Properties

Label 99.2.e
Level 99
Weight 2
Character orbit e
Rep. character \(\chi_{99}(34,\cdot)\)
Character field \(\Q(\zeta_{3})\)
Dimension 20
Newform subspaces 5
Sturm bound 24
Trace bound 2

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Defining parameters

Level: \( N \) = \( 99 = 3^{2} \cdot 11 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 99.e (of order \(3\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) = \( 9 \)
Character field: \(\Q(\zeta_{3})\)
Newform subspaces: \( 5 \)
Sturm bound: \(24\)
Trace bound: \(2\)
Distinguishing \(T_p\): \(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(99, [\chi])\).

Total New Old
Modular forms 28 20 8
Cusp forms 20 20 0
Eisenstein series 8 0 8

Trace form

\( 20q - q^{3} - 10q^{4} - 3q^{5} + 2q^{6} - 2q^{7} - 12q^{8} - 5q^{9} + O(q^{10}) \) \( 20q - q^{3} - 10q^{4} - 3q^{5} + 2q^{6} - 2q^{7} - 12q^{8} - 5q^{9} - 4q^{11} - 2q^{12} - 2q^{13} + 6q^{14} + 8q^{15} - 10q^{16} - 12q^{17} + 4q^{18} + 4q^{19} + 6q^{20} - 4q^{21} - 6q^{23} + 18q^{24} - 7q^{25} + 48q^{26} - 22q^{27} + 16q^{28} + 12q^{29} - 22q^{30} - 5q^{31} + 12q^{32} + 2q^{33} - 12q^{34} + 26q^{36} - 2q^{37} - 30q^{38} - 40q^{39} - 12q^{40} + 12q^{41} + 26q^{42} - 8q^{43} + 20q^{44} + 19q^{45} + 6q^{47} - 50q^{48} - 6q^{50} - 34q^{51} + 10q^{52} - 12q^{53} - 28q^{54} - 6q^{55} - 12q^{56} + 18q^{57} + 18q^{58} - 27q^{59} + 88q^{60} - 2q^{61} - 60q^{62} + 52q^{63} + 8q^{64} + 18q^{65} - 10q^{66} - 11q^{67} - 18q^{68} - 29q^{69} + 12q^{70} + 66q^{71} + 84q^{72} + 28q^{73} + 54q^{74} + 37q^{75} - 20q^{76} - 8q^{77} - 10q^{78} - 2q^{79} - 96q^{80} + 7q^{81} + 12q^{82} + 22q^{84} - 24q^{85} + 36q^{86} - 12q^{89} - 128q^{90} + 44q^{91} - 54q^{92} + 29q^{93} + 30q^{94} + 36q^{95} - 68q^{96} - 23q^{97} + 36q^{98} - 5q^{99} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(99, [\chi])\) into newform subspaces

Label Dim. \(A\) Field CM Traces $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\)
99.2.e.a \(2\) \(0.791\) \(\Q(\sqrt{-3}) \) None \(-1\) \(-3\) \(-1\) \(-4\) \(q+(-1+\zeta_{6})q^{2}+(-2+\zeta_{6})q^{3}+\zeta_{6}q^{4}+\cdots\)
99.2.e.b \(2\) \(0.791\) \(\Q(\sqrt{-3}) \) None \(0\) \(-3\) \(3\) \(4\) \(q+(-1-\zeta_{6})q^{3}+2\zeta_{6}q^{4}+3\zeta_{6}q^{5}+\cdots\)
99.2.e.c \(2\) \(0.791\) \(\Q(\sqrt{-3}) \) None \(2\) \(0\) \(2\) \(-4\) \(q+(2-2\zeta_{6})q^{2}+(1-2\zeta_{6})q^{3}-2\zeta_{6}q^{4}+\cdots\)
99.2.e.d \(6\) \(0.791\) \(\Q(\zeta_{18})\) None \(0\) \(0\) \(-3\) \(3\) \(q+(\zeta_{18}^{2}-\zeta_{18}^{3}-\zeta_{18}^{4}+\zeta_{18}^{5})q^{2}+\cdots\)
99.2.e.e \(8\) \(0.791\) 8.0.508277025.1 None \(-1\) \(5\) \(-4\) \(-1\) \(q+(-\beta _{2}-\beta _{7})q^{2}+(1+\beta _{7})q^{3}+(-2+\cdots)q^{4}+\cdots\)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ (\( 1 + T - T^{2} + 2 T^{3} + 4 T^{4} \))(\( 1 - 2 T^{2} + 4 T^{4} \))(\( 1 - 2 T + 2 T^{2} - 4 T^{3} + 4 T^{4} \))(\( 1 - 3 T^{2} + 2 T^{3} + 3 T^{4} - 3 T^{5} - T^{6} - 6 T^{7} + 12 T^{8} + 16 T^{9} - 48 T^{10} + 64 T^{12} \))(\( 1 + T + 2 T^{2} + 3 T^{3} + 2 T^{4} - 4 T^{5} + T^{6} - 6 T^{7} - 11 T^{8} - 12 T^{9} + 4 T^{10} - 32 T^{11} + 32 T^{12} + 96 T^{13} + 128 T^{14} + 128 T^{15} + 256 T^{16} \))
$3$ (\( 1 + 3 T + 3 T^{2} \))(\( 1 + 3 T + 3 T^{2} \))(\( 1 + 3 T^{2} \))(\( 1 + 9 T^{3} + 27 T^{6} \))(\( 1 - 5 T + 15 T^{2} - 35 T^{3} + 67 T^{4} - 105 T^{5} + 135 T^{6} - 135 T^{7} + 81 T^{8} \))
$5$ (\( 1 + T - 4 T^{2} + 5 T^{3} + 25 T^{4} \))(\( 1 - 3 T + 4 T^{2} - 15 T^{3} + 25 T^{4} \))(\( 1 - 2 T - T^{2} - 10 T^{3} + 25 T^{4} \))(\( 1 + 3 T - 6 T^{2} - 13 T^{3} + 63 T^{4} + 60 T^{5} - 259 T^{6} + 300 T^{7} + 1575 T^{8} - 1625 T^{9} - 3750 T^{10} + 9375 T^{11} + 15625 T^{12} \))(\( 1 + 4 T + 5 T^{2} - 18 T^{3} - 94 T^{4} - 232 T^{5} - 110 T^{6} + 1011 T^{7} + 3826 T^{8} + 5055 T^{9} - 2750 T^{10} - 29000 T^{11} - 58750 T^{12} - 56250 T^{13} + 78125 T^{14} + 312500 T^{15} + 390625 T^{16} \))
$7$ (\( ( 1 - T + 7 T^{2} )( 1 + 5 T + 7 T^{2} ) \))(\( ( 1 - 5 T + 7 T^{2} )( 1 + T + 7 T^{2} ) \))(\( ( 1 - T + 7 T^{2} )( 1 + 5 T + 7 T^{2} ) \))(\( 1 - 3 T - 12 T^{2} + 19 T^{3} + 171 T^{4} - 126 T^{5} - 1161 T^{6} - 882 T^{7} + 8379 T^{8} + 6517 T^{9} - 28812 T^{10} - 50421 T^{11} + 117649 T^{12} \))(\( 1 + T - 21 T^{2} - 10 T^{3} + 254 T^{4} + 36 T^{5} - 2408 T^{6} - 83 T^{7} + 18573 T^{8} - 581 T^{9} - 117992 T^{10} + 12348 T^{11} + 609854 T^{12} - 168070 T^{13} - 2470629 T^{14} + 823543 T^{15} + 5764801 T^{16} \))
$11$ (\( 1 + T + T^{2} \))(\( 1 + T + T^{2} \))(\( 1 + T + T^{2} \))(\( ( 1 - T + T^{2} )^{3} \))(\( ( 1 + T + T^{2} )^{4} \))
$13$ (\( ( 1 - 7 T + 13 T^{2} )( 1 + 5 T + 13 T^{2} ) \))(\( ( 1 - 5 T + 13 T^{2} )( 1 + 7 T + 13 T^{2} ) \))(\( 1 + 4 T + 3 T^{2} + 52 T^{3} + 169 T^{4} \))(\( 1 - 9 T + 27 T^{2} - 20 T^{3} + 117 T^{4} - 1467 T^{5} + 7086 T^{6} - 19071 T^{7} + 19773 T^{8} - 43940 T^{9} + 771147 T^{10} - 3341637 T^{11} + 4826809 T^{12} \))(\( 1 + 7 T + 12 T^{2} + 23 T^{3} + 77 T^{4} - 624 T^{5} - 1550 T^{6} + 622 T^{7} - 8472 T^{8} + 8086 T^{9} - 261950 T^{10} - 1370928 T^{11} + 2199197 T^{12} + 8539739 T^{13} + 57921708 T^{14} + 439239619 T^{15} + 815730721 T^{16} \))
$17$ (\( ( 1 - 4 T + 17 T^{2} )^{2} \))(\( ( 1 + 6 T + 17 T^{2} )^{2} \))(\( ( 1 - 4 T + 17 T^{2} )^{2} \))(\( ( 1 + 3 T + 45 T^{2} + 103 T^{3} + 765 T^{4} + 867 T^{5} + 4913 T^{6} )^{2} \))(\( ( 1 + 5 T + 44 T^{2} + 86 T^{3} + 682 T^{4} + 1462 T^{5} + 12716 T^{6} + 24565 T^{7} + 83521 T^{8} )^{2} \))
$19$ (\( ( 1 - 6 T + 19 T^{2} )^{2} \))(\( ( 1 - 2 T + 19 T^{2} )^{2} \))(\( ( 1 + 6 T + 19 T^{2} )^{2} \))(\( ( 1 + 9 T + 75 T^{2} + 333 T^{3} + 1425 T^{4} + 3249 T^{5} + 6859 T^{6} )^{2} \))(\( ( 1 - 9 T + 76 T^{2} - 432 T^{3} + 2112 T^{4} - 8208 T^{5} + 27436 T^{6} - 61731 T^{7} + 130321 T^{8} )^{2} \))
$23$ (\( 1 - 4 T - 7 T^{2} - 92 T^{3} + 529 T^{4} \))(\( 1 + 3 T - 14 T^{2} + 69 T^{3} + 529 T^{4} \))(\( 1 - T - 22 T^{2} - 23 T^{3} + 529 T^{4} \))(\( 1 - 6 T - 36 T^{2} + 82 T^{3} + 1986 T^{4} - 1806 T^{5} - 47893 T^{6} - 41538 T^{7} + 1050594 T^{8} + 997694 T^{9} - 10074276 T^{10} - 38618058 T^{11} + 148035889 T^{12} \))(\( 1 + 14 T + 53 T^{2} + 96 T^{3} + 1775 T^{4} + 8902 T^{5} - 9839 T^{6} + 41958 T^{7} + 1213519 T^{8} + 965034 T^{9} - 5204831 T^{10} + 108310634 T^{11} + 496717775 T^{12} + 617888928 T^{13} + 7845902117 T^{14} + 47667556258 T^{15} + 78310985281 T^{16} \))
$29$ (\( 1 + 6 T + 7 T^{2} + 174 T^{3} + 841 T^{4} \))(\( 1 - 6 T + 7 T^{2} - 174 T^{3} + 841 T^{4} \))(\( 1 - 29 T^{2} + 841 T^{4} \))(\( 1 - 6 T - 6 T^{2} + 18 T^{3} - 264 T^{4} + 3900 T^{5} - 14429 T^{6} + 113100 T^{7} - 222024 T^{8} + 439002 T^{9} - 4243686 T^{10} - 123066894 T^{11} + 594823321 T^{12} \))(\( 1 - 6 T - 71 T^{2} + 192 T^{3} + 4573 T^{4} - 4224 T^{5} - 186713 T^{6} + 77688 T^{7} + 5509213 T^{8} + 2252952 T^{9} - 157025633 T^{10} - 103019136 T^{11} + 3234396013 T^{12} + 3938140608 T^{13} - 42232455791 T^{14} - 103499257854 T^{15} + 500246412961 T^{16} \))
$31$ (\( ( 1 - 4 T + 31 T^{2} )( 1 + 11 T + 31 T^{2} ) \))(\( 1 + 8 T + 33 T^{2} + 248 T^{3} + 961 T^{4} \))(\( 1 + T - 30 T^{2} + 31 T^{3} + 961 T^{4} \))(\( 1 - 9 T - 18 T^{2} + 187 T^{3} + 1881 T^{4} - 6768 T^{5} - 31569 T^{6} - 209808 T^{7} + 1807641 T^{8} + 5570917 T^{9} - 16623378 T^{10} - 257662359 T^{11} + 887503681 T^{12} \))(\( 1 - 2 T - 99 T^{2} + 20 T^{3} + 5978 T^{4} + 2478 T^{5} - 249986 T^{6} - 33695 T^{7} + 8127744 T^{8} - 1044545 T^{9} - 240236546 T^{10} + 73822098 T^{11} + 5520808538 T^{12} + 572583020 T^{13} - 87862864419 T^{14} - 55025228222 T^{15} + 852891037441 T^{16} \))
$37$ (\( ( 1 - 3 T + 37 T^{2} )^{2} \))(\( ( 1 - 2 T + 37 T^{2} )^{2} \))(\( ( 1 - 3 T + 37 T^{2} )^{2} \))(\( ( 1 + 12 T + 138 T^{2} + 885 T^{3} + 5106 T^{4} + 16428 T^{5} + 50653 T^{6} )^{2} \))(\( ( 1 - 3 T + 67 T^{2} - 189 T^{3} + 2277 T^{4} - 6993 T^{5} + 91723 T^{6} - 151959 T^{7} + 1874161 T^{8} )^{2} \))
$41$ (\( 1 - 2 T - 37 T^{2} - 82 T^{3} + 1681 T^{4} \))(\( 1 - 41 T^{2} + 1681 T^{4} \))(\( 1 - 2 T - 37 T^{2} - 82 T^{3} + 1681 T^{4} \))(\( 1 - 6 T - 6 T^{2} - 202 T^{3} - 108 T^{4} + 9528 T^{5} + 5627 T^{6} + 390648 T^{7} - 181548 T^{8} - 13922042 T^{9} - 16954566 T^{10} - 695137206 T^{11} + 4750104241 T^{12} \))(\( 1 - 2 T - 97 T^{2} + 612 T^{3} + 4493 T^{4} - 38218 T^{5} - 22457 T^{6} + 901512 T^{7} - 2262647 T^{8} + 36961992 T^{9} - 37750217 T^{10} - 2634022778 T^{11} + 12696144173 T^{12} + 70903995012 T^{13} - 460760111377 T^{14} - 389508547762 T^{15} + 7984925229121 T^{16} \))
$43$ (\( 1 + 6 T - 7 T^{2} + 258 T^{3} + 1849 T^{4} \))(\( ( 1 - 5 T + 43 T^{2} )( 1 + 13 T + 43 T^{2} ) \))(\( 1 + 12 T + 101 T^{2} + 516 T^{3} + 1849 T^{4} \))(\( 1 + 3 T + 24 T^{2} + 525 T^{3} + 1725 T^{4} + 13692 T^{5} + 213923 T^{6} + 588756 T^{7} + 3189525 T^{8} + 41741175 T^{9} + 82051224 T^{10} + 441025329 T^{11} + 6321363049 T^{12} \))(\( 1 - 21 T + 113 T^{2} - 516 T^{3} + 15670 T^{4} - 131670 T^{5} + 283958 T^{6} - 4210359 T^{7} + 56707993 T^{8} - 181045437 T^{9} + 525038342 T^{10} - 10468686690 T^{11} + 53572611670 T^{12} - 75856356588 T^{13} + 714314024537 T^{14} - 5708190833247 T^{15} + 11688200277601 T^{16} \))
$47$ (\( 1 - 7 T + 2 T^{2} - 329 T^{3} + 2209 T^{4} \))(\( 1 + 3 T - 38 T^{2} + 141 T^{3} + 2209 T^{4} \))(\( 1 - 7 T + 2 T^{2} - 329 T^{3} + 2209 T^{4} \))(\( 1 + 12 T + 18 T^{2} - 386 T^{3} - 1434 T^{4} + 3954 T^{5} + 43979 T^{6} + 185838 T^{7} - 3167706 T^{8} - 40075678 T^{9} + 87834258 T^{10} + 2752140084 T^{11} + 10779215329 T^{12} \))(\( 1 - 7 T - 124 T^{2} + 759 T^{3} + 11189 T^{4} - 50870 T^{5} - 638711 T^{6} + 948885 T^{7} + 33885262 T^{8} + 44597595 T^{9} - 1410912599 T^{10} - 5281476010 T^{11} + 54598750709 T^{12} + 174072860313 T^{13} - 1336622700796 T^{14} - 3546361843241 T^{15} + 23811286661761 T^{16} \))
$53$ (\( ( 1 + 9 T + 53 T^{2} )^{2} \))(\( ( 1 - 3 T + 53 T^{2} )^{2} \))(\( ( 1 - 3 T + 53 T^{2} )^{2} \))(\( ( 1 - 3 T + 33 T^{2} - 261 T^{3} + 1749 T^{4} - 8427 T^{5} + 148877 T^{6} )^{2} \))(\( ( 1 + 6 T + 167 T^{2} + 789 T^{3} + 11961 T^{4} + 41817 T^{5} + 469103 T^{6} + 893262 T^{7} + 7890481 T^{8} )^{2} \))
$59$ (\( 1 - 7 T - 10 T^{2} - 413 T^{3} + 3481 T^{4} \))(\( 1 - 59 T^{2} + 3481 T^{4} \))(\( 1 + 11 T + 62 T^{2} + 649 T^{3} + 3481 T^{4} \))(\( 1 + 21 T + 198 T^{2} + 1081 T^{3} + 2481 T^{4} - 40380 T^{5} - 564133 T^{6} - 2382420 T^{7} + 8636361 T^{8} + 222014699 T^{9} + 2399237478 T^{10} + 15013410279 T^{11} + 42180533641 T^{12} \))(\( 1 + 2 T - 211 T^{2} - 180 T^{3} + 27194 T^{4} + 9886 T^{5} - 2406650 T^{6} - 211857 T^{7} + 162461212 T^{8} - 12499563 T^{9} - 8377548650 T^{10} + 2030376794 T^{11} + 329519515034 T^{12} - 128686373820 T^{13} - 8900092598251 T^{14} + 4977302969638 T^{15} + 146830437604321 T^{16} \))
$61$ (\( 1 - 61 T^{2} + 3721 T^{4} \))(\( 1 + 8 T + 3 T^{2} + 488 T^{3} + 3721 T^{4} \))(\( 1 - 61 T^{2} + 3721 T^{4} \))(\( 1 - 21 T + 114 T^{2} - 1101 T^{3} + 26439 T^{4} - 183108 T^{5} + 602021 T^{6} - 11169588 T^{7} + 98379519 T^{8} - 249906081 T^{9} + 1578425874 T^{10} - 17736522321 T^{11} + 51520374361 T^{12} \))(\( 1 + 15 T - 97 T^{2} - 990 T^{3} + 28072 T^{4} + 155670 T^{5} - 2089990 T^{6} + 70125 T^{7} + 207502789 T^{8} + 4277625 T^{9} - 7776852790 T^{10} + 35334132270 T^{11} + 388680448552 T^{12} - 836150337990 T^{13} - 4997476313017 T^{14} + 47141142540315 T^{15} + 191707312997281 T^{16} \))
$67$ (\( ( 1 - 5 T + 67 T^{2} )( 1 + 16 T + 67 T^{2} ) \))(\( 1 - 13 T + 102 T^{2} - 871 T^{3} + 4489 T^{4} \))(\( 1 - 4 T - 51 T^{2} - 268 T^{3} + 4489 T^{4} \))(\( 1 + 3 T - 66 T^{2} + 575 T^{3} + 1125 T^{4} - 29808 T^{5} + 214875 T^{6} - 1997136 T^{7} + 5050125 T^{8} + 172938725 T^{9} - 1329973986 T^{10} + 4050375321 T^{11} + 90458382169 T^{12} \))(\( 1 + 14 T - 51 T^{2} - 500 T^{3} + 11354 T^{4} + 6834 T^{5} - 1143698 T^{6} - 2330227 T^{7} + 43235628 T^{8} - 156125209 T^{9} - 5134060322 T^{10} + 2055414342 T^{11} + 228795827834 T^{12} - 675062553500 T^{13} - 4613377490619 T^{14} + 84849962474522 T^{15} + 406067677556641 T^{16} \))
$71$ (\( ( 1 - 9 T + 71 T^{2} )^{2} \))(\( ( 1 + 71 T^{2} )^{2} \))(\( ( 1 - 15 T + 71 T^{2} )^{2} \))(\( ( 1 - 12 T + 222 T^{2} - 1593 T^{3} + 15762 T^{4} - 60492 T^{5} + 357911 T^{6} )^{2} \))(\( ( 1 + 3 T + 197 T^{2} + 909 T^{3} + 17697 T^{4} + 64539 T^{5} + 993077 T^{6} + 1073733 T^{7} + 25411681 T^{8} )^{2} \))
$73$ (\( ( 1 - 4 T + 73 T^{2} )^{2} \))(\( ( 1 - 2 T + 73 T^{2} )^{2} \))(\( ( 1 + 8 T + 73 T^{2} )^{2} \))(\( ( 1 + 6 T + 114 T^{2} + 173 T^{3} + 8322 T^{4} + 31974 T^{5} + 389017 T^{6} )^{2} \))(\( ( 1 - 22 T + 421 T^{2} - 4807 T^{3} + 49780 T^{4} - 350911 T^{5} + 2243509 T^{6} - 8558374 T^{7} + 28398241 T^{8} )^{2} \))
$79$ (\( 1 + 8 T - 15 T^{2} + 632 T^{3} + 6241 T^{4} \))(\( 1 + 2 T - 75 T^{2} + 158 T^{3} + 6241 T^{4} \))(\( 1 - 10 T + 21 T^{2} - 790 T^{3} + 6241 T^{4} \))(\( 1 - 9 T - 180 T^{2} + 533 T^{3} + 34533 T^{4} - 61128 T^{5} - 2824521 T^{6} - 4829112 T^{7} + 215520453 T^{8} + 262789787 T^{9} - 7011014580 T^{10} - 27693507591 T^{11} + 243087455521 T^{12} \))(\( 1 + 11 T - 171 T^{2} - 1952 T^{3} + 23168 T^{4} + 216654 T^{5} - 1770938 T^{6} - 6655723 T^{7} + 156847173 T^{8} - 525802117 T^{9} - 11052424058 T^{10} + 106818871506 T^{11} + 902395476608 T^{12} - 6006414090848 T^{13} - 41567954894091 T^{14} + 211242998847749 T^{15} + 1517108809906561 T^{16} \))
$83$ (\( 1 - 12 T + 61 T^{2} - 996 T^{3} + 6889 T^{4} \))(\( 1 - 18 T + 241 T^{2} - 1494 T^{3} + 6889 T^{4} \))(\( 1 + 12 T + 61 T^{2} + 996 T^{3} + 6889 T^{4} \))(\( 1 - 222 T^{2} - 54 T^{3} + 30858 T^{4} + 5994 T^{5} - 2946269 T^{6} + 497502 T^{7} + 212580762 T^{8} - 30876498 T^{9} - 10535747262 T^{10} + 326940373369 T^{12} \))(\( 1 + 18 T - 5 T^{2} - 1620 T^{3} - 2021 T^{4} + 27828 T^{5} - 824807 T^{6} + 4969494 T^{7} + 193494259 T^{8} + 412468002 T^{9} - 5682095423 T^{10} + 15911688636 T^{11} - 95913266741 T^{12} - 6381245841660 T^{13} - 1634701866845 T^{14} + 488448917813286 T^{15} + 2252292232139041 T^{16} \))
$89$ (\( ( 1 - 6 T + 89 T^{2} )^{2} \))(\( ( 1 - 3 T + 89 T^{2} )^{2} \))(\( ( 1 - 3 T + 89 T^{2} )^{2} \))(\( ( 1 + 12 T + 159 T^{2} + 1728 T^{3} + 14151 T^{4} + 95052 T^{5} + 704969 T^{6} )^{2} \))(\( ( 1 + 6 T + 224 T^{2} + 1554 T^{3} + 26094 T^{4} + 138306 T^{5} + 1774304 T^{6} + 4229814 T^{7} + 62742241 T^{8} )^{2} \))
$97$ (\( ( 1 - 14 T + 97 T^{2} )( 1 - 5 T + 97 T^{2} ) \))(\( 1 + 2 T - 93 T^{2} + 194 T^{3} + 9409 T^{4} \))(\( 1 + 17 T + 192 T^{2} + 1649 T^{3} + 9409 T^{4} \))(\( 1 - 3 T - 141 T^{2} - 724 T^{3} + 8361 T^{4} + 79623 T^{5} - 423066 T^{6} + 7723431 T^{7} + 78668649 T^{8} - 660775252 T^{9} - 12482628621 T^{10} - 25762020771 T^{11} + 832972004929 T^{12} \))(\( 1 + 26 T + 132 T^{2} - 776 T^{3} + 16487 T^{4} + 336432 T^{5} + 568456 T^{6} + 7259774 T^{7} + 271106712 T^{8} + 704198078 T^{9} + 5348602504 T^{10} + 307052402736 T^{11} + 1459582255847 T^{12} - 6663776039432 T^{13} + 109952304650628 T^{14} + 2100755396430938 T^{15} + 7837433594376961 T^{16} \))
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