Properties

Label 45.2
Level 45
Weight 2
Dimension 39
Nonzero newspaces 6
Newform subspaces 7
Sturm bound 288
Trace bound 2

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Defining parameters

Level: \( N \) = \( 45 = 3^{2} \cdot 5 \)
Weight: \( k \) = \( 2 \)
Nonzero newspaces: \( 6 \)
Newform subspaces: \( 7 \)
Sturm bound: \(288\)
Trace bound: \(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_1(45))\).

Total New Old
Modular forms 104 65 39
Cusp forms 41 39 2
Eisenstein series 63 26 37

Trace form

\( 39q - 7q^{2} - 8q^{3} - 11q^{4} - 9q^{5} - 16q^{6} - 12q^{7} - 3q^{8} - 4q^{9} + O(q^{10}) \) \( 39q - 7q^{2} - 8q^{3} - 11q^{4} - 9q^{5} - 16q^{6} - 12q^{7} - 3q^{8} - 4q^{9} - 15q^{10} - 4q^{11} + 8q^{12} - 2q^{13} + 12q^{14} + 4q^{15} - 7q^{16} + 2q^{17} + 16q^{18} - 20q^{19} + 25q^{20} + 8q^{22} + 12q^{23} + 36q^{24} + 3q^{25} + 14q^{26} + 16q^{27} + 8q^{28} + 22q^{29} + 32q^{30} - 8q^{31} + 13q^{32} + 8q^{33} + 14q^{34} - 8q^{35} - 32q^{36} + 2q^{37} - 16q^{38} - 32q^{39} - 23q^{40} - 50q^{41} - 36q^{42} - 32q^{43} - 68q^{44} - 28q^{45} - 40q^{46} - 40q^{47} - 40q^{48} + 3q^{49} - 31q^{50} - 16q^{51} + 2q^{52} + 2q^{53} - 4q^{54} + 12q^{55} + 24q^{56} + 16q^{57} + 26q^{58} + 44q^{59} + 20q^{60} + 26q^{61} + 84q^{62} + 60q^{63} + 49q^{64} + 62q^{65} + 88q^{66} + 24q^{67} + 70q^{68} + 60q^{69} + 60q^{70} + 64q^{71} + 24q^{72} - 2q^{73} - 2q^{74} - 8q^{75} + 36q^{76} - 12q^{77} - 40q^{78} - 12q^{79} - 35q^{80} - 40q^{81} - 30q^{82} - 72q^{83} - 72q^{84} - 6q^{85} - 100q^{86} - 44q^{87} - 66q^{89} - 104q^{90} - 48q^{91} - 96q^{92} - 60q^{93} - 4q^{94} - 40q^{95} - 56q^{96} + 2q^{97} - 83q^{98} - 20q^{99} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_1(45))\)

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
45.2.a \(\chi_{45}(1, \cdot)\) 45.2.a.a 1 1
45.2.b \(\chi_{45}(19, \cdot)\) 45.2.b.a 2 1
45.2.e \(\chi_{45}(16, \cdot)\) 45.2.e.a 2 2
45.2.e.b 6
45.2.f \(\chi_{45}(8, \cdot)\) 45.2.f.a 4 2
45.2.j \(\chi_{45}(4, \cdot)\) 45.2.j.a 8 2
45.2.l \(\chi_{45}(2, \cdot)\) 45.2.l.a 16 4

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_1(45))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_1(45)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_1(15))\)\(^{\oplus 2}\)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ (\( 1 - T + 2 T^{2} \))(\( 1 + T^{2} + 4 T^{4} \))(\( 1 + T - T^{2} + 2 T^{3} + 4 T^{4} \))(\( 1 + T - T^{3} - 2 T^{4} + T^{5} + 11 T^{6} + 2 T^{7} - 8 T^{8} - 8 T^{9} + 32 T^{11} + 64 T^{12} \))(\( 1 + T^{4} + 16 T^{8} \))(\( 1 + 4 T^{2} + 7 T^{4} + 4 T^{6} + T^{8} + 16 T^{10} + 112 T^{12} + 256 T^{14} + 256 T^{16} \))(\( 1 + 6 T + 18 T^{2} + 36 T^{3} + 58 T^{4} + 78 T^{5} + 72 T^{6} - 149 T^{8} - 354 T^{9} - 576 T^{10} - 732 T^{11} - 530 T^{12} + 480 T^{13} + 2592 T^{14} + 5496 T^{15} + 8641 T^{16} + 10992 T^{17} + 10368 T^{18} + 3840 T^{19} - 8480 T^{20} - 23424 T^{21} - 36864 T^{22} - 45312 T^{23} - 38144 T^{24} + 73728 T^{26} + 159744 T^{27} + 237568 T^{28} + 294912 T^{29} + 294912 T^{30} + 196608 T^{31} + 65536 T^{32} \))
$3$ (\( \))(\( \))(\( 1 + 3 T + 3 T^{2} \))(\( 1 - T + 4 T^{2} - 3 T^{3} + 12 T^{4} - 9 T^{5} + 27 T^{6} \))(\( \))(\( 1 - 9 T^{4} + 81 T^{8} \))(\( 1 + 6 T + 18 T^{2} + 30 T^{3} + 30 T^{4} + 36 T^{5} + 126 T^{6} + 396 T^{7} + 819 T^{8} + 1188 T^{9} + 1134 T^{10} + 972 T^{11} + 2430 T^{12} + 7290 T^{13} + 13122 T^{14} + 13122 T^{15} + 6561 T^{16} \))
$5$ (\( 1 + T \))(\( 1 + 5 T^{2} \))(\( 1 - T + T^{2} \))(\( ( 1 + T + T^{2} )^{3} \))(\( 1 - 8 T^{2} + 25 T^{4} \))(\( 1 + 2 T^{2} - 21 T^{4} + 50 T^{6} + 625 T^{8} \))(\( 1 + 6 T + 16 T^{2} + 24 T^{3} + 22 T^{4} + 30 T^{5} + 136 T^{6} + 630 T^{7} + 1831 T^{8} + 3150 T^{9} + 3400 T^{10} + 3750 T^{11} + 13750 T^{12} + 75000 T^{13} + 250000 T^{14} + 468750 T^{15} + 390625 T^{16} \))
$7$ (\( 1 + 7 T^{2} \))(\( ( 1 - 7 T^{2} )^{2} \))(\( 1 - 3 T + 2 T^{2} - 21 T^{3} + 49 T^{4} \))(\( 1 + 5 T + T^{2} - 14 T^{3} + 73 T^{4} + 173 T^{5} - 82 T^{6} + 1211 T^{7} + 3577 T^{8} - 4802 T^{9} + 2401 T^{10} + 84035 T^{11} + 117649 T^{12} \))(\( ( 1 + 4 T + 8 T^{2} + 28 T^{3} + 49 T^{4} )^{2} \))(\( 1 + 16 T^{2} + 121 T^{4} + 592 T^{6} + 3280 T^{8} + 29008 T^{10} + 290521 T^{12} + 1882384 T^{14} + 5764801 T^{16} \))(\( 1 + 2 T + 2 T^{2} - 16 T^{3} - 66 T^{4} - 104 T^{5} + 52 T^{6} + 1072 T^{7} - 235 T^{8} + 2614 T^{9} - 2614 T^{10} + 626 T^{11} + 8430 T^{12} - 20146 T^{13} + 75264 T^{14} - 1168356 T^{15} + 4373152 T^{16} - 8178492 T^{17} + 3687936 T^{18} - 6910078 T^{19} + 20240430 T^{20} + 10521182 T^{21} - 307534486 T^{22} + 2152741402 T^{23} - 1354728235 T^{24} + 43259066704 T^{25} + 14688712948 T^{26} - 205641981272 T^{27} - 913524955266 T^{28} - 1550224166512 T^{29} + 1356446145698 T^{30} + 9495123019886 T^{31} + 33232930569601 T^{32} \))
$11$ (\( 1 - 4 T + 11 T^{2} \))(\( ( 1 + 11 T^{2} )^{2} \))(\( 1 - 2 T - 7 T^{2} - 22 T^{3} + 121 T^{4} \))(\( 1 - 2 T - 21 T^{2} + 14 T^{3} + 286 T^{4} + 58 T^{5} - 3673 T^{6} + 638 T^{7} + 34606 T^{8} + 18634 T^{9} - 307461 T^{10} - 322102 T^{11} + 1771561 T^{12} \))(\( ( 1 - 6 T + 11 T^{2} )^{2}( 1 + 6 T + 11 T^{2} )^{2} \))(\( ( 1 + 6 T + 8 T^{2} + 36 T^{3} + 267 T^{4} + 396 T^{5} + 968 T^{6} + 7986 T^{7} + 14641 T^{8} )^{2} \))(\( ( 1 + 34 T^{2} + 652 T^{4} - 414 T^{5} + 9016 T^{6} - 9072 T^{7} + 100687 T^{8} - 99792 T^{9} + 1090936 T^{10} - 551034 T^{11} + 9545932 T^{12} + 60233074 T^{14} + 214358881 T^{16} )^{2} \))
$13$ (\( 1 + 2 T + 13 T^{2} \))(\( ( 1 - 13 T^{2} )^{2} \))(\( ( 1 - 7 T + 13 T^{2} )( 1 + 5 T + 13 T^{2} ) \))(\( 1 + 4 T - 19 T^{2} - 60 T^{3} + 370 T^{4} + 536 T^{5} - 4235 T^{6} + 6968 T^{7} + 62530 T^{8} - 131820 T^{9} - 542659 T^{10} + 1485172 T^{11} + 4826809 T^{12} \))(\( ( 1 - 6 T + 13 T^{2} )^{2}( 1 + 4 T + 13 T^{2} )^{2} \))(\( ( 1 + 20 T^{2} + 231 T^{4} + 3380 T^{6} + 28561 T^{8} )^{2} \))(\( 1 + 2 T + 2 T^{2} + 56 T^{3} + 144 T^{4} + 658 T^{5} + 2596 T^{6} + 4150 T^{7} + 35702 T^{8} - 23108 T^{9} - 42562 T^{10} + 1025978 T^{11} - 7929096 T^{12} - 10433170 T^{13} - 40700658 T^{14} - 417980868 T^{15} - 885935405 T^{16} - 5433751284 T^{17} - 6878411202 T^{18} - 22921674490 T^{19} - 226462910856 T^{20} + 380938449554 T^{21} - 205438644658 T^{22} - 1449992730836 T^{23} + 29123218201142 T^{24} + 44008672397950 T^{25} + 357880644840004 T^{26} + 1179241539276346 T^{27} + 3354924257637264 T^{28} + 16961005969166168 T^{29} + 7874752771398578 T^{30} + 102371786028181514 T^{31} + 665416609183179841 T^{32} \))
$17$ (\( 1 + 2 T + 17 T^{2} \))(\( 1 - 14 T^{2} + 289 T^{4} \))(\( ( 1 - 4 T + 17 T^{2} )^{2} \))(\( ( 1 + 2 T + 43 T^{2} + 56 T^{3} + 731 T^{4} + 578 T^{5} + 4913 T^{6} )^{2} \))(\( 1 - 254 T^{4} + 83521 T^{8} \))(\( ( 1 - 40 T^{2} + 786 T^{4} - 11560 T^{6} + 83521 T^{8} )^{2} \))(\( 1 + 964 T^{4} + 237772 T^{8} - 70157828 T^{12} - 47638889354 T^{16} - 5859651952388 T^{20} + 1658639798261452 T^{24} + 561647836689489604 T^{28} + 48661191875666868481 T^{32} \))
$19$ (\( 1 - 4 T + 19 T^{2} \))(\( ( 1 + 4 T + 19 T^{2} )^{2} \))(\( ( 1 + 8 T + 19 T^{2} )^{2} \))(\( ( 1 - 4 T + 53 T^{2} - 148 T^{3} + 1007 T^{4} - 1444 T^{5} + 6859 T^{6} )^{2} \))(\( ( 1 - 19 T^{2} )^{4} \))(\( ( 1 + 2 T + 36 T^{2} + 38 T^{3} + 361 T^{4} )^{4} \))(\( ( 1 - 92 T^{2} + 4132 T^{4} - 123032 T^{6} + 2693650 T^{8} - 44414552 T^{10} + 538486372 T^{12} - 4328221052 T^{14} + 16983563041 T^{16} )^{2} \))
$23$ (\( 1 + 23 T^{2} \))(\( 1 + 34 T^{2} + 529 T^{4} \))(\( 1 + 3 T - 14 T^{2} + 69 T^{3} + 529 T^{4} \))(\( 1 + 3 T - 27 T^{2} + 66 T^{3} + 405 T^{4} - 2625 T^{5} - 9794 T^{6} - 60375 T^{7} + 214245 T^{8} + 803022 T^{9} - 7555707 T^{10} + 19309029 T^{11} + 148035889 T^{12} \))(\( 1 - 158 T^{4} + 279841 T^{8} \))(\( 1 + 88 T^{2} + 4753 T^{4} + 170104 T^{6} + 4569664 T^{8} + 89985016 T^{10} + 1330084273 T^{12} + 13027158232 T^{14} + 78310985281 T^{16} \))(\( 1 - 18 T + 162 T^{2} - 972 T^{3} + 4138 T^{4} - 11184 T^{5} + 3348 T^{6} + 119100 T^{7} - 338219 T^{8} - 2163498 T^{9} + 26656146 T^{10} - 182453766 T^{11} + 892508410 T^{12} - 1883089950 T^{13} - 11227491072 T^{14} + 141260851668 T^{15} - 837597200144 T^{16} + 3248999588364 T^{17} - 5939342777088 T^{18} - 22911555421650 T^{19} + 249760445962810 T^{20} - 1174335019617738 T^{21} + 3946066270423794 T^{22} - 7366333044933606 T^{23} - 26486263130754539 T^{24} + 214517281980243300 T^{25} + 138695959543296852 T^{26} - 10656224332509359568 T^{27} + 90682715899700088298 T^{28} - \)\(48\!\cdots\!76\)\( T^{29} + \)\(18\!\cdots\!58\)\( T^{30} - \)\(47\!\cdots\!26\)\( T^{31} + \)\(61\!\cdots\!61\)\( T^{32} \))
$29$ (\( 1 - 2 T + 29 T^{2} \))(\( ( 1 + 29 T^{2} )^{2} \))(\( 1 - T - 28 T^{2} - 29 T^{3} + 841 T^{4} \))(\( 1 - 7 T - 9 T^{2} + 304 T^{3} - 803 T^{4} - 2929 T^{5} + 36038 T^{6} - 84941 T^{7} - 675323 T^{8} + 7414256 T^{9} - 6365529 T^{10} - 143578043 T^{11} + 594823321 T^{12} \))(\( ( 1 + 40 T^{2} + 841 T^{4} )^{2} \))(\( ( 1 - 6 T - 19 T^{2} + 18 T^{3} + 1140 T^{4} + 522 T^{5} - 15979 T^{6} - 146334 T^{7} + 707281 T^{8} )^{2} \))(\( 1 - 148 T^{2} + 10806 T^{4} - 562376 T^{6} + 24743345 T^{8} - 973959144 T^{10} + 34914297094 T^{12} - 1155748123996 T^{14} + 35134519733316 T^{16} - 971984172280636 T^{18} + 24694218962941414 T^{20} - 579333612552397224 T^{22} + 12377769580906494545 T^{24} - \)\(23\!\cdots\!76\)\( T^{26} + \)\(38\!\cdots\!46\)\( T^{28} - \)\(44\!\cdots\!88\)\( T^{30} + \)\(25\!\cdots\!21\)\( T^{32} \))
$31$ (\( 1 + 31 T^{2} \))(\( ( 1 - 8 T + 31 T^{2} )^{2} \))(\( 1 - 31 T^{2} + 961 T^{4} \))(\( 1 + 8 T + 31 T^{2} + 208 T^{3} - 158 T^{4} - 7756 T^{5} - 34897 T^{6} - 240436 T^{7} - 151838 T^{8} + 6196528 T^{9} + 28629151 T^{10} + 229033208 T^{11} + 887503681 T^{12} \))(\( ( 1 + 4 T + 31 T^{2} )^{4} \))(\( ( 1 - 2 T - 56 T^{2} + 4 T^{3} + 2515 T^{4} + 124 T^{5} - 53816 T^{6} - 59582 T^{7} + 923521 T^{8} )^{2} \))(\( ( 1 + 2 T - 78 T^{2} + 76 T^{3} + 3500 T^{4} - 7572 T^{5} - 100340 T^{6} + 131474 T^{7} + 2627523 T^{8} + 4075694 T^{9} - 96426740 T^{10} - 225577452 T^{11} + 3232323500 T^{12} + 2175815476 T^{13} - 69225287118 T^{14} + 55025228222 T^{15} + 852891037441 T^{16} )^{2} \))
$37$ (\( 1 + 10 T + 37 T^{2} \))(\( ( 1 - 37 T^{2} )^{2} \))(\( ( 1 + 4 T + 37 T^{2} )^{2} \))(\( ( 1 - 6 T + 99 T^{2} - 440 T^{3} + 3663 T^{4} - 8214 T^{5} + 50653 T^{6} )^{2} \))(\( ( 1 - 2 T + 2 T^{2} - 74 T^{3} + 1369 T^{4} )^{2} \))(\( ( 1 - 56 T^{2} + 1369 T^{4} )^{4} \))(\( ( 1 - 2 T + 2 T^{2} + 50 T^{3} + 680 T^{4} - 6430 T^{5} + 12750 T^{6} - 67506 T^{7} - 1916078 T^{8} - 2497722 T^{9} + 17454750 T^{10} - 325698790 T^{11} + 1274429480 T^{12} + 3467197850 T^{13} + 5131452818 T^{14} - 189863754266 T^{15} + 3512479453921 T^{16} )^{2} \))
$41$ (\( 1 + 10 T + 41 T^{2} \))(\( ( 1 + 41 T^{2} )^{2} \))(\( 1 + 5 T - 16 T^{2} + 205 T^{3} + 1681 T^{4} \))(\( 1 - 13 T + 27 T^{2} + 292 T^{3} + 445 T^{4} - 22279 T^{5} + 169790 T^{6} - 913439 T^{7} + 748045 T^{8} + 20124932 T^{9} + 76295547 T^{10} - 1506130613 T^{11} + 4750104241 T^{12} \))(\( ( 1 - 80 T^{2} + 1681 T^{4} )^{2} \))(\( ( 1 + 12 T + 29 T^{2} + 396 T^{3} + 5640 T^{4} + 16236 T^{5} + 48749 T^{6} + 827052 T^{7} + 2825761 T^{8} )^{2} \))(\( ( 1 + 12 T + 190 T^{2} + 1704 T^{3} + 16861 T^{4} + 130290 T^{5} + 1067482 T^{6} + 7225794 T^{7} + 51027208 T^{8} + 296257554 T^{9} + 1794437242 T^{10} + 8979717090 T^{11} + 47645156221 T^{12} + 197418966504 T^{13} + 902519805790 T^{14} + 2337051286572 T^{15} + 7984925229121 T^{16} )^{2} \))
$43$ (\( 1 - 4 T + 43 T^{2} \))(\( ( 1 - 43 T^{2} )^{2} \))(\( ( 1 - 13 T + 43 T^{2} )( 1 + 5 T + 43 T^{2} ) \))(\( 1 + 10 T - 25 T^{2} - 462 T^{3} + 1690 T^{4} + 17990 T^{5} + 34975 T^{6} + 773570 T^{7} + 3124810 T^{8} - 36732234 T^{9} - 85470025 T^{10} + 1470084430 T^{11} + 6321363049 T^{12} \))(\( ( 1 + 16 T + 128 T^{2} + 688 T^{3} + 1849 T^{4} )^{2} \))(\( 1 + 88 T^{2} + 3838 T^{4} + 18304 T^{6} - 2480621 T^{8} + 33844096 T^{10} + 13121358238 T^{12} + 556279948312 T^{14} + 11688200277601 T^{16} \))(\( 1 + 2 T + 2 T^{2} - 112 T^{3} - 3840 T^{4} - 7982 T^{5} - 2012 T^{6} + 464698 T^{7} + 6098798 T^{8} + 19506388 T^{9} - 1100914 T^{10} - 890327938 T^{11} - 7456003800 T^{12} - 38444523010 T^{13} - 1944910962 T^{14} + 1611174599916 T^{15} + 14665036034491 T^{16} + 69280507796388 T^{17} - 3596140368738 T^{18} - 3056608690956070 T^{19} - 25490593247443800 T^{20} - 130885723924780534 T^{21} - 6959277079726786 T^{22} + 5302199293874251516 T^{23} + 71283972476632423598 T^{24} + \)\(23\!\cdots\!14\)\( T^{25} - 43482302414327908988 T^{26} - \)\(74\!\cdots\!74\)\( T^{27} - \)\(15\!\cdots\!40\)\( T^{28} - \)\(19\!\cdots\!16\)\( T^{29} + \)\(14\!\cdots\!98\)\( T^{30} + \)\(63\!\cdots\!14\)\( T^{31} + \)\(13\!\cdots\!01\)\( T^{32} \))
$47$ (\( 1 + 8 T + 47 T^{2} \))(\( 1 - 14 T^{2} + 2209 T^{4} \))(\( 1 + 7 T + 2 T^{2} + 329 T^{3} + 2209 T^{4} \))(\( 1 + 13 T + 39 T^{2} - 16 T^{3} - 299 T^{4} - 19253 T^{5} - 232366 T^{6} - 904891 T^{7} - 660491 T^{8} - 1661168 T^{9} + 190307559 T^{10} + 2981485091 T^{11} + 10779215329 T^{12} \))(\( 1 - 3518 T^{4} + 4879681 T^{8} \))(\( 1 + 160 T^{2} + 14809 T^{4} + 1019680 T^{6} + 55015600 T^{8} + 2252473120 T^{10} + 72263195929 T^{12} + 1724674452640 T^{14} + 23811286661761 T^{16} \))(\( 1 + 12 T + 72 T^{2} + 288 T^{3} - 134 T^{4} - 24762 T^{5} - 246024 T^{6} - 2286006 T^{7} - 18983651 T^{8} - 98916168 T^{9} - 204929766 T^{10} + 1977304884 T^{11} + 21327903874 T^{12} + 124668663210 T^{13} + 894258211950 T^{14} + 7685415723390 T^{15} + 46309407650224 T^{16} + 361214538999330 T^{17} + 1975416390197550 T^{18} + 12943474620451830 T^{19} + 104073367303784194 T^{20} + 453485002462114188 T^{21} - 2208982075035583014 T^{22} - 50113217696402345784 T^{23} - \)\(45\!\cdots\!11\)\( T^{24} - \)\(25\!\cdots\!02\)\( T^{25} - \)\(12\!\cdots\!76\)\( T^{26} - \)\(61\!\cdots\!86\)\( T^{27} - \)\(15\!\cdots\!94\)\( T^{28} + \)\(15\!\cdots\!76\)\( T^{29} + \)\(18\!\cdots\!68\)\( T^{30} + \)\(14\!\cdots\!16\)\( T^{31} + \)\(56\!\cdots\!21\)\( T^{32} \))
$53$ (\( 1 - 10 T + 53 T^{2} \))(\( 1 - 86 T^{2} + 2809 T^{4} \))(\( ( 1 + 2 T + 53 T^{2} )^{2} \))(\( ( 1 + 2 T + 139 T^{2} + 188 T^{3} + 7367 T^{4} + 5618 T^{5} + 148877 T^{6} )^{2} \))(\( ( 1 - 56 T^{2} + 2809 T^{4} )( 1 + 56 T^{2} + 2809 T^{4} ) \))(\( ( 1 - 196 T^{2} + 15174 T^{4} - 550564 T^{6} + 7890481 T^{8} )^{2} \))(\( 1 - 6680 T^{4} + 12266524 T^{8} + 25145244376 T^{12} - 161780081477882 T^{16} + 198408072989184856 T^{20} + \)\(76\!\cdots\!64\)\( T^{24} - \)\(32\!\cdots\!80\)\( T^{28} + \)\(38\!\cdots\!21\)\( T^{32} \))
$59$ (\( 1 - 4 T + 59 T^{2} \))(\( ( 1 + 59 T^{2} )^{2} \))(\( 1 - 14 T + 137 T^{2} - 826 T^{3} + 3481 T^{4} \))(\( 1 - 2 T - 153 T^{2} + 110 T^{3} + 14962 T^{4} - 3194 T^{5} - 1012513 T^{6} - 188446 T^{7} + 52082722 T^{8} + 22591690 T^{9} - 1853956233 T^{10} - 1429848598 T^{11} + 42180533641 T^{12} \))(\( ( 1 + 46 T^{2} + 3481 T^{4} )^{2} \))(\( ( 1 - 12 T + 2 T^{2} - 288 T^{3} + 8187 T^{4} - 16992 T^{5} + 6962 T^{6} - 2464548 T^{7} + 12117361 T^{8} )^{2} \))(\( 1 - 328 T^{2} + 54732 T^{4} - 6528848 T^{6} + 640897706 T^{8} - 55004293416 T^{10} + 4215408852016 T^{12} - 289798546717912 T^{14} + 17966936709291123 T^{16} - 1008788741125051672 T^{18} + 51079630822473449776 T^{20} - \)\(23\!\cdots\!56\)\( T^{22} + \)\(94\!\cdots\!26\)\( T^{24} - \)\(33\!\cdots\!48\)\( T^{26} + \)\(97\!\cdots\!92\)\( T^{28} - \)\(20\!\cdots\!08\)\( T^{30} + \)\(21\!\cdots\!41\)\( T^{32} \))
$61$ (\( 1 + 2 T + 61 T^{2} \))(\( ( 1 - 2 T + 61 T^{2} )^{2} \))(\( 1 + 7 T - 12 T^{2} + 427 T^{3} + 3721 T^{4} \))(\( 1 + T - 145 T^{2} - 240 T^{3} + 12217 T^{4} + 14087 T^{5} - 812786 T^{6} + 859307 T^{7} + 45459457 T^{8} - 54475440 T^{9} - 2007646945 T^{10} + 844596301 T^{11} + 51520374361 T^{12} \))(\( ( 1 - 8 T + 61 T^{2} )^{4} \))(\( ( 1 + 4 T - 35 T^{2} - 284 T^{3} - 2096 T^{4} - 17324 T^{5} - 130235 T^{6} + 907924 T^{7} + 13845841 T^{8} )^{2} \))(\( ( 1 - 4 T - 126 T^{2} + 1444 T^{3} + 6365 T^{4} - 126018 T^{5} + 435286 T^{6} + 4572842 T^{7} - 47796264 T^{8} + 278943362 T^{9} + 1619699206 T^{10} - 28603691658 T^{11} + 88128777965 T^{12} + 1219597058644 T^{13} - 6491567169486 T^{14} - 12570971344084 T^{15} + 191707312997281 T^{16} )^{2} \))
$67$ (\( 1 - 12 T + 67 T^{2} \))(\( ( 1 - 67 T^{2} )^{2} \))(\( 1 - 3 T - 58 T^{2} - 201 T^{3} + 4489 T^{4} \))(\( 1 + 11 T - 41 T^{2} - 152 T^{3} + 4753 T^{4} - 32755 T^{5} - 764902 T^{6} - 2194585 T^{7} + 21336217 T^{8} - 45715976 T^{9} - 826195961 T^{10} + 14851376177 T^{11} + 90458382169 T^{12} \))(\( ( 1 - 8 T + 32 T^{2} - 536 T^{3} + 4489 T^{4} )^{2} \))(\( 1 + 184 T^{2} + 16657 T^{4} + 1512664 T^{6} + 123674896 T^{8} + 6790348696 T^{10} + 335657222497 T^{12} + 16644342319096 T^{14} + 406067677556641 T^{16} \))(\( 1 - 4 T + 8 T^{2} + 1436 T^{3} - 5070 T^{4} + 45274 T^{5} + 890512 T^{6} - 2479154 T^{7} + 72952373 T^{8} + 410747644 T^{9} - 246911326 T^{10} + 51154442144 T^{11} + 220045950810 T^{12} + 1378484916350 T^{13} + 21635958751422 T^{14} + 133914238479582 T^{15} + 1373516219103616 T^{16} + 8972253978131994 T^{17} + 97123818835133358 T^{18} + 414597258896175050 T^{19} + 4434172580332358010 T^{20} + 69064896673193309408 T^{21} - 22335199089162546094 T^{22} + \)\(24\!\cdots\!12\)\( T^{23} + \)\(29\!\cdots\!93\)\( T^{24} - \)\(67\!\cdots\!38\)\( T^{25} + \)\(16\!\cdots\!88\)\( T^{26} + \)\(55\!\cdots\!42\)\( T^{27} - \)\(41\!\cdots\!70\)\( T^{28} + \)\(78\!\cdots\!32\)\( T^{29} + \)\(29\!\cdots\!32\)\( T^{30} - \)\(98\!\cdots\!72\)\( T^{31} + \)\(16\!\cdots\!81\)\( T^{32} \))
$71$ (\( 1 - 8 T + 71 T^{2} \))(\( ( 1 + 71 T^{2} )^{2} \))(\( ( 1 - 2 T + 71 T^{2} )^{2} \))(\( ( 1 + 10 T + 121 T^{2} + 712 T^{3} + 8591 T^{4} + 50410 T^{5} + 357911 T^{6} )^{2} \))(\( ( 1 - 110 T^{2} + 5041 T^{4} )^{2} \))(\( ( 1 - 18 T + 196 T^{2} - 1278 T^{3} + 5041 T^{4} )^{4} \))(\( ( 1 - 452 T^{2} + 95704 T^{4} - 12356144 T^{6} + 1062541066 T^{8} - 62287321904 T^{10} + 2431999518424 T^{12} - 57901328332292 T^{14} + 645753531245761 T^{16} )^{2} \))
$73$ (\( 1 - 10 T + 73 T^{2} \))(\( ( 1 - 73 T^{2} )^{2} \))(\( ( 1 - 4 T + 73 T^{2} )^{2} \))(\( ( 1 + 8 T + 155 T^{2} + 1040 T^{3} + 11315 T^{4} + 42632 T^{5} + 389017 T^{6} )^{2} \))(\( ( 1 - 2 T + 2 T^{2} - 146 T^{3} + 5329 T^{4} )^{2} \))(\( ( 1 - 74 T^{2} + 5329 T^{4} )^{4} \))(\( ( 1 + 4 T + 8 T^{2} - 292 T^{3} - 844 T^{4} + 28196 T^{5} + 162168 T^{6} - 417348 T^{7} - 41227034 T^{8} - 30466404 T^{9} + 864193272 T^{10} + 10968723332 T^{11} - 23968115404 T^{12} - 605336905156 T^{13} + 1210673810312 T^{14} + 44189594076388 T^{15} + 806460091894081 T^{16} )^{2} \))
$79$ (\( 1 + 79 T^{2} \))(\( ( 1 + 16 T + 79 T^{2} )^{2} \))(\( 1 - 6 T - 43 T^{2} - 474 T^{3} + 6241 T^{4} \))(\( 1 + 2 T - 149 T^{2} - 374 T^{3} + 10642 T^{4} + 16154 T^{5} - 772597 T^{6} + 1276166 T^{7} + 66416722 T^{8} - 184396586 T^{9} - 5803562069 T^{10} + 6154112798 T^{11} + 243087455521 T^{12} \))(\( ( 1 - 14 T^{2} + 6241 T^{4} )^{2} \))(\( ( 1 - 8 T - 98 T^{2} - 32 T^{3} + 14947 T^{4} - 2528 T^{5} - 611618 T^{6} - 3944312 T^{7} + 38950081 T^{8} )^{2} \))(\( 1 + 260 T^{2} + 23724 T^{4} + 1706392 T^{6} + 225017642 T^{8} + 14869625148 T^{10} - 181450880816 T^{12} - 19987311435172 T^{14} + 2160188956785123 T^{16} - 124740810666908452 T^{18} - 7067526505304546096 T^{20} + \)\(36\!\cdots\!08\)\( T^{22} + \)\(34\!\cdots\!62\)\( T^{24} + \)\(16\!\cdots\!92\)\( T^{26} + \)\(14\!\cdots\!84\)\( T^{28} + \)\(95\!\cdots\!60\)\( T^{30} + \)\(23\!\cdots\!21\)\( T^{32} \))
$83$ (\( 1 + 12 T + 83 T^{2} \))(\( 1 + 154 T^{2} + 6889 T^{4} \))(\( 1 + 9 T - 2 T^{2} + 747 T^{3} + 6889 T^{4} \))(\( 1 - 15 T - 51 T^{2} + 678 T^{3} + 17133 T^{4} - 80979 T^{5} - 925778 T^{6} - 6721257 T^{7} + 118029237 T^{8} + 387671586 T^{9} - 2420374371 T^{10} - 59085609645 T^{11} + 326940373369 T^{12} \))(\( 1 + 8722 T^{4} + 47458321 T^{8} \))(\( 1 + 160 T^{2} + 5929 T^{4} + 942880 T^{6} + 163131520 T^{8} + 6495500320 T^{10} + 281380385209 T^{12} + 52310459739040 T^{14} + 2252292232139041 T^{16} \))(\( 1 + 66 T + 2178 T^{2} + 47916 T^{3} + 791830 T^{4} + 10596156 T^{5} + 122712084 T^{6} + 1311937572 T^{7} + 13564902253 T^{8} + 136924099746 T^{9} + 1327585690602 T^{10} + 12258029803398 T^{11} + 109901521003222 T^{12} + 994600086849090 T^{13} + 9293464416138048 T^{14} + 88105947208965780 T^{15} + 817933709355268336 T^{16} + 7312793618344159740 T^{17} + 64022676362775012672 T^{18} + \)\(56\!\cdots\!30\)\( T^{19} + \)\(52\!\cdots\!62\)\( T^{20} + \)\(48\!\cdots\!14\)\( T^{21} + \)\(43\!\cdots\!38\)\( T^{22} + \)\(37\!\cdots\!42\)\( T^{23} + \)\(30\!\cdots\!73\)\( T^{24} + \)\(24\!\cdots\!16\)\( T^{25} + \)\(19\!\cdots\!16\)\( T^{26} + \)\(13\!\cdots\!52\)\( T^{27} + \)\(84\!\cdots\!30\)\( T^{28} + \)\(42\!\cdots\!08\)\( T^{29} + \)\(16\!\cdots\!62\)\( T^{30} + \)\(40\!\cdots\!62\)\( T^{31} + \)\(50\!\cdots\!81\)\( T^{32} \))
$89$ (\( 1 - 6 T + 89 T^{2} \))(\( ( 1 + 89 T^{2} )^{2} \))(\( ( 1 + 15 T + 89 T^{2} )^{2} \))(\( ( 1 + 3 T + 89 T^{2} )^{6} \))(\( ( 1 + 16 T^{2} + 7921 T^{4} )^{2} \))(\( ( 1 + 6 T + 79 T^{2} + 534 T^{3} + 7921 T^{4} )^{4} \))(\( ( 1 + 412 T^{2} + 69850 T^{4} + 6725680 T^{6} + 546021283 T^{8} + 53274111280 T^{10} + 4382545533850 T^{12} + 204756291875932 T^{14} + 3936588805702081 T^{16} )^{2} \))
$97$ (\( 1 - 2 T + 97 T^{2} \))(\( ( 1 - 97 T^{2} )^{2} \))(\( 1 + 2 T - 93 T^{2} + 194 T^{3} + 9409 T^{4} \))(\( 1 - 18 T + 69 T^{2} - 214 T^{3} + 906 T^{4} + 163158 T^{5} - 2743251 T^{6} + 15826326 T^{7} + 8524554 T^{8} - 195312022 T^{9} + 6108520389 T^{10} - 154572124626 T^{11} + 832972004929 T^{12} \))(\( ( 1 + 22 T + 242 T^{2} + 2134 T^{3} + 9409 T^{4} )^{2} \))(\( 1 + 52 T^{2} - 12902 T^{4} - 167024 T^{6} + 129576019 T^{8} - 1571528816 T^{10} - 1142204783462 T^{12} + 43314544256308 T^{14} + 7837433594376961 T^{16} \))(\( 1 - 28 T + 392 T^{2} + 872 T^{3} - 117924 T^{4} + 2250436 T^{5} - 16405808 T^{6} - 121301108 T^{7} + 5089087514 T^{8} - 63345173552 T^{9} + 270560580248 T^{10} + 4797083097932 T^{11} - 105427115995440 T^{12} + 964302703147436 T^{13} - 1291090828127400 T^{14} - 85355358908186064 T^{15} + 1288991316725561923 T^{16} - 8279469814094048208 T^{17} - 12147873601850706600 T^{18} + \)\(88\!\cdots\!28\)\( T^{19} - \)\(93\!\cdots\!40\)\( T^{20} + \)\(41\!\cdots\!24\)\( T^{21} + \)\(22\!\cdots\!92\)\( T^{22} - \)\(51\!\cdots\!76\)\( T^{23} + \)\(39\!\cdots\!54\)\( T^{24} - \)\(92\!\cdots\!36\)\( T^{25} - \)\(12\!\cdots\!92\)\( T^{26} + \)\(16\!\cdots\!08\)\( T^{27} - \)\(81\!\cdots\!84\)\( T^{28} + \)\(58\!\cdots\!44\)\( T^{29} + \)\(25\!\cdots\!48\)\( T^{30} - \)\(17\!\cdots\!04\)\( T^{31} + \)\(61\!\cdots\!21\)\( T^{32} \))
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