Properties

Label 35.2
Level 35
Weight 2
Dimension 25
Nonzero newspaces 6
Newform subspaces 8
Sturm bound 192
Trace bound 2

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Defining parameters

Level: \( N \) = \( 35 = 5 \cdot 7 \)
Weight: \( k \) = \( 2 \)
Nonzero newspaces: \( 6 \)
Newform subspaces: \( 8 \)
Sturm bound: \(192\)
Trace bound: \(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_1(35))\).

Total New Old
Modular forms 72 57 15
Cusp forms 25 25 0
Eisenstein series 47 32 15

Trace form

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

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_1(35))\)

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
35.2.a \(\chi_{35}(1, \cdot)\) 35.2.a.a 1 1
35.2.a.b 2
35.2.b \(\chi_{35}(29, \cdot)\) 35.2.b.a 2 1
35.2.e \(\chi_{35}(11, \cdot)\) 35.2.e.a 4 2
35.2.f \(\chi_{35}(13, \cdot)\) 35.2.f.a 4 2
35.2.j \(\chi_{35}(4, \cdot)\) 35.2.j.a 4 2
35.2.k \(\chi_{35}(3, \cdot)\) 35.2.k.a 4 4
35.2.k.b 4

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ (\( 1 + 2 T^{2} \))(\( 1 + T + 2 T^{3} + 4 T^{4} \))(\( ( 1 - 2 T + 2 T^{2} )( 1 + 2 T + 2 T^{2} ) \))(\( 1 + 2 T + T^{2} - 2 T^{3} - 3 T^{4} - 4 T^{5} + 4 T^{6} + 16 T^{7} + 16 T^{8} \))(\( ( 1 + 2 T + 2 T^{2} + 4 T^{3} + 4 T^{4} )^{2} \))(\( 1 + 3 T^{2} + 5 T^{4} + 12 T^{6} + 16 T^{8} \))(\( 1 + 4 T + 5 T^{2} - 2 T^{3} - 11 T^{4} - 4 T^{5} + 20 T^{6} + 32 T^{7} + 16 T^{8} \))(\( 1 - 2 T + 5 T^{2} - 8 T^{3} + 13 T^{4} - 16 T^{5} + 20 T^{6} - 16 T^{7} + 16 T^{8} \))
$3$ (\( 1 - T + 3 T^{2} \))(\( 1 + T + 2 T^{2} + 3 T^{3} + 9 T^{4} \))(\( 1 - 5 T^{2} + 9 T^{4} \))(\( 1 + 2 T - T^{2} - 2 T^{3} + 4 T^{4} - 6 T^{5} - 9 T^{6} + 54 T^{7} + 81 T^{8} \))(\( 1 - 17 T^{4} + 81 T^{8} \))(\( 1 + 5 T^{2} + 16 T^{4} + 45 T^{6} + 81 T^{8} \))(\( 1 + 2 T + 5 T^{2} + 10 T^{3} + 16 T^{4} + 30 T^{5} + 45 T^{6} + 54 T^{7} + 81 T^{8} \))(\( 1 + 4 T + 5 T^{2} - 4 T^{3} - 20 T^{4} - 12 T^{5} + 45 T^{6} + 108 T^{7} + 81 T^{8} \))
$5$ (\( 1 + T \))(\( ( 1 - T )^{2} \))(\( 1 + 4 T + 5 T^{2} \))(\( ( 1 + T + T^{2} )^{2} \))(\( 1 + 25 T^{4} \))(\( 1 + 2 T - T^{2} + 10 T^{3} + 25 T^{4} \))(\( 1 + 4 T + 11 T^{2} + 20 T^{3} + 25 T^{4} \))(\( 1 - 4 T + 11 T^{2} - 20 T^{3} + 25 T^{4} \))
$7$ (\( 1 - T \))(\( ( 1 + T )^{2} \))(\( 1 + T^{2} \))(\( 1 - 2 T - 3 T^{2} - 14 T^{3} + 49 T^{4} \))(\( 1 - 4 T + 8 T^{2} - 28 T^{3} + 49 T^{4} \))(\( 1 - 13 T^{2} + 49 T^{4} \))(\( 1 + 11 T^{2} + 49 T^{4} \))(\( ( 1 + 5 T + 7 T^{2} )^{2} \))
$11$ (\( 1 + 3 T + 11 T^{2} \))(\( 1 - T + 18 T^{2} - 11 T^{3} + 121 T^{4} \))(\( ( 1 + 3 T + 11 T^{2} )^{2} \))(\( 1 + 4 T - 2 T^{2} - 16 T^{3} + 27 T^{4} - 176 T^{5} - 242 T^{6} + 5324 T^{7} + 14641 T^{8} \))(\( ( 1 + T + 11 T^{2} )^{4} \))(\( ( 1 - 11 T^{2} + 121 T^{4} )^{2} \))(\( 1 - 2 T - 16 T^{2} + 4 T^{3} + 235 T^{4} + 44 T^{5} - 1936 T^{6} - 2662 T^{7} + 14641 T^{8} \))(\( 1 - 2 T - 16 T^{2} + 4 T^{3} + 235 T^{4} + 44 T^{5} - 1936 T^{6} - 2662 T^{7} + 14641 T^{8} \))
$13$ (\( 1 - 5 T + 13 T^{2} \))(\( 1 - 5 T + 28 T^{2} - 65 T^{3} + 169 T^{4} \))(\( 1 - 25 T^{2} + 169 T^{4} \))(\( ( 1 + 4 T + 22 T^{2} + 52 T^{3} + 169 T^{4} )^{2} \))(\( 1 + 103 T^{4} + 28561 T^{8} \))(\( ( 1 - 22 T^{2} + 169 T^{4} )^{2} \))(\( ( 1 - 4 T + 8 T^{2} - 52 T^{3} + 169 T^{4} )^{2} \))(\( ( 1 + 4 T + 8 T^{2} + 52 T^{3} + 169 T^{4} )^{2} \))
$17$ (\( 1 - 3 T + 17 T^{2} \))(\( 1 + 5 T + 36 T^{2} + 85 T^{3} + 289 T^{4} \))(\( 1 + 15 T^{2} + 289 T^{4} \))(\( 1 + 4 T - 14 T^{2} - 16 T^{3} + 339 T^{4} - 272 T^{5} - 4046 T^{6} + 19652 T^{7} + 83521 T^{8} \))(\( 1 + 263 T^{4} + 83521 T^{8} \))(\( ( 1 - 8 T + 47 T^{2} - 136 T^{3} + 289 T^{4} )( 1 + 8 T + 47 T^{2} + 136 T^{3} + 289 T^{4} ) \))(\( 1 - 4 T + 20 T^{2} - 100 T^{3} + 271 T^{4} - 1700 T^{5} + 5780 T^{6} - 19652 T^{7} + 83521 T^{8} \))(\( 1 - 8 T + 20 T^{2} + 52 T^{3} - 545 T^{4} + 884 T^{5} + 5780 T^{6} - 39304 T^{7} + 83521 T^{8} \))
$19$ (\( 1 - 2 T + 19 T^{2} \))(\( 1 + 6 T + 30 T^{2} + 114 T^{3} + 361 T^{4} \))(\( ( 1 + 19 T^{2} )^{2} \))(\( 1 - 30 T^{2} + 539 T^{4} - 10830 T^{6} + 130321 T^{8} \))(\( ( 1 + 28 T^{2} + 361 T^{4} )^{2} \))(\( ( 1 - 6 T + 17 T^{2} - 114 T^{3} + 361 T^{4} )^{2} \))(\( 1 - 2 T - 32 T^{2} + 4 T^{3} + 859 T^{4} + 76 T^{5} - 11552 T^{6} - 13718 T^{7} + 130321 T^{8} \))(\( 1 + 2 T - 32 T^{2} - 4 T^{3} + 859 T^{4} - 76 T^{5} - 11552 T^{6} + 13718 T^{7} + 130321 T^{8} \))
$23$ (\( 1 + 6 T + 23 T^{2} \))(\( 1 + 2 T + 30 T^{2} + 46 T^{3} + 529 T^{4} \))(\( 1 - 10 T^{2} + 529 T^{4} \))(\( 1 - 2 T - 41 T^{2} + 2 T^{3} + 1404 T^{4} + 46 T^{5} - 21689 T^{6} - 24334 T^{7} + 279841 T^{8} \))(\( ( 1 - 4 T + 8 T^{2} - 92 T^{3} + 529 T^{4} )^{2} \))(\( 1 + 37 T^{2} + 840 T^{4} + 19573 T^{6} + 279841 T^{8} \))(\( 1 + 4 T + 53 T^{2} + 244 T^{3} + 1588 T^{4} + 5612 T^{5} + 28037 T^{6} + 48668 T^{7} + 279841 T^{8} \))(\( 1 - 14 T + 53 T^{2} + 226 T^{3} - 2552 T^{4} + 5198 T^{5} + 28037 T^{6} - 170338 T^{7} + 279841 T^{8} \))
$29$ (\( 1 - 3 T + 29 T^{2} \))(\( 1 - T + 20 T^{2} - 29 T^{3} + 841 T^{4} \))(\( ( 1 - 5 T + 29 T^{2} )^{2} \))(\( ( 1 + T + 29 T^{2} )^{4} \))(\( ( 1 - 49 T^{2} + 841 T^{4} )^{2} \))(\( ( 1 + 7 T + 29 T^{2} )^{4} \))(\( ( 1 - 49 T^{2} + 841 T^{4} )^{2} \))(\( ( 1 - 49 T^{2} + 841 T^{4} )^{2} \))
$31$ (\( 1 + 4 T + 31 T^{2} \))(\( ( 1 + 31 T^{2} )^{2} \))(\( ( 1 - 2 T + 31 T^{2} )^{2} \))(\( ( 1 - 6 T + 5 T^{2} - 186 T^{3} + 961 T^{4} )^{2} \))(\( ( 1 - 52 T^{2} + 961 T^{4} )^{2} \))(\( ( 1 + 2 T - 27 T^{2} + 62 T^{3} + 961 T^{4} )^{2} \))(\( 1 + 12 T + 106 T^{2} + 696 T^{3} + 3891 T^{4} + 21576 T^{5} + 101866 T^{6} + 357492 T^{7} + 923521 T^{8} \))(\( 1 + 12 T + 106 T^{2} + 696 T^{3} + 3891 T^{4} + 21576 T^{5} + 101866 T^{6} + 357492 T^{7} + 923521 T^{8} \))
$37$ (\( 1 - 2 T + 37 T^{2} \))(\( ( 1 - 6 T + 37 T^{2} )^{2} \))(\( ( 1 - 12 T + 37 T^{2} )( 1 + 12 T + 37 T^{2} ) \))(\( ( 1 - 37 T^{2} + 1369 T^{4} )^{2} \))(\( ( 1 + 12 T + 72 T^{2} + 444 T^{3} + 1369 T^{4} )^{2} \))(\( 1 + 10 T^{2} - 1269 T^{4} + 13690 T^{6} + 1874161 T^{8} \))(\( 1 - 12 T + 72 T^{2} - 288 T^{3} + 983 T^{4} - 10656 T^{5} + 98568 T^{6} - 607836 T^{7} + 1874161 T^{8} \))(\( 1 + 12 T + 72 T^{2} + 288 T^{3} + 983 T^{4} + 10656 T^{5} + 98568 T^{6} + 607836 T^{7} + 1874161 T^{8} \))
$41$ (\( 1 + 12 T + 41 T^{2} \))(\( 1 - 2 T + 66 T^{2} - 82 T^{3} + 1681 T^{4} \))(\( ( 1 - 2 T + 41 T^{2} )^{2} \))(\( ( 1 + 10 T + 99 T^{2} + 410 T^{3} + 1681 T^{4} )^{2} \))(\( ( 1 + 8 T^{2} + 1681 T^{4} )^{2} \))(\( ( 1 - 5 T + 41 T^{2} )^{4} \))(\( 1 - 122 T^{2} + 6651 T^{4} - 205082 T^{6} + 2825761 T^{8} \))(\( 1 - 122 T^{2} + 6651 T^{4} - 205082 T^{6} + 2825761 T^{8} \))
$43$ (\( 1 + 10 T + 43 T^{2} \))(\( 1 - 10 T + 94 T^{2} - 430 T^{3} + 1849 T^{4} \))(\( 1 - 70 T^{2} + 1849 T^{4} \))(\( ( 1 - 10 T + 109 T^{2} - 430 T^{3} + 1849 T^{4} )^{2} \))(\( ( 1 + 6 T + 18 T^{2} + 258 T^{3} + 1849 T^{4} )^{2} \))(\( ( 1 - 37 T^{2} + 1849 T^{4} )^{2} \))(\( 1 + 6 T + 18 T^{2} + 60 T^{3} - 889 T^{4} + 2580 T^{5} + 33282 T^{6} + 477042 T^{7} + 3418801 T^{8} \))(\( 1 + 6 T + 18 T^{2} + 60 T^{3} - 889 T^{4} + 2580 T^{5} + 33282 T^{6} + 477042 T^{7} + 3418801 T^{8} \))
$47$ (\( 1 - 9 T + 47 T^{2} \))(\( 1 + 5 T + 62 T^{2} + 235 T^{3} + 2209 T^{4} \))(\( 1 - 85 T^{2} + 2209 T^{4} \))(\( ( 1 + 2 T - 43 T^{2} + 94 T^{3} + 2209 T^{4} )^{2} \))(\( 1 - 2017 T^{4} + 4879681 T^{8} \))(\( ( 1 + 47 T^{2} + 2209 T^{4} )^{2} \))(\( 1 - 18 T + 90 T^{2} + 528 T^{3} - 8377 T^{4} + 24816 T^{5} + 198810 T^{6} - 1868814 T^{7} + 4879681 T^{8} \))(\( 1 - 6 T + 90 T^{2} - 672 T^{3} + 5159 T^{4} - 31584 T^{5} + 198810 T^{6} - 622938 T^{7} + 4879681 T^{8} \))
$53$ (\( 1 - 12 T + 53 T^{2} \))(\( 1 + 2 T + 90 T^{2} + 106 T^{3} + 2809 T^{4} \))(\( 1 - 70 T^{2} + 2809 T^{4} \))(\( 1 - 8 T - 50 T^{2} - 64 T^{3} + 6795 T^{4} - 3392 T^{5} - 140450 T^{6} - 1191016 T^{7} + 7890481 T^{8} \))(\( ( 1 - 2 T + 2 T^{2} - 106 T^{3} + 2809 T^{4} )^{2} \))(\( 1 + 70 T^{2} + 2091 T^{4} + 196630 T^{6} + 7890481 T^{8} \))(\( ( 1 - 14 T + 143 T^{2} - 742 T^{3} + 2809 T^{4} )( 1 + 4 T - 37 T^{2} + 212 T^{3} + 2809 T^{4} ) \))(\( ( 1 - 14 T + 143 T^{2} - 742 T^{3} + 2809 T^{4} )( 1 + 4 T - 37 T^{2} + 212 T^{3} + 2809 T^{4} ) \))
$59$ (\( 1 + 59 T^{2} \))(\( ( 1 + 4 T + 59 T^{2} )^{2} \))(\( ( 1 + 10 T + 59 T^{2} )^{2} \))(\( 1 - 8 T + 2 T^{2} + 448 T^{3} - 3413 T^{4} + 26432 T^{5} + 6962 T^{6} - 1643032 T^{7} + 12117361 T^{8} \))(\( ( 1 + 28 T^{2} + 3481 T^{4} )^{2} \))(\( ( 1 - 10 T + 41 T^{2} - 590 T^{3} + 3481 T^{4} )^{2} \))(\( 1 - 6 T - 64 T^{2} + 108 T^{3} + 4395 T^{4} + 6372 T^{5} - 222784 T^{6} - 1232274 T^{7} + 12117361 T^{8} \))(\( 1 + 6 T - 64 T^{2} - 108 T^{3} + 4395 T^{4} - 6372 T^{5} - 222784 T^{6} + 1232274 T^{7} + 12117361 T^{8} \))
$61$ (\( 1 - 8 T + 61 T^{2} \))(\( 1 - 6 T - 22 T^{2} - 366 T^{3} + 3721 T^{4} \))(\( ( 1 + 8 T + 61 T^{2} )^{2} \))(\( 1 - 6 T - 23 T^{2} + 378 T^{3} - 2436 T^{4} + 23058 T^{5} - 85583 T^{6} - 1361886 T^{7} + 13845841 T^{8} \))(\( ( 1 - 82 T^{2} + 3721 T^{4} )^{2} \))(\( ( 1 + 7 T - 12 T^{2} + 427 T^{3} + 3721 T^{4} )^{2} \))(\( 1 + 12 T + 157 T^{2} + 1308 T^{3} + 11088 T^{4} + 79788 T^{5} + 584197 T^{6} + 2723772 T^{7} + 13845841 T^{8} \))(\( 1 + 12 T + 157 T^{2} + 1308 T^{3} + 11088 T^{4} + 79788 T^{5} + 584197 T^{6} + 2723772 T^{7} + 13845841 T^{8} \))
$67$ (\( 1 + 4 T + 67 T^{2} \))(\( 1 - 4 T + 70 T^{2} - 268 T^{3} + 4489 T^{4} \))(\( 1 - 130 T^{2} + 4489 T^{4} \))(\( 1 + 22 T + 231 T^{2} + 2618 T^{3} + 27092 T^{4} + 175406 T^{5} + 1036959 T^{6} + 6616786 T^{7} + 20151121 T^{8} \))(\( ( 1 + 2 T + 2 T^{2} + 134 T^{3} + 4489 T^{4} )^{2} \))(\( ( 1 - 13 T^{2} + 4489 T^{4} )( 1 + 122 T^{2} + 4489 T^{4} ) \))(\( 1 + 22 T + 137 T^{2} - 834 T^{3} - 16648 T^{4} - 55878 T^{5} + 614993 T^{6} + 6616786 T^{7} + 20151121 T^{8} \))(\( 1 - 8 T + 137 T^{2} - 1224 T^{3} + 11492 T^{4} - 82008 T^{5} + 614993 T^{6} - 2406104 T^{7} + 20151121 T^{8} \))
$71$ (\( 1 + 71 T^{2} \))(\( ( 1 - 8 T + 71 T^{2} )^{2} \))(\( ( 1 + 8 T + 71 T^{2} )^{2} \))(\( ( 1 + 8 T + 86 T^{2} + 568 T^{3} + 5041 T^{4} )^{2} \))(\( ( 1 + 6 T + 71 T^{2} )^{4} \))(\( ( 1 + 2 T + 71 T^{2} )^{4} \))(\( ( 1 - 6 T + 148 T^{2} - 426 T^{3} + 5041 T^{4} )^{2} \))(\( ( 1 - 6 T + 148 T^{2} - 426 T^{3} + 5041 T^{4} )^{2} \))
$73$ (\( 1 - 2 T + 73 T^{2} \))(\( 1 + 8 T + 94 T^{2} + 584 T^{3} + 5329 T^{4} \))(\( ( 1 - 16 T + 73 T^{2} )( 1 + 16 T + 73 T^{2} ) \))(\( 1 + 4 T - 126 T^{2} - 16 T^{3} + 13667 T^{4} - 1168 T^{5} - 671454 T^{6} + 1556068 T^{7} + 28398241 T^{8} \))(\( ( 1 + 5329 T^{4} )^{2} \))(\( ( 1 - 16 T + 183 T^{2} - 1168 T^{3} + 5329 T^{4} )( 1 + 16 T + 183 T^{2} + 1168 T^{3} + 5329 T^{4} ) \))(\( 1 + 24 T + 144 T^{2} - 1752 T^{3} - 31057 T^{4} - 127896 T^{5} + 767376 T^{6} + 9336408 T^{7} + 28398241 T^{8} \))(\( 1 + 144 T^{2} + 600 T^{3} + 10991 T^{4} + 43800 T^{5} + 767376 T^{6} + 28398241 T^{8} \))
$79$ (\( 1 + T + 79 T^{2} \))(\( 1 + 9 T + 174 T^{2} + 711 T^{3} + 6241 T^{4} \))(\( ( 1 - 5 T + 79 T^{2} )^{2} \))(\( 1 + 24 T + 282 T^{2} + 3264 T^{3} + 34691 T^{4} + 257856 T^{5} + 1759962 T^{6} + 11832936 T^{7} + 38950081 T^{8} \))(\( ( 1 + 11 T^{2} + 6241 T^{4} )^{2} \))(\( ( 1 + 2 T - 75 T^{2} + 158 T^{3} + 6241 T^{4} )^{2} \))(\( 1 - 6 T + 148 T^{2} - 816 T^{3} + 13203 T^{4} - 64464 T^{5} + 923668 T^{6} - 2958234 T^{7} + 38950081 T^{8} \))(\( 1 + 6 T + 148 T^{2} + 816 T^{3} + 13203 T^{4} + 64464 T^{5} + 923668 T^{6} + 2958234 T^{7} + 38950081 T^{8} \))
$83$ (\( 1 - 12 T + 83 T^{2} \))(\( ( 1 - 4 T + 83 T^{2} )^{2} \))(\( 1 - 150 T^{2} + 6889 T^{4} \))(\( ( 1 - 2 T + 5 T^{2} - 166 T^{3} + 6889 T^{4} )^{2} \))(\( 1 + 7538 T^{4} + 47458321 T^{8} \))(\( ( 1 - 45 T^{2} + 6889 T^{4} )^{2} \))(\( 1 + 2 T + 2 T^{2} + 140 T^{3} + 9631 T^{4} + 11620 T^{5} + 13778 T^{6} + 1143574 T^{7} + 47458321 T^{8} \))(\( 1 - 2 T + 2 T^{2} - 140 T^{3} + 9631 T^{4} - 11620 T^{5} + 13778 T^{6} - 1143574 T^{7} + 47458321 T^{8} \))
$89$ (\( 1 + 12 T + 89 T^{2} \))(\( 1 - 6 T + 170 T^{2} - 534 T^{3} + 7921 T^{4} \))(\( ( 1 + 89 T^{2} )^{2} \))(\( 1 - 6 T - 119 T^{2} + 138 T^{3} + 12900 T^{4} + 12282 T^{5} - 942599 T^{6} - 4229814 T^{7} + 62742241 T^{8} \))(\( ( 1 + 138 T^{2} + 7921 T^{4} )^{2} \))(\( ( 1 - 9 T - 8 T^{2} - 801 T^{3} + 7921 T^{4} )^{2} \))(\( ( 1 + 16 T + 89 T^{2} )^{2}( 1 - 16 T + 167 T^{2} - 1424 T^{3} + 7921 T^{4} ) \))(\( ( 1 - 16 T + 89 T^{2} )^{2}( 1 + 16 T + 167 T^{2} + 1424 T^{3} + 7921 T^{4} ) \))
$97$ (\( 1 + T + 97 T^{2} \))(\( 1 + 9 T + 108 T^{2} + 873 T^{3} + 9409 T^{4} \))(\( 1 - 145 T^{2} + 9409 T^{4} \))(\( ( 1 - 12 T + 198 T^{2} - 1164 T^{3} + 9409 T^{4} )^{2} \))(\( 1 + 16903 T^{4} + 88529281 T^{8} \))(\( ( 1 + 62 T^{2} + 9409 T^{4} )^{2} \))(\( 1 + 4 T + 8 T^{2} + 12 T^{3} - 8818 T^{4} + 1164 T^{5} + 75272 T^{6} + 3650692 T^{7} + 88529281 T^{8} \))(\( 1 - 4 T + 8 T^{2} - 12 T^{3} - 8818 T^{4} - 1164 T^{5} + 75272 T^{6} - 3650692 T^{7} + 88529281 T^{8} \))
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