( T^{24} - \)\(18\!\cdots\!50\)\( T^{26} + \)\(27\!\cdots\!05\)\( T^{28} - \)\(27\!\cdots\!30\)\( T^{30} + \)\(13\!\cdots\!01\)\( T^{32} \))
$47$ (\( 1 - 62 T^{2} + 2031 T^{4} - 368922 T^{6} + 20631365 T^{8} - 625319636 T^{10} + 67952187226 T^{12} - 3345437111480 T^{14} + 84733582253162 T^{16} - 7390070579259320 T^{18} + 331584996915154906 T^{20} - 6740455005895900244 T^{22} + \)\(49\!\cdots\!65\)\( T^{24} - \)\(19\!\cdots\!78\)\( T^{26} + \)\(23\!\cdots\!71\)\( T^{28} - \)\(15\!\cdots\!78\)\( T^{30} + \)\(56\!\cdots\!21\)\( T^{32} \))(\( 1 - 62 T^{2} + 2031 T^{4} - 368922 T^{6} + 20631365 T^{8} - 625319636 T^{10} + 67952187226 T^{12} - 3345437111480 T^{14} + 84733582253162 T^{16} - 7390070579259320 T^{18} + 331584996915154906 T^{20} - 6740455005895900244 T^{22} + \)\(49\!\cdots\!65\)\( T^{24} - \)\(19\!\cdots\!78\)\( T^{26} + \)\(23\!\cdots\!71\)\( T^{28} - \)\(15\!\cdots\!78\)\( T^{30} + \)\(56\!\cdots\!21\)\( T^{32} \))
$53$ (\( ( 1 + 189 T^{2} - 116 T^{3} + 18282 T^{4} - 21964 T^{5} + 1293871 T^{6} - 2238960 T^{7} + 74773842 T^{8} - 118664880 T^{9} + 3634483639 T^{10} - 3269934428 T^{11} + 144253773642 T^{12} - 48510677188 T^{13} + 4189064253381 T^{14} + 62259690411361 T^{16} )^{2} \))(\( ( 1 + 189 T^{2} - 116 T^{3} + 18282 T^{4} - 21964 T^{5} + 1293871 T^{6} - 2238960 T^{7} + 74773842 T^{8} - 118664880 T^{9} + 3634483639 T^{10} - 3269934428 T^{11} + 144253773642 T^{12} - 48510677188 T^{13} + 4189064253381 T^{14} + 62259690411361 T^{16} )^{2} \))
$59$ (\( 1 - 412 T^{2} + 84726 T^{4} - 11812648 T^{6} + 1278873825 T^{8} - 115825341832 T^{10} + 9135071092582 T^{12} - 639128365018580 T^{14} + 39901635887620324 T^{16} - 2224805838629676980 T^{18} + \)\(11\!\cdots\!02\)\( T^{20} - \)\(48\!\cdots\!12\)\( T^{22} + \)\(18\!\cdots\!25\)\( T^{24} - \)\(60\!\cdots\!48\)\( T^{26} + \)\(15\!\cdots\!06\)\( T^{28} - \)\(25\!\cdots\!32\)\( T^{30} + \)\(21\!\cdots\!41\)\( T^{32} \))(\( 1 - 412 T^{2} + 84726 T^{4} - 11812648 T^{6} + 1278873825 T^{8} - 115825341832 T^{10} + 9135071092582 T^{12} - 639128365018580 T^{14} + 39901635887620324 T^{16} - 2224805838629676980 T^{18} + \)\(11\!\cdots\!02\)\( T^{20} - \)\(48\!\cdots\!12\)\( T^{22} + \)\(18\!\cdots\!25\)\( T^{24} - \)\(60\!\cdots\!48\)\( T^{26} + \)\(15\!\cdots\!06\)\( T^{28} - \)\(25\!\cdots\!32\)\( T^{30} + \)\(21\!\cdots\!41\)\( T^{32} \))
$61$ (\( ( 1 + 28 T + 652 T^{2} + 10788 T^{3} + 153837 T^{4} + 1841484 T^{5} + 19466748 T^{6} + 180831988 T^{7} + 1498320772 T^{8} + 11030751268 T^{9} + 72435769308 T^{10} + 417981879804 T^{11} + 2130002641917 T^{12} + 9111504895188 T^{13} + 33591284083372 T^{14} + 87996799408588 T^{15} + 191707312997281 T^{16} )^{2} \))(\( ( 1 - 28 T + 652 T^{2} - 10788 T^{3} + 153837 T^{4} - 1841484 T^{5} + 19466748 T^{6} - 180831988 T^{7} + 1498320772 T^{8} - 11030751268 T^{9} + 72435769308 T^{10} - 417981879804 T^{11} + 2130002641917 T^{12} - 9111504895188 T^{13} + 33591284083372 T^{14} - 87996799408588 T^{15} + 191707312997281 T^{16} )^{2} \))
$67$ (\( ( 1 - 16 T + 442 T^{2} - 5580 T^{3} + 88929 T^{4} - 911040 T^{5} + 10786038 T^{6} - 91408468 T^{7} + 872545804 T^{8} - 6124367356 T^{9} + 48418524582 T^{10} - 274007123520 T^{11} + 1792019039409 T^{12} - 7533698097060 T^{13} + 39982604918698 T^{14} - 96971385685168 T^{15} + 406067677556641 T^{16} )^{2} \))(\( ( 1 - 16 T + 442 T^{2} - 5580 T^{3} + 88929 T^{4} - 911040 T^{5} + 10786038 T^{6} - 91408468 T^{7} + 872545804 T^{8} - 6124367356 T^{9} + 48418524582 T^{10} - 274007123520 T^{11} + 1792019039409 T^{12} - 7533698097060 T^{13} + 39982604918698 T^{14} - 96971385685168 T^{15} + 406067677556641 T^{16} )^{2} \))
$71$ (\( ( 1 + 28 T + 665 T^{2} + 9556 T^{3} + 124346 T^{4} + 1164764 T^{5} + 10825387 T^{6} + 79684020 T^{7} + 712940082 T^{8} + 5657565420 T^{9} + 54570775867 T^{10} + 416881848004 T^{11} + 3159840885626 T^{12} + 17241215678156 T^{13} + 85186688807465 T^{14} + 254663364434948 T^{15} + 645753531245761 T^{16} )^{2} \))(\( ( 1 + 28 T + 665 T^{2} + 9556 T^{3} + 124346 T^{4} + 1164764 T^{5} + 10825387 T^{6} + 79684020 T^{7} + 712940082 T^{8} + 5657565420 T^{9} + 54570775867 T^{10} + 416881848004 T^{11} + 3159840885626 T^{12} + 17241215678156 T^{13} + 85186688807465 T^{14} + 254663364434948 T^{15} + 645753531245761 T^{16} )^{2} \))
$73$ (\( ( 1 - 44 T + 1204 T^{2} - 23988 T^{3} + 389092 T^{4} - 5297980 T^{5} + 62491788 T^{6} - 643636388 T^{7} + 5856468854 T^{8} - 46985456324 T^{9} + 333018738252 T^{10} - 2061004285660 T^{11} + 11049528387172 T^{12} - 49728841372884 T^{13} + 182206408451956 T^{14} - 486085534840268 T^{15} + 806460091894081 T^{16} )^{2} \))(\( ( 1 + 44 T + 1204 T^{2} + 23988 T^{3} + 389092 T^{4} + 5297980 T^{5} + 62491788 T^{6} + 643636388 T^{7} + 5856468854 T^{8} + 46985456324 T^{9} + 333018738252 T^{10} + 2061004285660 T^{11} + 11049528387172 T^{12} + 49728841372884 T^{13} + 182206408451956 T^{14} + 486085534840268 T^{15} + 806460091894081 T^{16} )^{2} \))
$79$ (\( 1 - 538 T^{2} + 147807 T^{4} - 27647974 T^{6} + 3965128581 T^{8} - 467618207428 T^{10} + 47603630292250 T^{12} - 4323347326389920 T^{14} + 357043246909249546 T^{16} - 26982010663999490720 T^{18} + \)\(18\!\cdots\!50\)\( T^{20} - \)\(11\!\cdots\!88\)\( T^{22} + \)\(60\!\cdots\!41\)\( T^{24} - \)\(26\!\cdots\!74\)\( T^{26} + \)\(87\!\cdots\!87\)\( T^{28} - \)\(19\!\cdots\!78\)\( T^{30} + \)\(23\!\cdots\!21\)\( T^{32} \))(\( 1 - 538 T^{2} + 147807 T^{4} - 27647974 T^{6} + 3965128581 T^{8} - 467618207428 T^{10} + 47603630292250 T^{12} - 4323347326389920 T^{14} + 357043246909249546 T^{16} - 26982010663999490720 T^{18} + \)\(18\!\cdots\!50\)\( T^{20} - \)\(11\!\cdots\!88\)\( T^{22} + \)\(60\!\cdots\!41\)\( T^{24} - \)\(26\!\cdots\!74\)\( T^{26} + \)\(87\!\cdots\!87\)\( T^{28} - \)\(19\!\cdots\!78\)\( T^{30} + \)\(23\!\cdots\!21\)\( T^{32} \))
$83$ (\( ( 1 - 4 T + 523 T^{2} - 1544 T^{3} + 123705 T^{4} - 263372 T^{5} + 17791866 T^{6} - 28394068 T^{7} + 1752220430 T^{8} - 2356707644 T^{9} + 122568164874 T^{10} - 150592685764 T^{11} + 5870831599305 T^{12} - 6081878752792 T^{13} + 170989815271987 T^{14} - 108544203958508 T^{15} + 2252292232139041 T^{16} )^{2} \))(\( ( 1 + 4 T + 523 T^{2} + 1544 T^{3} + 123705 T^{4} + 263372 T^{5} + 17791866 T^{6} + 28394068 T^{7} + 1752220430 T^{8} + 2356707644 T^{9} + 122568164874 T^{10} + 150592685764 T^{11} + 5870831599305 T^{12} + 6081878752792 T^{13} + 170989815271987 T^{14} + 108544203958508 T^{15} + 2252292232139041 T^{16} )^{2} \))
$89$ (\( 1 - 640 T^{2} + 210552 T^{4} - 47499904 T^{6} + 8295415452 T^{8} - 1194559990144 T^{10} + 146915919459400 T^{12} - 15750756401654144 T^{14} + 1489545499043303110 T^{16} - \)\(12\!\cdots\!24\)\( T^{18} + \)\(92\!\cdots\!00\)\( T^{20} - \)\(59\!\cdots\!84\)\( T^{22} + \)\(32\!\cdots\!12\)\( T^{24} - \)\(14\!\cdots\!04\)\( T^{26} + \)\(52\!\cdots\!92\)\( T^{28} - \)\(12\!\cdots\!40\)\( T^{30} + \)\(15\!\cdots\!61\)\( T^{32} \))(\( 1 - 640 T^{2} + 210552 T^{4} - 47499904 T^{6} + 8295415452 T^{8} - 1194559990144 T^{10} + 146915919459400 T^{12} - 15750756401654144 T^{14} + 1489545499043303110 T^{16} - \)\(12\!\cdots\!24\)\( T^{18} + \)\(92\!\cdots\!00\)\( T^{20} - \)\(59\!\cdots\!84\)\( T^{22} + \)\(32\!\cdots\!12\)\( T^{24} - \)\(14\!\cdots\!04\)\( T^{26} + \)\(52\!\cdots\!92\)\( T^{28} - \)\(12\!\cdots\!40\)\( T^{30} + \)\(15\!\cdots\!61\)\( T^{32} \))
$97$ (\( 1 - 608 T^{2} + 187588 T^{4} - 40328000 T^{6} + 6818823050 T^{8} - 967972100768 T^{10} + 120672937256208 T^{12} - 13517338511335328 T^{14} + 1373778662576096019 T^{16} - \)\(12\!\cdots\!52\)\( T^{18} + \)\(10\!\cdots\!48\)\( T^{20} - \)\(80\!\cdots\!72\)\( T^{22} + \)\(53\!\cdots\!50\)\( T^{24} - \)\(29\!\cdots\!00\)\( T^{26} + \)\(13\!\cdots\!08\)\( T^{28} - \)\(39\!\cdots\!52\)\( T^{30} + \)\(61\!\cdots\!21\)\( T^{32} \))(\( 1 - 608 T^{2} + 187588 T^{4} - 40328000 T^{6} + 6818823050 T^{8} - 967972100768 T^{10} + 120672937256208 T^{12} - 13517338511335328 T^{14} + 1373778662576096019 T^{16} - \)\(12\!\cdots\!52\)\( T^{18} + \)\(10\!\cdots\!48\)\( T^{20} - \)\(80\!\cdots\!72\)\( T^{22} + \)\(53\!\cdots\!50\)\( T^{24} - \)\(29\!\cdots\!00\)\( T^{26} + \)\(13\!\cdots\!08\)\( T^{28} - \)\(39\!\cdots\!52\)\( T^{30} + \)\(61\!\cdots\!21\)\( T^{32} \))
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