Properties

Label 72.2
Level 72
Weight 2
Dimension 55
Nonzero newspaces 6
Newform subspaces 10
Sturm bound 576
Trace bound 2

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Defining parameters

Level: \( N \) = \( 72 = 2^{3} \cdot 3^{2} \)
Weight: \( k \) = \( 2 \)
Nonzero newspaces: \( 6 \)
Newform subspaces: \( 10 \)
Sturm bound: \(576\)
Trace bound: \(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_1(72))\).

Total New Old
Modular forms 192 73 119
Cusp forms 97 55 42
Eisenstein series 95 18 77

Trace form

\( 55q - 2q^{2} - 3q^{3} - 2q^{4} + 4q^{5} - 8q^{6} - 2q^{7} - 14q^{8} - 9q^{9} + O(q^{10}) \) \( 55q - 2q^{2} - 3q^{3} - 2q^{4} + 4q^{5} - 8q^{6} - 2q^{7} - 14q^{8} - 9q^{9} - 24q^{10} - 17q^{11} - 14q^{12} - 2q^{13} - 14q^{14} - 24q^{15} - 10q^{16} - 20q^{17} - 22q^{19} + 10q^{20} - 12q^{21} + 14q^{22} - 18q^{23} + 24q^{24} - 8q^{25} + 28q^{26} + 8q^{28} + 6q^{29} + 46q^{30} - 16q^{31} + 38q^{32} - 7q^{33} - 14q^{34} + 36q^{35} + 50q^{36} + 6q^{37} + 38q^{38} + 36q^{39} + 10q^{40} - 5q^{41} + 50q^{42} + 25q^{43} + 42q^{44} + 2q^{45} + 16q^{46} + 42q^{47} + 20q^{48} - 52q^{49} + 28q^{50} + 9q^{51} - 26q^{52} - 14q^{53} - 12q^{54} - 16q^{55} - 32q^{56} - 29q^{57} - 10q^{58} + q^{59} - 66q^{60} - 8q^{61} - 64q^{62} + 32q^{63} - 20q^{64} - 34q^{65} - 104q^{66} - 25q^{67} - 84q^{68} + 22q^{69} - 46q^{70} + 8q^{71} - 102q^{72} - 24q^{73} - 110q^{74} + 49q^{75} - 34q^{76} + 12q^{77} - 74q^{78} - 4q^{79} - 96q^{80} + 11q^{81} + 20q^{82} + 32q^{83} - 64q^{84} + 8q^{85} - 50q^{86} + 18q^{87} + 14q^{88} + 22q^{89} - 14q^{90} + 24q^{91} + 54q^{92} - 10q^{93} + 24q^{94} + 64q^{95} + 48q^{96} + 51q^{97} + 96q^{98} + 30q^{99} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_1(72))\)

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
72.2.a \(\chi_{72}(1, \cdot)\) 72.2.a.a 1 1
72.2.c \(\chi_{72}(71, \cdot)\) None 0 1
72.2.d \(\chi_{72}(37, \cdot)\) 72.2.d.a 2 1
72.2.d.b 2
72.2.f \(\chi_{72}(35, \cdot)\) 72.2.f.a 4 1
72.2.i \(\chi_{72}(25, \cdot)\) 72.2.i.a 2 2
72.2.i.b 4
72.2.l \(\chi_{72}(11, \cdot)\) 72.2.l.a 4 2
72.2.l.b 16
72.2.n \(\chi_{72}(13, \cdot)\) 72.2.n.a 4 2
72.2.n.b 16
72.2.o \(\chi_{72}(23, \cdot)\) None 0 2

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_1(72))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_1(72)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_1(18))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(24))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(36))\)\(^{\oplus 2}\)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ (\( 1 + 2 T^{2} \))(\( 1 - 2 T + 2 T^{2} \))(\( 1 - 2 T^{2} + 4 T^{4} \))(\( 1 - 2 T^{2} + 4 T^{4} \))(\( 1 + 3 T + 7 T^{2} + 12 T^{3} + 16 T^{4} + 12 T^{5} - 8 T^{6} - 36 T^{7} - 68 T^{8} - 72 T^{9} - 32 T^{10} + 96 T^{11} + 256 T^{12} + 384 T^{13} + 448 T^{14} + 384 T^{15} + 256 T^{16} \))(\( 1 + 2 T + 2 T^{2} + 4 T^{3} + 4 T^{4} \))(\( 1 - T + T^{2} + 2 T^{4} - 4 T^{5} - 8 T^{7} + 4 T^{8} - 16 T^{9} - 32 T^{11} + 32 T^{12} + 64 T^{14} - 128 T^{15} + 256 T^{16} \))
$3$ (\( 1 + 3 T^{2} \))(\( 1 - T - 2 T^{2} - 3 T^{3} + 9 T^{4} \))(\( 1 - 2 T + T^{2} - 6 T^{3} + 9 T^{4} \))(\( ( 1 + 3 T + 6 T^{2} + 15 T^{3} + 30 T^{4} + 45 T^{5} + 54 T^{6} + 81 T^{7} + 81 T^{8} )^{2} \))(\( 1 + 3 T^{2} + 9 T^{4} \))(\( 1 - T^{2} - 2 T^{4} + 9 T^{6} + 18 T^{8} + 81 T^{10} - 162 T^{12} - 729 T^{14} + 6561 T^{16} \))
$5$ (\( 1 - 2 T + 5 T^{2} \))(\( 1 - 2 T^{2} + 25 T^{4} \))(\( ( 1 - 4 T + 5 T^{2} )( 1 + 4 T + 5 T^{2} ) \))(\( ( 1 + 4 T^{2} + 25 T^{4} )^{2} \))(\( 1 - T - 4 T^{2} - 5 T^{3} + 25 T^{4} \))(\( 1 - T - T^{2} + 8 T^{3} - 26 T^{4} + 40 T^{5} - 25 T^{6} - 125 T^{7} + 625 T^{8} \))(\( ( 1 - 5 T^{2} + 25 T^{4} )^{2} \))(\( 1 - 13 T^{2} + 48 T^{4} + 103 T^{6} - 1099 T^{8} + 3648 T^{10} - 5222 T^{12} - 177298 T^{14} + 1667616 T^{16} - 4432450 T^{18} - 3263750 T^{20} + 57000000 T^{22} - 429296875 T^{24} + 1005859375 T^{26} + 11718750000 T^{28} - 79345703125 T^{30} + 152587890625 T^{32} \))(\( ( 1 - 4 T + 11 T^{2} - 20 T^{3} + 25 T^{4} )( 1 + 4 T + 11 T^{2} + 20 T^{3} + 25 T^{4} ) \))(\( 1 + 19 T^{2} + 176 T^{4} + 1031 T^{6} + 3893 T^{8} + 4928 T^{10} - 61926 T^{12} - 631122 T^{14} - 3717344 T^{16} - 15778050 T^{18} - 38703750 T^{20} + 77000000 T^{22} + 1520703125 T^{24} + 10068359375 T^{26} + 42968750000 T^{28} + 115966796875 T^{30} + 152587890625 T^{32} \))
$7$ (\( 1 + 7 T^{2} \))(\( ( 1 - 2 T + 7 T^{2} )^{2} \))(\( ( 1 + 2 T + 7 T^{2} )^{2} \))(\( ( 1 - 4 T + 7 T^{2} )^{2}( 1 + 4 T + 7 T^{2} )^{2} \))(\( 1 - 3 T + 2 T^{2} - 21 T^{3} + 49 T^{4} \))(\( 1 + 3 T + T^{2} - 18 T^{3} - 48 T^{4} - 126 T^{5} + 49 T^{6} + 1029 T^{7} + 2401 T^{8} \))(\( ( 1 + 7 T^{2} + 49 T^{4} )^{2} \))(\( 1 + 23 T^{2} + 204 T^{4} + 1399 T^{6} + 12029 T^{8} + 45000 T^{10} - 343382 T^{12} - 4039462 T^{14} - 24407256 T^{16} - 197933638 T^{18} - 824460182 T^{20} + 5294205000 T^{22} + 69344791229 T^{24} + 395182873351 T^{26} + 2823622589004 T^{28} + 15599130675527 T^{30} + 33232930569601 T^{32} \))(\( ( 1 - T + 7 T^{2} )^{2}( 1 + 5 T + 7 T^{2} )^{2} \))(\( ( 1 - 3 T - 14 T^{2} + 39 T^{3} + 139 T^{4} - 252 T^{5} - 1208 T^{6} + 666 T^{7} + 9424 T^{8} + 4662 T^{9} - 59192 T^{10} - 86436 T^{11} + 333739 T^{12} + 655473 T^{13} - 1647086 T^{14} - 2470629 T^{15} + 5764801 T^{16} )^{2} \))
$11$ (\( 1 + 4 T + 11 T^{2} \))(\( 1 + 10 T^{2} + 121 T^{4} \))(\( ( 1 - 11 T^{2} )^{2} \))(\( ( 1 - 6 T + 11 T^{2} )^{2}( 1 + 6 T + 11 T^{2} )^{2} \))(\( 1 + 5 T + 14 T^{2} + 55 T^{3} + 121 T^{4} \))(\( ( 1 + T - 10 T^{2} + 11 T^{3} + 121 T^{4} )^{2} \))(\( ( 1 + 6 T + 11 T^{2} )^{2}( 1 + 6 T + 25 T^{2} + 66 T^{3} + 121 T^{4} ) \))(\( ( 1 - 9 T + 41 T^{2} - 108 T^{3} + 276 T^{4} - 1188 T^{5} + 4961 T^{6} - 11979 T^{7} + 14641 T^{8} )^{2}( 1 + 3 T + 38 T^{2} + 87 T^{3} + 606 T^{4} + 957 T^{5} + 4598 T^{6} + 3993 T^{7} + 14641 T^{8} )^{2} \))(\( 1 + 13 T^{2} + 48 T^{4} + 1573 T^{6} + 14641 T^{8} \))(\( 1 + 48 T^{2} + 1090 T^{4} + 17592 T^{6} + 242041 T^{8} + 2632140 T^{10} + 21031138 T^{12} + 158095260 T^{14} + 1558598596 T^{16} + 19129526460 T^{18} + 307916891458 T^{20} + 4662996570540 T^{22} + 51883637916121 T^{24} + 456291173580792 T^{26} + 3420886930625890 T^{28} + 18227992011995568 T^{30} + 45949729863572161 T^{32} \))
$13$ (\( 1 + 2 T + 13 T^{2} \))(\( ( 1 - 13 T^{2} )^{2} \))(\( ( 1 - 6 T + 13 T^{2} )( 1 + 6 T + 13 T^{2} ) \))(\( ( 1 - 14 T^{2} + 169 T^{4} )^{2} \))(\( ( 1 - 7 T + 13 T^{2} )( 1 + 2 T + 13 T^{2} ) \))(\( 1 + 5 T + T^{2} - 10 T^{3} + 82 T^{4} - 130 T^{5} + 169 T^{6} + 10985 T^{7} + 28561 T^{8} \))(\( ( 1 + 13 T^{2} + 169 T^{4} )^{2} \))(\( 1 + 47 T^{2} + 1116 T^{4} + 17239 T^{6} + 168353 T^{8} + 463056 T^{10} - 26916578 T^{12} - 784598686 T^{14} - 12569281176 T^{16} - 132597177934 T^{18} - 768764384258 T^{20} + 2235082868304 T^{22} + 137330714072513 T^{24} + 2376542540984911 T^{26} + 26000662996688796 T^{28} + 185056690127866583 T^{30} + 665416609183179841 T^{32} \))(\( ( 1 - T^{2} + 169 T^{4} )( 1 + 23 T^{2} + 169 T^{4} ) \))(\( 1 + 51 T^{2} + 1288 T^{4} + 16911 T^{6} + 73645 T^{8} - 1059264 T^{10} - 5456606 T^{12} + 411982806 T^{14} + 9084740848 T^{16} + 69625094214 T^{18} - 155846123966 T^{20} - 5112865008576 T^{22} + 60074488948045 T^{24} + 2331324955658439 T^{26} + 30007933637755528 T^{28} + 200806195670663739 T^{30} + 665416609183179841 T^{32} \))
$17$ (\( 1 + 2 T + 17 T^{2} \))(\( ( 1 + 17 T^{2} )^{2} \))(\( ( 1 - 2 T + 17 T^{2} )^{2} \))(\( ( 1 - 32 T^{2} + 289 T^{4} )^{2} \))(\( ( 1 + 2 T + 17 T^{2} )^{2} \))(\( ( 1 + 5 T + 32 T^{2} + 85 T^{3} + 289 T^{4} )^{2} \))(\( ( 1 - 6 T + 19 T^{2} - 102 T^{3} + 289 T^{4} )( 1 + 6 T + 19 T^{2} + 102 T^{3} + 289 T^{4} ) \))(\( ( 1 - 101 T^{2} + 4882 T^{4} - 146891 T^{6} + 2998666 T^{8} - 42451499 T^{10} + 407749522 T^{12} - 2437894469 T^{14} + 6975757441 T^{16} )^{2} \))(\( ( 1 - 5 T + 17 T^{2} )^{4} \))(\( ( 1 + 7 T + 66 T^{2} + 309 T^{3} + 1702 T^{4} + 5253 T^{5} + 19074 T^{6} + 34391 T^{7} + 83521 T^{8} )^{4} \))
$19$ (\( 1 + 4 T + 19 T^{2} \))(\( ( 1 - 19 T^{2} )^{2} \))(\( 1 - 22 T^{2} + 361 T^{4} \))(\( ( 1 + 4 T + 19 T^{2} )^{4} \))(\( ( 1 + 4 T + 19 T^{2} )^{2} \))(\( ( 1 - 7 T + 42 T^{2} - 133 T^{3} + 361 T^{4} )^{2} \))(\( ( 1 + 2 T - 15 T^{2} + 38 T^{3} + 361 T^{4} )^{2} \))(\( ( 1 + T + 64 T^{2} + 49 T^{3} + 1726 T^{4} + 931 T^{5} + 23104 T^{6} + 6859 T^{7} + 130321 T^{8} )^{4} \))(\( ( 1 - 37 T^{2} + 361 T^{4} )^{2} \))(\( ( 1 - 69 T^{2} + 2306 T^{4} - 53763 T^{6} + 1069146 T^{8} - 19408443 T^{10} + 300520226 T^{12} - 3246165789 T^{14} + 16983563041 T^{16} )^{2} \))
$23$ (\( 1 - 8 T + 23 T^{2} \))(\( ( 1 + 23 T^{2} )^{2} \))(\( ( 1 + 4 T + 23 T^{2} )^{2} \))(\( ( 1 + 22 T^{2} + 529 T^{4} )^{2} \))(\( 1 - T - 22 T^{2} - 23 T^{3} + 529 T^{4} \))(\( 1 + 5 T - 19 T^{2} - 10 T^{3} + 832 T^{4} - 230 T^{5} - 10051 T^{6} + 60835 T^{7} + 279841 T^{8} \))(\( ( 1 - 23 T^{2} + 529 T^{4} )^{2} \))(\( 1 - 85 T^{2} + 3432 T^{4} - 80441 T^{6} + 1017209 T^{8} + 1764696 T^{10} - 386280290 T^{12} + 9323832998 T^{14} - 183562178736 T^{16} + 4932307655942 T^{18} - 108097062633890 T^{20} + 261238341174744 T^{22} + 79658639026700729 T^{24} - 3332389988537139209 T^{26} + 75210991050693741672 T^{28} - \)\(98\!\cdots\!65\)\( T^{30} + \)\(61\!\cdots\!61\)\( T^{32} \))(\( ( 1 + 2 T - 19 T^{2} + 46 T^{3} + 529 T^{4} )^{2} \))(\( ( 1 + 5 T - 32 T^{2} + 45 T^{3} + 1385 T^{4} - 3040 T^{5} - 11142 T^{6} + 48640 T^{7} - 123716 T^{8} + 1118720 T^{9} - 5894118 T^{10} - 36987680 T^{11} + 387579785 T^{12} + 289635435 T^{13} - 4737148448 T^{14} + 17024127235 T^{15} + 78310985281 T^{16} )^{2} \))
$29$ (\( 1 + 6 T + 29 T^{2} \))(\( 1 - 50 T^{2} + 841 T^{4} \))(\( 1 - 22 T^{2} + 841 T^{4} \))(\( ( 1 + 52 T^{2} + 841 T^{4} )^{2} \))(\( 1 - 9 T + 52 T^{2} - 261 T^{3} + 841 T^{4} \))(\( 1 - 3 T - 43 T^{2} + 18 T^{3} + 1602 T^{4} + 522 T^{5} - 36163 T^{6} - 73167 T^{7} + 707281 T^{8} \))(\( ( 1 - 29 T^{2} + 841 T^{4} )^{2} \))(\( 1 - 97 T^{2} + 3780 T^{4} - 80561 T^{6} + 1632089 T^{8} - 54786000 T^{10} + 1038848902 T^{12} + 18974156858 T^{14} - 1329857348520 T^{16} + 15957265917578 T^{18} + 734758090255462 T^{20} - 32587990464306000 T^{22} + 816446667883105529 T^{24} - 33892595421897492761 T^{26} + \)\(13\!\cdots\!80\)\( T^{28} - \)\(28\!\cdots\!57\)\( T^{30} + \)\(25\!\cdots\!21\)\( T^{32} \))(\( ( 1 + 29 T^{2} + 841 T^{4} )^{2} \))(\( 1 + 123 T^{2} + 7912 T^{4} + 340647 T^{6} + 10067965 T^{8} + 144056448 T^{10} - 3705324014 T^{12} - 328820871018 T^{14} - 12137186779472 T^{16} - 276538352526138 T^{18} - 2620705273945934 T^{20} + 85688134810823808 T^{22} + 5036463377066894365 T^{24} + \)\(14\!\cdots\!47\)\( T^{26} + \)\(27\!\cdots\!92\)\( T^{28} + \)\(36\!\cdots\!63\)\( T^{30} + \)\(25\!\cdots\!21\)\( T^{32} \))
$31$ (\( 1 - 8 T + 31 T^{2} \))(\( ( 1 + 10 T + 31 T^{2} )^{2} \))(\( ( 1 - 2 T + 31 T^{2} )^{2} \))(\( ( 1 - 50 T^{2} + 961 T^{4} )^{2} \))(\( 1 - T - 30 T^{2} - 31 T^{3} + 961 T^{4} \))(\( 1 + 7 T - 17 T^{2} + 28 T^{3} + 1876 T^{4} + 868 T^{5} - 16337 T^{6} + 208537 T^{7} + 923521 T^{8} \))(\( ( 1 + 31 T^{2} + 961 T^{4} )^{2} \))(\( 1 + 131 T^{2} + 7152 T^{4} + 312151 T^{6} + 16524593 T^{8} + 706250232 T^{10} + 22761450214 T^{12} + 818587141454 T^{14} + 29243993208000 T^{16} + 786662242937294 T^{18} + 21020677263083494 T^{20} + 626799680607103992 T^{22} + 14093677267060286513 T^{24} + \)\(25\!\cdots\!51\)\( T^{26} + \)\(56\!\cdots\!72\)\( T^{28} + \)\(99\!\cdots\!51\)\( T^{30} + \)\(72\!\cdots\!81\)\( T^{32} \))(\( ( 1 - 11 T + 31 T^{2} )^{2}( 1 + 7 T + 31 T^{2} )^{2} \))(\( ( 1 + 5 T - 48 T^{2} - 155 T^{3} + 1121 T^{4} - 2040 T^{5} - 44678 T^{6} + 79040 T^{7} + 1805724 T^{8} + 2450240 T^{9} - 42935558 T^{10} - 60773640 T^{11} + 1035267041 T^{12} - 4437518405 T^{13} - 42600176688 T^{14} + 137563070555 T^{15} + 852891037441 T^{16} )^{2} \))
$37$ (\( 1 - 6 T + 37 T^{2} \))(\( ( 1 - 37 T^{2} )^{2} \))(\( 1 - 10 T^{2} + 1369 T^{4} \))(\( ( 1 - 37 T^{2} )^{4} \))(\( ( 1 + 6 T + 37 T^{2} )^{2} \))(\( ( 1 - 6 T + 50 T^{2} - 222 T^{3} + 1369 T^{4} )^{2} \))(\( ( 1 - 37 T^{2} )^{4} \))(\( ( 1 - 140 T^{2} + 8596 T^{4} - 317540 T^{6} + 10470934 T^{8} - 434712260 T^{10} + 16110287956 T^{12} - 359201697260 T^{14} + 3512479453921 T^{16} )^{2} \))(\( ( 1 - 12 T + 37 T^{2} )^{2}( 1 + 12 T + 37 T^{2} )^{2} \))(\( ( 1 - 144 T^{2} + 12668 T^{4} - 733824 T^{6} + 31784838 T^{8} - 1004605056 T^{10} + 23741871548 T^{12} - 369464602896 T^{14} + 3512479453921 T^{16} )^{2} \))
$41$ (\( 1 - 6 T + 41 T^{2} \))(\( ( 1 + 41 T^{2} )^{2} \))(\( ( 1 + 2 T + 41 T^{2} )^{2} \))(\( ( 1 - 80 T^{2} + 1681 T^{4} )^{2} \))(\( 1 + 3 T - 32 T^{2} + 123 T^{3} + 1681 T^{4} \))(\( 1 - 12 T + 59 T^{2} - 36 T^{3} - 360 T^{4} - 1476 T^{5} + 99179 T^{6} - 827052 T^{7} + 2825761 T^{8} \))(\( ( 1 - 6 T + 41 T^{2} )^{2}( 1 - 6 T - 5 T^{2} - 246 T^{3} + 1681 T^{4} ) \))(\( ( 1 + 18 T + 292 T^{2} + 3312 T^{3} + 34963 T^{4} + 303732 T^{5} + 2503888 T^{6} + 17908938 T^{7} + 122450608 T^{8} + 734266458 T^{9} + 4209035728 T^{10} + 20933513172 T^{11} + 98797081843 T^{12} + 383715737712 T^{13} + 1387030438372 T^{14} + 3505576929858 T^{15} + 7984925229121 T^{16} )^{2} \))(\( ( 1 - 5 T - 16 T^{2} - 205 T^{3} + 1681 T^{4} )^{2} \))(\( ( 1 + 4 T - 74 T^{2} + 288 T^{3} + 4517 T^{4} - 22796 T^{5} - 11538 T^{6} + 738968 T^{7} - 2462804 T^{8} + 30297688 T^{9} - 19395378 T^{10} - 1571123116 T^{11} + 12763962437 T^{12} + 33366585888 T^{13} - 351507713834 T^{14} + 779017095524 T^{15} + 7984925229121 T^{16} )^{2} \))
$43$ (\( 1 - 4 T + 43 T^{2} \))(\( ( 1 - 43 T^{2} )^{2} \))(\( 1 - 70 T^{2} + 1849 T^{4} \))(\( ( 1 - 8 T + 43 T^{2} )^{4} \))(\( 1 + T - 42 T^{2} + 43 T^{3} + 1849 T^{4} \))(\( 1 + 8 T - 5 T^{2} - 136 T^{3} + 160 T^{4} - 5848 T^{5} - 9245 T^{6} + 636056 T^{7} + 3418801 T^{8} \))(\( ( 1 + 10 T + 43 T^{2} )^{2}( 1 - 10 T + 57 T^{2} - 430 T^{3} + 1849 T^{4} ) \))(\( ( 1 - 4 T - 96 T^{2} + 868 T^{3} + 4061 T^{4} - 55182 T^{5} + 78652 T^{6} + 1341518 T^{7} - 8451684 T^{8} + 57685274 T^{9} + 145427548 T^{10} - 4387355274 T^{11} + 13883750861 T^{12} + 127603328524 T^{13} - 606850852704 T^{14} - 1087274444428 T^{15} + 11688200277601 T^{16} )^{2} \))(\( 1 - 35 T^{2} - 624 T^{4} - 64715 T^{6} + 3418801 T^{8} \))(\( 1 + 324 T^{2} + 58258 T^{4} + 7305024 T^{6} + 705172105 T^{8} + 54954910764 T^{10} + 3558033934834 T^{12} + 194436327001344 T^{14} + 9048929543300068 T^{16} + 359512768625485056 T^{18} + 12164209974444414034 T^{20} + \)\(34\!\cdots\!36\)\( T^{22} + \)\(82\!\cdots\!05\)\( T^{24} + \)\(15\!\cdots\!76\)\( T^{26} + \)\(23\!\cdots\!58\)\( T^{28} + \)\(23\!\cdots\!76\)\( T^{30} + \)\(13\!\cdots\!01\)\( T^{32} \))
$47$ (\( 1 + 47 T^{2} \))(\( ( 1 + 47 T^{2} )^{2} \))(\( ( 1 - 12 T + 47 T^{2} )^{2} \))(\( ( 1 + 70 T^{2} + 2209 T^{4} )^{2} \))(\( 1 - 3 T - 38 T^{2} - 141 T^{3} + 2209 T^{4} \))(\( 1 + 3 T - 79 T^{2} - 18 T^{3} + 5112 T^{4} - 846 T^{5} - 174511 T^{6} + 311469 T^{7} + 4879681 T^{8} \))(\( ( 1 - 47 T^{2} + 2209 T^{4} )^{2} \))(\( 1 - 265 T^{2} + 38172 T^{4} - 3663305 T^{6} + 256619789 T^{8} - 13483362840 T^{10} + 543517455226 T^{12} - 17957421673750 T^{14} + 672442507889160 T^{16} - 39667944477313750 T^{18} + 2652191799434662906 T^{20} - \)\(14\!\cdots\!60\)\( T^{22} + \)\(61\!\cdots\!29\)\( T^{24} - \)\(19\!\cdots\!45\)\( T^{26} + \)\(44\!\cdots\!52\)\( T^{28} - \)\(68\!\cdots\!85\)\( T^{30} + \)\(56\!\cdots\!21\)\( T^{32} \))(\( ( 1 - 6 T - 11 T^{2} - 282 T^{3} + 2209 T^{4} )^{2} \))(\( ( 1 - 3 T - 98 T^{2} + 219 T^{3} + 3523 T^{4} + 1116 T^{5} - 253856 T^{6} - 226218 T^{7} + 18909448 T^{8} - 10632246 T^{9} - 560767904 T^{10} + 115866468 T^{11} + 17191116163 T^{12} + 50226556533 T^{13} - 1056363102242 T^{14} - 1519869361389 T^{15} + 23811286661761 T^{16} )^{2} \))
$53$ (\( 1 - 2 T + 53 T^{2} \))(\( 1 + 94 T^{2} + 2809 T^{4} \))(\( 1 - 70 T^{2} + 2809 T^{4} \))(\( ( 1 + 52 T^{2} + 2809 T^{4} )^{2} \))(\( ( 1 - 2 T + 53 T^{2} )^{2} \))(\( ( 1 + 10 T + 98 T^{2} + 530 T^{3} + 2809 T^{4} )^{2} \))(\( ( 1 + 53 T^{2} )^{4} \))(\( ( 1 + 196 T^{2} + 21556 T^{4} + 1665484 T^{6} + 98315734 T^{8} + 4678344556 T^{10} + 170087208436 T^{12} + 4344214781284 T^{14} + 62259690411361 T^{16} )^{2} \))(\( ( 1 - 53 T^{2} )^{4} \))(\( ( 1 - 264 T^{2} + 35516 T^{4} - 3123720 T^{6} + 194863110 T^{8} - 8774529480 T^{10} + 280238323196 T^{12} - 5851391338056 T^{14} + 62259690411361 T^{16} )^{2} \))
$59$ (\( 1 + 4 T + 59 T^{2} \))(\( 1 + 10 T^{2} + 3481 T^{4} \))(\( 1 - 102 T^{2} + 3481 T^{4} \))(\( ( 1 + 10 T^{2} + 3481 T^{4} )^{2} \))(\( 1 + 11 T + 62 T^{2} + 649 T^{3} + 3481 T^{4} \))(\( ( 1 + 7 T - 10 T^{2} + 413 T^{3} + 3481 T^{4} )^{2} \))(\( ( 1 - 6 T + 59 T^{2} )^{2}( 1 - 6 T - 23 T^{2} - 354 T^{3} + 3481 T^{4} ) \))(\( ( 1 - 6 T + 178 T^{2} - 996 T^{3} + 15871 T^{4} - 89238 T^{5} + 1194346 T^{6} - 6437052 T^{7} + 79012984 T^{8} - 379786068 T^{9} + 4157518426 T^{10} - 18327611202 T^{11} + 192314636431 T^{12} - 712064601804 T^{13} + 7508134988098 T^{14} - 14931908908914 T^{15} + 146830437604321 T^{16} )^{2} \))(\( 1 + 117 T^{2} + 10208 T^{4} + 407277 T^{6} + 12117361 T^{8} \))(\( 1 + 364 T^{2} + 70130 T^{4} + 9549968 T^{6} + 1028964713 T^{8} + 92991430988 T^{10} + 7291973853618 T^{12} + 506327334867240 T^{14} + 31497451193778532 T^{16} + 1762525452672862440 T^{18} + 88359479586850462098 T^{20} + \)\(39\!\cdots\!08\)\( T^{22} + \)\(15\!\cdots\!73\)\( T^{24} + \)\(48\!\cdots\!68\)\( T^{26} + \)\(12\!\cdots\!30\)\( T^{28} + \)\(22\!\cdots\!04\)\( T^{30} + \)\(21\!\cdots\!41\)\( T^{32} \))
$61$ (\( 1 + 2 T + 61 T^{2} \))(\( ( 1 - 61 T^{2} )^{2} \))(\( ( 1 - 61 T^{2} )^{2} \))(\( ( 1 + 70 T^{2} + 3721 T^{4} )^{2} \))(\( 1 + 7 T - 12 T^{2} + 427 T^{3} + 3721 T^{4} \))(\( 1 - T - 113 T^{2} + 8 T^{3} + 9214 T^{4} + 488 T^{5} - 420473 T^{6} - 226981 T^{7} + 13845841 T^{8} \))(\( ( 1 + 61 T^{2} + 3721 T^{4} )^{2} \))(\( 1 + 299 T^{2} + 43416 T^{4} + 4465735 T^{6} + 393470093 T^{8} + 31461221184 T^{10} + 2332722797890 T^{12} + 164022774523334 T^{14} + 10603540284680400 T^{16} + 610328744001325814 T^{18} + 32298508956660075490 T^{20} + \)\(16\!\cdots\!24\)\( T^{22} + \)\(75\!\cdots\!33\)\( T^{24} + \)\(31\!\cdots\!35\)\( T^{26} + \)\(11\!\cdots\!36\)\( T^{28} + \)\(29\!\cdots\!59\)\( T^{30} + \)\(36\!\cdots\!61\)\( T^{32} \))(\( ( 1 - 10 T + 39 T^{2} - 610 T^{3} + 3721 T^{4} )( 1 + 10 T + 39 T^{2} + 610 T^{3} + 3721 T^{4} ) \))(\( 1 + 323 T^{2} + 52152 T^{4} + 6035887 T^{6} + 584536925 T^{8} + 49803739968 T^{10} + 3801157073266 T^{12} + 264102902456582 T^{14} + 16823476283802768 T^{16} + 982726900040941622 T^{18} + 52630216452466386706 T^{20} + \)\(25\!\cdots\!48\)\( T^{22} + \)\(11\!\cdots\!25\)\( T^{24} + \)\(43\!\cdots\!87\)\( T^{26} + \)\(13\!\cdots\!92\)\( T^{28} + \)\(31\!\cdots\!43\)\( T^{30} + \)\(36\!\cdots\!61\)\( T^{32} \))
$67$ (\( 1 + 4 T + 67 T^{2} \))(\( ( 1 - 67 T^{2} )^{2} \))(\( 1 + 10 T^{2} + 4489 T^{4} \))(\( ( 1 + 4 T + 67 T^{2} )^{4} \))(\( 1 - T - 66 T^{2} - 67 T^{3} + 4489 T^{4} \))(\( 1 + 4 T - 89 T^{2} - 116 T^{3} + 5464 T^{4} - 7772 T^{5} - 399521 T^{6} + 1203052 T^{7} + 20151121 T^{8} \))(\( ( 1 - 14 T + 67 T^{2} )^{2}( 1 + 14 T + 129 T^{2} + 938 T^{3} + 4489 T^{4} ) \))(\( ( 1 + 8 T - 138 T^{2} - 1052 T^{3} + 11279 T^{4} + 57198 T^{5} - 930218 T^{6} - 1298482 T^{7} + 71382744 T^{8} - 86998294 T^{9} - 4175748602 T^{10} + 17203042074 T^{11} + 227284493759 T^{12} - 1420331612564 T^{13} - 12483256739322 T^{14} + 48485692842584 T^{15} + 406067677556641 T^{16} )^{2} \))(\( 1 + 125 T^{2} + 11136 T^{4} + 561125 T^{6} + 20151121 T^{8} \))(\( 1 + 296 T^{2} + 39450 T^{4} + 3796936 T^{6} + 362561921 T^{8} + 33090030780 T^{10} + 2631719645962 T^{12} + 193036342550180 T^{14} + 13423375489686516 T^{16} + 866540141707758020 T^{18} + 53032101023857423402 T^{20} + \)\(29\!\cdots\!20\)\( T^{22} + \)\(14\!\cdots\!61\)\( T^{24} + \)\(69\!\cdots\!64\)\( T^{26} + \)\(32\!\cdots\!50\)\( T^{28} + \)\(10\!\cdots\!84\)\( T^{30} + \)\(16\!\cdots\!81\)\( T^{32} \))
$71$ (\( 1 + 8 T + 71 T^{2} \))(\( ( 1 + 71 T^{2} )^{2} \))(\( ( 1 + 12 T + 71 T^{2} )^{2} \))(\( ( 1 - 74 T^{2} + 5041 T^{4} )^{2} \))(\( ( 1 - 4 T + 71 T^{2} )^{2} \))(\( ( 1 + 4 T + 71 T^{2} )^{4} \))(\( ( 1 + 71 T^{2} )^{4} \))(\( ( 1 + 400 T^{2} + 73324 T^{4} + 8374048 T^{6} + 685441990 T^{8} + 42213575968 T^{10} + 1863286097644 T^{12} + 51240113568400 T^{14} + 645753531245761 T^{16} )^{2} \))(\( ( 1 + 6 T + 71 T^{2} )^{4} \))(\( ( 1 - 18 T + 332 T^{2} - 3582 T^{3} + 36198 T^{4} - 254322 T^{5} + 1673612 T^{6} - 6442398 T^{7} + 25411681 T^{8} )^{4} \))
$73$ (\( 1 - 10 T + 73 T^{2} \))(\( ( 1 - 14 T + 73 T^{2} )^{2} \))(\( ( 1 + 6 T + 73 T^{2} )^{2} \))(\( ( 1 + 4 T + 73 T^{2} )^{4} \))(\( ( 1 + 2 T + 73 T^{2} )^{2} \))(\( ( 1 + 7 T + 84 T^{2} + 511 T^{3} + 5329 T^{4} )^{2} \))(\( ( 1 + 2 T - 69 T^{2} + 146 T^{3} + 5329 T^{4} )^{2} \))(\( ( 1 + T + 214 T^{2} - 5 T^{3} + 20758 T^{4} - 365 T^{5} + 1140406 T^{6} + 389017 T^{7} + 28398241 T^{8} )^{4} \))(\( ( 1 - 9 T + 73 T^{2} )^{4} \))(\( ( 1 + 11 T + 278 T^{2} + 2325 T^{3} + 29966 T^{4} + 169725 T^{5} + 1481462 T^{6} + 4279187 T^{7} + 28398241 T^{8} )^{4} \))
$79$ (\( 1 + 8 T + 79 T^{2} \))(\( ( 1 + 10 T + 79 T^{2} )^{2} \))(\( ( 1 - 10 T + 79 T^{2} )^{2} \))(\( ( 1 - 146 T^{2} + 6241 T^{4} )^{2} \))(\( 1 + T - 78 T^{2} + 79 T^{3} + 6241 T^{4} \))(\( 1 - 7 T - 113 T^{2} - 28 T^{3} + 16132 T^{4} - 2212 T^{5} - 705233 T^{6} - 3451273 T^{7} + 38950081 T^{8} \))(\( ( 1 + 79 T^{2} + 6241 T^{4} )^{2} \))(\( 1 + 383 T^{2} + 74724 T^{4} + 9593503 T^{6} + 916548293 T^{8} + 73505234952 T^{10} + 5745220732498 T^{12} + 475438917658202 T^{14} + 38877973133064792 T^{16} + 2967214285104838682 T^{18} + \)\(22\!\cdots\!38\)\( T^{20} + \)\(17\!\cdots\!92\)\( T^{22} + \)\(13\!\cdots\!73\)\( T^{24} + \)\(90\!\cdots\!03\)\( T^{26} + \)\(44\!\cdots\!84\)\( T^{28} + \)\(14\!\cdots\!23\)\( T^{30} + \)\(23\!\cdots\!21\)\( T^{32} \))(\( ( 1 - 14 T + 117 T^{2} - 1106 T^{3} + 6241 T^{4} )^{2} \))(\( ( 1 + 15 T - 80 T^{2} - 1305 T^{3} + 13273 T^{4} + 59400 T^{5} - 1907150 T^{6} - 913560 T^{7} + 190569148 T^{8} - 72171240 T^{9} - 11902523150 T^{10} + 29286516600 T^{11} + 516984425113 T^{12} - 4015558600695 T^{13} - 19446996441680 T^{14} + 288058634792385 T^{15} + 1517108809906561 T^{16} )^{2} \))
$83$ (\( 1 - 4 T + 83 T^{2} \))(\( 1 - 134 T^{2} + 6889 T^{4} \))(\( 1 + 90 T^{2} + 6889 T^{4} \))(\( ( 1 + 34 T^{2} + 6889 T^{4} )^{2} \))(\( 1 + T - 82 T^{2} + 83 T^{3} + 6889 T^{4} \))(\( 1 + 25 T + 311 T^{2} + 3700 T^{3} + 39832 T^{4} + 307100 T^{5} + 2142479 T^{6} + 14294675 T^{7} + 47458321 T^{8} \))(\( ( 1 - 18 T + 241 T^{2} - 1494 T^{3} + 6889 T^{4} )( 1 + 18 T + 241 T^{2} + 1494 T^{3} + 6889 T^{4} ) \))(\( ( 1 - 27 T + 544 T^{2} - 8127 T^{3} + 107593 T^{4} - 1304076 T^{5} + 14403022 T^{6} - 148987620 T^{7} + 1398163588 T^{8} - 12365972460 T^{9} + 99222418558 T^{10} - 745653703812 T^{11} + 5106183131353 T^{12} - 32012583305661 T^{13} + 177855563112736 T^{14} - 732673376719929 T^{15} + 2252292232139041 T^{16} )^{2} \))(\( 1 + 150 T^{2} + 15611 T^{4} + 1033350 T^{6} + 47458321 T^{8} \))(\( 1 + 559 T^{2} + 168260 T^{4} + 35967119 T^{6} + 6055752221 T^{8} + 843016361960 T^{10} + 99713806040442 T^{12} + 10182388433060610 T^{14} + 904920089638581976 T^{16} + 70146473915354542290 T^{18} + \)\(47\!\cdots\!82\)\( T^{20} + \)\(27\!\cdots\!40\)\( T^{22} + \)\(13\!\cdots\!61\)\( T^{24} + \)\(55\!\cdots\!31\)\( T^{26} + \)\(17\!\cdots\!60\)\( T^{28} + \)\(41\!\cdots\!11\)\( T^{30} + \)\(50\!\cdots\!81\)\( T^{32} \))
$89$ (\( 1 - 6 T + 89 T^{2} \))(\( ( 1 + 89 T^{2} )^{2} \))(\( ( 1 - 10 T + 89 T^{2} )^{2} \))(\( ( 1 - 128 T^{2} + 7921 T^{4} )^{2} \))(\( ( 1 + 18 T + 89 T^{2} )^{2} \))(\( ( 1 - 6 T + 89 T^{2} )^{4} \))(\( ( 1 - 18 T + 89 T^{2} )^{2}( 1 + 18 T + 89 T^{2} )^{2} \))(\( ( 1 - 440 T^{2} + 100348 T^{4} - 14946776 T^{6} + 1566592486 T^{8} - 118393412696 T^{10} + 6296058399868 T^{12} - 218671768022840 T^{14} + 3936588805702081 T^{16} )^{2} \))(\( ( 1 + 14 T + 89 T^{2} )^{4} \))(\( ( 1 - 16 T + 408 T^{2} - 4116 T^{3} + 56278 T^{4} - 366324 T^{5} + 3231768 T^{6} - 11279504 T^{7} + 62742241 T^{8} )^{4} \))
$97$ (\( 1 - 2 T + 97 T^{2} \))(\( ( 1 - 2 T + 97 T^{2} )^{2} \))(\( ( 1 + 2 T + 97 T^{2} )^{2} \))(\( ( 1 - 8 T + 97 T^{2} )^{4} \))(\( 1 - 13 T + 72 T^{2} - 1261 T^{3} + 9409 T^{4} \))(\( 1 - 8 T - 113 T^{2} + 136 T^{3} + 15712 T^{4} + 13192 T^{5} - 1063217 T^{6} - 7301384 T^{7} + 88529281 T^{8} \))(\( ( 1 + 10 T + 97 T^{2} )^{2}( 1 - 10 T + 3 T^{2} - 970 T^{3} + 9409 T^{4} ) \))(\( ( 1 - 4 T - 198 T^{2} - 320 T^{3} + 21017 T^{4} + 106308 T^{5} - 1078406 T^{6} - 6931984 T^{7} + 64836612 T^{8} - 672402448 T^{9} - 10146722054 T^{10} + 97024441284 T^{11} + 1860619898777 T^{12} - 2747948882240 T^{13} - 164928456975942 T^{14} - 323193137912452 T^{15} + 7837433594376961 T^{16} )^{2} \))(\( ( 1 + T - 96 T^{2} + 97 T^{3} + 9409 T^{4} )^{2} \))(\( ( 1 - 254 T^{2} + 1560 T^{3} + 35209 T^{4} - 273780 T^{5} - 2055806 T^{6} + 15575820 T^{7} + 104325124 T^{8} + 1510854540 T^{9} - 19343078654 T^{10} - 249871613940 T^{11} + 3117027454729 T^{12} + 13396250800920 T^{13} - 211574889251966 T^{14} + 7837433594376961 T^{16} )^{2} \))
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