Properties

Label 756.2.l
Level 756
Weight 2
Character orbit l
Rep. character \(\chi_{756}(289,\cdot)\)
Character field \(\Q(\zeta_{3})\)
Dimension 16
Newform subspaces 2
Sturm bound 288
Trace bound 1

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Defining parameters

Level: \( N \) = \( 756 = 2^{2} \cdot 3^{3} \cdot 7 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 756.l (of order \(3\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) = \( 63 \)
Character field: \(\Q(\zeta_{3})\)
Newform subspaces: \( 2 \)
Sturm bound: \(288\)
Trace bound: \(1\)
Distinguishing \(T_p\): \(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(756, [\chi])\).

Total New Old
Modular forms 324 16 308
Cusp forms 252 16 236
Eisenstein series 72 0 72

Trace form

\( 16q - 8q^{5} + q^{7} + O(q^{10}) \) \( 16q - 8q^{5} + q^{7} - 4q^{11} - q^{13} + 5q^{17} + 2q^{19} + 14q^{23} + 16q^{25} - 2q^{29} + 2q^{31} + 11q^{35} - q^{37} + 24q^{41} + 2q^{43} + 6q^{47} - 11q^{49} + 18q^{53} - 12q^{55} + 7q^{59} - 13q^{61} - 9q^{65} - 7q^{67} + 14q^{71} + 14q^{73} - 35q^{77} - q^{79} + 26q^{83} - 6q^{85} + 21q^{89} + 5q^{91} + 38q^{95} - q^{97} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(756, [\chi])\) into newform subspaces

Label Dim. \(A\) Field CM Traces $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\)
756.2.l.a \(2\) \(6.037\) \(\Q(\sqrt{-3}) \) None \(0\) \(0\) \(-4\) \(4\) \(q-2q^{5}+(3-2\zeta_{6})q^{7}-4q^{11}+(-3+\cdots)q^{13}+\cdots\)
756.2.l.b \(14\) \(6.037\) \(\mathbb{Q}[x]/(x^{14} - \cdots)\) None \(0\) \(0\) \(-4\) \(-3\) \(q+(\beta _{3}-\beta _{7})q^{5}-\beta _{5}q^{7}+(\beta _{9}-\beta _{13})q^{11}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(756, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(756, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(63, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(126, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(189, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(252, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(378, [\chi])\)\(^{\oplus 2}\)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ 1
$3$ 1
$5$ (\( ( 1 + 2 T + 5 T^{2} )^{2} \))(\( ( 1 + 2 T + 15 T^{2} + 48 T^{3} + 154 T^{4} + 429 T^{5} + 1202 T^{6} + 2471 T^{7} + 6010 T^{8} + 10725 T^{9} + 19250 T^{10} + 30000 T^{11} + 46875 T^{12} + 31250 T^{13} + 78125 T^{14} )^{2} \))
$7$ (\( 1 - 4 T + 7 T^{2} \))(\( 1 + 3 T + 11 T^{2} - 5 T^{3} - 15 T^{4} - 88 T^{5} + 312 T^{6} + 753 T^{7} + 2184 T^{8} - 4312 T^{9} - 5145 T^{10} - 12005 T^{11} + 184877 T^{12} + 352947 T^{13} + 823543 T^{14} \))
$11$ (\( ( 1 + 4 T + 11 T^{2} )^{2} \))(\( ( 1 - 2 T + 36 T^{2} - 57 T^{3} + 460 T^{4} - 402 T^{5} + 2501 T^{6} - 455 T^{7} + 27511 T^{8} - 48642 T^{9} + 612260 T^{10} - 834537 T^{11} + 5797836 T^{12} - 3543122 T^{13} + 19487171 T^{14} )^{2} \))
$13$ (\( 1 + 3 T - 4 T^{2} + 39 T^{3} + 169 T^{4} \))(\( 1 - 2 T - 25 T^{2} - 20 T^{3} + 172 T^{4} + 1281 T^{5} + 1882 T^{6} - 1142 T^{7} - 27931 T^{8} - 309997 T^{9} + 9092 T^{10} + 1859069 T^{11} + 4530175 T^{12} + 5185603 T^{13} - 73974588 T^{14} + 67412839 T^{15} + 765599575 T^{16} + 4084374593 T^{17} + 259676612 T^{18} - 115099716121 T^{19} - 134817602179 T^{20} - 71658806414 T^{21} + 1535205216922 T^{22} + 13584363696813 T^{23} + 23711660598028 T^{24} - 35843207880740 T^{25} - 582452128062025 T^{26} - 605750213184506 T^{27} + 3937376385699289 T^{28} \))
$17$ (\( 1 - 7 T + 32 T^{2} - 119 T^{3} + 289 T^{4} \))(\( 1 + 2 T - 65 T^{2} - 210 T^{3} + 2087 T^{4} + 9143 T^{5} - 37340 T^{6} - 240381 T^{7} + 293834 T^{8} + 4176065 T^{9} + 3382763 T^{10} - 47662233 T^{11} - 157347285 T^{12} + 273515658 T^{13} + 3170983122 T^{14} + 4649766186 T^{15} - 45473365365 T^{16} - 234164550729 T^{17} + 282531748523 T^{18} + 5929415122705 T^{19} + 7092438449546 T^{20} - 98637620554413 T^{21} - 260474782846940 T^{22} + 1084248954812071 T^{23} + 4207379270237063 T^{24} - 7197098224602930 T^{25} - 37870445419934465 T^{26} + 19809156065811874 T^{27} + 168377826559400929 T^{28} \))
$19$ (\( 1 + 5 T + 6 T^{2} + 95 T^{3} + 361 T^{4} \))(\( 1 - 7 T - 54 T^{2} + 381 T^{3} + 1875 T^{4} - 9873 T^{5} - 65652 T^{6} + 221430 T^{7} + 1870425 T^{8} - 4319703 T^{9} - 46476858 T^{10} + 61637031 T^{11} + 1073881146 T^{12} - 457871775 T^{13} - 21789737442 T^{14} - 8699563725 T^{15} + 387671093706 T^{16} + 422768395629 T^{17} - 6056910611418 T^{18} - 10696012278597 T^{19} + 87995791969425 T^{20} + 197930019166770 T^{21} - 1115004880767732 T^{22} - 3185895640172067 T^{23} + 11495749233376875 T^{24} + 44382788640221439 T^{25} - 119519005629572694 T^{26} - 294370884235799413 T^{27} + 799006685782884121 T^{28} \))
$23$ (\( ( 1 + 4 T + 23 T^{2} )^{2} \))(\( ( 1 - 11 T + 129 T^{2} - 876 T^{3} + 6604 T^{4} - 35691 T^{5} + 217568 T^{6} - 992939 T^{7} + 5004064 T^{8} - 18880539 T^{9} + 80350868 T^{10} - 245140716 T^{11} + 830288247 T^{12} - 1628394779 T^{13} + 3404825447 T^{14} )^{2} \))
$29$ (\( 1 + T - 28 T^{2} + 29 T^{3} + 841 T^{4} \))(\( 1 + T - 89 T^{2} - 606 T^{3} + 3413 T^{4} + 45595 T^{5} + 49603 T^{6} - 1643802 T^{7} - 7893892 T^{8} + 19552444 T^{9} + 271585946 T^{10} + 420585531 T^{11} - 3441402105 T^{12} - 10956432363 T^{13} + 10513309734 T^{14} - 317736538527 T^{15} - 2894219170305 T^{16} + 10257660515559 T^{17} + 192087579472826 T^{18} + 401043092198156 T^{19} - 4695471055055332 T^{20} - 28355381176486818 T^{21} + 24813722822104483 T^{22} + 661453320769747055 T^{23} + 1435873787253586013 T^{24} - 7393508918017732374 T^{25} - 31489515705286744649 T^{26} + 10260628712958602189 T^{27} + \)\(29\!\cdots\!81\)\( T^{28} \))
$31$ (\( 1 - 3 T - 22 T^{2} - 93 T^{3} + 961 T^{4} \))(\( 1 + T - 85 T^{2} - 302 T^{3} + 2323 T^{4} + 16596 T^{5} - 8720 T^{6} - 121502 T^{7} + 336059 T^{8} - 5031700 T^{9} - 52734649 T^{10} - 249456712 T^{11} - 356372186 T^{12} + 8588183707 T^{13} + 82956488229 T^{14} + 266233694917 T^{15} - 342473670746 T^{16} - 7431564907192 T^{17} - 48701555779129 T^{18} - 144053299086700 T^{19} + 298253599533179 T^{20} - 3342837639714722 T^{21} - 7437209846485520 T^{22} + 438791969378495916 T^{23} + 1903996510656400723 T^{24} - 7673360022714258962 T^{25} - 66951336622026729685 T^{26} + 24417546297445042591 T^{27} + \)\(75\!\cdots\!21\)\( T^{28} \))
$37$ (\( ( 1 + T + 37 T^{2} )( 1 + 10 T + 37 T^{2} ) \))(\( 1 - 10 T - 84 T^{2} + 1296 T^{3} + 1134 T^{4} - 66630 T^{5} + 77382 T^{6} + 1851174 T^{7} - 1251993 T^{8} - 21837810 T^{9} - 268575252 T^{10} - 774884982 T^{11} + 27666538119 T^{12} + 26159905206 T^{13} - 1381876923558 T^{14} + 967916492622 T^{15} + 37875490684911 T^{16} - 39250248993246 T^{17} - 503353262863572 T^{18} - 1514320157614170 T^{19} - 3212271503983137 T^{20} + 175735422719804142 T^{21} + 271802685103314822 T^{22} - 8659350722545980510 T^{23} + 5452934678321840766 T^{24} + \)\(23\!\cdots\!48\)\( T^{25} - \)\(55\!\cdots\!04\)\( T^{26} - \)\(24\!\cdots\!70\)\( T^{27} + \)\(90\!\cdots\!89\)\( T^{28} \))
$41$ (\( 1 + 9 T + 40 T^{2} + 369 T^{3} + 1681 T^{4} \))(\( 1 - 33 T + 463 T^{2} - 3882 T^{3} + 26359 T^{4} - 177381 T^{5} + 987377 T^{6} - 3338436 T^{7} - 1489480 T^{8} + 146370792 T^{9} - 1532374696 T^{10} + 10553354379 T^{11} - 66691501599 T^{12} + 460808734581 T^{13} - 3110830401306 T^{14} + 18893158117821 T^{15} - 112108414187919 T^{16} + 727347737155059 T^{17} - 4330124653343656 T^{18} + 16957963898481192 T^{19} - 7075185264884680 T^{20} - 650174679078190116 T^{21} + 7884131517953805617 T^{22} - 58071334904735196141 T^{23} + \)\(35\!\cdots\!59\)\( T^{24} - \)\(21\!\cdots\!62\)\( T^{25} + \)\(10\!\cdots\!03\)\( T^{26} - \)\(30\!\cdots\!93\)\( T^{27} + \)\(37\!\cdots\!61\)\( T^{28} \))
$43$ (\( ( 1 - 8 T + 43 T^{2} )( 1 + 13 T + 43 T^{2} ) \))(\( 1 - 7 T - 222 T^{2} + 1221 T^{3} + 30003 T^{4} - 121065 T^{5} - 2964828 T^{6} + 8400318 T^{7} + 232132089 T^{8} - 430114695 T^{9} - 14993117802 T^{10} + 15328887375 T^{11} + 817956723570 T^{12} - 259275458991 T^{13} - 38016795252930 T^{14} - 11148844736613 T^{15} + 1512401981880930 T^{16} + 1218753848524125 T^{17} - 51258486134595402 T^{18} - 63230491623369885 T^{19} + 1467391209891779361 T^{20} + 2283362771617132026 T^{21} - 34653503452639217628 T^{22} - 60846374564133897795 T^{23} + \)\(64\!\cdots\!47\)\( T^{24} + \)\(11\!\cdots\!47\)\( T^{25} - \)\(88\!\cdots\!22\)\( T^{26} - \)\(12\!\cdots\!01\)\( T^{27} + \)\(73\!\cdots\!49\)\( T^{28} \))
$47$ (\( 1 - 3 T - 38 T^{2} - 141 T^{3} + 2209 T^{4} \))(\( 1 - 3 T - 215 T^{2} + 750 T^{3} + 23197 T^{4} - 82998 T^{5} - 1793695 T^{6} + 5388753 T^{7} + 119401781 T^{8} - 227010657 T^{9} - 7448336422 T^{10} + 6436178352 T^{11} + 428559183834 T^{12} - 95001497937 T^{13} - 21681767924409 T^{14} - 4465070403039 T^{15} + 946687237089306 T^{16} + 668223345039696 T^{17} - 36345505720041382 T^{18} - 52063760718739599 T^{19} + 1287057508065100949 T^{20} + 2730066860264352639 T^{21} - 42710185828767396895 T^{22} - 92885591006583455466 T^{23} + \)\(12\!\cdots\!53\)\( T^{24} + \)\(18\!\cdots\!50\)\( T^{25} - \)\(24\!\cdots\!15\)\( T^{26} - \)\(16\!\cdots\!81\)\( T^{27} + \)\(25\!\cdots\!69\)\( T^{28} \))
$53$ (\( 1 - 3 T - 44 T^{2} - 159 T^{3} + 2809 T^{4} \))(\( 1 - 15 T - 44 T^{2} + 1563 T^{3} + 1621 T^{4} - 132831 T^{5} + 280796 T^{6} + 6875916 T^{7} - 32544895 T^{8} - 334617081 T^{9} + 3332374934 T^{10} + 9404405181 T^{11} - 219667861866 T^{12} - 268474558113 T^{13} + 14089335458730 T^{14} - 14229151579989 T^{15} - 617047023981594 T^{16} + 1400099630131737 T^{17} + 26294041101603254 T^{18} - 139935355155015933 T^{19} - 721336805685386455 T^{20} + 8077215121783465692 T^{21} + 17482272028748523356 T^{22} - \)\(43\!\cdots\!23\)\( T^{23} + \)\(28\!\cdots\!29\)\( T^{24} + \)\(14\!\cdots\!11\)\( T^{25} - \)\(21\!\cdots\!04\)\( T^{26} - \)\(39\!\cdots\!95\)\( T^{27} + \)\(13\!\cdots\!69\)\( T^{28} \))
$59$ (\( 1 + 7 T - 10 T^{2} + 413 T^{3} + 3481 T^{4} \))(\( 1 - 14 T - 41 T^{2} + 1932 T^{3} - 9268 T^{4} - 67958 T^{5} + 1000462 T^{6} - 4853283 T^{7} - 9914932 T^{8} + 321187219 T^{9} - 1782148489 T^{10} + 6021692682 T^{11} - 34607406657 T^{12} - 638777810472 T^{13} + 11047710270819 T^{14} - 37687890817848 T^{15} - 120468382573017 T^{16} + 1236729221336478 T^{17} - 21594936596817529 T^{18} + 229624547391334481 T^{19} - 418217122774227412 T^{20} - 12078129944196810777 T^{21} + \)\(14\!\cdots\!02\)\( T^{22} - \)\(58\!\cdots\!62\)\( T^{23} - \)\(47\!\cdots\!68\)\( T^{24} + \)\(58\!\cdots\!88\)\( T^{25} - \)\(72\!\cdots\!21\)\( T^{26} - \)\(14\!\cdots\!06\)\( T^{27} + \)\(61\!\cdots\!61\)\( T^{28} \))
$61$ (\( 1 + 3 T - 52 T^{2} + 183 T^{3} + 3721 T^{4} \))(\( 1 + 10 T - 277 T^{2} - 2192 T^{3} + 49786 T^{4} + 275394 T^{5} - 6629804 T^{6} - 25018331 T^{7} + 694393718 T^{8} + 1681932119 T^{9} - 61000901305 T^{10} - 82163567860 T^{11} + 4597121803909 T^{12} + 1969709923870 T^{13} - 300133324238415 T^{14} + 120152305356070 T^{15} + 17105890232345389 T^{16} - 18649568796430660 T^{17} - 844608780325722505 T^{18} + 1420553646240491819 T^{19} + 35775424305286664198 T^{20} - 78626180519452100951 T^{21} - \)\(12\!\cdots\!24\)\( T^{22} + \)\(32\!\cdots\!54\)\( T^{23} + \)\(35\!\cdots\!86\)\( T^{24} - \)\(95\!\cdots\!12\)\( T^{25} - \)\(73\!\cdots\!17\)\( T^{26} + \)\(16\!\cdots\!10\)\( T^{27} + \)\(98\!\cdots\!41\)\( T^{28} \))
$67$ (\( 1 + 13 T + 102 T^{2} + 871 T^{3} + 4489 T^{4} \))(\( 1 - 6 T - 308 T^{2} + 164 T^{3} + 62908 T^{4} + 146213 T^{5} - 7475396 T^{6} - 42326464 T^{7} + 603793243 T^{8} + 5085070936 T^{9} - 25351846493 T^{10} - 374911832519 T^{11} + 131329441787 T^{12} + 10392590626498 T^{13} + 54642703536353 T^{14} + 696303571975366 T^{15} + 589537864181843 T^{16} - 112759607483911997 T^{17} - 510868126253868653 T^{18} + 6865481941569590152 T^{19} + 54618159926353884067 T^{20} - \)\(25\!\cdots\!72\)\( T^{21} - \)\(30\!\cdots\!36\)\( T^{22} + \)\(39\!\cdots\!11\)\( T^{23} + \)\(11\!\cdots\!92\)\( T^{24} + \)\(20\!\cdots\!12\)\( T^{25} - \)\(25\!\cdots\!88\)\( T^{26} - \)\(32\!\cdots\!22\)\( T^{27} + \)\(36\!\cdots\!29\)\( T^{28} \))
$71$ (\( ( 1 - 8 T + 71 T^{2} )^{2} \))(\( ( 1 + T + 381 T^{2} + 417 T^{3} + 66850 T^{4} + 68550 T^{5} + 7142942 T^{6} + 6246700 T^{7} + 507148882 T^{8} + 345560550 T^{9} + 23926350350 T^{10} + 10596670977 T^{11} + 687411382731 T^{12} + 128100283921 T^{13} + 9095120158391 T^{14} )^{2} \))
$73$ (\( ( 1 - 10 T + 73 T^{2} )( 1 + 17 T + 73 T^{2} ) \))(\( 1 - 21 T - 107 T^{2} + 4532 T^{3} + 8593 T^{4} - 574123 T^{5} - 1508435 T^{6} + 52395764 T^{7} + 342478696 T^{8} - 4227610868 T^{9} - 46283154524 T^{10} + 253115011813 T^{11} + 4673586532751 T^{12} - 7744045394141 T^{13} - 369527474872486 T^{14} - 565315313772293 T^{15} + 24905542633030079 T^{16} + 98466042550457821 T^{17} - 1314360176412792284 T^{18} - 8764139996708872724 T^{19} + 51828748479625639144 T^{20} + \)\(57\!\cdots\!08\)\( T^{21} - \)\(12\!\cdots\!35\)\( T^{22} - \)\(33\!\cdots\!99\)\( T^{23} + \)\(36\!\cdots\!57\)\( T^{24} + \)\(14\!\cdots\!64\)\( T^{25} - \)\(24\!\cdots\!47\)\( T^{26} - \)\(35\!\cdots\!93\)\( T^{27} + \)\(12\!\cdots\!09\)\( T^{28} \))
$79$ (\( 1 - 9 T + 2 T^{2} - 711 T^{3} + 6241 T^{4} \))(\( 1 + 10 T - 226 T^{2} - 4280 T^{3} + 9610 T^{4} + 654627 T^{5} + 3201292 T^{6} - 42192356 T^{7} - 519045919 T^{8} - 901423894 T^{9} + 24953123963 T^{10} + 320449224329 T^{11} + 1694205545953 T^{12} - 14615843505140 T^{13} - 287108375549019 T^{14} - 1154651636906060 T^{15} + 10573536812292673 T^{16} + 157993965113945831 T^{17} + 971926199561891003 T^{18} - 2773732161244197706 T^{19} - \)\(12\!\cdots\!99\)\( T^{20} - \)\(81\!\cdots\!04\)\( T^{21} + \)\(48\!\cdots\!12\)\( T^{22} + \)\(78\!\cdots\!13\)\( T^{23} + \)\(90\!\cdots\!10\)\( T^{24} - \)\(32\!\cdots\!20\)\( T^{25} - \)\(13\!\cdots\!66\)\( T^{26} + \)\(46\!\cdots\!90\)\( T^{27} + \)\(36\!\cdots\!81\)\( T^{28} \))
$83$ (\( 1 - T - 82 T^{2} - 83 T^{3} + 6889 T^{4} \))(\( 1 - 25 T + 157 T^{2} + 750 T^{3} - 6622 T^{4} - 69964 T^{5} + 1466905 T^{6} - 20424981 T^{7} + 74872112 T^{8} + 997051343 T^{9} + 7203133487 T^{10} - 269189728764 T^{11} + 1307717205141 T^{12} + 2467734607815 T^{13} - 39825488981322 T^{14} + 204821972448645 T^{15} + 9008863826216349 T^{16} - 153919187440781268 T^{17} + 341848621231895327 T^{18} + 3927425763234733549 T^{19} + 24478716252205585328 T^{20} - \)\(55\!\cdots\!87\)\( T^{21} + \)\(33\!\cdots\!05\)\( T^{22} - \)\(13\!\cdots\!92\)\( T^{23} - \)\(10\!\cdots\!78\)\( T^{24} + \)\(96\!\cdots\!50\)\( T^{25} + \)\(16\!\cdots\!77\)\( T^{26} - \)\(22\!\cdots\!75\)\( T^{27} + \)\(73\!\cdots\!29\)\( T^{28} \))
$89$ (\( 1 - 15 T + 136 T^{2} - 1335 T^{3} + 7921 T^{4} \))(\( 1 - 6 T - 395 T^{2} + 1770 T^{3} + 83413 T^{4} - 244017 T^{5} - 12791578 T^{6} + 19742595 T^{7} + 1607888462 T^{8} - 933579267 T^{9} - 173735706451 T^{10} + 11863751967 T^{11} + 16888438123377 T^{12} + 583089996186 T^{13} - 1538438146646466 T^{14} + 51895009660554 T^{15} + 133773318375269217 T^{16} + 8363577360424023 T^{17} - 10900567564453896691 T^{18} - 5213162127281843883 T^{19} + \)\(79\!\cdots\!82\)\( T^{20} + \)\(87\!\cdots\!55\)\( T^{21} - \)\(50\!\cdots\!18\)\( T^{22} - \)\(85\!\cdots\!53\)\( T^{23} + \)\(26\!\cdots\!13\)\( T^{24} + \)\(49\!\cdots\!30\)\( T^{25} - \)\(97\!\cdots\!95\)\( T^{26} - \)\(13\!\cdots\!14\)\( T^{27} + \)\(19\!\cdots\!41\)\( T^{28} \))
$97$ (\( 1 - 17 T + 192 T^{2} - 1649 T^{3} + 9409 T^{4} \))(\( 1 + 18 T - 332 T^{2} - 7282 T^{3} + 70792 T^{4} + 1694642 T^{5} - 11251793 T^{6} - 262778647 T^{7} + 1667779429 T^{8} + 30411590872 T^{9} - 218639545013 T^{10} - 2436064838528 T^{11} + 25890765026051 T^{12} + 94489464293929 T^{13} - 2645836565169718 T^{14} + 9165478036511113 T^{15} + 243606208130113859 T^{16} - 2223330604373865344 T^{17} - 19356001718168025653 T^{18} + \)\(26\!\cdots\!04\)\( T^{19} + \)\(13\!\cdots\!41\)\( T^{20} - \)\(21\!\cdots\!11\)\( T^{21} - \)\(88\!\cdots\!73\)\( T^{22} + \)\(12\!\cdots\!14\)\( T^{23} + \)\(52\!\cdots\!08\)\( T^{24} - \)\(52\!\cdots\!46\)\( T^{25} - \)\(23\!\cdots\!12\)\( T^{26} + \)\(12\!\cdots\!86\)\( T^{27} + \)\(65\!\cdots\!69\)\( T^{28} \))
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