Properties

Label 43.2
Level 43
Weight 2
Dimension 57
Nonzero newspaces 4
Newform subspaces 7
Sturm bound 308
Trace bound 1

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Defining parameters

Level: \( N \) = \( 43 \)
Weight: \( k \) = \( 2 \)
Nonzero newspaces: \( 4 \)
Newform subspaces: \( 7 \)
Sturm bound: \(308\)
Trace bound: \(1\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_1(43))\).

Total New Old
Modular forms 98 98 0
Cusp forms 57 57 0
Eisenstein series 41 41 0

Trace form

\( 57q - 18q^{2} - 17q^{3} - 14q^{4} - 15q^{5} - 9q^{6} - 13q^{7} - 6q^{8} - 8q^{9} + O(q^{10}) \) \( 57q - 18q^{2} - 17q^{3} - 14q^{4} - 15q^{5} - 9q^{6} - 13q^{7} - 6q^{8} - 8q^{9} - 3q^{10} - 9q^{11} + 7q^{12} - 7q^{13} + 3q^{14} + 3q^{15} + 10q^{16} - 3q^{17} + 18q^{18} - q^{19} + 21q^{20} + 11q^{21} + 15q^{22} + 3q^{23} + 39q^{24} + 10q^{25} + 21q^{26} + 19q^{27} + 35q^{28} + 9q^{29} + 51q^{30} + 4q^{31} - 15q^{33} - 30q^{34} - 15q^{35} - 42q^{36} - 25q^{37} - 45q^{38} - 14q^{39} - 99q^{40} - 30q^{42} - 83q^{43} - 42q^{44} - 48q^{45} - 33q^{46} + 6q^{47} - 65q^{48} - 13q^{49} - 12q^{50} + 9q^{51} - 35q^{52} - 9q^{53} + 36q^{54} + 9q^{55} + 57q^{56} + 52q^{57} + 69q^{58} + 39q^{59} + 147q^{60} + 41q^{61} + 75q^{62} + 83q^{63} + 106q^{64} + 63q^{65} + 123q^{66} + 47q^{67} + 105q^{68} + 33q^{69} + 39q^{70} + 9q^{71} - 36q^{72} + 11q^{73} - 54q^{74} - 44q^{75} - 28q^{76} - 51q^{77} - 126q^{78} - 25q^{79} - 3q^{80} - 68q^{81} - 105q^{82} - 21q^{83} - 217q^{84} - 18q^{85} - 60q^{86} - 132q^{87} - 51q^{88} - 15q^{89} - 165q^{90} + 7q^{91} - 63q^{92} - 61q^{93} - 45q^{94} + 15q^{95} - 42q^{96} - 49q^{97} + 3q^{98} - 12q^{99} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_1(43))\)

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
43.2.a \(\chi_{43}(1, \cdot)\) 43.2.a.a 1 1
43.2.a.b 2
43.2.c \(\chi_{43}(6, \cdot)\) 43.2.c.a 2 2
43.2.c.b 4
43.2.e \(\chi_{43}(4, \cdot)\) 43.2.e.a 6 6
43.2.e.b 6
43.2.g \(\chi_{43}(9, \cdot)\) 43.2.g.a 36 12

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ (\( 1 + 2 T + 2 T^{2} \))(\( 1 + 2 T^{2} + 4 T^{4} \))(\( ( 1 - T + 2 T^{2} )^{2} \))(\( ( 1 + 3 T + 5 T^{2} + 6 T^{3} + 4 T^{4} )^{2} \))(\( 1 + 5 T + 9 T^{2} + 7 T^{3} + 3 T^{4} + T^{5} - T^{6} + 2 T^{7} + 12 T^{8} + 56 T^{9} + 144 T^{10} + 160 T^{11} + 64 T^{12} \))(\( 1 - 3 T + 7 T^{2} - 15 T^{3} + 31 T^{4} - 49 T^{5} + 71 T^{6} - 98 T^{7} + 124 T^{8} - 120 T^{9} + 112 T^{10} - 96 T^{11} + 64 T^{12} \))
$3$ (\( 1 + 2 T + 3 T^{2} \))(\( 1 + 4 T^{2} + 9 T^{4} \))(\( 1 + T - 2 T^{2} + 3 T^{3} + 9 T^{4} \))(\( 1 - 3 T + 2 T^{2} - 3 T^{3} + 13 T^{4} - 9 T^{5} + 18 T^{6} - 81 T^{7} + 81 T^{8} \))(\( 1 - 2 T + T^{2} + 4 T^{3} + 3 T^{4} - 4 T^{5} + 13 T^{6} - 12 T^{7} + 27 T^{8} + 108 T^{9} + 81 T^{10} - 486 T^{11} + 729 T^{12} \))(\( 1 + 3 T + 6 T^{2} + 2 T^{3} - 5 T^{4} - 21 T^{5} - 20 T^{6} - 63 T^{7} - 45 T^{8} + 54 T^{9} + 486 T^{10} + 729 T^{11} + 729 T^{12} \))
$5$ (\( 1 + 4 T + 5 T^{2} \))(\( 1 - 4 T + 12 T^{2} - 20 T^{3} + 25 T^{4} \))(\( 1 - T - 4 T^{2} - 5 T^{3} + 25 T^{4} \))(\( 1 - 2 T - 2 T^{2} + 8 T^{3} - 9 T^{4} + 40 T^{5} - 50 T^{6} - 250 T^{7} + 625 T^{8} \))(\( 1 + 3 T + 4 T^{2} + 18 T^{3} + 76 T^{4} + 145 T^{5} + 251 T^{6} + 725 T^{7} + 1900 T^{8} + 2250 T^{9} + 2500 T^{10} + 9375 T^{11} + 15625 T^{12} \))(\( 1 - 2 T - T^{2} + 12 T^{3} - 19 T^{4} - 22 T^{5} + 139 T^{6} - 110 T^{7} - 475 T^{8} + 1500 T^{9} - 625 T^{10} - 6250 T^{11} + 15625 T^{12} \))
$7$ (\( 1 + 7 T^{2} \))(\( 1 + 4 T + 16 T^{2} + 28 T^{3} + 49 T^{4} \))(\( 1 + 3 T + 2 T^{2} + 21 T^{3} + 49 T^{4} \))(\( 1 - 4 T + 3 T^{2} + 4 T^{3} + 8 T^{4} + 28 T^{5} + 147 T^{6} - 1372 T^{7} + 2401 T^{8} \))(\( ( 1 + 14 T^{2} + 7 T^{3} + 98 T^{4} + 343 T^{6} )^{2} \))(\( ( 1 + 8 T + 40 T^{2} + 125 T^{3} + 280 T^{4} + 392 T^{5} + 343 T^{6} )^{2} \))
$11$ (\( 1 - 3 T + 11 T^{2} \))(\( 1 + 2 T + 15 T^{2} + 22 T^{3} + 121 T^{4} \))(\( ( 1 + 11 T^{2} )^{2} \))(\( ( 1 + 5 T + 27 T^{2} + 55 T^{3} + 121 T^{4} )^{2} \))(\( 1 + 10 T + 26 T^{2} - 81 T^{3} - 529 T^{4} - 199 T^{5} + 3535 T^{6} - 2189 T^{7} - 64009 T^{8} - 107811 T^{9} + 380666 T^{10} + 1610510 T^{11} + 1771561 T^{12} \))(\( 1 - 14 T + 80 T^{2} - 231 T^{3} + 149 T^{4} + 2135 T^{5} - 11369 T^{6} + 23485 T^{7} + 18029 T^{8} - 307461 T^{9} + 1171280 T^{10} - 2254714 T^{11} + 1771561 T^{12} \))
$13$ (\( 1 + 5 T + 13 T^{2} \))(\( 1 - 2 T + 19 T^{2} - 26 T^{3} + 169 T^{4} \))(\( ( 1 - 7 T + 13 T^{2} )( 1 + 2 T + 13 T^{2} ) \))(\( 1 + 5 T - 6 T^{2} + 25 T^{3} + 467 T^{4} + 325 T^{5} - 1014 T^{6} + 10985 T^{7} + 28561 T^{8} \))(\( 1 - 5 T + 40 T^{2} - 212 T^{3} + 953 T^{4} - 4319 T^{5} + 14876 T^{6} - 56147 T^{7} + 161057 T^{8} - 465764 T^{9} + 1142440 T^{10} - 1856465 T^{11} + 4826809 T^{12} \))(\( 1 + 9 T + 12 T^{2} - 142 T^{3} - 524 T^{4} + 1141 T^{5} + 11103 T^{6} + 14833 T^{7} - 88556 T^{8} - 311974 T^{9} + 342732 T^{10} + 3341637 T^{11} + 4826809 T^{12} \))
$17$ (\( 1 + 3 T + 17 T^{2} \))(\( 1 - 10 T + 51 T^{2} - 170 T^{3} + 289 T^{4} \))(\( 1 + 3 T - 8 T^{2} + 51 T^{3} + 289 T^{4} \))(\( 1 + T - 2 T^{2} - 31 T^{3} - 297 T^{4} - 527 T^{5} - 578 T^{6} + 4913 T^{7} + 83521 T^{8} \))(\( 1 + 4 T - T^{2} - 72 T^{3} + 93 T^{4} - 42 T^{5} - 293 T^{6} - 714 T^{7} + 26877 T^{8} - 353736 T^{9} - 83521 T^{10} + 5679428 T^{11} + 24137569 T^{12} \))(\( 1 - 8 T + 19 T^{2} - 30 T^{3} - 55 T^{4} + 2588 T^{5} - 16885 T^{6} + 43996 T^{7} - 15895 T^{8} - 147390 T^{9} + 1586899 T^{10} - 11358856 T^{11} + 24137569 T^{12} \))
$19$ (\( 1 + 2 T + 19 T^{2} \))(\( 1 + 4 T + 34 T^{2} + 76 T^{3} + 361 T^{4} \))(\( ( 1 - 7 T + 19 T^{2} )( 1 + 8 T + 19 T^{2} ) \))(\( 1 + 2 T - 30 T^{2} - 8 T^{3} + 719 T^{4} - 152 T^{5} - 10830 T^{6} + 13718 T^{7} + 130321 T^{8} \))(\( 1 - 2 T + 13 T^{2} + 40 T^{3} + 23 T^{4} + 1756 T^{5} - 1723 T^{6} + 33364 T^{7} + 8303 T^{8} + 274360 T^{9} + 1694173 T^{10} - 4952198 T^{11} + 47045881 T^{12} \))(\( 1 + 4 T - 24 T^{2} - 207 T^{3} - 659 T^{4} + 2459 T^{5} + 28923 T^{6} + 46721 T^{7} - 237899 T^{8} - 1419813 T^{9} - 3127704 T^{10} + 9904396 T^{11} + 47045881 T^{12} \))
$23$ (\( 1 + T + 23 T^{2} \))(\( 1 - 2 T + 15 T^{2} - 46 T^{3} + 529 T^{4} \))(\( 1 - 7 T + 26 T^{2} - 161 T^{3} + 529 T^{4} \))(\( 1 + 11 T + 46 T^{2} + 319 T^{3} + 2313 T^{4} + 7337 T^{5} + 24334 T^{6} + 133837 T^{7} + 279841 T^{8} \))(\( 1 - T - 8 T^{2} + 255 T^{3} - 57 T^{4} - 1090 T^{5} + 35133 T^{6} - 25070 T^{7} - 30153 T^{8} + 3102585 T^{9} - 2238728 T^{10} - 6436343 T^{11} + 148035889 T^{12} \))(\( 1 - T - 8 T^{2} - 60 T^{3} - 169 T^{4} + 1465 T^{5} - 1792 T^{6} + 33695 T^{7} - 89401 T^{8} - 730020 T^{9} - 2238728 T^{10} - 6436343 T^{11} + 148035889 T^{12} \))
$29$ (\( 1 + 6 T + 29 T^{2} \))(\( 1 + 40 T^{2} + 841 T^{4} \))(\( 1 + 3 T - 20 T^{2} + 87 T^{3} + 841 T^{4} \))(\( ( 1 + 3 T - 20 T^{2} + 87 T^{3} + 841 T^{4} )^{2} \))(\( 1 - 6 T + 7 T^{2} + 132 T^{3} - 995 T^{4} + 2142 T^{5} + 16003 T^{6} + 62118 T^{7} - 836795 T^{8} + 3219348 T^{9} + 4950967 T^{10} - 123066894 T^{11} + 594823321 T^{12} \))(\( 1 - 9 T + 24 T^{2} - 242 T^{3} + 1881 T^{4} - 5333 T^{5} + 16380 T^{6} - 154657 T^{7} + 1581921 T^{8} - 5902138 T^{9} + 16974744 T^{10} - 184600341 T^{11} + 594823321 T^{12} \))
$31$ (\( 1 + T + 31 T^{2} \))(\( ( 1 + 3 T + 31 T^{2} )^{2} \))(\( 1 + 5 T - 6 T^{2} + 155 T^{3} + 961 T^{4} \))(\( ( 1 - 31 T^{2} + 961 T^{4} )^{2} \))(\( 1 + 6 T - 23 T^{2} - 478 T^{3} - 1763 T^{4} + 9154 T^{5} + 96655 T^{6} + 283774 T^{7} - 1694243 T^{8} - 14240098 T^{9} - 21240983 T^{10} + 171774906 T^{11} + 887503681 T^{12} \))(\( 1 + 18 T + 125 T^{2} + 572 T^{3} + 3467 T^{4} + 20608 T^{5} + 104889 T^{6} + 638848 T^{7} + 3331787 T^{8} + 17040452 T^{9} + 115440125 T^{10} + 515324718 T^{11} + 887503681 T^{12} \))
$37$ (\( 1 + 37 T^{2} \))(\( 1 + 2 T^{2} + 1369 T^{4} \))(\( 1 - 9 T + 44 T^{2} - 333 T^{3} + 1369 T^{4} \))(\( 1 - 3 T - 56 T^{2} + 27 T^{3} + 2523 T^{4} + 999 T^{5} - 76664 T^{6} - 151959 T^{7} + 1874161 T^{8} \))(\( ( 1 + 7 T + 97 T^{2} + 427 T^{3} + 3589 T^{4} + 9583 T^{5} + 50653 T^{6} )^{2} \))(\( ( 1 + 2 T + 26 T^{2} + 399 T^{3} + 962 T^{4} + 2738 T^{5} + 50653 T^{6} )^{2} \))
$41$ (\( 1 - 5 T + 41 T^{2} \))(\( 1 + 2 T + 75 T^{2} + 82 T^{3} + 1681 T^{4} \))(\( ( 1 + 10 T + 41 T^{2} )^{2} \))(\( ( 1 - 10 T + 87 T^{2} - 410 T^{3} + 1681 T^{4} )^{2} \))(\( 1 - 2 T + 19 T^{2} + 30 T^{3} - 307 T^{4} - 4018 T^{5} + 60803 T^{6} - 164738 T^{7} - 516067 T^{8} + 2067630 T^{9} + 53689459 T^{10} - 231712402 T^{11} + 4750104241 T^{12} \))(\( 1 - 23 T + 208 T^{2} - 1146 T^{3} + 6840 T^{4} - 44849 T^{5} + 273267 T^{6} - 1838809 T^{7} + 11498040 T^{8} - 78983466 T^{9} + 587758288 T^{10} - 2664692623 T^{11} + 4750104241 T^{12} \))
$43$ (\( 1 + T \))(\( ( 1 - T )^{2} \))(\( 1 + 8 T + 43 T^{2} \))(\( ( 1 + 13 T + 43 T^{2} )^{2} \))(\( 1 + 13 T + 141 T^{2} + 895 T^{3} + 6063 T^{4} + 24037 T^{5} + 79507 T^{6} \))(\( 1 + 29 T + 393 T^{2} + 3221 T^{3} + 16899 T^{4} + 53621 T^{5} + 79507 T^{6} \))
$47$ (\( 1 - 4 T + 47 T^{2} \))(\( ( 1 - 6 T + 47 T^{2} )^{2} \))(\( ( 1 + 8 T + 47 T^{2} )^{2} \))(\( ( 1 - 9 T + 103 T^{2} - 423 T^{3} + 2209 T^{4} )^{2} \))(\( 1 - 17 T + 158 T^{2} - 816 T^{3} + 3471 T^{4} - 14523 T^{5} + 108772 T^{6} - 682581 T^{7} + 7667439 T^{8} - 84719568 T^{9} + 770989598 T^{10} - 3898865119 T^{11} + 10779215329 T^{12} \))(\( 1 - T - 46 T^{2} + 93 T^{3} + 2069 T^{4} - 6440 T^{5} - 90803 T^{6} - 302680 T^{7} + 4570421 T^{8} + 9655539 T^{9} - 224465326 T^{10} - 229345007 T^{11} + 10779215329 T^{12} \))
$53$ (\( 1 + 5 T + 53 T^{2} \))(\( 1 - 22 T + 219 T^{2} - 1166 T^{3} + 2809 T^{4} \))(\( 1 - 5 T - 28 T^{2} - 265 T^{3} + 2809 T^{4} \))(\( 1 - 5 T - 86 T^{2} - 25 T^{3} + 8187 T^{4} - 1325 T^{5} - 241574 T^{6} - 744385 T^{7} + 7890481 T^{8} \))(\( 1 + 2 T + 56 T^{2} + 531 T^{3} + 6599 T^{4} + 27685 T^{5} + 387913 T^{6} + 1467305 T^{7} + 18536591 T^{8} + 79053687 T^{9} + 441866936 T^{10} + 836390986 T^{11} + 22164361129 T^{12} \))(\( 1 + 10 T + 47 T^{2} + 528 T^{3} + 8081 T^{4} + 52826 T^{5} + 297535 T^{6} + 2799778 T^{7} + 22699529 T^{8} + 78607056 T^{9} + 370852607 T^{10} + 4181954930 T^{11} + 22164361129 T^{12} \))
$59$ (\( 1 + 12 T + 59 T^{2} \))(\( 1 + 4 T + 114 T^{2} + 236 T^{3} + 3481 T^{4} \))(\( ( 1 - 12 T + 59 T^{2} )^{2} \))(\( ( 1 + T + 87 T^{2} + 59 T^{3} + 3481 T^{4} )^{2} \))(\( 1 - 12 T - 20 T^{2} + 1053 T^{3} - 3791 T^{4} - 30145 T^{5} + 385979 T^{6} - 1778555 T^{7} - 13196471 T^{8} + 216264087 T^{9} - 242347220 T^{10} - 8579091588 T^{11} + 42180533641 T^{12} \))(\( 1 - 22 T + 152 T^{2} + 201 T^{3} - 5011 T^{4} - 58697 T^{5} + 1026535 T^{6} - 3463123 T^{7} - 17443291 T^{8} + 41281179 T^{9} + 1841838872 T^{10} - 15728334578 T^{11} + 42180533641 T^{12} \))
$61$ (\( 1 - 2 T + 61 T^{2} \))(\( 1 - 8 T + 120 T^{2} - 488 T^{3} + 3721 T^{4} \))(\( ( 1 - 14 T + 61 T^{2} )( 1 + T + 61 T^{2} ) \))(\( 1 - T - 110 T^{2} + 11 T^{3} + 8539 T^{4} + 671 T^{5} - 409310 T^{6} - 226981 T^{7} + 13845841 T^{8} \))(\( 1 - 12 T + 27 T^{2} - 404 T^{3} + 3229 T^{4} + 16766 T^{5} - 226185 T^{6} + 1022726 T^{7} + 12015109 T^{8} - 91700324 T^{9} + 373837707 T^{10} - 10135155612 T^{11} + 51520374361 T^{12} \))(\( 1 - 19 T + 90 T^{2} + 436 T^{3} - 5311 T^{4} + 40503 T^{5} - 386744 T^{6} + 2470683 T^{7} - 19762231 T^{8} + 98963716 T^{9} + 1246125690 T^{10} - 16047329719 T^{11} + 51520374361 T^{12} \))
$67$ (\( 1 + 3 T + 67 T^{2} \))(\( 1 - 2 T + 63 T^{2} - 134 T^{3} + 4489 T^{4} \))(\( 1 - 3 T - 58 T^{2} - 201 T^{3} + 4489 T^{4} \))(\( 1 - T - 122 T^{2} + 11 T^{3} + 10573 T^{4} + 737 T^{5} - 547658 T^{6} - 300763 T^{7} + 20151121 T^{8} \))(\( 1 + 18 T + 117 T^{2} + 704 T^{3} + 12085 T^{4} + 129244 T^{5} + 998865 T^{6} + 8659348 T^{7} + 54249565 T^{8} + 211737152 T^{9} + 2357681157 T^{10} + 24302251926 T^{11} + 90458382169 T^{12} \))(\( 1 + 4 T - 30 T^{2} - 745 T^{3} + 2831 T^{4} + 213 T^{5} + 116347 T^{6} + 14271 T^{7} + 12708359 T^{8} - 224068435 T^{9} - 604533630 T^{10} + 5400500428 T^{11} + 90458382169 T^{12} \))
$71$ (\( 1 - 2 T + 71 T^{2} \))(\( 1 + 12 T + 170 T^{2} + 852 T^{3} + 5041 T^{4} \))(\( 1 - T - 70 T^{2} - 71 T^{3} + 5041 T^{4} \))(\( 1 + 3 T + 16 T^{2} - 447 T^{3} - 5631 T^{4} - 31737 T^{5} + 80656 T^{6} + 1073733 T^{7} + 25411681 T^{8} \))(\( 1 - 26 T + 101 T^{2} + 3378 T^{3} - 35947 T^{4} - 149374 T^{5} + 4056325 T^{6} - 10605554 T^{7} - 181208827 T^{8} + 1209023358 T^{9} + 2566579781 T^{10} - 46909963126 T^{11} + 128100283921 T^{12} \))(\( 1 - 28 T + 293 T^{2} - 1470 T^{3} + 953 T^{4} + 112168 T^{5} - 1557515 T^{6} + 7963928 T^{7} + 4804073 T^{8} - 526129170 T^{9} + 7445622533 T^{10} - 50518421828 T^{11} + 128100283921 T^{12} \))
$73$ (\( 1 - 2 T + 73 T^{2} \))(\( 1 + 24 T + 272 T^{2} + 1752 T^{3} + 5329 T^{4} \))(\( 1 + 11 T + 48 T^{2} + 803 T^{3} + 5329 T^{4} \))(\( 1 - 3 T - 128 T^{2} + 27 T^{3} + 12783 T^{4} + 1971 T^{5} - 682112 T^{6} - 1167051 T^{7} + 28398241 T^{8} \))(\( 1 + 9 T - 6 T^{2} + 430 T^{3} - 1187 T^{4} - 23719 T^{5} + 416688 T^{6} - 1731487 T^{7} - 6325523 T^{8} + 167277310 T^{9} - 170389446 T^{10} + 18657644337 T^{11} + 151334226289 T^{12} \))(\( 1 - 21 T + 277 T^{2} - 3395 T^{3} + 42226 T^{4} - 408520 T^{5} + 3543793 T^{6} - 29821960 T^{7} + 225022354 T^{8} - 1320712715 T^{9} + 7866312757 T^{10} - 43534503453 T^{11} + 151334226289 T^{12} \))
$79$ (\( 1 + 8 T + 79 T^{2} \))(\( 1 - 4 T + 154 T^{2} - 316 T^{3} + 6241 T^{4} \))(\( 1 - 5 T - 54 T^{2} - 395 T^{3} + 6241 T^{4} \))(\( 1 - 5 T - 138 T^{2} - 25 T^{3} + 18353 T^{4} - 1975 T^{5} - 861258 T^{6} - 2465195 T^{7} + 38950081 T^{8} \))(\( ( 1 + 10 T + 254 T^{2} + 1581 T^{3} + 20066 T^{4} + 62410 T^{5} + 493039 T^{6} )^{2} \))(\( ( 1 - 3 T + 212 T^{2} - 391 T^{3} + 16748 T^{4} - 18723 T^{5} + 493039 T^{6} )^{2} \))
$83$ (\( 1 - 15 T + 83 T^{2} \))(\( 1 - 18 T + 215 T^{2} - 1494 T^{3} + 6889 T^{4} \))(\( 1 + 9 T - 2 T^{2} + 747 T^{3} + 6889 T^{4} \))(\( 1 + 3 T + 52 T^{2} - 627 T^{3} - 5787 T^{4} - 52041 T^{5} + 358228 T^{6} + 1715361 T^{7} + 47458321 T^{8} \))(\( 1 - 20 T + 72 T^{2} + 1865 T^{3} - 18531 T^{4} - 47725 T^{5} + 1743083 T^{6} - 3961175 T^{7} - 127660059 T^{8} + 1066382755 T^{9} + 3416999112 T^{10} - 78780812860 T^{11} + 326940373369 T^{12} \))(\( 1 + 39 T + 640 T^{2} + 6519 T^{3} + 67141 T^{4} + 848516 T^{5} + 9110429 T^{6} + 70426828 T^{7} + 462534349 T^{8} + 3727479453 T^{9} + 30373325440 T^{10} + 153622585077 T^{11} + 326940373369 T^{12} \))
$89$ (\( 1 + 4 T + 89 T^{2} \))(\( 1 + 12 T + 196 T^{2} + 1068 T^{3} + 7921 T^{4} \))(\( 1 - T - 88 T^{2} - 89 T^{3} + 7921 T^{4} \))(\( 1 + 3 T - 160 T^{2} - 27 T^{3} + 19839 T^{4} - 2403 T^{5} - 1267360 T^{6} + 2114907 T^{7} + 62742241 T^{8} \))(\( 1 - 11 T + 81 T^{2} + 431 T^{3} + 2694 T^{4} - 97288 T^{5} + 1979249 T^{6} - 8658632 T^{7} + 21339174 T^{8} + 303841639 T^{9} + 5082121521 T^{10} - 61424653939 T^{11} + 496981290961 T^{12} \))(\( 1 - 11 T - 143 T^{2} + 3063 T^{3} - 12244 T^{4} - 185572 T^{5} + 3036725 T^{6} - 16515908 T^{7} - 96984724 T^{8} + 2159320047 T^{9} - 8972140463 T^{10} - 61424653939 T^{11} + 496981290961 T^{12} \))
$97$ (\( 1 - 7 T + 97 T^{2} \))(\( 1 + 2 T + 187 T^{2} + 194 T^{3} + 9409 T^{4} \))(\( ( 1 + 2 T + 97 T^{2} )^{2} \))(\( ( 1 + 14 T + 238 T^{2} + 1358 T^{3} + 9409 T^{4} )^{2} \))(\( 1 - 28 T + 449 T^{2} - 5950 T^{3} + 80095 T^{4} - 914662 T^{5} + 9384215 T^{6} - 88722214 T^{7} + 753613855 T^{8} - 5430404350 T^{9} + 39749647169 T^{10} - 240445527196 T^{11} + 832972004929 T^{12} \))(\( 1 + 19 T + 145 T^{2} + 793 T^{3} + 5720 T^{4} - 26012 T^{5} - 1115155 T^{6} - 2523164 T^{7} + 53819480 T^{8} + 723749689 T^{9} + 12836745745 T^{10} + 163159464883 T^{11} + 832972004929 T^{12} \))
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