Properties

Label 324.4
Level 324
Weight 4
Dimension 3976
Nonzero newspaces 8
Sturm bound 23328
Trace bound 1

Downloads

Learn more about

Defining parameters

Level: \( N \) = \( 324\( 324 = 2^{2} \cdot 3^{4} \) \)
Weight: \( k \) = \( 4 \)
Nonzero newspaces: \( 8 \)
Sturm bound: \(23328\)
Trace bound: \(1\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_1(324))\).

Total New Old
Modular forms 9018 4088 4930
Cusp forms 8478 3976 4502
Eisenstein series 540 112 428

Trace form

\( 3976q - 12q^{2} - 20q^{4} - 36q^{5} - 18q^{6} + 18q^{7} - 9q^{8} - 36q^{9} + O(q^{10}) \) \( 3976q - 12q^{2} - 20q^{4} - 36q^{5} - 18q^{6} + 18q^{7} - 9q^{8} - 36q^{9} - 13q^{10} + 87q^{11} - 18q^{12} - 58q^{13} - 87q^{14} - 200q^{16} - 222q^{17} - 18q^{18} + 270q^{19} - 243q^{20} + 72q^{21} - 474q^{23} - 18q^{24} - 60q^{25} - 27q^{26} - 702q^{27} - 153q^{28} - 798q^{29} - 18q^{30} - 216q^{31} + 678q^{32} + 396q^{33} + 518q^{34} + 1236q^{35} - 18q^{36} + 284q^{37} + 882q^{38} - 625q^{40} - 2895q^{41} - 3573q^{42} - 567q^{43} - 5337q^{44} - 1062q^{45} - 1161q^{46} + 2142q^{47} + 2259q^{48} + 1960q^{49} + 7608q^{50} + 2961q^{51} + 3065q^{52} + 3876q^{53} + 6660q^{54} + 3348q^{55} + 8829q^{56} + 2178q^{57} + 743q^{58} + 2667q^{59} + 1269q^{60} + 140q^{61} - 3771q^{62} - 1998q^{63} - 3863q^{64} - 9450q^{65} - 7434q^{66} - 189q^{67} - 9294q^{68} - 7146q^{69} + 369q^{70} - 600q^{71} - 18q^{72} + 950q^{73} + 5865q^{74} + 4500q^{75} + 3042q^{76} + 7272q^{77} + 225q^{78} - 3528q^{79} + 5724q^{81} - 1930q^{82} + 864q^{83} - 261q^{84} - 7226q^{85} - 8340q^{86} + 5472q^{87} - 1980q^{88} - 13902q^{89} + 12519q^{90} - 5634q^{91} + 15207q^{92} - 18486q^{93} + 2205q^{94} - 10356q^{95} + 2907q^{96} + 3569q^{97} - 5823q^{98} - 7542q^{99} + O(q^{100}) \)

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_1(324))\)

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
324.4.a \(\chi_{324}(1, \cdot)\) 324.4.a.a 1 1
324.4.a.b 1
324.4.a.c 3
324.4.a.d 3
324.4.a.e 4
324.4.b \(\chi_{324}(323, \cdot)\) 324.4.b.a 4 1
324.4.b.b 8
324.4.b.c 24
324.4.b.d 32
324.4.e \(\chi_{324}(109, \cdot)\) 324.4.e.a 2 2
324.4.e.b 2
324.4.e.c 2
324.4.e.d 2
324.4.e.e 2
324.4.e.f 2
324.4.e.g 2
324.4.e.h 2
324.4.e.i 8
324.4.h \(\chi_{324}(107, \cdot)\) n/a 140 2
324.4.i \(\chi_{324}(37, \cdot)\) 324.4.i.a 54 6
324.4.l \(\chi_{324}(35, \cdot)\) n/a 312 6
324.4.m \(\chi_{324}(13, \cdot)\) n/a 486 18
324.4.p \(\chi_{324}(11, \cdot)\) n/a 2880 18

"n/a" means that newforms for that character have not been added to the database yet

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_1(324))\) into lower level spaces

\( S_{4}^{\mathrm{old}}(\Gamma_1(324)) \cong \) \(S_{4}^{\mathrm{new}}(\Gamma_1(6))\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_1(9))\)\(^{\oplus 9}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_1(12))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_1(18))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_1(27))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_1(36))\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_1(54))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_1(81))\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_1(108))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_1(162))\)\(^{\oplus 2}\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ (\( ( 1 + 8 T^{2} )^{2} \))(\( 1 + 11 T^{2} + 96 T^{4} + 704 T^{6} + 4096 T^{8} \))
$3$ 1
$5$ (\( 1 + 3 T + 125 T^{2} \))(\( 1 - 3 T + 125 T^{2} \))(\( 1 + 6 T + 204 T^{2} + 636 T^{3} + 25500 T^{4} + 93750 T^{5} + 1953125 T^{6} \))(\( 1 - 6 T + 204 T^{2} - 636 T^{3} + 25500 T^{4} - 93750 T^{5} + 1953125 T^{6} \))(\( 1 - 58 T^{2} + 4875 T^{4} - 906250 T^{6} + 244140625 T^{8} \))(\( 1 - 88 T^{2} - 7881 T^{4} - 1375000 T^{6} + 244140625 T^{8} \))(\( ( 1 - 277 T^{2} + 42900 T^{4} - 4328125 T^{6} + 244140625 T^{8} )^{2} \))(\( 1 + 18 T + 199 T^{2} + 2250 T^{3} + 15625 T^{4} \))(\( 1 + 9 T - 44 T^{2} + 1125 T^{3} + 15625 T^{4} \))(\( 1 + 3 T - 116 T^{2} + 375 T^{3} + 15625 T^{4} \))(\( 1 - 125 T^{2} + 15625 T^{4} \))(\( 1 - 125 T^{2} + 15625 T^{4} \))(\( 1 - 3 T - 116 T^{2} - 375 T^{3} + 15625 T^{4} \))(\( 1 - 9 T - 44 T^{2} - 1125 T^{3} + 15625 T^{4} \))(\( 1 - 18 T + 199 T^{2} - 2250 T^{3} + 15625 T^{4} \))(\( 1 + 58 T^{2} - 1511 T^{4} - 1529750 T^{6} - 272937500 T^{8} - 23902343750 T^{10} - 368896484375 T^{12} + 221252441406250 T^{14} + 59604644775390625 T^{16} \))
$7$ (\( 1 + 4 T + 343 T^{2} \))(\( 1 + 4 T + 343 T^{2} \))(\( 1 - 6 T + 420 T^{2} - 8714 T^{3} + 144060 T^{4} - 705894 T^{5} + 40353607 T^{6} \))(\( 1 - 6 T + 420 T^{2} - 8714 T^{3} + 144060 T^{4} - 705894 T^{5} + 40353607 T^{6} \))(\( ( 1 + 8 T - 54 T^{2} + 2744 T^{3} + 117649 T^{4} )^{2} \))(\( ( 1 - 343 T^{2} )^{4} \))(\( ( 1 - 763 T^{2} + 295620 T^{4} - 89766187 T^{6} + 13841287201 T^{8} )^{2} \))(\( 1 + 8 T - 279 T^{2} + 2744 T^{3} + 117649 T^{4} \))(\( 1 - T - 342 T^{2} - 343 T^{3} + 117649 T^{4} \))(\( 1 - 4 T - 327 T^{2} - 1372 T^{3} + 117649 T^{4} \))(\( ( 1 - 20 T + 343 T^{2} )( 1 + 37 T + 343 T^{2} ) \))(\( ( 1 - 20 T + 343 T^{2} )( 1 - 17 T + 343 T^{2} ) \))(\( 1 - 4 T - 327 T^{2} - 1372 T^{3} + 117649 T^{4} \))(\( 1 - T - 342 T^{2} - 343 T^{3} + 117649 T^{4} \))(\( 1 + 8 T - 279 T^{2} + 2744 T^{3} + 117649 T^{4} \))(\( ( 1 - 8 T + 118 T^{2} + 5920 T^{3} - 136685 T^{4} + 2030560 T^{5} + 13882582 T^{6} - 322828856 T^{7} + 13841287201 T^{8} )^{2} \))
$11$ (\( 1 - 24 T + 1331 T^{2} \))(\( 1 + 24 T + 1331 T^{2} \))(\( 1 + 51 T + 2796 T^{2} + 69531 T^{3} + 3721476 T^{4} + 90349611 T^{5} + 2357947691 T^{6} \))(\( 1 - 51 T + 2796 T^{2} - 69531 T^{3} + 3721476 T^{4} - 90349611 T^{5} + 2357947691 T^{6} \))(\( 1 + 3956 T^{2} + 7020150 T^{4} + 7008295316 T^{6} + 3138428376721 T^{8} \))(\( ( 1 + 1331 T^{2} )^{4} \))(\( ( 1 + 3248 T^{2} + 5313729 T^{4} + 5754030128 T^{6} + 3138428376721 T^{8} )^{2} \))(\( 1 - 36 T - 35 T^{2} - 47916 T^{3} + 1771561 T^{4} \))(\( 1 + 63 T + 2638 T^{2} + 83853 T^{3} + 1771561 T^{4} \))(\( 1 - 24 T - 755 T^{2} - 31944 T^{3} + 1771561 T^{4} \))(\( 1 - 1331 T^{2} + 1771561 T^{4} \))(\( 1 - 1331 T^{2} + 1771561 T^{4} \))(\( 1 + 24 T - 755 T^{2} + 31944 T^{3} + 1771561 T^{4} \))(\( 1 - 63 T + 2638 T^{2} - 83853 T^{3} + 1771561 T^{4} \))(\( 1 + 36 T - 35 T^{2} + 47916 T^{3} + 1771561 T^{4} \))(\( 1 - 3956 T^{2} + 8629786 T^{4} - 13755122768 T^{6} + 18419261375683 T^{8} - 24368039046000848 T^{10} + 27083965267429611706 T^{12} - \)\(21\!\cdots\!36\)\( T^{14} + \)\(98\!\cdots\!41\)\( T^{16} \))
$13$ (\( 1 + 25 T + 2197 T^{2} \))(\( 1 + 25 T + 2197 T^{2} \))(\( 1 + 12 T + 1320 T^{2} - 11738 T^{3} + 2900040 T^{4} + 57921708 T^{5} + 10604499373 T^{6} \))(\( 1 + 12 T + 1320 T^{2} - 11738 T^{3} + 2900040 T^{4} + 57921708 T^{5} + 10604499373 T^{6} \))(\( ( 1 - 58 T + 4479 T^{2} - 127426 T^{3} + 4826809 T^{4} )^{2} \))(\( ( 1 + 92 T + 6267 T^{2} + 202124 T^{3} + 4826809 T^{4} )^{2} \))(\( ( 1 + 107 T + 7194 T^{2} + 235079 T^{3} + 4826809 T^{4} )^{4} \))(\( 1 - 10 T - 2097 T^{2} - 21970 T^{3} + 4826809 T^{4} \))(\( 1 - 28 T - 1413 T^{2} - 61516 T^{3} + 4826809 T^{4} \))(\( 1 - 25 T - 1572 T^{2} - 54925 T^{3} + 4826809 T^{4} \))(\( ( 1 + 19 T + 2197 T^{2} )( 1 + 70 T + 2197 T^{2} ) \))(\( ( 1 - 89 T + 2197 T^{2} )( 1 + 70 T + 2197 T^{2} ) \))(\( 1 - 25 T - 1572 T^{2} - 54925 T^{3} + 4826809 T^{4} \))(\( 1 - 28 T - 1413 T^{2} - 61516 T^{3} + 4826809 T^{4} \))(\( 1 - 10 T - 2097 T^{2} - 21970 T^{3} + 4826809 T^{4} \))(\( ( 1 + 58 T - 1115 T^{2} + 4930 T^{3} + 7843924 T^{4} + 10831210 T^{5} - 5381892035 T^{6} + 615060963634 T^{7} + 23298085122481 T^{8} )^{2} \))
$17$ (\( 1 - 21 T + 4913 T^{2} \))(\( 1 + 21 T + 4913 T^{2} \))(\( 1 + 111 T + 8115 T^{2} + 513210 T^{3} + 39868995 T^{4} + 2679270159 T^{5} + 118587876497 T^{6} \))(\( 1 - 111 T + 8115 T^{2} - 513210 T^{3} + 39868995 T^{4} - 2679270159 T^{5} + 118587876497 T^{6} \))(\( 1 + 8510 T^{2} + 47982147 T^{4} + 205410712190 T^{6} + 582622237229761 T^{8} \))(\( 1 + 9776 T^{2} + 71432607 T^{4} + 235968874544 T^{6} + 582622237229761 T^{8} \))(\( ( 1 - 9325 T^{2} + 46774812 T^{4} - 225082830925 T^{6} + 582622237229761 T^{8} )^{2} \))(\( ( 1 + 18 T + 4913 T^{2} )^{2} \))(\( ( 1 - 72 T + 4913 T^{2} )^{2} \))(\( ( 1 + 21 T + 4913 T^{2} )^{2} \))(\( ( 1 + 4913 T^{2} )^{2} \))(\( ( 1 + 4913 T^{2} )^{2} \))(\( ( 1 - 21 T + 4913 T^{2} )^{2} \))(\( ( 1 + 72 T + 4913 T^{2} )^{2} \))(\( ( 1 - 18 T + 4913 T^{2} )^{2} \))(\( ( 1 + 8510 T^{2} + 47982147 T^{4} + 205410712190 T^{6} + 582622237229761 T^{8} )^{2} \))
$19$ (\( 1 + 52 T + 6859 T^{2} \))(\( 1 + 52 T + 6859 T^{2} \))(\( 1 - 15 T + 13065 T^{2} - 422138 T^{3} + 89612835 T^{4} - 705688215 T^{5} + 322687697779 T^{6} \))(\( 1 - 15 T + 13065 T^{2} - 422138 T^{3} + 89612835 T^{4} - 705688215 T^{5} + 322687697779 T^{6} \))(\( ( 1 - 112 T + 16098 T^{2} - 768208 T^{3} + 47045881 T^{4} )^{2} \))(\( ( 1 - 6859 T^{2} )^{4} \))(\( ( 1 + 239 T^{2} + 51737136 T^{4} + 11243965559 T^{6} + 2213314919066161 T^{8} )^{2} \))(\( ( 1 + 100 T + 6859 T^{2} )^{2} \))(\( ( 1 - 98 T + 6859 T^{2} )^{2} \))(\( ( 1 + 52 T + 6859 T^{2} )^{2} \))(\( ( 1 - 107 T + 6859 T^{2} )^{2} \))(\( ( 1 + 163 T + 6859 T^{2} )^{2} \))(\( ( 1 + 52 T + 6859 T^{2} )^{2} \))(\( ( 1 - 98 T + 6859 T^{2} )^{2} \))(\( ( 1 + 100 T + 6859 T^{2} )^{2} \))(\( ( 1 - 112 T + 16098 T^{2} - 768208 T^{3} + 47045881 T^{4} )^{4} \))
$23$ (\( 1 + 168 T + 12167 T^{2} \))(\( 1 - 168 T + 12167 T^{2} \))(\( 1 + 210 T + 46860 T^{2} + 5112462 T^{3} + 570145620 T^{4} + 31087536690 T^{5} + 1801152661463 T^{6} \))(\( 1 - 210 T + 46860 T^{2} - 5112462 T^{3} + 570145620 T^{4} - 31087536690 T^{5} + 1801152661463 T^{6} \))(\( 1 + 20516 T^{2} + 315948966 T^{4} + 3037104298724 T^{6} + 21914624432020321 T^{8} \))(\( ( 1 + 12167 T^{2} )^{4} \))(\( ( 1 + 37877 T^{2} + 652699188 T^{4} + 5607155367653 T^{6} + 21914624432020321 T^{8} )^{2} \))(\( 1 - 72 T - 6983 T^{2} - 876024 T^{3} + 148035889 T^{4} \))(\( 1 + 126 T + 3709 T^{2} + 1533042 T^{3} + 148035889 T^{4} \))(\( 1 + 168 T + 16057 T^{2} + 2044056 T^{3} + 148035889 T^{4} \))(\( 1 - 12167 T^{2} + 148035889 T^{4} \))(\( 1 - 12167 T^{2} + 148035889 T^{4} \))(\( 1 - 168 T + 16057 T^{2} - 2044056 T^{3} + 148035889 T^{4} \))(\( 1 - 126 T + 3709 T^{2} - 1533042 T^{3} + 148035889 T^{4} \))(\( 1 + 72 T - 6983 T^{2} + 876024 T^{3} + 148035889 T^{4} \))(\( 1 - 20516 T^{2} + 104957290 T^{4} - 407800389008 T^{6} + 15599892891827251 T^{8} - 60369093121345108112 T^{10} + \)\(23\!\cdots\!90\)\( T^{12} - \)\(66\!\cdots\!04\)\( T^{14} + \)\(48\!\cdots\!41\)\( T^{16} \))
$29$ (\( 1 - 177 T + 24389 T^{2} \))(\( 1 + 177 T + 24389 T^{2} \))(\( 1 + 456 T + 137904 T^{2} + 25122426 T^{3} + 3363340656 T^{4} + 271239434376 T^{5} + 14507145975869 T^{6} \))(\( 1 - 456 T + 137904 T^{2} - 25122426 T^{3} + 3363340656 T^{4} - 271239434376 T^{5} + 14507145975869 T^{6} \))(\( 1 + 33062 T^{2} + 809056203 T^{4} + 19666048638902 T^{6} + 353814783205469041 T^{8} \))(\( 1 + 36920 T^{2} + 768263079 T^{4} + 21960877011320 T^{6} + 353814783205469041 T^{8} \))(\( ( 1 - 65269 T^{2} + 2027800740 T^{4} - 38823523338349 T^{6} + 353814783205469041 T^{8} )^{2} \))(\( 1 + 234 T + 30367 T^{2} + 5707026 T^{3} + 594823321 T^{4} \))(\( 1 - 126 T - 8513 T^{2} - 3073014 T^{3} + 594823321 T^{4} \))(\( 1 - 177 T + 6940 T^{2} - 4316853 T^{3} + 594823321 T^{4} \))(\( 1 - 24389 T^{2} + 594823321 T^{4} \))(\( 1 - 24389 T^{2} + 594823321 T^{4} \))(\( 1 + 177 T + 6940 T^{2} + 4316853 T^{3} + 594823321 T^{4} \))(\( 1 + 126 T - 8513 T^{2} + 3073014 T^{3} + 594823321 T^{4} \))(\( 1 - 234 T + 30367 T^{2} - 5707026 T^{3} + 594823321 T^{4} \))(\( 1 - 33062 T^{2} + 284039641 T^{4} + 12583081094218 T^{6} - 349441743692069756 T^{8} + \)\(74\!\cdots\!78\)\( T^{10} + \)\(10\!\cdots\!81\)\( T^{12} - \)\(69\!\cdots\!82\)\( T^{14} + \)\(12\!\cdots\!81\)\( T^{16} \))
$31$ (\( 1 + 124 T + 29791 T^{2} \))(\( 1 + 124 T + 29791 T^{2} \))(\( 1 + 48 T + 54420 T^{2} - 194852 T^{3} + 1621226220 T^{4} + 42600176688 T^{5} + 26439622160671 T^{6} \))(\( 1 + 48 T + 54420 T^{2} - 194852 T^{3} + 1621226220 T^{4} + 42600176688 T^{5} + 26439622160671 T^{6} \))(\( ( 1 - 184 T + 40830 T^{2} - 5481544 T^{3} + 887503681 T^{4} )^{2} \))(\( ( 1 - 29791 T^{2} )^{4} \))(\( ( 1 - 72991 T^{2} + 2848123200 T^{4} - 64779781179871 T^{6} + 787662783788549761 T^{8} )^{2} \))(\( 1 - 16 T - 29535 T^{2} - 476656 T^{3} + 887503681 T^{4} \))(\( 1 - 259 T + 37290 T^{2} - 7715869 T^{3} + 887503681 T^{4} \))(\( ( 1 - 341 T + 29791 T^{2} )( 1 + 217 T + 29791 T^{2} ) \))(\( ( 1 + 19 T + 29791 T^{2} )( 1 + 289 T + 29791 T^{2} ) \))(\( ( 1 + 19 T + 29791 T^{2} )( 1 + 289 T + 29791 T^{2} ) \))(\( ( 1 - 341 T + 29791 T^{2} )( 1 + 217 T + 29791 T^{2} ) \))(\( 1 - 259 T + 37290 T^{2} - 7715869 T^{3} + 887503681 T^{4} \))(\( 1 - 16 T - 29535 T^{2} - 476656 T^{3} + 887503681 T^{4} \))(\( ( 1 + 184 T - 6974 T^{2} - 3450368 T^{3} - 229018877 T^{4} - 102789913088 T^{5} - 6189450671294 T^{6} + 4864890477563464 T^{7} + 787662783788549761 T^{8} )^{2} \))
$37$ (\( 1 + 265 T + 50653 T^{2} \))(\( 1 + 265 T + 50653 T^{2} \))(\( 1 + 48 T + 131451 T^{2} + 4180336 T^{3} + 6658387503 T^{4} + 123154867632 T^{5} + 129961739795077 T^{6} \))(\( 1 + 48 T + 131451 T^{2} + 4180336 T^{3} + 6658387503 T^{4} + 123154867632 T^{5} + 129961739795077 T^{6} \))(\( ( 1 - 334 T + 92151 T^{2} - 16918102 T^{3} + 2565726409 T^{4} )^{2} \))(\( ( 1 - 214 T - 4857 T^{2} - 10839742 T^{3} + 2565726409 T^{4} )^{2} \))(\( ( 1 + 314 T + 125706 T^{2} + 15905042 T^{3} + 2565726409 T^{4} )^{4} \))(\( ( 1 + 226 T + 50653 T^{2} )^{2} \))(\( ( 1 - 386 T + 50653 T^{2} )^{2} \))(\( ( 1 + 265 T + 50653 T^{2} )^{2} \))(\( ( 1 + 433 T + 50653 T^{2} )^{2} \))(\( ( 1 - 323 T + 50653 T^{2} )^{2} \))(\( ( 1 + 265 T + 50653 T^{2} )^{2} \))(\( ( 1 - 386 T + 50653 T^{2} )^{2} \))(\( ( 1 + 226 T + 50653 T^{2} )^{2} \))(\( ( 1 - 334 T + 92151 T^{2} - 16918102 T^{3} + 2565726409 T^{4} )^{4} \))
$41$ (\( 1 + 426 T + 68921 T^{2} \))(\( 1 - 426 T + 68921 T^{2} \))(\( 1 + 897 T + 430314 T^{2} + 135440817 T^{3} + 29657671194 T^{4} + 4260843504177 T^{5} + 327381934393961 T^{6} \))(\( 1 - 897 T + 430314 T^{2} - 135440817 T^{3} + 29657671194 T^{4} - 4260843504177 T^{5} + 327381934393961 T^{6} \))(\( 1 + 104324 T^{2} + 6758722662 T^{4} + 495549874838084 T^{6} + 22563490300366186081 T^{8} \))(\( ( 1 - 108560 T^{2} + 4750104241 T^{4} )^{2} \))(\( ( 1 - 104518 T^{2} + 5236801563 T^{4} - 496471395060838 T^{6} + 22563490300366186081 T^{8} )^{2} \))(\( 1 - 90 T - 60821 T^{2} - 6202890 T^{3} + 4750104241 T^{4} \))(\( 1 - 450 T + 133579 T^{2} - 31014450 T^{3} + 4750104241 T^{4} \))(\( 1 + 426 T + 112555 T^{2} + 29360346 T^{3} + 4750104241 T^{4} \))(\( 1 - 68921 T^{2} + 4750104241 T^{4} \))(\( 1 - 68921 T^{2} + 4750104241 T^{4} \))(\( 1 - 426 T + 112555 T^{2} - 29360346 T^{3} + 4750104241 T^{4} \))(\( 1 + 450 T + 133579 T^{2} + 31014450 T^{3} + 4750104241 T^{4} \))(\( 1 + 90 T - 60821 T^{2} + 6202890 T^{3} + 4750104241 T^{4} \))(\( 1 - 104324 T^{2} + 4124774314 T^{4} + 286002766685680 T^{6} - 28580903421142095053 T^{8} + \)\(13\!\cdots\!80\)\( T^{10} + \)\(93\!\cdots\!34\)\( T^{12} - \)\(11\!\cdots\!04\)\( T^{14} + \)\(50\!\cdots\!61\)\( T^{16} \))
$43$ (\( 1 + 160 T + 79507 T^{2} \))(\( 1 + 160 T + 79507 T^{2} \))(\( 1 + 129 T + 51612 T^{2} + 19087657 T^{3} + 4103515284 T^{4} + 815455833321 T^{5} + 502592611936843 T^{6} \))(\( 1 + 129 T + 51612 T^{2} + 19087657 T^{3} + 4103515284 T^{4} + 815455833321 T^{5} + 502592611936843 T^{6} \))(\( ( 1 - 328 T + 124674 T^{2} - 26078296 T^{3} + 6321363049 T^{4} )^{2} \))(\( ( 1 - 79507 T^{2} )^{4} \))(\( ( 1 - 261520 T^{2} + 29250769857 T^{4} - 1653162864574480 T^{6} + 39959630797262576401 T^{8} )^{2} \))(\( 1 + 452 T + 124797 T^{2} + 35937164 T^{3} + 6321363049 T^{4} \))(\( 1 - 34 T - 78351 T^{2} - 2703238 T^{3} + 6321363049 T^{4} \))(\( 1 - 160 T - 53907 T^{2} - 12721120 T^{3} + 6321363049 T^{4} \))(\( ( 1 - 449 T + 79507 T^{2} )( 1 - 71 T + 79507 T^{2} ) \))(\( ( 1 - 449 T + 79507 T^{2} )( 1 - 71 T + 79507 T^{2} ) \))(\( 1 - 160 T - 53907 T^{2} - 12721120 T^{3} + 6321363049 T^{4} \))(\( 1 - 34 T - 78351 T^{2} - 2703238 T^{3} + 6321363049 T^{4} \))(\( 1 + 452 T + 124797 T^{2} + 35937164 T^{3} + 6321363049 T^{4} \))(\( ( 1 + 328 T - 17090 T^{2} - 11263520 T^{3} + 668562139 T^{4} - 895528684640 T^{5} - 108032094507410 T^{6} + 164850376715284504 T^{7} + 39959630797262576401 T^{8} )^{2} \))
$47$ (\( 1 - 540 T + 103823 T^{2} \))(\( 1 + 540 T + 103823 T^{2} \))(\( 1 + 522 T + 150036 T^{2} + 43832430 T^{3} + 15577187628 T^{4} + 5626750401738 T^{5} + 1119130473102767 T^{6} \))(\( 1 - 522 T + 150036 T^{2} - 43832430 T^{3} + 15577187628 T^{4} - 5626750401738 T^{5} + 1119130473102767 T^{6} \))(\( ( 1 + 110446 T^{2} + 10779215329 T^{4} )^{2} \))(\( ( 1 + 103823 T^{2} )^{4} \))(\( ( 1 + 58493 T^{2} + 21321442788 T^{4} + 630508642239197 T^{6} + \)\(11\!\cdots\!41\)\( T^{8} )^{2} \))(\( 1 - 432 T + 82801 T^{2} - 44851536 T^{3} + 10779215329 T^{4} \))(\( 1 - 54 T - 100907 T^{2} - 5606442 T^{3} + 10779215329 T^{4} \))(\( 1 - 540 T + 187777 T^{2} - 56064420 T^{3} + 10779215329 T^{4} \))(\( 1 - 103823 T^{2} + 10779215329 T^{4} \))(\( 1 - 103823 T^{2} + 10779215329 T^{4} \))(\( 1 + 540 T + 187777 T^{2} + 56064420 T^{3} + 10779215329 T^{4} \))(\( 1 + 54 T - 100907 T^{2} + 5606442 T^{3} + 10779215329 T^{4} \))(\( 1 + 432 T + 82801 T^{2} + 44851536 T^{3} + 10779215329 T^{4} \))(\( ( 1 - 110446 T^{2} + 1419103587 T^{4} - 1190521216226734 T^{6} + \)\(11\!\cdots\!41\)\( T^{8} )^{2} \))
$53$ (\( 1 - 258 T + 148877 T^{2} \))(\( 1 + 258 T + 148877 T^{2} \))(\( 1 + 1104 T + 764331 T^{2} + 340574064 T^{3} + 113791306287 T^{4} + 24469454686416 T^{5} + 3299763591802133 T^{6} \))(\( 1 - 1104 T + 764331 T^{2} - 340574064 T^{3} + 113791306287 T^{4} - 24469454686416 T^{5} + 3299763591802133 T^{6} \))(\( 1 + 101876 T^{2} - 13986441354 T^{4} + 2258016454378004 T^{6} + \)\(49\!\cdots\!41\)\( T^{8} \))(\( ( 1 + 296296 T^{2} + 22164361129 T^{4} )^{2} \))(\( ( 1 - 363880 T^{2} + 76436876862 T^{4} - 8065167727620520 T^{6} + \)\(49\!\cdots\!41\)\( T^{8} )^{2} \))(\( ( 1 + 414 T + 148877 T^{2} )^{2} \))(\( ( 1 + 693 T + 148877 T^{2} )^{2} \))(\( ( 1 + 258 T + 148877 T^{2} )^{2} \))(\( ( 1 + 148877 T^{2} )^{2} \))(\( ( 1 + 148877 T^{2} )^{2} \))(\( ( 1 - 258 T + 148877 T^{2} )^{2} \))(\( ( 1 - 693 T + 148877 T^{2} )^{2} \))(\( ( 1 - 414 T + 148877 T^{2} )^{2} \))(\( ( 1 + 101876 T^{2} - 13986441354 T^{4} + 2258016454378004 T^{6} + \)\(49\!\cdots\!41\)\( T^{8} )^{2} \))
$59$ (\( 1 + 528 T + 205379 T^{2} \))(\( 1 - 528 T + 205379 T^{2} \))(\( 1 + 453 T + 324852 T^{2} + 89180661 T^{3} + 66717778908 T^{4} + 19107781739373 T^{5} + 8662995818654939 T^{6} \))(\( 1 - 453 T + 324852 T^{2} - 89180661 T^{3} + 66717778908 T^{4} - 19107781739373 T^{5} + 8662995818654939 T^{6} \))(\( 1 + 467852 T^{2} + 136295522358 T^{4} + 19734247025009132 T^{6} + \)\(17\!\cdots\!81\)\( T^{8} \))(\( ( 1 + 205379 T^{2} )^{4} \))(\( ( 1 + 717032 T^{2} + 211655426649 T^{4} + 30244792397673512 T^{6} + \)\(17\!\cdots\!81\)\( T^{8} )^{2} \))(\( 1 + 684 T + 262477 T^{2} + 140479236 T^{3} + 42180533641 T^{4} \))(\( 1 + 180 T - 172979 T^{2} + 36968220 T^{3} + 42180533641 T^{4} \))(\( 1 + 528 T + 73405 T^{2} + 108440112 T^{3} + 42180533641 T^{4} \))(\( 1 - 205379 T^{2} + 42180533641 T^{4} \))(\( 1 - 205379 T^{2} + 42180533641 T^{4} \))(\( 1 - 528 T + 73405 T^{2} - 108440112 T^{3} + 42180533641 T^{4} \))(\( 1 - 180 T - 172979 T^{2} - 36968220 T^{3} + 42180533641 T^{4} \))(\( 1 - 684 T + 262477 T^{2} - 140479236 T^{3} + 42180533641 T^{4} \))(\( 1 - 467852 T^{2} + 82589971546 T^{4} - 24297638676216752 T^{6} + \)\(75\!\cdots\!19\)\( T^{8} - \)\(10\!\cdots\!32\)\( T^{10} + \)\(14\!\cdots\!26\)\( T^{12} - \)\(35\!\cdots\!92\)\( T^{14} + \)\(31\!\cdots\!61\)\( T^{16} \))
$61$ (\( 1 + 505 T + 226981 T^{2} \))(\( 1 + 505 T + 226981 T^{2} \))(\( 1 - 402 T + 688884 T^{2} - 181282988 T^{3} + 156363579204 T^{4} - 20711190493122 T^{5} + 11694146092834141 T^{6} \))(\( 1 - 402 T + 688884 T^{2} - 181282988 T^{3} + 156363579204 T^{4} - 20711190493122 T^{5} + 11694146092834141 T^{6} \))(\( ( 1 - 502 T + 455727 T^{2} - 113944462 T^{3} + 51520374361 T^{4} )^{2} \))(\( ( 1 + 830 T + 461919 T^{2} + 188394230 T^{3} + 51520374361 T^{4} )^{2} \))(\( ( 1 + 131 T + 144450 T^{2} + 29734511 T^{3} + 51520374361 T^{4} )^{4} \))(\( 1 + 422 T - 48897 T^{2} + 95785982 T^{3} + 51520374361 T^{4} \))(\( 1 - 280 T - 148581 T^{2} - 63554680 T^{3} + 51520374361 T^{4} \))(\( 1 - 505 T + 28044 T^{2} - 114625405 T^{3} + 51520374361 T^{4} \))(\( ( 1 - 719 T + 226981 T^{2} )( 1 - 182 T + 226981 T^{2} ) \))(\( ( 1 - 182 T + 226981 T^{2} )( 1 + 901 T + 226981 T^{2} ) \))(\( 1 - 505 T + 28044 T^{2} - 114625405 T^{3} + 51520374361 T^{4} \))(\( 1 - 280 T - 148581 T^{2} - 63554680 T^{3} + 51520374361 T^{4} \))(\( 1 + 422 T - 48897 T^{2} + 95785982 T^{3} + 51520374361 T^{4} \))(\( ( 1 + 502 T - 203723 T^{2} + 886030 T^{3} + 98966604244 T^{4} + 201111975430 T^{5} - 10495885225946003 T^{6} + 5870461338602738782 T^{7} + \)\(26\!\cdots\!21\)\( T^{8} )^{2} \))
$67$ (\( 1 + 244 T + 300763 T^{2} \))(\( 1 + 244 T + 300763 T^{2} \))(\( 1 - 213 T + 822156 T^{2} - 131220101 T^{3} + 247274105028 T^{4} - 19267635401997 T^{5} + 27206534396294947 T^{6} \))(\( 1 - 213 T + 822156 T^{2} - 131220101 T^{3} + 247274105028 T^{4} - 19267635401997 T^{5} + 27206534396294947 T^{6} \))(\( ( 1 - 160 T + 437826 T^{2} - 48122080 T^{3} + 90458382169 T^{4} )^{2} \))(\( ( 1 - 300763 T^{2} )^{4} \))(\( ( 1 - 1070656 T^{2} + 463177520241 T^{4} - 96849809619532864 T^{6} + \)\(81\!\cdots\!61\)\( T^{8} )^{2} \))(\( 1 + 332 T - 190539 T^{2} + 99853316 T^{3} + 90458382169 T^{4} \))(\( 1 - 586 T + 42633 T^{2} - 176247118 T^{3} + 90458382169 T^{4} \))(\( 1 - 244 T - 241227 T^{2} - 73386172 T^{3} + 90458382169 T^{4} \))(\( ( 1 + 127 T + 300763 T^{2} )( 1 + 880 T + 300763 T^{2} ) \))(\( ( 1 - 1007 T + 300763 T^{2} )( 1 + 880 T + 300763 T^{2} ) \))(\( 1 - 244 T - 241227 T^{2} - 73386172 T^{3} + 90458382169 T^{4} \))(\( 1 - 586 T + 42633 T^{2} - 176247118 T^{3} + 90458382169 T^{4} \))(\( 1 + 332 T - 190539 T^{2} + 99853316 T^{3} + 90458382169 T^{4} \))(\( ( 1 + 160 T - 412226 T^{2} - 26192000 T^{3} + 93533691307 T^{4} - 7877584496000 T^{5} - 37289297047998194 T^{6} + 4353045503407191520 T^{7} + \)\(81\!\cdots\!61\)\( T^{8} )^{2} \))
$71$ (\( 1 + 204 T + 357911 T^{2} \))(\( 1 - 204 T + 357911 T^{2} \))(\( 1 - 60 T + 650661 T^{2} + 70262328 T^{3} + 232878729171 T^{4} - 7686017035260 T^{5} + 45848500718449031 T^{6} \))(\( 1 + 60 T + 650661 T^{2} - 70262328 T^{3} + 232878729171 T^{4} + 7686017035260 T^{5} + 45848500718449031 T^{6} \))(\( 1 + 625028 T^{2} + 209892928038 T^{4} + 80066264258574788 T^{6} + \)\(16\!\cdots\!41\)\( T^{8} \))(\( ( 1 + 357911 T^{2} )^{4} \))(\( ( 1 + 1260428 T^{2} + 653321166774 T^{4} + 161461184661978188 T^{6} + \)\(16\!\cdots\!41\)\( T^{8} )^{2} \))(\( ( 1 - 360 T + 357911 T^{2} )^{2} \))(\( ( 1 - 504 T + 357911 T^{2} )^{2} \))(\( ( 1 - 204 T + 357911 T^{2} )^{2} \))(\( ( 1 + 357911 T^{2} )^{2} \))(\( ( 1 + 357911 T^{2} )^{2} \))(\( ( 1 + 204 T + 357911 T^{2} )^{2} \))(\( ( 1 + 504 T + 357911 T^{2} )^{2} \))(\( ( 1 + 360 T + 357911 T^{2} )^{2} \))(\( ( 1 + 625028 T^{2} + 209892928038 T^{4} + 80066264258574788 T^{6} + \)\(16\!\cdots\!41\)\( T^{8} )^{2} \))
$73$ (\( 1 + 397 T + 389017 T^{2} \))(\( 1 + 397 T + 389017 T^{2} \))(\( 1 - 375 T + 786003 T^{2} - 133393466 T^{3} + 305768529051 T^{4} - 56750334858375 T^{5} + 58871586708267913 T^{6} \))(\( 1 - 375 T + 786003 T^{2} - 133393466 T^{3} + 305768529051 T^{4} - 56750334858375 T^{5} + 58871586708267913 T^{6} \))(\( ( 1 - 305 T + 389017 T^{2} )^{4} \))(\( ( 1 - 592 T - 38553 T^{2} - 230298064 T^{3} + 151334226289 T^{4} )^{2} \))(\( ( 1 - 985 T + 570834 T^{2} - 383181745 T^{3} + 151334226289 T^{4} )^{4} \))(\( ( 1 - 26 T + 389017 T^{2} )^{2} \))(\( ( 1 - 161 T + 389017 T^{2} )^{2} \))(\( ( 1 + 397 T + 389017 T^{2} )^{2} \))(\( ( 1 + 271 T + 389017 T^{2} )^{2} \))(\( ( 1 + 919 T + 389017 T^{2} )^{2} \))(\( ( 1 + 397 T + 389017 T^{2} )^{2} \))(\( ( 1 - 161 T + 389017 T^{2} )^{2} \))(\( ( 1 - 26 T + 389017 T^{2} )^{2} \))(\( ( 1 - 305 T + 389017 T^{2} )^{8} \))
$79$ (\( 1 - 200 T + 493039 T^{2} \))(\( 1 - 200 T + 493039 T^{2} \))(\( 1 + 552 T + 1457268 T^{2} + 526466284 T^{3} + 718489957452 T^{4} + 134184275447592 T^{5} + 119851595982618319 T^{6} \))(\( 1 + 552 T + 1457268 T^{2} + 526466284 T^{3} + 718489957452 T^{4} + 134184275447592 T^{5} + 119851595982618319 T^{6} \))(\( ( 1 + 32 T + 513834 T^{2} + 15777248 T^{3} + 243087455521 T^{4} )^{2} \))(\( ( 1 - 493039 T^{2} )^{4} \))(\( ( 1 - 259483 T^{2} + 356344618932 T^{4} - 63077062220955643 T^{6} + \)\(59\!\cdots\!41\)\( T^{8} )^{2} \))(\( 1 + 512 T - 230895 T^{2} + 252435968 T^{3} + 243087455521 T^{4} \))(\( 1 + 440 T - 299439 T^{2} + 216937160 T^{3} + 243087455521 T^{4} \))(\( 1 + 200 T - 453039 T^{2} + 98607800 T^{3} + 243087455521 T^{4} \))(\( ( 1 - 884 T + 493039 T^{2} )( 1 + 1387 T + 493039 T^{2} ) \))(\( ( 1 - 884 T + 493039 T^{2} )( 1 - 503 T + 493039 T^{2} ) \))(\( 1 + 200 T - 453039 T^{2} + 98607800 T^{3} + 243087455521 T^{4} \))(\( 1 + 440 T - 299439 T^{2} + 216937160 T^{3} + 243087455521 T^{4} \))(\( 1 + 512 T - 230895 T^{2} + 252435968 T^{3} + 243087455521 T^{4} \))(\( ( 1 - 32 T - 512810 T^{2} + 15111808 T^{3} + 20433052099 T^{4} + 7450710704512 T^{5} - 124657678065724010 T^{6} - 3835251071443786208 T^{7} + \)\(59\!\cdots\!41\)\( T^{8} )^{2} \))
$83$ (\( 1 - 540 T + 571787 T^{2} \))(\( 1 + 540 T + 571787 T^{2} \))(\( 1 - 612 T + 1597668 T^{2} - 696388500 T^{3} + 913525792716 T^{4} - 200087508501828 T^{5} + 186940255267540403 T^{6} \))(\( 1 + 612 T + 1597668 T^{2} + 696388500 T^{3} + 913525792716 T^{4} + 200087508501828 T^{5} + 186940255267540403 T^{6} \))(\( 1 + 436460 T^{2} + 141527823702 T^{4} + 142696395360633740 T^{6} + \)\(10\!\cdots\!61\)\( T^{8} \))(\( ( 1 + 571787 T^{2} )^{4} \))(\( ( 1 + 1120157 T^{2} + 964066176588 T^{4} + 366224547811898933 T^{6} + \)\(10\!\cdots\!61\)\( T^{8} )^{2} \))(\( 1 + 1188 T + 839557 T^{2} + 679282956 T^{3} + 326940373369 T^{4} \))(\( 1 + 999 T + 426214 T^{2} + 571215213 T^{3} + 326940373369 T^{4} \))(\( 1 - 540 T - 280187 T^{2} - 308764980 T^{3} + 326940373369 T^{4} \))(\( 1 - 571787 T^{2} + 326940373369 T^{4} \))(\( 1 - 571787 T^{2} + 326940373369 T^{4} \))(\( 1 + 540 T - 280187 T^{2} + 308764980 T^{3} + 326940373369 T^{4} \))(\( 1 - 999 T + 426214 T^{2} - 571215213 T^{3} + 326940373369 T^{4} \))(\( 1 - 1188 T + 839557 T^{2} - 679282956 T^{3} + 326940373369 T^{4} \))(\( 1 - 436460 T^{2} + 48969507898 T^{4} + 223621556788292560 T^{6} - \)\(14\!\cdots\!57\)\( T^{8} + \)\(73\!\cdots\!40\)\( T^{10} + \)\(52\!\cdots\!78\)\( T^{12} - \)\(15\!\cdots\!40\)\( T^{14} + \)\(11\!\cdots\!21\)\( T^{16} \))
$89$ (\( 1 - 453 T + 704969 T^{2} \))(\( 1 + 453 T + 704969 T^{2} \))(\( 1 + 462 T + 1340727 T^{2} + 821513604 T^{3} + 945170972463 T^{4} + 229605356423982 T^{5} + 350356403707485209 T^{6} \))(\( 1 - 462 T + 1340727 T^{2} - 821513604 T^{3} + 945170972463 T^{4} - 229605356423982 T^{5} + 350356403707485209 T^{6} \))(\( 1 - 1173874 T^{2} + 984168880467 T^{4} - 583393415945552914 T^{6} + \)\(24\!\cdots\!21\)\( T^{8} \))(\( 1 - 293920 T^{2} - 410592324561 T^{4} - 146072741039257120 T^{6} + \)\(24\!\cdots\!21\)\( T^{8} \))(\( ( 1 - 2694616 T^{2} + 2805913145742 T^{4} - 1339173738324165976 T^{6} + \)\(24\!\cdots\!21\)\( T^{8} )^{2} \))(\( ( 1 - 630 T + 704969 T^{2} )^{2} \))(\( ( 1 - 882 T + 704969 T^{2} )^{2} \))(\( ( 1 + 453 T + 704969 T^{2} )^{2} \))(\( ( 1 + 704969 T^{2} )^{2} \))(\( ( 1 + 704969 T^{2} )^{2} \))(\( ( 1 - 453 T + 704969 T^{2} )^{2} \))(\( ( 1 + 882 T + 704969 T^{2} )^{2} \))(\( ( 1 + 630 T + 704969 T^{2} )^{2} \))(\( ( 1 - 1173874 T^{2} + 984168880467 T^{4} - 583393415945552914 T^{6} + \)\(24\!\cdots\!21\)\( T^{8} )^{2} \))
$97$ (\( 1 - 290 T + 912673 T^{2} \))(\( 1 - 290 T + 912673 T^{2} \))(\( 1 + 93 T + 1698378 T^{2} + 256157701 T^{3} + 1550063744394 T^{4} + 77466396458397 T^{5} + 760231058654565217 T^{6} \))(\( 1 + 93 T + 1698378 T^{2} + 256157701 T^{3} + 1550063744394 T^{4} + 77466396458397 T^{5} + 760231058654565217 T^{6} \))(\( ( 1 - 1012 T + 2069286 T^{2} - 923625076 T^{3} + 832972004929 T^{4} )^{2} \))(\( ( 1 + 1816 T + 912673 T^{2} )^{4} \))(\( ( 1 - 286 T + 1318911 T^{2} - 261024478 T^{3} + 832972004929 T^{4} )^{4} \))(\( 1 - 1054 T + 198243 T^{2} - 961957342 T^{3} + 832972004929 T^{4} \))(\( 1 - 721 T - 392832 T^{2} - 658037233 T^{3} + 832972004929 T^{4} \))(\( 1 + 290 T - 828573 T^{2} + 264675170 T^{3} + 832972004929 T^{4} \))(\( ( 1 + 523 T + 912673 T^{2} )( 1 + 1330 T + 912673 T^{2} ) \))(\( ( 1 - 1853 T + 912673 T^{2} )( 1 + 1330 T + 912673 T^{2} ) \))(\( 1 + 290 T - 828573 T^{2} + 264675170 T^{3} + 832972004929 T^{4} \))(\( 1 - 721 T - 392832 T^{2} - 658037233 T^{3} + 832972004929 T^{4} \))(\( 1 - 1054 T + 198243 T^{2} - 961957342 T^{3} + 832972004929 T^{4} \))(\( ( 1 + 1012 T - 1045142 T^{2} + 246867280 T^{3} + 2514263967955 T^{4} + 225309101039440 T^{5} - 870574027175504918 T^{6} + \)\(76\!\cdots\!04\)\( T^{7} + \)\(69\!\cdots\!41\)\( T^{8} )^{2} \))
show more
show less