Properties

Label 102.2
Level 102
Weight 2
Dimension 73
Nonzero newspaces 5
Newform subspaces 9
Sturm bound 1152
Trace bound 1

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Defining parameters

Level: \( N \) = \( 102 = 2 \cdot 3 \cdot 17 \)
Weight: \( k \) = \( 2 \)
Nonzero newspaces: \( 5 \)
Newform subspaces: \( 9 \)
Sturm bound: \(1152\)
Trace bound: \(1\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_1(102))\).

Total New Old
Modular forms 352 73 279
Cusp forms 225 73 152
Eisenstein series 127 0 127

Trace form

\( 73q + q^{2} + q^{3} + q^{4} + 6q^{5} + q^{6} + 8q^{7} + q^{8} + q^{9} + O(q^{10}) \) \( 73q + q^{2} + q^{3} + q^{4} + 6q^{5} + q^{6} + 8q^{7} + q^{8} + q^{9} - 2q^{10} - 20q^{11} - 7q^{12} - 18q^{13} - 24q^{14} - 42q^{15} - 7q^{16} - 15q^{17} - 31q^{18} - 12q^{19} - 2q^{20} - 40q^{21} - 20q^{22} - 8q^{23} - 7q^{24} - 41q^{25} + 6q^{26} + q^{27} + 8q^{28} - 10q^{29} + 6q^{30} - 32q^{31} + q^{32} - 20q^{33} + 17q^{34} - 16q^{35} + q^{36} - 26q^{37} + 20q^{38} + 14q^{39} + 6q^{40} + 2q^{41} + 56q^{42} - 4q^{43} + 12q^{44} + 62q^{45} + 24q^{46} + 48q^{47} + q^{48} + 57q^{49} + 31q^{50} + 81q^{51} + 14q^{52} + 46q^{53} + 57q^{54} + 40q^{55} + 8q^{56} + 92q^{57} + 30q^{58} + 28q^{59} + 38q^{60} + 30q^{61} + 32q^{62} + 8q^{63} + q^{64} + 76q^{65} + 20q^{66} + 36q^{67} + 9q^{68} - 40q^{69} - 16q^{70} - 24q^{71} - 7q^{72} - 94q^{73} - 34q^{74} - 81q^{75} + 20q^{76} - 64q^{77} - 50q^{78} - 80q^{79} - 26q^{80} - 7q^{81} - 94q^{82} - 44q^{83} - 24q^{84} - 98q^{85} - 20q^{86} - 18q^{87} - 52q^{88} - 6q^{89} - 2q^{90} - 80q^{91} - 8q^{92} + 32q^{93} - 80q^{94} - 8q^{95} + q^{96} - 30q^{97} - 15q^{98} + 20q^{99} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_1(102))\)

We only show spaces with even parity, since no modular forms exist when this condition is not satisfied. Within each space \( S_k^{\mathrm{new}}(N, \chi) \) we list the newforms together with their dimension.

Label \(\chi\) Newforms Dimension \(\chi\) degree
102.2.a \(\chi_{102}(1, \cdot)\) 102.2.a.a 1 1
102.2.a.b 1
102.2.a.c 1
102.2.b \(\chi_{102}(67, \cdot)\) 102.2.b.a 2 1
102.2.f \(\chi_{102}(13, \cdot)\) 102.2.f.a 4 2
102.2.h \(\chi_{102}(19, \cdot)\) 102.2.h.a 8 4
102.2.h.b 8
102.2.i \(\chi_{102}(5, \cdot)\) 102.2.i.a 24 8
102.2.i.b 24

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_1(102))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_1(102)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_1(17))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(34))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(51))\)\(^{\oplus 2}\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ (\( 1 + T \))(\( 1 + T \))(\( 1 - T \))(\( ( 1 - T )^{2} \))(\( ( 1 + T^{2} )^{2} \))(\( ( 1 + T^{4} )^{2} \))(\( ( 1 + T^{4} )^{2} \))
$3$ (\( 1 + T \))(\( 1 - T \))(\( 1 - T \))(\( 1 + T^{2} \))(\( 1 + T^{4} \))(\( 1 + T^{8} \))(\( 1 + T^{8} \))
$5$ (\( 1 + 4 T + 5 T^{2} \))(\( 1 + 5 T^{2} \))(\( 1 + 2 T + 5 T^{2} \))(\( ( 1 - 4 T + 5 T^{2} )( 1 + 4 T + 5 T^{2} ) \))(\( 1 + 4 T + 8 T^{2} + 12 T^{3} + 14 T^{4} + 60 T^{5} + 200 T^{6} + 500 T^{7} + 625 T^{8} \))(\( 1 - 8 T + 28 T^{2} - 56 T^{3} + 72 T^{4} - 72 T^{5} + 4 T^{6} + 584 T^{7} - 2146 T^{8} + 2920 T^{9} + 100 T^{10} - 9000 T^{11} + 45000 T^{12} - 175000 T^{13} + 437500 T^{14} - 625000 T^{15} + 390625 T^{16} \))(\( 1 - 8 T + 44 T^{2} - 152 T^{3} + 392 T^{4} - 552 T^{5} - 76 T^{6} + 3592 T^{7} - 10594 T^{8} + 17960 T^{9} - 1900 T^{10} - 69000 T^{11} + 245000 T^{12} - 475000 T^{13} + 687500 T^{14} - 625000 T^{15} + 390625 T^{16} \))
$7$ (\( 1 + 2 T + 7 T^{2} \))(\( 1 - 2 T + 7 T^{2} \))(\( 1 + 7 T^{2} \))(\( 1 - 10 T^{2} + 49 T^{4} \))(\( 1 - 8 T + 32 T^{2} - 88 T^{3} + 226 T^{4} - 616 T^{5} + 1568 T^{6} - 2744 T^{7} + 2401 T^{8} \))(\( 1 - 16 T^{2} + 128 T^{4} - 272 T^{6} - 446 T^{8} - 13328 T^{10} + 307328 T^{12} - 1882384 T^{14} + 5764801 T^{16} \))(\( 1 + 16 T^{2} - 16 T^{3} + 128 T^{4} - 368 T^{5} + 656 T^{6} - 4224 T^{7} + 3586 T^{8} - 29568 T^{9} + 32144 T^{10} - 126224 T^{11} + 307328 T^{12} - 268912 T^{13} + 1882384 T^{14} + 5764801 T^{16} \))
$11$ (\( 1 + 11 T^{2} \))(\( 1 + 11 T^{2} \))(\( 1 + 4 T + 11 T^{2} \))(\( ( 1 - 11 T^{2} )^{2} \))(\( 1 - 206 T^{4} + 14641 T^{8} \))(\( 1 + 16 T^{2} - 48 T^{3} + 128 T^{4} - 784 T^{5} + 2512 T^{6} - 10880 T^{7} + 30946 T^{8} - 119680 T^{9} + 303952 T^{10} - 1043504 T^{11} + 1874048 T^{12} - 7730448 T^{13} + 28344976 T^{14} + 214358881 T^{16} \))(\( 1 + 16 T + 144 T^{2} + 832 T^{3} + 3200 T^{4} + 6208 T^{5} - 12464 T^{6} - 159696 T^{7} - 696478 T^{8} - 1756656 T^{9} - 1508144 T^{10} + 8262848 T^{11} + 46851200 T^{12} + 133994432 T^{13} + 255104784 T^{14} + 311794736 T^{15} + 214358881 T^{16} \))
$13$ (\( 1 + 6 T + 13 T^{2} \))(\( 1 - 2 T + 13 T^{2} \))(\( 1 + 2 T + 13 T^{2} \))(\( ( 1 + 6 T + 13 T^{2} )^{2} \))(\( ( 1 + 18 T^{2} + 169 T^{4} )^{2} \))(\( 1 - 80 T^{2} + 2996 T^{4} - 69104 T^{6} + 1078470 T^{8} - 11678576 T^{10} + 85568756 T^{12} - 386144720 T^{14} + 815730721 T^{16} \))(\( 1 - 16 T^{2} + 436 T^{4} - 5424 T^{6} + 106694 T^{8} - 916656 T^{10} + 12452596 T^{12} - 77228944 T^{14} + 815730721 T^{16} \))
$17$ (\( 1 + T \))(\( 1 + T \))(\( 1 - T \))(\( 1 + 2 T + 17 T^{2} \))(\( ( 1 + 6 T + 17 T^{2} )^{2} \))(\( 1 + 8 T + 12 T^{2} - 136 T^{3} - 922 T^{4} - 2312 T^{5} + 3468 T^{6} + 39304 T^{7} + 83521 T^{8} \))(\( 1 - 8 T + 12 T^{2} + 136 T^{3} - 922 T^{4} + 2312 T^{5} + 3468 T^{6} - 39304 T^{7} + 83521 T^{8} \))
$19$ (\( 1 - 4 T + 19 T^{2} \))(\( 1 + 4 T + 19 T^{2} \))(\( 1 - 4 T + 19 T^{2} \))(\( ( 1 + 19 T^{2} )^{2} \))(\( 1 - 28 T^{2} + 406 T^{4} - 10108 T^{6} + 130321 T^{8} \))(\( 1 - 32 T^{3} - 396 T^{4} + 288 T^{5} + 512 T^{6} + 1856 T^{7} + 42694 T^{8} + 35264 T^{9} + 184832 T^{10} + 1975392 T^{11} - 51607116 T^{12} - 79235168 T^{13} + 16983563041 T^{16} \))(\( 1 + 16 T + 128 T^{2} + 784 T^{3} + 4916 T^{4} + 30032 T^{5} + 158592 T^{6} + 733904 T^{7} + 3223110 T^{8} + 13944176 T^{9} + 57251712 T^{10} + 205989488 T^{11} + 640658036 T^{12} + 1941261616 T^{13} + 6021872768 T^{14} + 14301947824 T^{15} + 16983563041 T^{16} \))
$23$ (\( 1 - 6 T + 23 T^{2} \))(\( 1 + 6 T + 23 T^{2} \))(\( 1 + 23 T^{2} \))(\( 1 - 10 T^{2} + 529 T^{4} \))(\( 1 - 8 T + 32 T^{2} + 40 T^{3} - 734 T^{4} + 920 T^{5} + 16928 T^{6} - 97336 T^{7} + 279841 T^{8} \))(\( 1 + 16 T + 128 T^{2} + 640 T^{3} + 2048 T^{4} + 1152 T^{5} - 34816 T^{6} - 255632 T^{7} - 1288126 T^{8} - 5879536 T^{9} - 18417664 T^{10} + 14016384 T^{11} + 573114368 T^{12} + 4119259520 T^{13} + 18948593792 T^{14} + 54477207152 T^{15} + 78310985281 T^{16} \))(\( 1 + 211202 T^{8} + 78310985281 T^{16} \))
$29$ (\( 1 + 4 T + 29 T^{2} \))(\( 1 + 29 T^{2} \))(\( 1 + 10 T + 29 T^{2} \))(\( 1 - 22 T^{2} + 841 T^{4} \))(\( 1 - 4 T + 8 T^{2} - 108 T^{3} + 1454 T^{4} - 3132 T^{5} + 6728 T^{6} - 97556 T^{7} + 707281 T^{8} \))(\( 1 - 12 T^{2} - 240 T^{3} + 72 T^{4} + 4848 T^{5} + 25836 T^{6} - 73152 T^{7} - 889762 T^{8} - 2121408 T^{9} + 21728076 T^{10} + 118237872 T^{11} + 50924232 T^{12} - 4922675760 T^{13} - 7137879852 T^{14} + 500246412961 T^{16} \))(\( 1 + 100 T^{2} - 240 T^{3} + 5000 T^{4} - 23632 T^{5} + 171964 T^{6} - 1177856 T^{7} + 4983646 T^{8} - 34157824 T^{9} + 144621724 T^{10} - 576360848 T^{11} + 3536405000 T^{12} - 4922675760 T^{13} + 59482332100 T^{14} + 500246412961 T^{16} \))
$31$ (\( 1 + 6 T + 31 T^{2} \))(\( 1 + 10 T + 31 T^{2} \))(\( 1 - 8 T + 31 T^{2} \))(\( 1 + 38 T^{2} + 961 T^{4} \))(\( 1 - 8 T + 32 T^{2} - 280 T^{3} + 2434 T^{4} - 8680 T^{5} + 30752 T^{6} - 238328 T^{7} + 923521 T^{8} \))(\( 1 - 32 T^{2} + 512 T^{4} + 27104 T^{6} - 1834046 T^{8} + 26046944 T^{10} + 472842752 T^{12} - 28400117792 T^{14} + 852891037441 T^{16} \))(\( 1 + 32 T^{2} - 272 T^{3} + 512 T^{4} - 9968 T^{5} + 30880 T^{6} - 15872 T^{7} + 2371330 T^{8} - 492032 T^{9} + 29675680 T^{10} - 296956688 T^{11} + 472842752 T^{12} - 7787129072 T^{13} + 28400117792 T^{14} + 852891037441 T^{16} \))
$37$ (\( 1 + 4 T + 37 T^{2} \))(\( 1 - 8 T + 37 T^{2} \))(\( 1 + 2 T + 37 T^{2} \))(\( ( 1 - 12 T + 37 T^{2} )( 1 + 12 T + 37 T^{2} ) \))(\( 1 + 12 T + 72 T^{2} + 612 T^{3} + 5006 T^{4} + 22644 T^{5} + 98568 T^{6} + 607836 T^{7} + 1874161 T^{8} \))(\( 1 - 44 T^{2} - 144 T^{3} + 968 T^{4} + 1296 T^{5} + 51404 T^{6} - 109632 T^{7} - 2564066 T^{8} - 4056384 T^{9} + 70372076 T^{10} + 65646288 T^{11} + 1814187848 T^{12} - 9985529808 T^{13} - 112891961996 T^{14} + 3512479453921 T^{16} \))(\( 1 - 16 T + 100 T^{2} - 544 T^{3} + 4488 T^{4} - 35872 T^{5} + 253052 T^{6} - 1161072 T^{7} + 4338974 T^{8} - 42959664 T^{9} + 346428188 T^{10} - 1817024416 T^{11} + 8411234568 T^{12} - 37723112608 T^{13} + 256572640900 T^{14} - 1518910034128 T^{15} + 3512479453921 T^{16} \))
$41$ (\( 1 + 10 T + 41 T^{2} \))(\( 1 - 6 T + 41 T^{2} \))(\( 1 - 10 T + 41 T^{2} \))(\( ( 1 - 41 T^{2} )^{2} \))(\( 1 + 20 T + 200 T^{2} + 1500 T^{3} + 10094 T^{4} + 61500 T^{5} + 336200 T^{6} + 1378420 T^{7} + 2825761 T^{8} \))(\( 1 - 16 T + 36 T^{2} + 656 T^{3} - 2680 T^{4} - 21584 T^{5} + 112492 T^{6} + 787664 T^{7} - 9578674 T^{8} + 32294224 T^{9} + 189099052 T^{10} - 1487590864 T^{11} - 7573039480 T^{12} + 76001667856 T^{13} + 171003752676 T^{14} - 3116068382096 T^{15} + 7984925229121 T^{16} \))(\( 1 - 28 T^{2} + 288 T^{3} + 392 T^{4} + 1184 T^{5} + 117676 T^{6} + 186176 T^{7} - 2187442 T^{8} + 7633216 T^{9} + 197813356 T^{10} + 81602464 T^{11} + 1107698312 T^{12} + 33366585888 T^{13} - 133002918748 T^{14} + 7984925229121 T^{16} \))
$43$ (\( 1 + 4 T + 43 T^{2} \))(\( 1 + 4 T + 43 T^{2} \))(\( 1 - 12 T + 43 T^{2} \))(\( ( 1 - 4 T + 43 T^{2} )^{2} \))(\( 1 - 76 T^{2} + 3094 T^{4} - 140524 T^{6} + 3418801 T^{8} \))(\( 1 + 16 T + 128 T^{2} + 1328 T^{3} + 7716 T^{4} - 5584 T^{5} - 195200 T^{6} - 2990960 T^{7} - 32399194 T^{8} - 128611280 T^{9} - 360924800 T^{10} - 443967088 T^{11} + 26379468516 T^{12} + 195227212304 T^{13} + 809134470272 T^{14} + 4349097777712 T^{15} + 11688200277601 T^{16} \))(\( 1 - 16 T + 128 T^{2} - 1200 T^{3} + 13988 T^{4} - 111280 T^{5} + 710016 T^{6} - 5643664 T^{7} + 43768230 T^{8} - 242677552 T^{9} + 1312819584 T^{10} - 8847538960 T^{11} + 47822188388 T^{12} - 176410131600 T^{13} + 809134470272 T^{14} - 4349097777712 T^{15} + 11688200277601 T^{16} \))
$47$ (\( 1 - 4 T + 47 T^{2} \))(\( 1 - 12 T + 47 T^{2} \))(\( 1 + 47 T^{2} \))(\( ( 1 - 8 T + 47 T^{2} )^{2} \))(\( ( 1 - 8 T + 38 T^{2} - 376 T^{3} + 2209 T^{4} )^{2} \))(\( 1 - 264 T^{2} + 32908 T^{4} - 2576888 T^{6} + 141960614 T^{8} - 5692345592 T^{10} + 160580542348 T^{12} - 2845712846856 T^{14} + 23811286661761 T^{16} \))(\( 1 - 232 T^{2} + 25676 T^{4} - 1829656 T^{6} + 97106982 T^{8} - 4041710104 T^{10} + 125290689356 T^{12} - 2500777956328 T^{14} + 23811286661761 T^{16} \))
$53$ (\( 1 + 2 T + 53 T^{2} \))(\( 1 - 6 T + 53 T^{2} \))(\( 1 - 6 T + 53 T^{2} \))(\( ( 1 + 6 T + 53 T^{2} )^{2} \))(\( ( 1 - 98 T^{2} + 2809 T^{4} )^{2} \))(\( 1 - 8 T + 32 T^{2} + 40 T^{3} + 76 T^{4} + 4632 T^{5} - 38688 T^{6} + 1404040 T^{7} - 13025274 T^{8} + 74414120 T^{9} - 108674592 T^{10} + 689598264 T^{11} + 599676556 T^{12} + 16727819720 T^{13} + 709259556128 T^{14} - 9397689118696 T^{15} + 62259690411361 T^{16} \))(\( 1 - 40 T + 800 T^{2} - 11384 T^{3} + 133324 T^{4} - 1351624 T^{5} + 12203488 T^{6} - 100697304 T^{7} + 764953350 T^{8} - 5336957112 T^{9} + 34279597792 T^{10} - 201225726248 T^{11} + 1051990488844 T^{12} - 4760737492312 T^{13} + 17731488903200 T^{14} - 46988445593480 T^{15} + 62259690411361 T^{16} \))
$59$ (\( 1 - 12 T + 59 T^{2} \))(\( 1 + 12 T + 59 T^{2} \))(\( 1 - 12 T + 59 T^{2} \))(\( ( 1 + 59 T^{2} )^{2} \))(\( 1 - 60 T^{2} + 3254 T^{4} - 208860 T^{6} + 12117361 T^{8} \))(\( 1 + 224 T^{3} - 3788 T^{4} - 14560 T^{5} + 25088 T^{6} - 507584 T^{7} + 9001734 T^{8} - 29947456 T^{9} + 87331328 T^{10} - 2990318240 T^{11} - 45900563468 T^{12} + 160143042976 T^{13} + 146830437604321 T^{16} \))(\( 1 - 16 T + 128 T^{2} - 592 T^{3} + 3700 T^{4} - 37008 T^{5} + 293760 T^{6} + 258608 T^{7} - 14233722 T^{8} + 15257872 T^{9} + 1022578560 T^{10} - 7600666032 T^{11} + 44834235700 T^{12} - 423235185008 T^{13} + 5399108306048 T^{14} - 39818423757104 T^{15} + 146830437604321 T^{16} \))
$61$ (\( 1 + 4 T + 61 T^{2} \))(\( 1 - 8 T + 61 T^{2} \))(\( 1 + 10 T + 61 T^{2} \))(\( ( 1 - 12 T + 61 T^{2} )( 1 + 12 T + 61 T^{2} ) \))(\( 1 - 20 T + 200 T^{2} - 1500 T^{3} + 11054 T^{4} - 91500 T^{5} + 744200 T^{6} - 4539620 T^{7} + 13845841 T^{8} \))(\( 1 + 32 T + 484 T^{2} + 5040 T^{3} + 44936 T^{4} + 394512 T^{5} + 3653692 T^{6} + 34598368 T^{7} + 295186014 T^{8} + 2110500448 T^{9} + 13595387932 T^{10} + 89546728272 T^{11} + 622176711176 T^{12} + 4256765357040 T^{13} + 24935861190724 T^{14} + 100567770752672 T^{15} + 191707312997281 T^{16} \))(\( 1 - 16 T + 52 T^{2} + 704 T^{3} - 6840 T^{4} - 960 T^{5} + 400684 T^{6} - 1150768 T^{7} - 10205858 T^{8} - 70196848 T^{9} + 1490945164 T^{10} - 217901760 T^{11} - 94705552440 T^{12} + 594595795904 T^{13} + 2679059466772 T^{14} - 50283885376336 T^{15} + 191707312997281 T^{16} \))
$67$ (\( 1 + 12 T + 67 T^{2} \))(\( 1 + 4 T + 67 T^{2} \))(\( 1 + 12 T + 67 T^{2} \))(\( ( 1 - 8 T + 67 T^{2} )^{2} \))(\( ( 1 - 8 T + 142 T^{2} - 536 T^{3} + 4489 T^{4} )^{2} \))(\( ( 1 - 24 T + 332 T^{2} - 3640 T^{3} + 33198 T^{4} - 243880 T^{5} + 1490348 T^{6} - 7218312 T^{7} + 20151121 T^{8} )^{2} \))(\( ( 1 + 8 T + 156 T^{2} + 1096 T^{3} + 11790 T^{4} + 73432 T^{5} + 700284 T^{6} + 2406104 T^{7} + 20151121 T^{8} )^{2} \))
$71$ (\( 1 + 6 T + 71 T^{2} \))(\( 1 - 6 T + 71 T^{2} \))(\( 1 + 71 T^{2} \))(\( 1 - 42 T^{2} + 5041 T^{4} \))(\( 1 - 24 T + 288 T^{2} - 3336 T^{3} + 34018 T^{4} - 236856 T^{5} + 1451808 T^{6} - 8589864 T^{7} + 25411681 T^{8} \))(\( 1 + 16 T - 32 T^{2} - 1280 T^{3} + 4608 T^{4} + 133888 T^{5} + 581984 T^{6} - 1077392 T^{7} - 26627902 T^{8} - 76494832 T^{9} + 2933781344 T^{10} + 47919987968 T^{11} + 117097026048 T^{12} - 2309413569280 T^{13} - 4099209085472 T^{14} + 145521922534256 T^{15} + 645753531245761 T^{16} \))(\( 1 + 32 T + 480 T^{2} + 5792 T^{3} + 71168 T^{4} + 782176 T^{5} + 7592416 T^{6} + 73042400 T^{7} + 661816706 T^{8} + 5186010400 T^{9} + 38273369056 T^{10} + 279949394336 T^{11} + 1808498513408 T^{12} + 10450096400992 T^{13} + 61488136282080 T^{14} + 291043845068512 T^{15} + 645753531245761 T^{16} \))
$73$ (\( 1 - 2 T + 73 T^{2} \))(\( 1 - 2 T + 73 T^{2} \))(\( 1 - 10 T + 73 T^{2} \))(\( ( 1 - 6 T + 73 T^{2} )( 1 + 6 T + 73 T^{2} ) \))(\( 1 - 4 T + 8 T^{2} - 44 T^{3} - 3602 T^{4} - 3212 T^{5} + 42632 T^{6} - 1556068 T^{7} + 28398241 T^{8} \))(\( 1 + 24 T + 284 T^{2} + 3688 T^{3} + 48200 T^{4} + 505048 T^{5} + 5141524 T^{6} + 49029160 T^{7} + 421421134 T^{8} + 3579128680 T^{9} + 27399181396 T^{10} + 196472257816 T^{11} + 1368795216200 T^{12} + 7645488034984 T^{13} + 42978920266076 T^{14} + 265137564458328 T^{15} + 806460091894081 T^{16} \))(\( 1 - 40 T + 700 T^{2} - 7000 T^{3} + 45000 T^{4} - 241000 T^{5} + 1930100 T^{6} - 20701080 T^{7} + 194933582 T^{8} - 1511178840 T^{9} + 10285502900 T^{10} - 93753097000 T^{11} + 1277920845000 T^{12} - 14511501151000 T^{13} + 105933958402300 T^{14} - 441895940763880 T^{15} + 806460091894081 T^{16} \))
$79$ (\( 1 - 10 T + 79 T^{2} \))(\( 1 + 10 T + 79 T^{2} \))(\( 1 + 8 T + 79 T^{2} \))(\( 1 - 122 T^{2} + 6241 T^{4} \))(\( 1 + 24 T + 288 T^{2} + 2760 T^{3} + 25154 T^{4} + 218040 T^{5} + 1797408 T^{6} + 11832936 T^{7} + 38950081 T^{8} \))(\( 1 + 208 T^{2} + 320 T^{3} + 21632 T^{4} + 30912 T^{5} + 1379536 T^{6} + 1172992 T^{7} + 86323458 T^{8} + 92666368 T^{9} + 8609684176 T^{10} + 15240821568 T^{11} + 842568152192 T^{12} + 984658047680 T^{13} + 50562190748368 T^{14} + 1517108809906561 T^{16} \))(\( 1 - 16 T + 176 T^{2} - 1216 T^{3} + 6272 T^{4} + 7744 T^{5} - 718032 T^{6} + 14979088 T^{7} - 166226238 T^{8} + 1183347952 T^{9} - 4481237712 T^{10} + 3818094016 T^{11} + 244294908032 T^{12} - 3741700581184 T^{13} + 42783392171696 T^{14} - 307262543778544 T^{15} + 1517108809906561 T^{16} \))
$83$ (\( 1 + 12 T + 83 T^{2} \))(\( 1 - 12 T + 83 T^{2} \))(\( 1 - 4 T + 83 T^{2} \))(\( ( 1 + 16 T + 83 T^{2} )^{2} \))(\( 1 - 156 T^{2} + 15254 T^{4} - 1074684 T^{6} + 47458321 T^{8} \))(\( 1 + 32 T + 512 T^{2} + 7552 T^{3} + 116212 T^{4} + 1429568 T^{5} + 14761984 T^{6} + 156654048 T^{7} + 1569808326 T^{8} + 13002285984 T^{9} + 101695307776 T^{10} + 817408398016 T^{11} + 5515226400052 T^{12} + 29747634935936 T^{13} + 167393471164928 T^{14} + 868353631668064 T^{15} + 2252292232139041 T^{16} \))(\( 1 - 16 T + 128 T^{2} - 1552 T^{3} + 10292 T^{4} + 22960 T^{5} - 480384 T^{6} + 9938736 T^{7} - 157082042 T^{8} + 824915088 T^{9} - 3309365376 T^{10} + 13128229520 T^{11} + 488441039732 T^{12} - 6113391077936 T^{13} + 41848367791232 T^{14} - 434176815834032 T^{15} + 2252292232139041 T^{16} \))
$89$ (\( 1 + 2 T + 89 T^{2} \))(\( 1 + 18 T + 89 T^{2} \))(\( 1 + 6 T + 89 T^{2} \))(\( ( 1 - 10 T + 89 T^{2} )^{2} \))(\( ( 1 + 89 T^{2} )^{4} \))(\( 1 - 352 T^{2} + 62084 T^{4} - 8237600 T^{6} + 853576390 T^{8} - 65250029600 T^{10} + 3895289290244 T^{12} - 174937414418272 T^{14} + 3936588805702081 T^{16} \))(\( 1 - 224 T^{2} + 26756 T^{4} - 1826976 T^{6} + 135182022 T^{8} - 14471476896 T^{10} + 1678731400196 T^{12} - 111323809175264 T^{14} + 3936588805702081 T^{16} \))
$97$ (\( 1 - 6 T + 97 T^{2} \))(\( 1 - 14 T + 97 T^{2} \))(\( 1 + 14 T + 97 T^{2} \))(\( 1 - 50 T^{2} + 9409 T^{4} \))(\( 1 + 4 T + 8 T^{2} - 180 T^{3} - 16114 T^{4} - 17460 T^{5} + 75272 T^{6} + 3650692 T^{7} + 88529281 T^{8} \))(\( 1 - 16 T + 228 T^{2} - 976 T^{3} - 376 T^{4} + 26448 T^{5} + 597548 T^{6} - 26226544 T^{7} + 266177550 T^{8} - 2543974768 T^{9} + 5622329132 T^{10} + 24138375504 T^{11} - 33287009656 T^{12} - 8381244090832 T^{13} + 189917617123812 T^{14} - 1292772551649808 T^{15} + 7837433594376961 T^{16} \))(\( 1 + 16 T + 196 T^{2} - 1008 T^{3} - 30712 T^{4} - 477840 T^{5} - 773300 T^{6} + 24024176 T^{7} + 558101262 T^{8} + 2330345072 T^{9} - 7275979700 T^{10} - 436111666320 T^{11} - 2718911278072 T^{12} - 8656038979056 T^{13} + 163262512966084 T^{14} + 1292772551649808 T^{15} + 7837433594376961 T^{16} \))
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