Properties

Label 1339.2.a
Level 1339
Weight 2
Character orbit a
Rep. character \(\chi_{1339}(1,\cdot)\)
Character field \(\Q\)
Dimension 103
Newform subspaces 7
Sturm bound 242
Trace bound 3

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Defining parameters

Level: \( N \) \(=\) \( 1339 = 13 \cdot 103 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1339.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 7 \)
Sturm bound: \(242\)
Trace bound: \(3\)
Distinguishing \(T_p\): \(2\), \(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(1339))\).

Total New Old
Modular forms 122 103 19
Cusp forms 119 103 16
Eisenstein series 3 0 3

The following table gives the dimensions of the cuspidal new subspaces with specified eigenvalues for the Atkin-Lehner operators and the Fricke involution.

\(13\)\(103\)FrickeDim.
\(+\)\(+\)\(+\)\(22\)
\(+\)\(-\)\(-\)\(30\)
\(-\)\(+\)\(-\)\(29\)
\(-\)\(-\)\(+\)\(22\)
Plus space\(+\)\(44\)
Minus space\(-\)\(59\)

Trace form

\( 103q - 3q^{2} + 97q^{4} + 2q^{5} - 3q^{8} + 103q^{9} + O(q^{10}) \) \( 103q - 3q^{2} + 97q^{4} + 2q^{5} - 3q^{8} + 103q^{9} + 10q^{10} - 4q^{11} + 16q^{12} - q^{13} + 16q^{14} + 4q^{15} + 81q^{16} - 14q^{17} + 9q^{18} - 16q^{19} + 26q^{20} + 8q^{21} - 4q^{22} - 16q^{23} + 24q^{24} + 125q^{25} - 3q^{26} - 12q^{27} - 20q^{28} + 6q^{29} - 12q^{30} + 16q^{31} - 19q^{32} - 12q^{33} - 54q^{34} - 16q^{35} + 73q^{36} + 6q^{37} + 36q^{38} + 4q^{39} + 10q^{40} - 18q^{41} - 48q^{42} - 20q^{43} - 36q^{44} + 14q^{45} - 24q^{46} + 24q^{47} + 20q^{48} + 111q^{49} - 53q^{50} - 24q^{51} + q^{52} + 2q^{53} - 52q^{54} - 4q^{55} - 12q^{56} - 36q^{57} - 14q^{58} - 16q^{59} - 60q^{60} - 14q^{61} - 88q^{62} - 12q^{63} + 5q^{64} + 2q^{65} - 84q^{66} - 16q^{67} - 18q^{68} + 32q^{69} + 48q^{70} + 16q^{71} - 39q^{72} + 10q^{73} + 6q^{74} + 4q^{75} - 64q^{76} - 40q^{77} + 12q^{78} + 4q^{79} + 86q^{80} + 111q^{81} + 22q^{82} - 4q^{83} - 100q^{84} - 24q^{86} + 20q^{87} - 16q^{88} + 78q^{89} + 22q^{90} - 8q^{91} - 76q^{92} + 76q^{93} - 36q^{94} + 16q^{95} - 4q^{96} - 6q^{97} + 17q^{98} - 28q^{99} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(1339))\) into newform subspaces

Label Dim. \(A\) Field CM Traces A-L signs $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\) 13 103
1339.2.a.a \(1\) \(10.692\) \(\Q\) None \(1\) \(-1\) \(1\) \(4\) \(+\) \(+\) \(q+q^{2}-q^{3}-q^{4}+q^{5}-q^{6}+4q^{7}+\cdots\)
1339.2.a.b \(1\) \(10.692\) \(\Q\) None \(1\) \(0\) \(0\) \(-4\) \(-\) \(+\) \(q+q^{2}-q^{4}-4q^{7}-3q^{8}-3q^{9}+6q^{11}+\cdots\)
1339.2.a.c \(3\) \(10.692\) 3.3.148.1 None \(-1\) \(-1\) \(-7\) \(2\) \(-\) \(-\) \(q-\beta _{1}q^{2}+(-\beta _{1}-\beta _{2})q^{3}+(\beta _{1}+\beta _{2})q^{4}+\cdots\)
1339.2.a.d \(19\) \(10.692\) \(\mathbb{Q}[x]/(x^{19} - \cdots)\) None \(-9\) \(-2\) \(-18\) \(-8\) \(-\) \(-\) \(q-\beta _{1}q^{2}-\beta _{14}q^{3}+(\beta _{1}+\beta _{2})q^{4}+(-1+\cdots)q^{5}+\cdots\)
1339.2.a.e \(21\) \(10.692\) None \(-1\) \(-6\) \(-18\) \(-2\) \(+\) \(+\)
1339.2.a.f \(28\) \(10.692\) None \(6\) \(5\) \(27\) \(6\) \(-\) \(+\)
1339.2.a.g \(30\) \(10.692\) None \(0\) \(5\) \(17\) \(2\) \(+\) \(-\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(1339))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(1339)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(103))\)\(^{\oplus 2}\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ (\( 1 - T + 2 T^{2} \))(\( 1 - T + 2 T^{2} \))(\( 1 + T + 3 T^{2} + 3 T^{3} + 6 T^{4} + 4 T^{5} + 8 T^{6} \))(\( 1 + 9 T + 51 T^{2} + 218 T^{3} + 775 T^{4} + 2395 T^{5} + 6631 T^{6} + 16763 T^{7} + 39252 T^{8} + 86013 T^{9} + 177832 T^{10} + 349029 T^{11} + 653521 T^{12} + 1171711 T^{13} + 2017506 T^{14} + 3343319 T^{15} + 5340883 T^{16} + 8233894 T^{17} + 12259899 T^{18} + 17637423 T^{19} + 24519798 T^{20} + 32935576 T^{21} + 42727064 T^{22} + 53493104 T^{23} + 64560192 T^{24} + 74989504 T^{25} + 83650688 T^{26} + 89351424 T^{27} + 91049984 T^{28} + 88077312 T^{29} + 80388096 T^{30} + 68661248 T^{31} + 54321152 T^{32} + 39239680 T^{33} + 25395200 T^{34} + 14286848 T^{35} + 6684672 T^{36} + 2359296 T^{37} + 524288 T^{38} \))
$3$ (\( 1 + T + 3 T^{2} \))(\( 1 + 3 T^{2} \))(\( 1 + T + 4 T^{2} + 5 T^{3} + 12 T^{4} + 9 T^{5} + 27 T^{6} \))(\( 1 + 2 T + 27 T^{2} + 49 T^{3} + 367 T^{4} + 599 T^{5} + 3359 T^{6} + 4938 T^{7} + 23407 T^{8} + 31253 T^{9} + 133113 T^{10} + 163160 T^{11} + 644570 T^{12} + 732606 T^{13} + 2725815 T^{14} + 2896510 T^{15} + 10213924 T^{16} + 10211683 T^{17} + 34183497 T^{18} + 32306849 T^{19} + 102550491 T^{20} + 91905147 T^{21} + 275775948 T^{22} + 234617310 T^{23} + 662373045 T^{24} + 534069774 T^{25} + 1409674590 T^{26} + 1070492760 T^{27} + 2620063179 T^{28} + 1845458397 T^{29} + 4146479829 T^{30} + 2624255658 T^{31} + 5355330957 T^{32} + 2864998431 T^{33} + 5266048869 T^{34} + 2109289329 T^{35} + 3486784401 T^{36} + 774840978 T^{37} + 1162261467 T^{38} \))
$5$ (\( 1 - T + 5 T^{2} \))(\( 1 + 5 T^{2} \))(\( 1 + 7 T + 28 T^{2} + 75 T^{3} + 140 T^{4} + 175 T^{5} + 125 T^{6} \))(\( 1 + 18 T + 198 T^{2} + 1614 T^{3} + 10808 T^{4} + 62204 T^{5} + 317677 T^{6} + 1466774 T^{7} + 6211678 T^{8} + 24361049 T^{9} + 89168408 T^{10} + 306318752 T^{11} + 992201397 T^{12} + 3040669028 T^{13} + 8841466196 T^{14} + 24443775380 T^{15} + 64364984588 T^{16} + 161611789234 T^{17} + 387282779146 T^{18} + 886146712711 T^{19} + 1936413895730 T^{20} + 4040294730850 T^{21} + 8045623073500 T^{22} + 15277359612500 T^{23} + 27629581862500 T^{24} + 47510453562500 T^{25} + 77515734140625 T^{26} + 119655762500000 T^{27} + 174157046875000 T^{28} + 237900869140625 T^{29} + 303304589843750 T^{30} + 358099121093750 T^{31} + 387789306640625 T^{32} + 379663085937500 T^{33} + 329833984375000 T^{34} + 246276855468750 T^{35} + 151062011718750 T^{36} + 68664550781250 T^{37} + 19073486328125 T^{38} \))
$7$ (\( 1 - 4 T + 7 T^{2} \))(\( 1 + 4 T + 7 T^{2} \))(\( 1 - 2 T + 17 T^{2} - 24 T^{3} + 119 T^{4} - 98 T^{5} + 343 T^{6} \))(\( 1 + 8 T + 95 T^{2} + 571 T^{3} + 4076 T^{4} + 20029 T^{5} + 109297 T^{6} + 460860 T^{7} + 2104889 T^{8} + 7864691 T^{9} + 31542666 T^{10} + 106877131 T^{11} + 387697905 T^{12} + 1211158179 T^{13} + 4048265203 T^{14} + 11784473502 T^{15} + 36684532288 T^{16} + 100051268292 T^{17} + 291598402452 T^{18} + 746445593183 T^{19} + 2041188817164 T^{20} + 4902512146308 T^{21} + 12582794574784 T^{22} + 28294520878302 T^{23} + 68039193266821 T^{24} + 142491548601171 T^{25} + 319285895777415 T^{26} + 616125391665931 T^{27} + 1272860347496262 T^{28} + 2221580548533059 T^{29} + 4162053310746527 T^{30} + 6378895619452860 T^{31} + 10589678170453879 T^{32} + 13584129926092621 T^{33} + 19351060714527668 T^{34} + 18976003355242171 T^{35} + 22099898828784665 T^{36} + 13027308783283592 T^{37} + 11398895185373143 T^{38} \))
$11$ (\( 1 + 4 T + 11 T^{2} \))(\( 1 - 6 T + 11 T^{2} \))(\( 1 + 10 T + 57 T^{2} + 216 T^{3} + 627 T^{4} + 1210 T^{5} + 1331 T^{6} \))(\( 1 + T + 102 T^{2} + 20 T^{3} + 5325 T^{4} - 2238 T^{5} + 191228 T^{6} - 172743 T^{7} + 5296369 T^{8} - 6689313 T^{9} + 119990771 T^{10} - 180484782 T^{11} + 2297389068 T^{12} - 3764423433 T^{13} + 37873614375 T^{14} - 63756252789 T^{15} + 543203736624 T^{16} - 900459102983 T^{17} + 6815121096055 T^{18} - 10750324909520 T^{19} + 74966332056605 T^{20} - 108955551460943 T^{21} + 723004173446544 T^{22} - 933455297083749 T^{23} + 6099583468708125 T^{24} - 6668905741388913 T^{25} + 44769613621646628 T^{26} - 38688515907048942 T^{27} + 282931961420759761 T^{28} - 173503551569989113 T^{29} + 1511115887562311459 T^{30} - 542141533079915703 T^{31} + 6601709197859637268 T^{32} - 849880127559293358 T^{33} + 22243846502138341575 T^{34} + 918994597271443220 T^{35} + 51555596906927964642 T^{36} + 5559917313492231481 T^{37} + 61159090448414546291 T^{38} \))
$13$ (\( 1 + T \))(\( 1 - T \))(\( ( 1 - T )^{3} \))(\( ( 1 - T )^{19} \))
$17$ (\( 1 - 3 T + 17 T^{2} \))(\( 1 - 6 T + 17 T^{2} \))(\( 1 - T + 30 T^{2} - 21 T^{3} + 510 T^{4} - 289 T^{5} + 4913 T^{6} \))(\( 1 + 16 T + 237 T^{2} + 2387 T^{3} + 22324 T^{4} + 171768 T^{5} + 1251784 T^{6} + 8021223 T^{7} + 49508712 T^{8} + 278086011 T^{9} + 1528413633 T^{10} + 7817335897 T^{11} + 39671992438 T^{12} + 190172110843 T^{13} + 912524107173 T^{14} + 4164222470908 T^{15} + 19049662981015 T^{16} + 82877635457990 T^{17} + 360409944876147 T^{18} + 1487644492080962 T^{19} + 6126969062894499 T^{20} + 23951636647359110 T^{21} + 93590994225726695 T^{22} + 347800024992707068 T^{23} + 1295653741238334261 T^{24} + 4590292447348560667 T^{25} + 16278952732274954774 T^{26} + 54531839052294159577 T^{27} + \)\(18\!\cdots\!01\)\( T^{28} + \)\(56\!\cdots\!39\)\( T^{29} + \)\(16\!\cdots\!96\)\( T^{30} + \)\(46\!\cdots\!03\)\( T^{31} + \)\(12\!\cdots\!08\)\( T^{32} + \)\(28\!\cdots\!72\)\( T^{33} + \)\(63\!\cdots\!32\)\( T^{34} + \)\(11\!\cdots\!47\)\( T^{35} + \)\(19\!\cdots\!49\)\( T^{36} + \)\(22\!\cdots\!44\)\( T^{37} + \)\(23\!\cdots\!53\)\( T^{38} \))
$19$ (\( 1 + 5 T + 19 T^{2} \))(\( 1 - 4 T + 19 T^{2} \))(\( 1 + 5 T + 44 T^{2} + 123 T^{3} + 836 T^{4} + 1805 T^{5} + 6859 T^{6} \))(\( 1 + 10 T + 227 T^{2} + 1845 T^{3} + 23672 T^{4} + 163985 T^{5} + 1554982 T^{6} + 9528630 T^{7} + 74258347 T^{8} + 414683275 T^{9} + 2808265383 T^{10} + 14604064023 T^{11} + 88732459543 T^{12} + 435306881316 T^{13} + 2418896170307 T^{14} + 11258826849766 T^{15} + 57955252169470 T^{16} + 256162313195053 T^{17} + 1233417003633455 T^{18} + 5164377869779015 T^{19} + 23434923069035645 T^{20} + 92474595063414133 T^{21} + 397515074630394730 T^{22} + 1467261573888354886 T^{23} + 5989426388400992393 T^{24} + 20479395736873659396 T^{25} + 79315437917448555277 T^{26} + \)\(24\!\cdots\!43\)\( T^{27} + \)\(90\!\cdots\!57\)\( T^{28} + \)\(25\!\cdots\!75\)\( T^{29} + \)\(86\!\cdots\!93\)\( T^{30} + \)\(21\!\cdots\!30\)\( T^{31} + \)\(65\!\cdots\!38\)\( T^{32} + \)\(13\!\cdots\!85\)\( T^{33} + \)\(35\!\cdots\!28\)\( T^{34} + \)\(53\!\cdots\!45\)\( T^{35} + \)\(12\!\cdots\!53\)\( T^{36} + \)\(10\!\cdots\!10\)\( T^{37} + \)\(19\!\cdots\!79\)\( T^{38} \))
$23$ (\( 1 + 23 T^{2} \))(\( 1 + 23 T^{2} \))(\( 1 - 4 T - 11 T^{2} + 216 T^{3} - 253 T^{4} - 2116 T^{5} + 12167 T^{6} \))(\( 1 + 14 T + 331 T^{2} + 3530 T^{3} + 48881 T^{4} + 429270 T^{5} + 4479056 T^{6} + 33901160 T^{7} + 293378798 T^{8} + 1971113777 T^{9} + 14868705538 T^{10} + 90488082120 T^{11} + 613003027963 T^{12} + 3428113140801 T^{13} + 21264972897606 T^{14} + 110411454229169 T^{15} + 635275663902722 T^{16} + 3083914591895232 T^{17} + 16590554705250677 T^{18} + 75576994646226703 T^{19} + 381582758220765571 T^{20} + 1631390819112577728 T^{21} + 7729399002704418574 T^{22} + 30897651762944882129 T^{23} + \)\(13\!\cdots\!58\)\( T^{24} + \)\(50\!\cdots\!89\)\( T^{25} + \)\(20\!\cdots\!61\)\( T^{26} + \)\(70\!\cdots\!20\)\( T^{27} + \)\(26\!\cdots\!94\)\( T^{28} + \)\(81\!\cdots\!73\)\( T^{29} + \)\(27\!\cdots\!46\)\( T^{30} + \)\(74\!\cdots\!60\)\( T^{31} + \)\(22\!\cdots\!48\)\( T^{32} + \)\(49\!\cdots\!30\)\( T^{33} + \)\(13\!\cdots\!67\)\( T^{34} + \)\(21\!\cdots\!30\)\( T^{35} + \)\(46\!\cdots\!93\)\( T^{36} + \)\(45\!\cdots\!66\)\( T^{37} + \)\(74\!\cdots\!87\)\( T^{38} \))
$29$ (\( 1 + 3 T + 29 T^{2} \))(\( 1 - 6 T + 29 T^{2} \))(\( 1 + 17 T + 178 T^{2} + 1137 T^{3} + 5162 T^{4} + 14297 T^{5} + 24389 T^{6} \))(\( 1 + 22 T + 540 T^{2} + 8186 T^{3} + 121345 T^{4} + 1431795 T^{5} + 16088985 T^{6} + 156972174 T^{7} + 1451579214 T^{8} + 12137034530 T^{9} + 96331162859 T^{10} + 707549615806 T^{11} + 4954535363860 T^{12} + 32641088584875 T^{13} + 206426609056547 T^{14} + 1246567426174602 T^{15} + 7295654087388475 T^{16} + 41359631525384506 T^{17} + 229531297445482986 T^{18} + 1246418917742693551 T^{19} + 6656407625919006594 T^{20} + 34783450112848369546 T^{21} + \)\(17\!\cdots\!75\)\( T^{22} + \)\(88\!\cdots\!62\)\( T^{23} + \)\(42\!\cdots\!03\)\( T^{24} + \)\(19\!\cdots\!75\)\( T^{25} + \)\(85\!\cdots\!40\)\( T^{26} + \)\(35\!\cdots\!66\)\( T^{27} + \)\(13\!\cdots\!71\)\( T^{28} + \)\(51\!\cdots\!30\)\( T^{29} + \)\(17\!\cdots\!06\)\( T^{30} + \)\(55\!\cdots\!34\)\( T^{31} + \)\(16\!\cdots\!65\)\( T^{32} + \)\(42\!\cdots\!95\)\( T^{33} + \)\(10\!\cdots\!05\)\( T^{34} + \)\(20\!\cdots\!06\)\( T^{35} + \)\(39\!\cdots\!60\)\( T^{36} + \)\(46\!\cdots\!42\)\( T^{37} + \)\(61\!\cdots\!69\)\( T^{38} \))
$31$ (\( 1 + 31 T^{2} \))(\( 1 - 6 T + 31 T^{2} \))(\( 1 + 8 T + 61 T^{2} + 224 T^{3} + 1891 T^{4} + 7688 T^{5} + 29791 T^{6} \))(\( 1 + 3 T + 253 T^{2} + 640 T^{3} + 33363 T^{4} + 81675 T^{5} + 3055527 T^{6} + 7831686 T^{7} + 216386675 T^{8} + 600299111 T^{9} + 12554953422 T^{10} + 37926394245 T^{11} + 619593171382 T^{12} + 2011968340151 T^{13} + 26676397096080 T^{14} + 90725449910352 T^{15} + 1019771506063363 T^{16} + 3509117681597213 T^{17} + 35009319078125048 T^{18} + 117045637610599947 T^{19} + 1085288891421876488 T^{20} + 3372262092014921693 T^{21} + 30380012937133647133 T^{22} + 83786858226658189392 T^{23} + \)\(76\!\cdots\!80\)\( T^{24} + \)\(17\!\cdots\!31\)\( T^{25} + \)\(17\!\cdots\!02\)\( T^{26} + \)\(32\!\cdots\!45\)\( T^{27} + \)\(33\!\cdots\!62\)\( T^{28} + \)\(49\!\cdots\!11\)\( T^{29} + \)\(54\!\cdots\!25\)\( T^{30} + \)\(61\!\cdots\!46\)\( T^{31} + \)\(74\!\cdots\!57\)\( T^{32} + \)\(61\!\cdots\!75\)\( T^{33} + \)\(78\!\cdots\!13\)\( T^{34} + \)\(46\!\cdots\!40\)\( T^{35} + \)\(57\!\cdots\!83\)\( T^{36} + \)\(20\!\cdots\!23\)\( T^{37} + \)\(21\!\cdots\!71\)\( T^{38} \))
$37$ (\( 1 + 5 T + 37 T^{2} \))(\( 1 - 4 T + 37 T^{2} \))(\( 1 - 3 T + 80 T^{2} - 251 T^{3} + 2960 T^{4} - 4107 T^{5} + 50653 T^{6} \))(\( 1 + 19 T + 491 T^{2} + 6616 T^{3} + 101441 T^{4} + 1094503 T^{5} + 12741626 T^{6} + 117650153 T^{7} + 1144023030 T^{8} + 9413283687 T^{9} + 80288050644 T^{10} + 603447940356 T^{11} + 4647014085427 T^{12} + 32444198183946 T^{13} + 230442803839096 T^{14} + 1516351289851025 T^{15} + 10121875623025439 T^{16} + 63575162012785243 T^{17} + 404839650047853747 T^{18} + 2445727359407079655 T^{19} + 14979067051770588639 T^{20} + 87034396795502997667 T^{21} + \)\(51\!\cdots\!67\)\( T^{22} + \)\(28\!\cdots\!25\)\( T^{23} + \)\(15\!\cdots\!72\)\( T^{24} + \)\(83\!\cdots\!14\)\( T^{25} + \)\(44\!\cdots\!91\)\( T^{26} + \)\(21\!\cdots\!76\)\( T^{27} + \)\(10\!\cdots\!88\)\( T^{28} + \)\(45\!\cdots\!63\)\( T^{29} + \)\(20\!\cdots\!90\)\( T^{30} + \)\(77\!\cdots\!93\)\( T^{31} + \)\(31\!\cdots\!22\)\( T^{32} + \)\(98\!\cdots\!67\)\( T^{33} + \)\(33\!\cdots\!13\)\( T^{34} + \)\(81\!\cdots\!56\)\( T^{35} + \)\(22\!\cdots\!47\)\( T^{36} + \)\(32\!\cdots\!51\)\( T^{37} + \)\(62\!\cdots\!73\)\( T^{38} \))
$41$ (\( 1 + 41 T^{2} \))(\( 1 + 10 T + 41 T^{2} \))(\( 1 + 8 T + 107 T^{2} + 496 T^{3} + 4387 T^{4} + 13448 T^{5} + 68921 T^{6} \))(\( 1 + 52 T + 1810 T^{2} + 46437 T^{3} + 985714 T^{4} + 17849959 T^{5} + 285501303 T^{6} + 4094702052 T^{7} + 53491353711 T^{8} + 642062053101 T^{9} + 7141858587350 T^{10} + 74019764499369 T^{11} + 718475303462794 T^{12} + 6554265725851788 T^{13} + 56372661730637615 T^{14} + 458151324923142410 T^{15} + 3525196730700775759 T^{16} + 25712523033730701312 T^{17} + \)\(17\!\cdots\!63\)\( T^{18} + \)\(11\!\cdots\!04\)\( T^{19} + \)\(72\!\cdots\!83\)\( T^{20} + \)\(43\!\cdots\!72\)\( T^{21} + \)\(24\!\cdots\!39\)\( T^{22} + \)\(12\!\cdots\!10\)\( T^{23} + \)\(65\!\cdots\!15\)\( T^{24} + \)\(31\!\cdots\!08\)\( T^{25} + \)\(13\!\cdots\!14\)\( T^{26} + \)\(59\!\cdots\!49\)\( T^{27} + \)\(23\!\cdots\!50\)\( T^{28} + \)\(86\!\cdots\!01\)\( T^{29} + \)\(29\!\cdots\!51\)\( T^{30} + \)\(92\!\cdots\!12\)\( T^{31} + \)\(26\!\cdots\!63\)\( T^{32} + \)\(67\!\cdots\!99\)\( T^{33} + \)\(15\!\cdots\!14\)\( T^{34} + \)\(29\!\cdots\!17\)\( T^{35} + \)\(47\!\cdots\!10\)\( T^{36} + \)\(55\!\cdots\!92\)\( T^{37} + \)\(43\!\cdots\!61\)\( T^{38} \))
$43$ (\( 1 + 12 T + 43 T^{2} \))(\( 1 - 8 T + 43 T^{2} \))(\( 1 - 8 T + 137 T^{2} - 672 T^{3} + 5891 T^{4} - 14792 T^{5} + 79507 T^{6} \))(\( 1 + 2 T + 383 T^{2} + 499 T^{3} + 72479 T^{4} + 54797 T^{5} + 9254511 T^{6} + 3648207 T^{7} + 912332337 T^{8} + 177826268 T^{9} + 74327957909 T^{10} + 6676426712 T^{11} + 5180277499429 T^{12} + 122770366735 T^{13} + 315196434995199 T^{14} - 5854795133954 T^{15} + 16977218156735673 T^{16} - 633445494713960 T^{17} + 815911630068533768 T^{18} - 32657981578676819 T^{19} + 35084200092946952024 T^{20} - 1171240719726112040 T^{21} + \)\(13\!\cdots\!11\)\( T^{22} - 20016379458757069154 T^{23} + \)\(46\!\cdots\!57\)\( T^{24} + \)\(77\!\cdots\!15\)\( T^{25} + \)\(14\!\cdots\!03\)\( T^{26} + \)\(78\!\cdots\!12\)\( T^{27} + \)\(37\!\cdots\!87\)\( T^{28} + \)\(38\!\cdots\!32\)\( T^{29} + \)\(84\!\cdots\!59\)\( T^{30} + \)\(14\!\cdots\!07\)\( T^{31} + \)\(15\!\cdots\!73\)\( T^{32} + \)\(40\!\cdots\!53\)\( T^{33} + \)\(23\!\cdots\!53\)\( T^{34} + \)\(68\!\cdots\!99\)\( T^{35} + \)\(22\!\cdots\!69\)\( T^{36} + \)\(50\!\cdots\!98\)\( T^{37} + \)\(10\!\cdots\!07\)\( T^{38} \))
$47$ (\( 1 - 6 T + 47 T^{2} \))(\( 1 - 6 T + 47 T^{2} \))(\( 1 + 4 T + 69 T^{2} + 108 T^{3} + 3243 T^{4} + 8836 T^{5} + 103823 T^{6} \))(\( 1 + 24 T + 736 T^{2} + 12508 T^{3} + 229100 T^{4} + 3086296 T^{5} + 42868352 T^{6} + 486306407 T^{7} + 5611672880 T^{8} + 55741857359 T^{9} + 561561912097 T^{10} + 5023694239757 T^{11} + 45619469431483 T^{12} + 375249712914668 T^{13} + 3138006134470401 T^{14} + 24070514052589038 T^{15} + 187849034602241882 T^{16} + 1354348522899912987 T^{17} + 9934013028503139578 T^{18} + 67525744118247356353 T^{19} + \)\(46\!\cdots\!66\)\( T^{20} + \)\(29\!\cdots\!83\)\( T^{21} + \)\(19\!\cdots\!86\)\( T^{22} + \)\(11\!\cdots\!78\)\( T^{23} + \)\(71\!\cdots\!07\)\( T^{24} + \)\(40\!\cdots\!72\)\( T^{25} + \)\(23\!\cdots\!29\)\( T^{26} + \)\(11\!\cdots\!77\)\( T^{27} + \)\(62\!\cdots\!99\)\( T^{28} + \)\(29\!\cdots\!91\)\( T^{29} + \)\(13\!\cdots\!40\)\( T^{30} + \)\(56\!\cdots\!87\)\( T^{31} + \)\(23\!\cdots\!04\)\( T^{32} + \)\(79\!\cdots\!24\)\( T^{33} + \)\(27\!\cdots\!00\)\( T^{34} + \)\(70\!\cdots\!68\)\( T^{35} + \)\(19\!\cdots\!32\)\( T^{36} + \)\(30\!\cdots\!36\)\( T^{37} + \)\(58\!\cdots\!83\)\( T^{38} \))
$53$ (\( 1 + 12 T + 53 T^{2} \))(\( 1 + 2 T + 53 T^{2} \))(\( 1 + 12 T + 179 T^{2} + 1172 T^{3} + 9487 T^{4} + 33708 T^{5} + 148877 T^{6} \))(\( 1 + 11 T + 346 T^{2} + 3101 T^{3} + 58649 T^{4} + 423672 T^{5} + 6448166 T^{6} + 37221598 T^{7} + 524392233 T^{8} + 2258001028 T^{9} + 33490123240 T^{10} + 80013840075 T^{11} + 1667255041334 T^{12} - 1345414395294 T^{13} + 60022530683787 T^{14} - 475207785558431 T^{15} + 1241988887046962 T^{16} - 43340472265016245 T^{17} - 2993396699206258 T^{18} - 2666790097581265998 T^{19} - 158650025057931674 T^{20} - \)\(12\!\cdots\!05\)\( T^{21} + \)\(18\!\cdots\!74\)\( T^{22} - \)\(37\!\cdots\!11\)\( T^{23} + \)\(25\!\cdots\!91\)\( T^{24} - \)\(29\!\cdots\!26\)\( T^{25} + \)\(19\!\cdots\!58\)\( T^{26} + \)\(49\!\cdots\!75\)\( T^{27} + \)\(11\!\cdots\!20\)\( T^{28} + \)\(39\!\cdots\!72\)\( T^{29} + \)\(48\!\cdots\!01\)\( T^{30} + \)\(18\!\cdots\!18\)\( T^{31} + \)\(16\!\cdots\!18\)\( T^{32} + \)\(58\!\cdots\!68\)\( T^{33} + \)\(42\!\cdots\!93\)\( T^{34} + \)\(12\!\cdots\!21\)\( T^{35} + \)\(71\!\cdots\!98\)\( T^{36} + \)\(11\!\cdots\!79\)\( T^{37} + \)\(57\!\cdots\!17\)\( T^{38} \))
$59$ (\( 1 + 5 T + 59 T^{2} \))(\( 1 + 4 T + 59 T^{2} \))(\( 1 + 11 T + 98 T^{2} + 475 T^{3} + 5782 T^{4} + 38291 T^{5} + 205379 T^{6} \))(\( 1 + 52 T + 1833 T^{2} + 47729 T^{3} + 1032141 T^{4} + 19098634 T^{5} + 313379464 T^{6} + 4632546659 T^{7} + 62746478699 T^{8} + 786525069382 T^{9} + 9215801486064 T^{10} + 101629912303778 T^{11} + 1061738143585558 T^{12} + 10557201164192470 T^{13} + 100334808606830921 T^{14} + 914064493119892775 T^{15} + 8001838078369489353 T^{16} + 67408678550362988011 T^{17} + \)\(54\!\cdots\!10\)\( T^{18} + \)\(42\!\cdots\!67\)\( T^{19} + \)\(32\!\cdots\!90\)\( T^{20} + \)\(23\!\cdots\!91\)\( T^{21} + \)\(16\!\cdots\!87\)\( T^{22} + \)\(11\!\cdots\!75\)\( T^{23} + \)\(71\!\cdots\!79\)\( T^{24} + \)\(44\!\cdots\!70\)\( T^{25} + \)\(26\!\cdots\!02\)\( T^{26} + \)\(14\!\cdots\!38\)\( T^{27} + \)\(79\!\cdots\!96\)\( T^{28} + \)\(40\!\cdots\!82\)\( T^{29} + \)\(18\!\cdots\!41\)\( T^{30} + \)\(82\!\cdots\!79\)\( T^{31} + \)\(32\!\cdots\!56\)\( T^{32} + \)\(11\!\cdots\!74\)\( T^{33} + \)\(37\!\cdots\!59\)\( T^{34} + \)\(10\!\cdots\!89\)\( T^{35} + \)\(23\!\cdots\!27\)\( T^{36} + \)\(39\!\cdots\!92\)\( T^{37} + \)\(44\!\cdots\!39\)\( T^{38} \))
$61$ (\( 1 + 5 T + 61 T^{2} \))(\( 1 - 2 T + 61 T^{2} \))(\( 1 - 9 T + 162 T^{2} - 1001 T^{3} + 9882 T^{4} - 33489 T^{5} + 226981 T^{6} \))(\( 1 + 25 T + 788 T^{2} + 14186 T^{3} + 274407 T^{4} + 3969590 T^{5} + 59172970 T^{6} + 725803271 T^{7} + 9050446396 T^{8} + 97062790561 T^{9} + 1055748693085 T^{10} + 10107098503889 T^{11} + 98581868115422 T^{12} + 857664462279724 T^{13} + 7686856190526509 T^{14} + 62047091242085191 T^{15} + 524987853136928919 T^{16} + 4037647559460115732 T^{17} + 33174419612092286184 T^{18} + \)\(24\!\cdots\!53\)\( T^{19} + \)\(20\!\cdots\!24\)\( T^{20} + \)\(15\!\cdots\!72\)\( T^{21} + \)\(11\!\cdots\!39\)\( T^{22} + \)\(85\!\cdots\!31\)\( T^{23} + \)\(64\!\cdots\!09\)\( T^{24} + \)\(44\!\cdots\!64\)\( T^{25} + \)\(30\!\cdots\!62\)\( T^{26} + \)\(19\!\cdots\!09\)\( T^{27} + \)\(12\!\cdots\!85\)\( T^{28} + \)\(69\!\cdots\!61\)\( T^{29} + \)\(39\!\cdots\!56\)\( T^{30} + \)\(19\!\cdots\!91\)\( T^{31} + \)\(95\!\cdots\!70\)\( T^{32} + \)\(39\!\cdots\!90\)\( T^{33} + \)\(16\!\cdots\!07\)\( T^{34} + \)\(52\!\cdots\!46\)\( T^{35} + \)\(17\!\cdots\!48\)\( T^{36} + \)\(34\!\cdots\!25\)\( T^{37} + \)\(83\!\cdots\!41\)\( T^{38} \))
$67$ (\( 1 + 4 T + 67 T^{2} \))(\( 1 + 2 T + 67 T^{2} \))(\( 1 - 8 T + 169 T^{2} - 944 T^{3} + 11323 T^{4} - 35912 T^{5} + 300763 T^{6} \))(\( 1 + 2 T + 868 T^{2} + 2853 T^{3} + 364854 T^{4} + 1616840 T^{5} + 99909914 T^{6} + 534048354 T^{7} + 20180882955 T^{8} + 120314182078 T^{9} + 3212581366838 T^{10} + 20143838084142 T^{11} + 418325138096849 T^{12} + 2642468537361910 T^{13} + 45504654788990413 T^{14} + 280772572166108335 T^{15} + 4186007272566093824 T^{16} + 24644805687647069749 T^{17} + \)\(32\!\cdots\!08\)\( T^{18} + \)\(18\!\cdots\!45\)\( T^{19} + \)\(21\!\cdots\!36\)\( T^{20} + \)\(11\!\cdots\!61\)\( T^{21} + \)\(12\!\cdots\!12\)\( T^{22} + \)\(56\!\cdots\!35\)\( T^{23} + \)\(61\!\cdots\!91\)\( T^{24} + \)\(23\!\cdots\!90\)\( T^{25} + \)\(25\!\cdots\!27\)\( T^{26} + \)\(81\!\cdots\!22\)\( T^{27} + \)\(87\!\cdots\!86\)\( T^{28} + \)\(21\!\cdots\!22\)\( T^{29} + \)\(24\!\cdots\!65\)\( T^{30} + \)\(43\!\cdots\!94\)\( T^{31} + \)\(54\!\cdots\!18\)\( T^{32} + \)\(59\!\cdots\!60\)\( T^{33} + \)\(89\!\cdots\!22\)\( T^{34} + \)\(47\!\cdots\!93\)\( T^{35} + \)\(95\!\cdots\!36\)\( T^{36} + \)\(14\!\cdots\!18\)\( T^{37} + \)\(49\!\cdots\!03\)\( T^{38} \))
$71$ (\( 1 + 6 T + 71 T^{2} \))(\( 1 + 2 T + 71 T^{2} \))(\( 1 - 10 T + 97 T^{2} - 324 T^{3} + 6887 T^{4} - 50410 T^{5} + 357911 T^{6} \))(\( 1 + 44 T + 1569 T^{2} + 41292 T^{3} + 942998 T^{4} + 18695128 T^{5} + 334976798 T^{6} + 5472812256 T^{7} + 82751616460 T^{8} + 1166224261931 T^{9} + 15432560275106 T^{10} + 192646148591907 T^{11} + 2278709585632134 T^{12} + 25614036867383486 T^{13} + 274402521434013453 T^{14} + 2806579876908761827 T^{15} + 27456070636853745989 T^{16} + \)\(25\!\cdots\!96\)\( T^{17} + \)\(23\!\cdots\!04\)\( T^{18} + \)\(19\!\cdots\!73\)\( T^{19} + \)\(16\!\cdots\!84\)\( T^{20} + \)\(12\!\cdots\!36\)\( T^{21} + \)\(98\!\cdots\!79\)\( T^{22} + \)\(71\!\cdots\!87\)\( T^{23} + \)\(49\!\cdots\!03\)\( T^{24} + \)\(32\!\cdots\!06\)\( T^{25} + \)\(20\!\cdots\!94\)\( T^{26} + \)\(12\!\cdots\!27\)\( T^{27} + \)\(70\!\cdots\!86\)\( T^{28} + \)\(37\!\cdots\!31\)\( T^{29} + \)\(19\!\cdots\!60\)\( T^{30} + \)\(89\!\cdots\!96\)\( T^{31} + \)\(39\!\cdots\!78\)\( T^{32} + \)\(15\!\cdots\!68\)\( T^{33} + \)\(55\!\cdots\!98\)\( T^{34} + \)\(17\!\cdots\!32\)\( T^{35} + \)\(46\!\cdots\!79\)\( T^{36} + \)\(92\!\cdots\!84\)\( T^{37} + \)\(14\!\cdots\!31\)\( T^{38} \))
$73$ (\( 1 - 5 T + 73 T^{2} \))(\( 1 - 8 T + 73 T^{2} \))(\( 1 - 9 T - 16 T^{2} + 1075 T^{3} - 1168 T^{4} - 47961 T^{5} + 389017 T^{6} \))(\( 1 + 39 T + 1112 T^{2} + 24931 T^{3} + 493693 T^{4} + 8689833 T^{5} + 140554412 T^{6} + 2104339625 T^{7} + 29567117227 T^{8} + 391478589428 T^{9} + 4919293990386 T^{10} + 58843806089045 T^{11} + 672740598448990 T^{12} + 7364523462401056 T^{13} + 77390887280479850 T^{14} + 781593317122606142 T^{15} + 7597892314073826869 T^{16} + 71132010810500523366 T^{17} + \)\(64\!\cdots\!23\)\( T^{18} + \)\(55\!\cdots\!13\)\( T^{19} + \)\(46\!\cdots\!79\)\( T^{20} + \)\(37\!\cdots\!14\)\( T^{21} + \)\(29\!\cdots\!73\)\( T^{22} + \)\(22\!\cdots\!22\)\( T^{23} + \)\(16\!\cdots\!50\)\( T^{24} + \)\(11\!\cdots\!84\)\( T^{25} + \)\(74\!\cdots\!30\)\( T^{26} + \)\(47\!\cdots\!45\)\( T^{27} + \)\(28\!\cdots\!18\)\( T^{28} + \)\(16\!\cdots\!72\)\( T^{29} + \)\(92\!\cdots\!79\)\( T^{30} + \)\(48\!\cdots\!25\)\( T^{31} + \)\(23\!\cdots\!96\)\( T^{32} + \)\(10\!\cdots\!97\)\( T^{33} + \)\(43\!\cdots\!01\)\( T^{34} + \)\(16\!\cdots\!91\)\( T^{35} + \)\(52\!\cdots\!36\)\( T^{36} + \)\(13\!\cdots\!91\)\( T^{37} + \)\(25\!\cdots\!37\)\( T^{38} \))
$79$ (\( 1 - 6 T + 79 T^{2} \))(\( 1 + 8 T + 79 T^{2} \))(\( 1 - 6 T + 129 T^{2} - 732 T^{3} + 10191 T^{4} - 37446 T^{5} + 493039 T^{6} \))(\( 1 - T + 577 T^{2} + 596 T^{3} + 175861 T^{4} + 388455 T^{5} + 38203016 T^{6} + 110694848 T^{7} + 6538158329 T^{8} + 21888460413 T^{9} + 930494535617 T^{10} + 3386357535816 T^{11} + 114035864303267 T^{12} + 433973964247942 T^{13} + 12305046386628366 T^{14} + 47612801071271311 T^{15} + 1186719959617893527 T^{16} + 4551239196498462969 T^{17} + \)\(10\!\cdots\!89\)\( T^{18} + \)\(38\!\cdots\!90\)\( T^{19} + \)\(81\!\cdots\!31\)\( T^{20} + \)\(28\!\cdots\!29\)\( T^{21} + \)\(58\!\cdots\!53\)\( T^{22} + \)\(18\!\cdots\!91\)\( T^{23} + \)\(37\!\cdots\!34\)\( T^{24} + \)\(10\!\cdots\!82\)\( T^{25} + \)\(21\!\cdots\!53\)\( T^{26} + \)\(51\!\cdots\!76\)\( T^{27} + \)\(11\!\cdots\!23\)\( T^{28} + \)\(20\!\cdots\!13\)\( T^{29} + \)\(48\!\cdots\!91\)\( T^{30} + \)\(65\!\cdots\!68\)\( T^{31} + \)\(17\!\cdots\!24\)\( T^{32} + \)\(14\!\cdots\!55\)\( T^{33} + \)\(51\!\cdots\!39\)\( T^{34} + \)\(13\!\cdots\!16\)\( T^{35} + \)\(10\!\cdots\!43\)\( T^{36} - \)\(14\!\cdots\!61\)\( T^{37} + \)\(11\!\cdots\!19\)\( T^{38} \))
$83$ (\( 1 + T + 83 T^{2} \))(\( 1 + 8 T + 83 T^{2} \))(\( 1 + 11 T + 230 T^{2} + 1831 T^{3} + 19090 T^{4} + 75779 T^{5} + 571787 T^{6} \))(\( 1 + 27 T + 1061 T^{2} + 21652 T^{3} + 529707 T^{4} + 9076993 T^{5} + 173187629 T^{6} + 2600117442 T^{7} + 41947522047 T^{8} + 565262861053 T^{9} + 8025293161899 T^{10} + 98500674340986 T^{11} + 1258813864291879 T^{12} + 14204784751823140 T^{13} + 165694154500131550 T^{14} + 1729082023699801251 T^{15} + 18570173171829956670 T^{16} + \)\(17\!\cdots\!63\)\( T^{17} + \)\(17\!\cdots\!87\)\( T^{18} + \)\(16\!\cdots\!47\)\( T^{19} + \)\(14\!\cdots\!21\)\( T^{20} + \)\(12\!\cdots\!07\)\( T^{21} + \)\(10\!\cdots\!90\)\( T^{22} + \)\(82\!\cdots\!71\)\( T^{23} + \)\(65\!\cdots\!50\)\( T^{24} + \)\(46\!\cdots\!60\)\( T^{25} + \)\(34\!\cdots\!33\)\( T^{26} + \)\(22\!\cdots\!26\)\( T^{27} + \)\(15\!\cdots\!97\)\( T^{28} + \)\(87\!\cdots\!97\)\( T^{29} + \)\(54\!\cdots\!49\)\( T^{30} + \)\(27\!\cdots\!62\)\( T^{31} + \)\(15\!\cdots\!27\)\( T^{32} + \)\(66\!\cdots\!97\)\( T^{33} + \)\(32\!\cdots\!49\)\( T^{34} + \)\(10\!\cdots\!12\)\( T^{35} + \)\(44\!\cdots\!03\)\( T^{36} + \)\(94\!\cdots\!43\)\( T^{37} + \)\(29\!\cdots\!47\)\( T^{38} \))
$89$ (\( 1 + 9 T + 89 T^{2} \))(\( 1 + 12 T + 89 T^{2} \))(\( 1 + T + 108 T^{2} + 345 T^{3} + 9612 T^{4} + 7921 T^{5} + 704969 T^{6} \))(\( 1 + 86 T + 4808 T^{2} + 198690 T^{3} + 6733357 T^{4} + 194194126 T^{5} + 4916860345 T^{6} + 111181073018 T^{7} + 2277381535342 T^{8} + 42660026197638 T^{9} + 736545846785157 T^{10} + 11788777804402798 T^{11} + 175751139094968731 T^{12} + 2449424687074653956 T^{13} + 32008239306383930487 T^{14} + \)\(39\!\cdots\!23\)\( T^{15} + \)\(45\!\cdots\!14\)\( T^{16} + \)\(49\!\cdots\!14\)\( T^{17} + \)\(50\!\cdots\!63\)\( T^{18} + \)\(49\!\cdots\!41\)\( T^{19} + \)\(45\!\cdots\!07\)\( T^{20} + \)\(39\!\cdots\!94\)\( T^{21} + \)\(32\!\cdots\!66\)\( T^{22} + \)\(24\!\cdots\!43\)\( T^{23} + \)\(17\!\cdots\!63\)\( T^{24} + \)\(12\!\cdots\!16\)\( T^{25} + \)\(77\!\cdots\!99\)\( T^{26} + \)\(46\!\cdots\!38\)\( T^{27} + \)\(25\!\cdots\!13\)\( T^{28} + \)\(13\!\cdots\!38\)\( T^{29} + \)\(63\!\cdots\!38\)\( T^{30} + \)\(27\!\cdots\!78\)\( T^{31} + \)\(10\!\cdots\!05\)\( T^{32} + \)\(37\!\cdots\!66\)\( T^{33} + \)\(11\!\cdots\!93\)\( T^{34} + \)\(30\!\cdots\!90\)\( T^{35} + \)\(66\!\cdots\!32\)\( T^{36} + \)\(10\!\cdots\!66\)\( T^{37} + \)\(10\!\cdots\!09\)\( T^{38} \))
$97$ (\( 1 - 10 T + 97 T^{2} \))(\( 1 - 2 T + 97 T^{2} \))(\( 1 - 22 T + 415 T^{2} - 4468 T^{3} + 40255 T^{4} - 206998 T^{5} + 912673 T^{6} \))(\( 1 + 20 T + 1242 T^{2} + 20932 T^{3} + 726793 T^{4} + 10658516 T^{5} + 270594271 T^{6} + 3529219650 T^{7} + 72782811720 T^{8} + 858250232818 T^{9} + 15213391467171 T^{10} + 164375428993321 T^{11} + 2594760312225812 T^{12} + 25971791784188106 T^{13} + 373876286436159441 T^{14} + 3496162540583458560 T^{15} + 46627609607879495423 T^{16} + \)\(40\!\cdots\!24\)\( T^{17} + \)\(51\!\cdots\!56\)\( T^{18} + \)\(42\!\cdots\!27\)\( T^{19} + \)\(49\!\cdots\!32\)\( T^{20} + \)\(38\!\cdots\!16\)\( T^{21} + \)\(42\!\cdots\!79\)\( T^{22} + \)\(30\!\cdots\!60\)\( T^{23} + \)\(32\!\cdots\!37\)\( T^{24} + \)\(21\!\cdots\!74\)\( T^{25} + \)\(20\!\cdots\!56\)\( T^{26} + \)\(12\!\cdots\!81\)\( T^{27} + \)\(11\!\cdots\!07\)\( T^{28} + \)\(63\!\cdots\!82\)\( T^{29} + \)\(52\!\cdots\!60\)\( T^{30} + \)\(24\!\cdots\!50\)\( T^{31} + \)\(18\!\cdots\!67\)\( T^{32} + \)\(69\!\cdots\!04\)\( T^{33} + \)\(46\!\cdots\!49\)\( T^{34} + \)\(12\!\cdots\!72\)\( T^{35} + \)\(74\!\cdots\!54\)\( T^{36} + \)\(11\!\cdots\!80\)\( T^{37} + \)\(56\!\cdots\!33\)\( T^{38} \))
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