Properties

Label 70.2
Level 70
Weight 2
Dimension 45
Nonzero newspaces 6
Newform subspaces 10
Sturm bound 576
Trace bound 4

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Defining parameters

Level: \( N \) = \( 70 = 2 \cdot 5 \cdot 7 \)
Weight: \( k \) = \( 2 \)
Nonzero newspaces: \( 6 \)
Newform subspaces: \( 10 \)
Sturm bound: \(576\)
Trace bound: \(4\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_1(70))\).

Total New Old
Modular forms 192 45 147
Cusp forms 97 45 52
Eisenstein series 95 0 95

Trace form

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

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_1(70))\)

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
70.2.a \(\chi_{70}(1, \cdot)\) 70.2.a.a 1 1
70.2.c \(\chi_{70}(29, \cdot)\) 70.2.c.a 4 1
70.2.e \(\chi_{70}(11, \cdot)\) 70.2.e.a 2 2
70.2.e.b 2
70.2.e.c 2
70.2.e.d 2
70.2.g \(\chi_{70}(13, \cdot)\) 70.2.g.a 8 2
70.2.i \(\chi_{70}(9, \cdot)\) 70.2.i.a 4 2
70.2.i.b 4
70.2.k \(\chi_{70}(3, \cdot)\) 70.2.k.a 16 4

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_1(70))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_1(70)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_1(14))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(35))\)\(^{\oplus 2}\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ (\( 1 - T \))(\( ( 1 + T^{2} )^{2} \))(\( 1 + T + T^{2} \))(\( 1 + T + T^{2} \))(\( 1 - T + T^{2} \))(\( 1 - T + T^{2} \))(\( ( 1 + T^{4} )^{2} \))(\( 1 - T^{2} + T^{4} \))(\( 1 - T^{2} + T^{4} \))(\( ( 1 - T^{4} + T^{8} )^{2} \))
$3$ (\( 1 + 3 T^{2} \))(\( ( 1 + 9 T^{4} )^{2} \))(\( ( 1 + 3 T^{2} )( 1 + 3 T + 3 T^{2} ) \))(\( 1 - 2 T + T^{2} - 6 T^{3} + 9 T^{4} \))(\( 1 + T - 2 T^{2} + 3 T^{3} + 9 T^{4} \))(\( 1 - 2 T + T^{2} - 6 T^{3} + 9 T^{4} \))(\( ( 1 - 8 T^{2} + 32 T^{4} - 72 T^{6} + 81 T^{8} )( 1 + 8 T^{2} + 32 T^{4} + 72 T^{6} + 81 T^{8} ) \))(\( ( 1 - 3 T^{2} )^{2}( 1 + 3 T^{2} + 9 T^{4} ) \))(\( ( 1 + 3 T^{2} + 9 T^{4} )^{2} \))(\( 1 + 3 T^{4} - 24 T^{5} + 48 T^{7} + 5 T^{8} - 72 T^{9} + 288 T^{10} + 120 T^{11} - 570 T^{12} - 720 T^{13} + 2016 T^{14} - 2136 T^{15} - 2570 T^{16} - 6408 T^{17} + 18144 T^{18} - 19440 T^{19} - 46170 T^{20} + 29160 T^{21} + 209952 T^{22} - 157464 T^{23} + 32805 T^{24} + 944784 T^{25} - 4251528 T^{27} + 1594323 T^{28} + 43046721 T^{32} \))
$5$ (\( 1 + T \))(\( 1 - 4 T + 8 T^{2} - 20 T^{3} + 25 T^{4} \))(\( 1 - T + T^{2} \))(\( 1 - T + T^{2} \))(\( 1 - T + T^{2} \))(\( 1 + T + T^{2} \))(\( 1 - 48 T^{4} + 625 T^{8} \))(\( 1 + 2 T - T^{2} + 10 T^{3} + 25 T^{4} \))(\( 1 - 4 T + 11 T^{2} - 20 T^{3} + 25 T^{4} \))(\( 1 + 12 T + 66 T^{2} + 216 T^{3} + 450 T^{4} + 468 T^{5} - 864 T^{6} - 6252 T^{7} - 18241 T^{8} - 31260 T^{9} - 21600 T^{10} + 58500 T^{11} + 281250 T^{12} + 675000 T^{13} + 1031250 T^{14} + 937500 T^{15} + 390625 T^{16} \))
$7$ (\( 1 + T \))(\( ( 1 + T^{2} )^{2} \))(\( 1 - T + 7 T^{2} \))(\( 1 + 4 T + 7 T^{2} \))(\( 1 + T + 7 T^{2} \))(\( 1 + 4 T + 7 T^{2} \))(\( 1 + 8 T + 32 T^{2} + 88 T^{3} + 226 T^{4} + 616 T^{5} + 1568 T^{6} + 2744 T^{7} + 2401 T^{8} \))(\( 1 + 11 T^{2} + 49 T^{4} \))(\( 1 + 2 T^{2} + 49 T^{4} \))(\( 1 - 8 T + 32 T^{2} - 80 T^{3} + 73 T^{4} + 160 T^{5} - 416 T^{6} - 696 T^{7} + 4944 T^{8} - 4872 T^{9} - 20384 T^{10} + 54880 T^{11} + 175273 T^{12} - 1344560 T^{13} + 3764768 T^{14} - 6588344 T^{15} + 5764801 T^{16} \))
$11$ (\( 1 - 4 T + 11 T^{2} \))(\( ( 1 - 2 T^{2} + 121 T^{4} )^{2} \))(\( 1 - 2 T - 7 T^{2} - 22 T^{3} + 121 T^{4} \))(\( 1 + 3 T - 2 T^{2} + 33 T^{3} + 121 T^{4} \))(\( 1 - 6 T + 25 T^{2} - 66 T^{3} + 121 T^{4} \))(\( 1 + 3 T - 2 T^{2} + 33 T^{3} + 121 T^{4} \))(\( ( 1 + 14 T^{2} + 121 T^{4} )^{4} \))(\( ( 1 - 11 T^{2} + 121 T^{4} )^{2} \))(\( ( 1 + 3 T - 2 T^{2} + 33 T^{3} + 121 T^{4} )^{2} \))(\( ( 1 + 6 T + 5 T^{2} - 18 T^{3} - 29 T^{4} - 252 T^{5} - 544 T^{6} - 516 T^{7} - 6650 T^{8} - 5676 T^{9} - 65824 T^{10} - 335412 T^{11} - 424589 T^{12} - 2898918 T^{13} + 8857805 T^{14} + 116923026 T^{15} + 214358881 T^{16} )^{2} \))
$13$ (\( 1 + 6 T + 13 T^{2} \))(\( 1 - 32 T^{2} + 498 T^{4} - 5408 T^{6} + 28561 T^{8} \))(\( ( 1 + 13 T^{2} )^{2} \))(\( ( 1 - 5 T + 13 T^{2} )^{2} \))(\( ( 1 + 4 T + 13 T^{2} )^{2} \))(\( ( 1 + T + 13 T^{2} )^{2} \))(\( ( 1 - 240 T^{4} + 28561 T^{8} )^{2} \))(\( ( 1 - 22 T^{2} + 169 T^{4} )^{2} \))(\( ( 1 - T^{2} + 169 T^{4} )^{2} \))(\( 1 + 714 T^{4} + 266977 T^{8} + 68653746 T^{12} + 13237862628 T^{16} + 1960819639506 T^{20} + 217781340700417 T^{24} + 16634832777451434 T^{28} + 665416609183179841 T^{32} \))
$17$ (\( 1 - 2 T + 17 T^{2} \))(\( ( 1 - 8 T + 17 T^{2} )^{2}( 1 + 8 T + 17 T^{2} )^{2} \))(\( 1 - 4 T - T^{2} - 68 T^{3} + 289 T^{4} \))(\( 1 + 6 T + 19 T^{2} + 102 T^{3} + 289 T^{4} \))(\( 1 - 17 T^{2} + 289 T^{4} \))(\( 1 - 6 T + 19 T^{2} - 102 T^{3} + 289 T^{4} \))(\( 1 - 252 T^{4} + 17030 T^{8} - 21047292 T^{12} + 6975757441 T^{16} \))(\( ( 1 - 8 T + 47 T^{2} - 136 T^{3} + 289 T^{4} )( 1 + 8 T + 47 T^{2} + 136 T^{3} + 289 T^{4} ) \))(\( ( 1 - 8 T + 47 T^{2} - 136 T^{3} + 289 T^{4} )( 1 + 8 T + 47 T^{2} + 136 T^{3} + 289 T^{4} ) \))(\( 1 + 36 T + 648 T^{2} + 7776 T^{3} + 70080 T^{4} + 507420 T^{5} + 3088368 T^{6} + 16357284 T^{7} + 77768530 T^{8} + 343436904 T^{9} + 1468660248 T^{10} + 6351048828 T^{11} + 28493424864 T^{12} + 131692528476 T^{13} + 608521516920 T^{14} + 2723900163624 T^{15} + 11569567164915 T^{16} + 46306302781608 T^{17} + 175862718389880 T^{18} + 647005392402588 T^{19} + 2379799338066144 T^{20} + 9017581135777596 T^{21} + 35449888073657112 T^{22} + 140925443446588392 T^{23} + 542494401823131730 T^{24} + 1939775574818354148 T^{25} + 6226131050341877232 T^{26} + 17390245624419136860 T^{27} + 40830166385061650880 T^{28} + 77017998783876566112 T^{29} + \)\(10\!\cdots\!92\)\( T^{30} + \)\(10\!\cdots\!48\)\( T^{31} + 48661191875666868481 T^{32} \))
$19$ (\( 1 + 19 T^{2} \))(\( ( 1 + 8 T + 48 T^{2} + 152 T^{3} + 361 T^{4} )^{2} \))(\( 1 - 6 T + 17 T^{2} - 114 T^{3} + 361 T^{4} \))(\( ( 1 - 8 T + 19 T^{2} )( 1 + 7 T + 19 T^{2} ) \))(\( 1 + 2 T - 15 T^{2} + 38 T^{3} + 361 T^{4} \))(\( ( 1 - 8 T + 19 T^{2} )( 1 + 7 T + 19 T^{2} ) \))(\( ( 1 + 24 T^{2} + 288 T^{4} + 8664 T^{6} + 130321 T^{8} )^{2} \))(\( ( 1 + 2 T - 15 T^{2} + 38 T^{3} + 361 T^{4} )^{2} \))(\( ( 1 + 5 T + 6 T^{2} + 95 T^{3} + 361 T^{4} )^{2} \))(\( 1 - 90 T^{2} + 4011 T^{4} - 123414 T^{6} + 3034057 T^{8} - 61998180 T^{10} + 1094493006 T^{12} - 18780034728 T^{14} + 344074548210 T^{16} - 6779592536808 T^{18} + 142635423034926 T^{20} - 2916758998496580 T^{22} + 51529098329487337 T^{24} - 756659411140252614 T^{26} + 8877606140374371771 T^{28} - 71910601720459570890 T^{30} + \)\(28\!\cdots\!81\)\( T^{32} \))
$23$ (\( 1 + 23 T^{2} \))(\( 1 - 36 T^{2} + 998 T^{4} - 19044 T^{6} + 279841 T^{8} \))(\( 1 + 3 T - 14 T^{2} + 69 T^{3} + 529 T^{4} \))(\( 1 + 3 T - 14 T^{2} + 69 T^{3} + 529 T^{4} \))(\( 1 - 3 T - 14 T^{2} - 69 T^{3} + 529 T^{4} \))(\( 1 + 9 T + 58 T^{2} + 207 T^{3} + 529 T^{4} \))(\( ( 1 + 4 T + 8 T^{2} + 84 T^{3} + 878 T^{4} + 1932 T^{5} + 4232 T^{6} + 48668 T^{7} + 279841 T^{8} )^{2} \))(\( 1 + 45 T^{2} + 1496 T^{4} + 23805 T^{6} + 279841 T^{8} \))(\( 1 - 3 T^{2} - 520 T^{4} - 1587 T^{6} + 279841 T^{8} \))(\( 1 + 4 T + 8 T^{2} - 128 T^{3} - 198 T^{4} - 292 T^{5} + 8608 T^{6} + 3564 T^{7} - 17335 T^{8} + 1017836 T^{9} + 5500848 T^{10} - 425132 T^{11} + 20115594 T^{12} + 929862264 T^{13} + 4560471720 T^{14} - 14572270428 T^{15} - 102219015884 T^{16} - 335162219844 T^{17} + 2412489539880 T^{18} + 11313634166088 T^{19} + 5629167940554 T^{20} - 2736295372276 T^{21} + 814322923933872 T^{22} + 3465553913672692 T^{23} - 1357520929846135 T^{24} + 6419308085454132 T^{25} + 356599408527090592 T^{26} - 278220449310866684 T^{27} - 4339095637540023558 T^{28} - 64516654327867825024 T^{29} + 92742690596309998472 T^{30} + \)\(10\!\cdots\!28\)\( T^{31} + \)\(61\!\cdots\!61\)\( T^{32} \))
$29$ (\( 1 - 6 T + 29 T^{2} \))(\( ( 1 + 4 T + 38 T^{2} + 116 T^{3} + 841 T^{4} )^{2} \))(\( ( 1 - 9 T + 29 T^{2} )^{2} \))(\( ( 1 + 6 T + 29 T^{2} )^{2} \))(\( ( 1 + 3 T + 29 T^{2} )^{2} \))(\( ( 1 - 6 T + 29 T^{2} )^{2} \))(\( ( 1 - 92 T^{2} + 3670 T^{4} - 77372 T^{6} + 707281 T^{8} )^{2} \))(\( ( 1 - T + 29 T^{2} )^{4} \))(\( ( 1 - 4 T + 29 T^{2} )^{4} \))(\( ( 1 - 70 T^{2} + 2685 T^{4} - 80906 T^{6} + 2505752 T^{8} - 68041946 T^{10} + 1899049485 T^{12} - 41637632470 T^{14} + 500246412961 T^{16} )^{2} \))
$31$ (\( 1 - 8 T + 31 T^{2} \))(\( ( 1 - 8 T + 54 T^{2} - 248 T^{3} + 961 T^{4} )^{2} \))(\( ( 1 - 11 T + 31 T^{2} )( 1 + 7 T + 31 T^{2} ) \))(\( ( 1 - 11 T + 31 T^{2} )( 1 + 7 T + 31 T^{2} ) \))(\( 1 + 8 T + 33 T^{2} + 248 T^{3} + 961 T^{4} \))(\( 1 + 8 T + 33 T^{2} + 248 T^{3} + 961 T^{4} \))(\( ( 1 - 108 T^{2} + 4806 T^{4} - 103788 T^{6} + 923521 T^{8} )^{2} \))(\( ( 1 + 10 T + 69 T^{2} + 310 T^{3} + 961 T^{4} )^{2} \))(\( ( 1 - 2 T - 27 T^{2} - 62 T^{3} + 961 T^{4} )^{2} \))(\( ( 1 - 12 T + 156 T^{2} - 1296 T^{3} + 10686 T^{4} - 72996 T^{5} + 489600 T^{6} - 2919012 T^{7} + 17046947 T^{8} - 90489372 T^{9} + 470505600 T^{10} - 2174623836 T^{11} + 9868745406 T^{12} - 37103379696 T^{13} + 138450574236 T^{14} - 330151369332 T^{15} + 852891037441 T^{16} )^{2} \))
$37$ (\( 1 + 10 T + 37 T^{2} \))(\( ( 1 - 12 T + 37 T^{2} )^{2}( 1 + 12 T + 37 T^{2} )^{2} \))(\( 1 - 4 T - 21 T^{2} - 148 T^{3} + 1369 T^{4} \))(\( ( 1 + T + 37 T^{2} )( 1 + 10 T + 37 T^{2} ) \))(\( 1 - 4 T - 21 T^{2} - 148 T^{3} + 1369 T^{4} \))(\( 1 - 7 T + 12 T^{2} - 259 T^{3} + 1369 T^{4} \))(\( ( 1 - 16 T + 128 T^{2} - 1040 T^{3} + 7666 T^{4} - 38480 T^{5} + 175232 T^{6} - 810448 T^{7} + 1874161 T^{8} )^{2} \))(\( 1 + 10 T^{2} - 1269 T^{4} + 13690 T^{6} + 1874161 T^{8} \))(\( ( 1 + 26 T^{2} + 1369 T^{4} )( 1 + 47 T^{2} + 1369 T^{4} ) \))(\( 1 - 4 T + 8 T^{2} + 240 T^{3} - 3049 T^{4} + 13544 T^{5} - 984 T^{6} - 514972 T^{7} + 3800209 T^{8} - 12256568 T^{9} - 16248928 T^{10} + 295990824 T^{11} + 119383314 T^{12} - 6433305488 T^{13} + 16607500784 T^{14} + 478842888760 T^{15} - 3828375485890 T^{16} + 17717186884120 T^{17} + 22735668573296 T^{18} - 325866222883664 T^{19} + 223743551149554 T^{20} + 20525174971850568 T^{21} - 41690303687539552 T^{22} - 1163539007448259544 T^{23} + 13348156033105669489 T^{24} - 66926657065750392844 T^{25} - 4731647022459163416 T^{26} + \)\(24\!\cdots\!72\)\( T^{27} - \)\(20\!\cdots\!69\)\( T^{28} + \)\(58\!\cdots\!80\)\( T^{29} + \)\(72\!\cdots\!12\)\( T^{30} - \)\(13\!\cdots\!72\)\( T^{31} + \)\(12\!\cdots\!41\)\( T^{32} \))
$41$ (\( 1 - 2 T + 41 T^{2} \))(\( ( 1 + 12 T + 94 T^{2} + 492 T^{3} + 1681 T^{4} )^{2} \))(\( ( 1 + 7 T + 41 T^{2} )^{2} \))(\( ( 1 - 3 T + 41 T^{2} )^{2} \))(\( ( 1 - 9 T + 41 T^{2} )^{2} \))(\( ( 1 - 3 T + 41 T^{2} )^{2} \))(\( ( 1 - 148 T^{2} + 8806 T^{4} - 248788 T^{6} + 2825761 T^{8} )^{2} \))(\( ( 1 + 3 T + 41 T^{2} )^{4} \))(\( ( 1 - 3 T + 41 T^{2} )^{4} \))(\( ( 1 - 188 T^{2} + 17802 T^{4} - 1146544 T^{6} + 54447827 T^{8} - 1927340464 T^{10} + 50304197322 T^{12} - 893019597308 T^{14} + 7984925229121 T^{16} )^{2} \))
$43$ (\( 1 - 4 T + 43 T^{2} \))(\( 1 - 92 T^{2} + 4278 T^{4} - 170108 T^{6} + 3418801 T^{8} \))(\( ( 1 + 5 T + 43 T^{2} )^{2} \))(\( ( 1 + 10 T + 43 T^{2} )^{2} \))(\( ( 1 + 7 T + 43 T^{2} )^{2} \))(\( ( 1 - 2 T + 43 T^{2} )^{2} \))(\( ( 1 - 8 T + 32 T^{2} - 344 T^{3} + 1849 T^{4} )^{4} \))(\( ( 1 - 61 T^{2} + 1849 T^{4} )^{2} \))(\( ( 1 - 82 T^{2} + 1849 T^{4} )^{2} \))(\( ( 1 + 4 T + 8 T^{2} + 32 T^{3} + 3425 T^{4} + 20696 T^{5} + 55896 T^{6} + 651228 T^{7} + 6527152 T^{8} + 28002804 T^{9} + 103351704 T^{10} + 1645476872 T^{11} + 11709393425 T^{12} + 4704270176 T^{13} + 50570904392 T^{14} + 1087274444428 T^{15} + 11688200277601 T^{16} )^{2} \))
$47$ (\( 1 - 8 T + 47 T^{2} \))(\( 1 - 108 T^{2} + 5798 T^{4} - 238572 T^{6} + 4879681 T^{8} \))(\( 1 + 8 T + 17 T^{2} + 376 T^{3} + 2209 T^{4} \))(\( 1 + 3 T - 38 T^{2} + 141 T^{3} + 2209 T^{4} \))(\( 1 - 47 T^{2} + 2209 T^{4} \))(\( 1 + 9 T + 34 T^{2} + 423 T^{3} + 2209 T^{4} \))(\( 1 + 6020 T^{4} + 17872774 T^{8} + 29375679620 T^{12} + 23811286661761 T^{16} \))(\( 1 + 30 T^{2} - 1309 T^{4} + 66270 T^{6} + 4879681 T^{8} \))(\( 1 + 45 T^{2} - 184 T^{4} + 99405 T^{6} + 4879681 T^{8} \))(\( 1 - 12 T + 72 T^{2} - 288 T^{3} - 449 T^{4} + 2880 T^{5} + 39240 T^{6} - 500820 T^{7} + 44673 T^{8} + 37644384 T^{9} - 288946944 T^{10} + 1723474464 T^{11} + 9984376994 T^{12} - 179746348152 T^{13} + 1044056404368 T^{14} - 4096130163936 T^{15} + 8350203627806 T^{16} - 192518117704992 T^{17} + 2306320597248912 T^{18} - 18661805104185096 T^{19} + 48720574714458914 T^{20} + 395270263010401248 T^{21} - 3114621328032504576 T^{22} + 19071515289987429792 T^{23} + 1063721609040849153 T^{24} - \)\(56\!\cdots\!40\)\( T^{25} + \)\(20\!\cdots\!60\)\( T^{26} + \)\(71\!\cdots\!40\)\( T^{27} - \)\(52\!\cdots\!09\)\( T^{28} - \)\(15\!\cdots\!76\)\( T^{29} + \)\(18\!\cdots\!68\)\( T^{30} - \)\(14\!\cdots\!16\)\( T^{31} + \)\(56\!\cdots\!21\)\( T^{32} \))
$53$ (\( 1 + 2 T + 53 T^{2} \))(\( 1 - 92 T^{2} + 4278 T^{4} - 258428 T^{6} + 7890481 T^{8} \))(\( 1 - 2 T - 49 T^{2} - 106 T^{3} + 2809 T^{4} \))(\( 1 + 3 T - 44 T^{2} + 159 T^{3} + 2809 T^{4} \))(\( 1 - 6 T - 17 T^{2} - 318 T^{3} + 2809 T^{4} \))(\( 1 + 9 T + 28 T^{2} + 477 T^{3} + 2809 T^{4} \))(\( ( 1 + 16 T + 128 T^{2} + 784 T^{3} + 4786 T^{4} + 41552 T^{5} + 359552 T^{6} + 2382032 T^{7} + 7890481 T^{8} )^{2} \))(\( 1 + 70 T^{2} + 2091 T^{4} + 196630 T^{6} + 7890481 T^{8} \))(\( 1 + 25 T^{2} - 2184 T^{4} + 70225 T^{6} + 7890481 T^{8} \))(\( 1 + 28 T + 392 T^{2} + 1808 T^{3} - 25321 T^{4} - 567704 T^{5} - 4335448 T^{6} + 1435172 T^{7} + 392677969 T^{8} + 4105212392 T^{9} + 14977011104 T^{10} - 124625519288 T^{11} - 2189877391278 T^{12} - 14281594759536 T^{13} - 14623477702416 T^{14} + 648224117593688 T^{15} + 7103959219709822 T^{16} + 34355878232465464 T^{17} - 41077348866086544 T^{18} - 2126200983015441072 T^{19} - 17279185948208624718 T^{20} - 52117830479026168984 T^{21} + \)\(33\!\cdots\!16\)\( T^{22} + \)\(48\!\cdots\!04\)\( T^{23} + \)\(24\!\cdots\!09\)\( T^{24} + \)\(47\!\cdots\!76\)\( T^{25} - \)\(75\!\cdots\!52\)\( T^{26} - \)\(52\!\cdots\!88\)\( T^{27} - \)\(12\!\cdots\!61\)\( T^{28} + \)\(47\!\cdots\!84\)\( T^{29} + \)\(54\!\cdots\!48\)\( T^{30} + \)\(20\!\cdots\!96\)\( T^{31} + \)\(38\!\cdots\!21\)\( T^{32} \))
$59$ (\( 1 + 8 T + 59 T^{2} \))(\( ( 1 - 8 T + 128 T^{2} - 472 T^{3} + 3481 T^{4} )^{2} \))(\( 1 + 10 T + 41 T^{2} + 590 T^{3} + 3481 T^{4} \))(\( 1 - 59 T^{2} + 3481 T^{4} \))(\( 1 - 6 T - 23 T^{2} - 354 T^{3} + 3481 T^{4} \))(\( 1 - 59 T^{2} + 3481 T^{4} \))(\( ( 1 + 136 T^{2} + 10336 T^{4} + 473416 T^{6} + 12117361 T^{8} )^{2} \))(\( ( 1 - 2 T - 55 T^{2} - 118 T^{3} + 3481 T^{4} )^{2} \))(\( ( 1 + 4 T - 43 T^{2} + 236 T^{3} + 3481 T^{4} )^{2} \))(\( 1 - 320 T^{2} + 54244 T^{4} - 6209920 T^{6} + 532238186 T^{8} - 36082424000 T^{10} + 2045329177232 T^{12} - 105925693186240 T^{14} + 5808829694261683 T^{16} - 368727337981301440 T^{18} + 24783992004353124752 T^{20} - \)\(15\!\cdots\!00\)\( T^{22} + \)\(78\!\cdots\!06\)\( T^{24} - \)\(31\!\cdots\!20\)\( T^{26} + \)\(96\!\cdots\!64\)\( T^{28} - \)\(19\!\cdots\!20\)\( T^{30} + \)\(21\!\cdots\!41\)\( T^{32} \))
$61$ (\( 1 + 14 T + 61 T^{2} \))(\( ( 1 - 12 T + 152 T^{2} - 732 T^{3} + 3721 T^{4} )^{2} \))(\( ( 1 - 13 T + 61 T^{2} )( 1 + 14 T + 61 T^{2} ) \))(\( 1 - 4 T - 45 T^{2} - 244 T^{3} + 3721 T^{4} \))(\( 1 + 5 T - 36 T^{2} + 305 T^{3} + 3721 T^{4} \))(\( 1 + 8 T + 3 T^{2} + 488 T^{3} + 3721 T^{4} \))(\( ( 1 - 96 T^{2} + 8688 T^{4} - 357216 T^{6} + 13845841 T^{8} )^{2} \))(\( ( 1 - 9 T + 20 T^{2} - 549 T^{3} + 3721 T^{4} )^{2} \))(\( ( 1 + 6 T - 25 T^{2} + 366 T^{3} + 3721 T^{4} )^{2} \))(\( ( 1 + 6 T + 207 T^{2} + 1170 T^{3} + 23079 T^{4} + 119796 T^{5} + 1969284 T^{6} + 9244740 T^{7} + 135493478 T^{8} + 563929140 T^{9} + 7327705764 T^{10} + 27191415876 T^{11} + 319548164439 T^{12} + 988177672170 T^{13} + 10664717492727 T^{14} + 18856457016126 T^{15} + 191707312997281 T^{16} )^{2} \))
$67$ (\( 1 + 12 T + 67 T^{2} \))(\( ( 1 - 70 T^{2} + 4489 T^{4} )^{2} \))(\( 1 - 9 T + 14 T^{2} - 603 T^{3} + 4489 T^{4} \))(\( 1 - 4 T - 51 T^{2} - 268 T^{3} + 4489 T^{4} \))(\( ( 1 - 11 T + 67 T^{2} )( 1 + 16 T + 67 T^{2} ) \))(\( 1 + 8 T - 3 T^{2} + 536 T^{3} + 4489 T^{4} \))(\( ( 1 - 8 T + 32 T^{2} + 552 T^{3} - 8974 T^{4} + 36984 T^{5} + 143648 T^{6} - 2406104 T^{7} + 20151121 T^{8} )^{2} \))(\( 1 + 85 T^{2} + 2736 T^{4} + 381565 T^{6} + 20151121 T^{8} \))(\( 1 + 130 T^{2} + 12411 T^{4} + 583570 T^{6} + 20151121 T^{8} \))(\( 1 - 32 T + 512 T^{2} - 3896 T^{3} - 10509 T^{4} + 663104 T^{5} - 8249312 T^{6} + 44834240 T^{7} + 165963557 T^{8} - 5505231928 T^{9} + 49681319072 T^{10} - 159136747544 T^{11} - 1412717145882 T^{12} + 21627811976512 T^{13} - 103082665870656 T^{14} - 335467607852040 T^{15} + 7552610469365206 T^{16} - 22476329726086680 T^{17} - 462738087093374784 T^{18} + 6504845613491678656 T^{19} - 28467834145442833722 T^{20} - \)\(21\!\cdots\!08\)\( T^{21} + \)\(44\!\cdots\!68\)\( T^{22} - \)\(33\!\cdots\!44\)\( T^{23} + \)\(67\!\cdots\!37\)\( T^{24} + \)\(12\!\cdots\!80\)\( T^{25} - \)\(15\!\cdots\!88\)\( T^{26} + \)\(80\!\cdots\!32\)\( T^{27} - \)\(85\!\cdots\!49\)\( T^{28} - \)\(21\!\cdots\!52\)\( T^{29} + \)\(18\!\cdots\!48\)\( T^{30} - \)\(78\!\cdots\!76\)\( T^{31} + \)\(16\!\cdots\!81\)\( T^{32} \))
$71$ (\( 1 + 16 T + 71 T^{2} \))(\( ( 1 + 12 T + 154 T^{2} + 852 T^{3} + 5041 T^{4} )^{2} \))(\( ( 1 - 2 T + 71 T^{2} )^{2} \))(\( ( 1 - 12 T + 71 T^{2} )^{2} \))(\( ( 1 + 6 T + 71 T^{2} )^{2} \))(\( ( 1 + 71 T^{2} )^{2} \))(\( ( 1 + 4 T + 144 T^{2} + 284 T^{3} + 5041 T^{4} )^{4} \))(\( ( 1 - 6 T + 71 T^{2} )^{4} \))(\( ( 1 + 6 T + 71 T^{2} )^{4} \))(\( ( 1 - 4 T + 94 T^{2} + 964 T^{3} - 1158 T^{4} + 68444 T^{5} + 473854 T^{6} - 1431644 T^{7} + 25411681 T^{8} )^{4} \))
$73$ (\( 1 - 2 T + 73 T^{2} \))(\( 1 - 236 T^{2} + 24198 T^{4} - 1257644 T^{6} + 28398241 T^{8} \))(\( 1 - 4 T - 57 T^{2} - 292 T^{3} + 5329 T^{4} \))(\( 1 - 4 T - 57 T^{2} - 292 T^{3} + 5329 T^{4} \))(\( 1 - 16 T + 183 T^{2} - 1168 T^{3} + 5329 T^{4} \))(\( 1 - 4 T - 57 T^{2} - 292 T^{3} + 5329 T^{4} \))(\( 1 + 1220 T^{4} + 55730374 T^{8} + 34645854020 T^{12} + 806460091894081 T^{16} \))(\( ( 1 - 97 T^{2} + 5329 T^{4} )( 1 + 143 T^{2} + 5329 T^{4} ) \))(\( ( 1 - 6 T - 37 T^{2} - 438 T^{3} + 5329 T^{4} )( 1 + 6 T - 37 T^{2} + 438 T^{3} + 5329 T^{4} ) \))(\( 1 + 12 T + 72 T^{2} + 288 T^{3} + 12064 T^{4} + 83700 T^{5} + 177264 T^{6} - 3613524 T^{7} + 19255890 T^{8} - 96865032 T^{9} - 2225167272 T^{10} - 39441388620 T^{11} + 6056513120 T^{12} - 1050679100940 T^{13} - 2551121523144 T^{14} - 71385024146568 T^{15} + 865631164724531 T^{16} - 5211106762699464 T^{17} - 13594926596834376 T^{18} - 408732031810375980 T^{19} + 171994319201421920 T^{20} - 81764822336595471660 T^{21} - \)\(33\!\cdots\!08\)\( T^{22} - \)\(10\!\cdots\!04\)\( T^{23} + \)\(15\!\cdots\!90\)\( T^{24} - \)\(21\!\cdots\!12\)\( T^{25} + \)\(76\!\cdots\!36\)\( T^{26} + \)\(26\!\cdots\!00\)\( T^{27} + \)\(27\!\cdots\!44\)\( T^{28} + \)\(48\!\cdots\!04\)\( T^{29} + \)\(87\!\cdots\!48\)\( T^{30} + \)\(10\!\cdots\!84\)\( T^{31} + \)\(65\!\cdots\!61\)\( T^{32} \))
$79$ (\( 1 + 8 T + 79 T^{2} \))(\( ( 1 + 4 T + 138 T^{2} + 316 T^{3} + 6241 T^{4} )^{2} \))(\( 1 + 10 T + 21 T^{2} + 790 T^{3} + 6241 T^{4} \))(\( 1 - 10 T + 21 T^{2} - 790 T^{3} + 6241 T^{4} \))(\( 1 + 2 T - 75 T^{2} + 158 T^{3} + 6241 T^{4} \))(\( 1 - 10 T + 21 T^{2} - 790 T^{3} + 6241 T^{4} \))(\( ( 1 - 208 T^{2} + 22498 T^{4} - 1298128 T^{6} + 38950081 T^{8} )^{2} \))(\( ( 1 + 10 T + 21 T^{2} + 790 T^{3} + 6241 T^{4} )^{2} \))(\( ( 1 - 14 T + 117 T^{2} - 1106 T^{3} + 6241 T^{4} )^{2} \))(\( 1 + 344 T^{2} + 56836 T^{4} + 6614224 T^{6} + 630756554 T^{8} + 45125337896 T^{10} + 2052833339792 T^{12} + 54520165812136 T^{14} + 1660868031271123 T^{16} + 340260354833540776 T^{18} + 79958024864398923152 T^{20} + \)\(10\!\cdots\!16\)\( T^{22} + \)\(95\!\cdots\!94\)\( T^{24} + \)\(62\!\cdots\!24\)\( T^{26} + \)\(33\!\cdots\!76\)\( T^{28} + \)\(12\!\cdots\!64\)\( T^{30} + \)\(23\!\cdots\!21\)\( T^{32} \))
$83$ (\( 1 - 8 T + 83 T^{2} \))(\( ( 1 - 160 T^{2} + 6889 T^{4} )^{2} \))(\( ( 1 + 7 T + 83 T^{2} )^{2} \))(\( ( 1 + 12 T + 83 T^{2} )^{2} \))(\( ( 1 - 3 T + 83 T^{2} )^{2} \))(\( ( 1 + 83 T^{2} )^{2} \))(\( 1 - 4224 T^{4} + 99283874 T^{8} - 200463947904 T^{12} + 2252292232139041 T^{16} \))(\( ( 1 - 85 T^{2} + 6889 T^{4} )^{2} \))(\( ( 1 - 130 T^{2} + 6889 T^{4} )^{2} \))(\( 1 - 25638 T^{4} + 318852241 T^{8} - 2957648820318 T^{12} + 22747905510477444 T^{16} - \)\(14\!\cdots\!78\)\( T^{20} + \)\(71\!\cdots\!81\)\( T^{24} - \)\(27\!\cdots\!18\)\( T^{28} + \)\(50\!\cdots\!81\)\( T^{32} \))
$89$ (\( 1 - 10 T + 89 T^{2} \))(\( ( 1 - 10 T + 89 T^{2} )^{4} \))(\( 1 + T - 88 T^{2} + 89 T^{3} + 7921 T^{4} \))(\( 1 + 6 T - 53 T^{2} + 534 T^{3} + 7921 T^{4} \))(\( 1 - 15 T + 136 T^{2} - 1335 T^{3} + 7921 T^{4} \))(\( 1 + 6 T - 53 T^{2} + 534 T^{3} + 7921 T^{4} \))(\( ( 1 - 60 T^{2} + 10470 T^{4} - 475260 T^{6} + 62742241 T^{8} )^{2} \))(\( ( 1 + 7 T - 40 T^{2} + 623 T^{3} + 7921 T^{4} )^{2} \))(\( ( 1 - 2 T - 85 T^{2} - 178 T^{3} + 7921 T^{4} )^{2} \))(\( 1 - 522 T^{2} + 144915 T^{4} - 27754278 T^{6} + 4091335849 T^{8} - 494840689332 T^{10} + 51768593923950 T^{12} - 4926383904774360 T^{14} + 445675116984224850 T^{16} - 39021886909717705560 T^{18} + \)\(32\!\cdots\!50\)\( T^{20} - \)\(24\!\cdots\!52\)\( T^{22} + \)\(16\!\cdots\!69\)\( T^{24} - \)\(86\!\cdots\!78\)\( T^{26} + \)\(35\!\cdots\!15\)\( T^{28} - \)\(10\!\cdots\!02\)\( T^{30} + \)\(15\!\cdots\!61\)\( T^{32} \))
$97$ (\( 1 - 2 T + 97 T^{2} \))(\( 1 - 124 T^{2} + 8838 T^{4} - 1166716 T^{6} + 88529281 T^{8} \))(\( ( 1 - 14 T + 97 T^{2} )^{2} \))(\( ( 1 - 14 T + 97 T^{2} )^{2} \))(\( ( 1 - 14 T + 97 T^{2} )^{2} \))(\( ( 1 + 10 T + 97 T^{2} )^{2} \))(\( 1 - 11900 T^{4} + 108781062 T^{8} - 1053498443900 T^{12} + 7837433594376961 T^{16} \))(\( ( 1 - 97 T^{2} )^{4} \))(\( ( 1 - 50 T^{2} + 9409 T^{4} )^{2} \))(\( ( 1 + 3868 T^{4} + 90482118 T^{8} + 342431258908 T^{12} + 7837433594376961 T^{16} )^{2} \))
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