Properties

Label 1344.2.w
Level 1344
Weight 2
Character orbit w
Rep. character \(\chi_{1344}(337,\cdot)\)
Character field \(\Q(\zeta_{4})\)
Dimension 48
Newform subspaces 2
Sturm bound 512
Trace bound 1

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Defining parameters

Level: \( N \) \(=\) \( 1344 = 2^{6} \cdot 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1344.w (of order \(4\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 16 \)
Character field: \(\Q(i)\)
Newform subspaces: \( 2 \)
Sturm bound: \(512\)
Trace bound: \(1\)
Distinguishing \(T_p\): \(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(1344, [\chi])\).

Total New Old
Modular forms 544 48 496
Cusp forms 480 48 432
Eisenstein series 64 0 64

Trace form

\( 48q + O(q^{10}) \) \( 48q - 8q^{11} - 16q^{15} - 16q^{19} + 16q^{29} + 16q^{37} - 24q^{43} - 48q^{49} + 16q^{51} - 16q^{53} - 32q^{61} + 8q^{63} + 32q^{65} + 8q^{67} - 32q^{69} + 32q^{75} - 16q^{77} - 48q^{79} - 48q^{81} - 80q^{83} + 32q^{85} - 8q^{99} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(1344, [\chi])\) into newform subspaces

Label Dim. \(A\) Field CM Traces $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\)
1344.2.w.a \(20\) \(10.732\) \(\mathbb{Q}[x]/(x^{20} - \cdots)\) None \(0\) \(0\) \(0\) \(0\) \(q-\beta _{13}q^{3}+\beta _{11}q^{5}-\beta _{15}q^{7}-\beta _{15}q^{9}+\cdots\)
1344.2.w.b \(28\) \(10.732\) None \(0\) \(0\) \(0\) \(0\)

Decomposition of \(S_{2}^{\mathrm{old}}(1344, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(1344, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(16, [\chi])\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(48, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(64, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(112, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(192, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(336, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(448, [\chi])\)\(^{\oplus 2}\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ (\( ( 1 + T^{4} )^{5} \))
$5$ (\( 1 - 24 T^{3} + 10 T^{4} + 24 T^{5} + 288 T^{6} - 320 T^{7} + 485 T^{8} - 2720 T^{9} + 5088 T^{10} - 20208 T^{11} + 28392 T^{12} - 21904 T^{13} + 280352 T^{14} - 374560 T^{15} + 85402 T^{16} - 2818304 T^{17} + 5181408 T^{18} - 7099744 T^{19} + 30439356 T^{20} - 35498720 T^{21} + 129535200 T^{22} - 352288000 T^{23} + 53376250 T^{24} - 1170500000 T^{25} + 4380500000 T^{26} - 1711250000 T^{27} + 11090625000 T^{28} - 39468750000 T^{29} + 49687500000 T^{30} - 132812500000 T^{31} + 118408203125 T^{32} - 390625000000 T^{33} + 1757812500000 T^{34} + 732421875000 T^{35} + 1525878906250 T^{36} - 18310546875000 T^{37} + 95367431640625 T^{40} \))
$7$ (\( ( 1 + T^{2} )^{10} \))
$11$ (\( 1 + 12 T + 72 T^{2} + 404 T^{3} + 2214 T^{4} + 9404 T^{5} + 35048 T^{6} + 135156 T^{7} + 456709 T^{8} + 1318720 T^{9} + 4165952 T^{10} + 13629648 T^{11} + 41111256 T^{12} + 141076784 T^{13} + 543949056 T^{14} + 1968344544 T^{15} + 6732047034 T^{16} + 23311377560 T^{17} + 80768058832 T^{18} + 261514302488 T^{19} + 834968255588 T^{20} + 2876657327368 T^{21} + 9772935118672 T^{22} + 31027443532360 T^{23} + 98563900624794 T^{24} + 317003857155744 T^{25} + 963638933596416 T^{26} + 2749187413938064 T^{27} + 8812562832664536 T^{28} + 32137997030742768 T^{29} + 108054065891385152 T^{30} + 376246206268137920 T^{31} + 1433348485503871189 T^{32} + 4665951682525138236 T^{33} + 13309472167425430568 T^{34} + 39282841785184782004 T^{35} + \)\(10\!\cdots\!54\)\( T^{36} + \)\(20\!\cdots\!84\)\( T^{37} + \)\(40\!\cdots\!32\)\( T^{38} + \)\(73\!\cdots\!92\)\( T^{39} + \)\(67\!\cdots\!01\)\( T^{40} \))
$13$ (\( 1 - 80 T^{3} - 118 T^{4} + 1616 T^{5} + 3200 T^{6} + 22560 T^{7} - 156451 T^{8} - 297120 T^{9} - 121472 T^{10} + 6082400 T^{11} + 43092280 T^{12} - 92564192 T^{13} - 225951616 T^{14} - 2287891296 T^{15} + 3788315570 T^{16} + 30959844064 T^{17} + 31244658048 T^{18} + 7310121920 T^{19} - 2235843760900 T^{20} + 95031584960 T^{21} + 5280347210112 T^{22} + 68018777408608 T^{23} + 108198080994770 T^{24} - 849478022965728 T^{25} - 1090625293673344 T^{26} - 5808265775303264 T^{27} + 35151696633933880 T^{28} + 64500806986335200 T^{29} - 16745946721881728 T^{30} - 532486696276273440 T^{31} - 3645008715497274931 T^{32} + 6832862404721227680 T^{33} + 12599604434237724800 T^{34} + 82716403110770663312 T^{35} - 78519159883615221238 T^{36} - \)\(69\!\cdots\!40\)\( T^{37} + \)\(19\!\cdots\!01\)\( T^{40} \))
$17$ (\( ( 1 + 94 T^{2} + 88 T^{3} + 4293 T^{4} + 6432 T^{5} + 134312 T^{6} + 219984 T^{7} + 3234206 T^{8} + 4965576 T^{9} + 61650188 T^{10} + 84414792 T^{11} + 934685534 T^{12} + 1080781392 T^{13} + 11217872552 T^{14} + 9132520224 T^{15} + 103622583717 T^{16} + 36109803224 T^{17} + 655721199454 T^{18} + 2015993900449 T^{20} )^{2} \))
$19$ (\( 1 + 8 T + 32 T^{2} + 120 T^{3} + 170 T^{4} - 664 T^{5} - 3552 T^{6} - 30376 T^{7} - 137251 T^{8} - 141472 T^{9} + 439424 T^{10} + 1977376 T^{11} + 92609976 T^{12} + 734762848 T^{13} + 3144939136 T^{14} + 11850068256 T^{15} + 18020441650 T^{16} - 30523407824 T^{17} - 82150980672 T^{18} - 1163681013680 T^{19} - 8113217288772 T^{20} - 22109939259920 T^{21} - 29656504022592 T^{22} - 209360054264816 T^{23} + 2348441976269650 T^{24} + 29341942158613344 T^{25} + 147956432344498816 T^{26} + 656783744694352672 T^{27} + 1572847365621497016 T^{28} + 638074909083447904 T^{29} + 2694137659267946624 T^{30} - 16480109906848838368 T^{31} - \)\(30\!\cdots\!11\)\( T^{32} - \)\(12\!\cdots\!84\)\( T^{33} - \)\(28\!\cdots\!92\)\( T^{34} - \)\(10\!\cdots\!36\)\( T^{35} + \)\(49\!\cdots\!70\)\( T^{36} + \)\(65\!\cdots\!80\)\( T^{37} + \)\(33\!\cdots\!12\)\( T^{38} + \)\(15\!\cdots\!32\)\( T^{39} + \)\(37\!\cdots\!01\)\( T^{40} \))
$23$ (\( 1 - 228 T^{2} + 24654 T^{4} - 1671172 T^{6} + 79227605 T^{8} - 2786970288 T^{10} + 76106467832 T^{12} - 1708777858416 T^{14} + 34629617977562 T^{16} - 710121266972024 T^{18} + 15689370725592020 T^{20} - 375654150228200696 T^{22} + 9690786924458927642 T^{24} - \)\(25\!\cdots\!24\)\( T^{26} + \)\(59\!\cdots\!92\)\( T^{28} - \)\(11\!\cdots\!12\)\( T^{30} + \)\(17\!\cdots\!05\)\( T^{32} - \)\(19\!\cdots\!48\)\( T^{34} + \)\(15\!\cdots\!94\)\( T^{36} - \)\(73\!\cdots\!32\)\( T^{38} + \)\(17\!\cdots\!01\)\( T^{40} \))
$29$ (\( 1 - 12 T + 72 T^{2} - 660 T^{3} + 5062 T^{4} - 19812 T^{5} + 91080 T^{6} - 552988 T^{7} + 898845 T^{8} + 3230928 T^{9} - 3305184 T^{10} + 180759600 T^{11} - 2401479032 T^{12} + 11590609840 T^{13} - 61731406432 T^{14} + 542524821712 T^{15} - 2180038368078 T^{16} + 2872412612888 T^{17} - 25762237441424 T^{18} + 2546561760168 T^{19} + 1284480158865124 T^{20} + 73850291044872 T^{21} - 21666041688237584 T^{22} + 70055271215725432 T^{23} - 1541899717012575918 T^{24} + 11127807454333267088 T^{25} - 36719280183883000672 T^{26} + \)\(19\!\cdots\!60\)\( T^{27} - \)\(12\!\cdots\!52\)\( T^{28} + \)\(26\!\cdots\!00\)\( T^{29} - \)\(13\!\cdots\!84\)\( T^{30} + \)\(39\!\cdots\!12\)\( T^{31} + \)\(31\!\cdots\!45\)\( T^{32} - \)\(56\!\cdots\!32\)\( T^{33} + \)\(27\!\cdots\!80\)\( T^{34} - \)\(17\!\cdots\!88\)\( T^{35} + \)\(12\!\cdots\!02\)\( T^{36} - \)\(47\!\cdots\!40\)\( T^{37} + \)\(15\!\cdots\!92\)\( T^{38} - \)\(73\!\cdots\!28\)\( T^{39} + \)\(17\!\cdots\!01\)\( T^{40} \))
$31$ (\( ( 1 + 134 T^{2} + 64 T^{3} + 7637 T^{4} - 64 T^{5} + 250776 T^{6} - 647232 T^{7} + 5787898 T^{8} - 46731200 T^{9} + 141742500 T^{10} - 1448667200 T^{11} + 5562169978 T^{12} - 19281688512 T^{13} + 231596902296 T^{14} - 1832265664 T^{15} + 6777865611797 T^{16} + 1760807303104 T^{17} + 114287399017094 T^{18} + 819628286980801 T^{20} )^{2} \))
$37$ (\( 1 - 12 T + 72 T^{2} - 692 T^{3} + 4646 T^{4} - 25508 T^{5} + 211016 T^{6} - 1455420 T^{7} + 12729021 T^{8} - 102763568 T^{9} + 668772640 T^{10} - 4907491152 T^{11} + 30685175816 T^{12} - 181058288208 T^{13} + 1246622676896 T^{14} - 7968453274800 T^{15} + 52782260071730 T^{16} - 346974533424104 T^{17} + 2138919388551792 T^{18} - 13427579452955416 T^{19} + 82729716000045604 T^{20} - 496820439759350392 T^{21} + 2928180642927403248 T^{22} - 17575301041531139912 T^{23} + 98922453318293568530 T^{24} - \)\(55\!\cdots\!00\)\( T^{25} + \)\(31\!\cdots\!64\)\( T^{26} - \)\(17\!\cdots\!64\)\( T^{27} + \)\(10\!\cdots\!36\)\( T^{28} - \)\(63\!\cdots\!04\)\( T^{29} + \)\(32\!\cdots\!60\)\( T^{30} - \)\(18\!\cdots\!84\)\( T^{31} + \)\(83\!\cdots\!01\)\( T^{32} - \)\(35\!\cdots\!40\)\( T^{33} + \)\(19\!\cdots\!24\)\( T^{34} - \)\(85\!\cdots\!44\)\( T^{35} + \)\(57\!\cdots\!86\)\( T^{36} - \)\(31\!\cdots\!64\)\( T^{37} + \)\(12\!\cdots\!88\)\( T^{38} - \)\(74\!\cdots\!76\)\( T^{39} + \)\(23\!\cdots\!01\)\( T^{40} \))
$41$ (\( 1 - 444 T^{2} + 98574 T^{4} - 14512124 T^{6} + 1588488149 T^{8} - 137675000112 T^{10} + 9846636632440 T^{12} - 599501624736304 T^{14} + 31880388630283994 T^{16} - 1512585137889846120 T^{18} + 64990434418739739988 T^{20} - \)\(25\!\cdots\!20\)\( T^{22} + \)\(90\!\cdots\!34\)\( T^{24} - \)\(28\!\cdots\!64\)\( T^{26} + \)\(78\!\cdots\!40\)\( T^{28} - \)\(18\!\cdots\!12\)\( T^{30} + \)\(35\!\cdots\!69\)\( T^{32} - \)\(55\!\cdots\!64\)\( T^{34} + \)\(62\!\cdots\!34\)\( T^{36} - \)\(47\!\cdots\!24\)\( T^{38} + \)\(18\!\cdots\!01\)\( T^{40} \))
$43$ (\( 1 + 4 T + 8 T^{2} - 300 T^{3} + 7078 T^{4} + 44284 T^{5} + 165512 T^{6} - 1626836 T^{7} + 19994909 T^{8} + 198352400 T^{9} + 930543520 T^{10} - 1629993648 T^{11} + 32879446344 T^{12} + 523812434096 T^{13} + 2532563214112 T^{14} + 8844397053040 T^{15} + 52614175288210 T^{16} + 1034528298704760 T^{17} + 4610401791613808 T^{18} + 37116758465801176 T^{19} + 100269344691342500 T^{20} + 1596020614029450568 T^{21} + 8524632912693930992 T^{22} + 82252241445119353320 T^{23} + \)\(17\!\cdots\!10\)\( T^{24} + \)\(13\!\cdots\!20\)\( T^{25} + \)\(16\!\cdots\!88\)\( T^{26} + \)\(14\!\cdots\!72\)\( T^{27} + \)\(38\!\cdots\!44\)\( T^{28} - \)\(81\!\cdots\!64\)\( T^{29} + \)\(20\!\cdots\!80\)\( T^{30} + \)\(18\!\cdots\!00\)\( T^{31} + \)\(79\!\cdots\!09\)\( T^{32} - \)\(27\!\cdots\!48\)\( T^{33} + \)\(12\!\cdots\!88\)\( T^{34} + \)\(14\!\cdots\!88\)\( T^{35} + \)\(96\!\cdots\!78\)\( T^{36} - \)\(17\!\cdots\!00\)\( T^{37} + \)\(20\!\cdots\!92\)\( T^{38} + \)\(43\!\cdots\!28\)\( T^{39} + \)\(46\!\cdots\!01\)\( T^{40} \))
$47$ (\( ( 1 + 254 T^{2} + 128 T^{3} + 33997 T^{4} + 26752 T^{5} + 3110696 T^{6} + 2769024 T^{7} + 212743186 T^{8} + 186077312 T^{9} + 11293892148 T^{10} + 8745633664 T^{11} + 469949697874 T^{12} + 287488378752 T^{13} + 15179204167976 T^{14} + 6135437627264 T^{15} + 366460983540013 T^{16} + 64847759419264 T^{17} + 6048066812087294 T^{18} + 52599132235830049 T^{20} )^{2} \))
$53$ (\( 1 + 36 T + 648 T^{2} + 8284 T^{3} + 81222 T^{4} + 595308 T^{5} + 3111560 T^{6} + 8277492 T^{7} - 42534915 T^{8} - 754029808 T^{9} - 6543023328 T^{10} - 50633847440 T^{11} - 420252241016 T^{12} - 3590853938064 T^{13} - 27232240956512 T^{14} - 163759336672624 T^{15} - 644108178966222 T^{16} - 225724213191624 T^{17} + 21359012626695920 T^{18} + 241077022660459848 T^{19} + 1941342482920284900 T^{20} + 12777082201004371944 T^{21} + 59997466468388839280 T^{22} - 33605143687329406248 T^{23} - \)\(50\!\cdots\!82\)\( T^{24} - \)\(68\!\cdots\!32\)\( T^{25} - \)\(60\!\cdots\!48\)\( T^{26} - \)\(42\!\cdots\!68\)\( T^{27} - \)\(26\!\cdots\!76\)\( T^{28} - \)\(16\!\cdots\!20\)\( T^{29} - \)\(11\!\cdots\!72\)\( T^{30} - \)\(69\!\cdots\!76\)\( T^{31} - \)\(20\!\cdots\!15\)\( T^{32} + \)\(21\!\cdots\!16\)\( T^{33} + \)\(42\!\cdots\!40\)\( T^{34} + \)\(43\!\cdots\!56\)\( T^{35} + \)\(31\!\cdots\!62\)\( T^{36} + \)\(17\!\cdots\!92\)\( T^{37} + \)\(70\!\cdots\!72\)\( T^{38} + \)\(20\!\cdots\!12\)\( T^{39} + \)\(30\!\cdots\!01\)\( T^{40} \))
$59$ (\( 1 - 64 T^{3} + 17498 T^{4} - 16320 T^{5} + 2048 T^{6} - 964352 T^{7} + 139305821 T^{8} - 265011968 T^{9} + 159053824 T^{10} - 7117236224 T^{11} + 675196772280 T^{12} - 1865453307392 T^{13} + 2177063692288 T^{14} - 46993822767872 T^{15} + 2378495821379602 T^{16} - 7365185499789056 T^{17} + 13796808199690240 T^{18} - 308926250030895744 T^{19} + 7830691115365441372 T^{20} - 18226648751822848896 T^{21} + 48026689343121725440 T^{22} - \)\(15\!\cdots\!24\)\( T^{23} + \)\(28\!\cdots\!22\)\( T^{24} - \)\(33\!\cdots\!28\)\( T^{25} + \)\(91\!\cdots\!08\)\( T^{26} - \)\(46\!\cdots\!48\)\( T^{27} + \)\(99\!\cdots\!80\)\( T^{28} - \)\(61\!\cdots\!36\)\( T^{29} + \)\(81\!\cdots\!24\)\( T^{30} - \)\(79\!\cdots\!12\)\( T^{31} + \)\(24\!\cdots\!01\)\( T^{32} - \)\(10\!\cdots\!08\)\( T^{33} + \)\(12\!\cdots\!28\)\( T^{34} - \)\(59\!\cdots\!80\)\( T^{35} + \)\(37\!\cdots\!18\)\( T^{36} - \)\(81\!\cdots\!16\)\( T^{37} + \)\(26\!\cdots\!01\)\( T^{40} \))
$61$ (\( 1 - 8 T + 32 T^{2} - 24 T^{3} - 22 T^{4} + 28776 T^{5} - 229216 T^{6} + 1701688 T^{7} - 7327587 T^{8} + 193356480 T^{9} - 944223232 T^{10} + 7441249856 T^{11} + 25030907320 T^{12} - 252322194176 T^{13} + 6350522522880 T^{14} - 30700611174144 T^{15} + 72405460654514 T^{16} + 1378721064691056 T^{17} + 10443615781913408 T^{18} + 4859962566983248 T^{19} + 374225284282585404 T^{20} + 296457716585978128 T^{21} + 38860694324499791168 T^{22} + \)\(31\!\cdots\!36\)\( T^{23} + \)\(10\!\cdots\!74\)\( T^{24} - \)\(25\!\cdots\!44\)\( T^{25} + \)\(32\!\cdots\!80\)\( T^{26} - \)\(79\!\cdots\!96\)\( T^{27} + \)\(47\!\cdots\!20\)\( T^{28} + \)\(87\!\cdots\!96\)\( T^{29} - \)\(67\!\cdots\!32\)\( T^{30} + \)\(84\!\cdots\!80\)\( T^{31} - \)\(19\!\cdots\!27\)\( T^{32} + \)\(27\!\cdots\!28\)\( T^{33} - \)\(22\!\cdots\!56\)\( T^{34} + \)\(17\!\cdots\!76\)\( T^{35} - \)\(80\!\cdots\!42\)\( T^{36} - \)\(53\!\cdots\!04\)\( T^{37} + \)\(43\!\cdots\!92\)\( T^{38} - \)\(66\!\cdots\!28\)\( T^{39} + \)\(50\!\cdots\!01\)\( T^{40} \))
$67$ (\( 1 - 12 T + 72 T^{2} - 444 T^{3} + 4102 T^{4} - 54484 T^{5} + 457032 T^{6} - 4150628 T^{7} - 7647939 T^{8} + 292442192 T^{9} - 1506268000 T^{10} + 8570912144 T^{11} - 68934819128 T^{12} + 1058425781488 T^{13} - 9188787191264 T^{14} + 93290480178608 T^{15} + 105617726671378 T^{16} - 7399255991086056 T^{17} + 46962879527876336 T^{18} - 365051212101087880 T^{19} + 2665710348810285028 T^{20} - 24458431210772887960 T^{21} + \)\(21\!\cdots\!04\)\( T^{22} - \)\(22\!\cdots\!28\)\( T^{23} + \)\(21\!\cdots\!38\)\( T^{24} + \)\(12\!\cdots\!56\)\( T^{25} - \)\(83\!\cdots\!16\)\( T^{26} + \)\(64\!\cdots\!24\)\( T^{27} - \)\(27\!\cdots\!48\)\( T^{28} + \)\(23\!\cdots\!68\)\( T^{29} - \)\(27\!\cdots\!00\)\( T^{30} + \)\(35\!\cdots\!36\)\( T^{31} - \)\(62\!\cdots\!79\)\( T^{32} - \)\(22\!\cdots\!36\)\( T^{33} + \)\(16\!\cdots\!28\)\( T^{34} - \)\(13\!\cdots\!12\)\( T^{35} + \)\(67\!\cdots\!62\)\( T^{36} - \)\(49\!\cdots\!88\)\( T^{37} + \)\(53\!\cdots\!48\)\( T^{38} - \)\(59\!\cdots\!36\)\( T^{39} + \)\(33\!\cdots\!01\)\( T^{40} \))
$71$ (\( 1 - 588 T^{2} + 187310 T^{4} - 41853292 T^{6} + 7286062485 T^{8} - 1044227363664 T^{10} + 127464606182264 T^{12} - 13544991508155920 T^{14} + 1271322412181875034 T^{16} - \)\(10\!\cdots\!40\)\( T^{18} + \)\(79\!\cdots\!36\)\( T^{20} - \)\(53\!\cdots\!40\)\( T^{22} + \)\(32\!\cdots\!54\)\( T^{24} - \)\(17\!\cdots\!20\)\( T^{26} + \)\(82\!\cdots\!04\)\( T^{28} - \)\(33\!\cdots\!64\)\( T^{30} + \)\(11\!\cdots\!85\)\( T^{32} - \)\(34\!\cdots\!52\)\( T^{34} + \)\(78\!\cdots\!10\)\( T^{36} - \)\(12\!\cdots\!68\)\( T^{38} + \)\(10\!\cdots\!01\)\( T^{40} \))
$73$ (\( 1 - 588 T^{2} + 185438 T^{4} - 41247404 T^{6} + 7184467213 T^{8} - 1034846312688 T^{10} + 127434000595944 T^{12} - 13711360420506544 T^{14} + 1308031080629694098 T^{16} - \)\(11\!\cdots\!16\)\( T^{18} + \)\(85\!\cdots\!04\)\( T^{20} - \)\(59\!\cdots\!64\)\( T^{22} + \)\(37\!\cdots\!18\)\( T^{24} - \)\(20\!\cdots\!16\)\( T^{26} + \)\(10\!\cdots\!64\)\( T^{28} - \)\(44\!\cdots\!12\)\( T^{30} + \)\(16\!\cdots\!73\)\( T^{32} - \)\(50\!\cdots\!36\)\( T^{34} + \)\(12\!\cdots\!18\)\( T^{36} - \)\(20\!\cdots\!72\)\( T^{38} + \)\(18\!\cdots\!01\)\( T^{40} \))
$79$ (\( ( 1 + 12 T + 410 T^{2} + 3636 T^{3} + 71661 T^{4} + 457104 T^{5} + 6962296 T^{6} + 26461552 T^{7} + 422059634 T^{8} + 425195336 T^{9} + 24696310492 T^{10} + 33590431544 T^{11} + 2634074175794 T^{12} + 13046577136528 T^{13} + 271181993145976 T^{14} + 1406534788208496 T^{15} + 17419890150090381 T^{16} + 69825413073674124 T^{17} + 622014612061690010 T^{18} + 1438219151791419828 T^{19} + 9468276082626847201 T^{20} )^{2} \))
$83$ (\( 1 + 40 T + 800 T^{2} + 11352 T^{3} + 130714 T^{4} + 1275720 T^{5} + 10891552 T^{6} + 81158136 T^{7} + 533032573 T^{8} + 3591219744 T^{9} + 28582322304 T^{10} + 250649734880 T^{11} + 2277897381816 T^{12} + 18560960241440 T^{13} + 98547001707136 T^{14} - 326327140527904 T^{15} - 18693002705618542 T^{16} - 320247589539147216 T^{17} - 4017303386407089728 T^{18} - 43457553116792444976 T^{19} - \)\(41\!\cdots\!72\)\( T^{20} - \)\(36\!\cdots\!08\)\( T^{21} - \)\(27\!\cdots\!92\)\( T^{22} - \)\(18\!\cdots\!92\)\( T^{23} - \)\(88\!\cdots\!82\)\( T^{24} - \)\(12\!\cdots\!72\)\( T^{25} + \)\(32\!\cdots\!84\)\( T^{26} + \)\(50\!\cdots\!80\)\( T^{27} + \)\(51\!\cdots\!56\)\( T^{28} + \)\(46\!\cdots\!40\)\( T^{29} + \)\(44\!\cdots\!96\)\( T^{30} + \)\(46\!\cdots\!48\)\( T^{31} + \)\(56\!\cdots\!53\)\( T^{32} + \)\(72\!\cdots\!68\)\( T^{33} + \)\(80\!\cdots\!08\)\( T^{34} + \)\(77\!\cdots\!40\)\( T^{35} + \)\(66\!\cdots\!34\)\( T^{36} + \)\(47\!\cdots\!96\)\( T^{37} + \)\(27\!\cdots\!00\)\( T^{38} + \)\(11\!\cdots\!80\)\( T^{39} + \)\(24\!\cdots\!01\)\( T^{40} \))
$89$ (\( 1 - 700 T^{2} + 256046 T^{4} - 64249660 T^{6} + 12305003221 T^{8} - 1904804480368 T^{10} + 247648665045240 T^{12} - 27922948227608560 T^{14} + 2822449388027483546 T^{16} - \)\(26\!\cdots\!68\)\( T^{18} + \)\(23\!\cdots\!96\)\( T^{20} - \)\(21\!\cdots\!28\)\( T^{22} + \)\(17\!\cdots\!86\)\( T^{24} - \)\(13\!\cdots\!60\)\( T^{26} + \)\(97\!\cdots\!40\)\( T^{28} - \)\(59\!\cdots\!68\)\( T^{30} + \)\(30\!\cdots\!41\)\( T^{32} - \)\(12\!\cdots\!60\)\( T^{34} + \)\(39\!\cdots\!06\)\( T^{36} - \)\(85\!\cdots\!00\)\( T^{38} + \)\(97\!\cdots\!01\)\( T^{40} \))
$97$ (\( ( 1 + 36 T + 1006 T^{2} + 17860 T^{3} + 282157 T^{4} + 3451280 T^{5} + 41738920 T^{6} + 425104976 T^{7} + 4638925202 T^{8} + 43768470968 T^{9} + 461725986516 T^{10} + 4245541683896 T^{11} + 43647647225618 T^{12} + 387981833760848 T^{13} + 3695116577316520 T^{14} + 29637315682178960 T^{15} + 235028881994751853 T^{16} + 1443057360779098180 T^{17} + 7884458195943222766 T^{18} + 27368318111564347812 T^{19} + 73742412689492826049 T^{20} )^{2} \))
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