Properties

Label 256.2
Level 256
Weight 2
Dimension 1100
Nonzero newspaces 6
Newform subspaces 16
Sturm bound 8192
Trace bound 9

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Defining parameters

Level: \( N \) = \( 256 = 2^{8} \)
Weight: \( k \) = \( 2 \)
Nonzero newspaces: \( 6 \)
Newform subspaces: \( 16 \)
Sturm bound: \(8192\)
Trace bound: \(9\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_1(256))\).

Total New Old
Modular forms 2224 1204 1020
Cusp forms 1873 1100 773
Eisenstein series 351 104 247

Trace form

\( 1100q - 32q^{2} - 24q^{3} - 32q^{4} - 32q^{5} - 32q^{6} - 24q^{7} - 32q^{8} - 40q^{9} + O(q^{10}) \) \( 1100q - 32q^{2} - 24q^{3} - 32q^{4} - 32q^{5} - 32q^{6} - 24q^{7} - 32q^{8} - 40q^{9} - 32q^{10} - 24q^{11} - 32q^{12} - 32q^{13} - 32q^{14} - 24q^{15} - 32q^{16} - 48q^{17} - 32q^{18} - 24q^{19} - 32q^{20} - 32q^{21} - 32q^{22} - 24q^{23} - 32q^{24} - 40q^{25} - 32q^{26} - 24q^{27} - 32q^{28} - 32q^{29} - 32q^{30} - 32q^{31} - 32q^{32} - 56q^{33} - 32q^{34} - 24q^{35} - 32q^{36} - 32q^{37} - 32q^{38} - 24q^{39} - 32q^{40} - 40q^{41} - 32q^{42} - 24q^{43} - 32q^{44} - 56q^{45} - 32q^{46} - 24q^{47} - 32q^{48} - 76q^{49} - 32q^{50} - 56q^{51} - 32q^{52} - 64q^{53} - 32q^{54} - 88q^{55} - 32q^{56} - 104q^{57} - 32q^{58} - 88q^{59} - 32q^{60} - 96q^{61} - 32q^{62} - 96q^{63} - 32q^{64} - 136q^{65} - 32q^{66} - 104q^{67} - 32q^{68} - 96q^{69} - 32q^{70} - 88q^{71} - 32q^{72} - 104q^{73} - 32q^{74} - 88q^{75} - 32q^{76} - 64q^{77} - 32q^{78} - 56q^{79} - 32q^{80} - 84q^{81} - 32q^{82} - 24q^{83} - 32q^{84} - 72q^{85} - 32q^{86} - 24q^{87} - 32q^{88} - 40q^{89} - 32q^{90} - 24q^{91} - 32q^{92} + 16q^{93} - 32q^{94} - 32q^{95} - 32q^{96} - 56q^{97} - 32q^{98} - 48q^{99} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_1(256))\)

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
256.2.a \(\chi_{256}(1, \cdot)\) 256.2.a.a 1 1
256.2.a.b 1
256.2.a.c 1
256.2.a.d 1
256.2.a.e 2
256.2.b \(\chi_{256}(129, \cdot)\) 256.2.b.a 2 1
256.2.b.b 2
256.2.b.c 2
256.2.e \(\chi_{256}(65, \cdot)\) 256.2.e.a 8 2
256.2.e.b 8
256.2.g \(\chi_{256}(33, \cdot)\) 256.2.g.a 4 4
256.2.g.b 4
256.2.g.c 8
256.2.g.d 8
256.2.i \(\chi_{256}(17, \cdot)\) 256.2.i.a 56 8
256.2.k \(\chi_{256}(9, \cdot)\) None 0 16
256.2.m \(\chi_{256}(5, \cdot)\) 256.2.m.a 992 32

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_1(256))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_1(256)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_1(16))\)\(^{\oplus 5}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(32))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(64))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(128))\)\(^{\oplus 2}\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ (\( 1 + 2 T + 3 T^{2} \))(\( 1 + 3 T^{2} \))(\( 1 + 3 T^{2} \))(\( 1 - 2 T + 3 T^{2} \))(\( 1 - 2 T^{2} + 9 T^{4} \))(\( 1 - 2 T^{2} + 9 T^{4} \))(\( ( 1 - 3 T^{2} )^{2} \))(\( 1 - 2 T^{2} + 9 T^{4} \))(\( 1 - 4 T^{4} - 26 T^{8} - 324 T^{12} + 6561 T^{16} \))(\( 1 - 4 T^{4} - 26 T^{8} - 324 T^{12} + 6561 T^{16} \))(\( ( 1 + 2 T + 3 T^{2} )^{2}( 1 - 4 T + 8 T^{2} - 12 T^{3} + 9 T^{4} ) \))(\( ( 1 - 2 T + 3 T^{2} )^{2}( 1 + 4 T + 8 T^{2} + 12 T^{3} + 9 T^{4} ) \))(\( 1 + 4 T + 8 T^{2} + 12 T^{3} + 16 T^{4} + 12 T^{5} + 24 T^{6} + 68 T^{7} + 142 T^{8} + 204 T^{9} + 216 T^{10} + 324 T^{11} + 1296 T^{12} + 2916 T^{13} + 5832 T^{14} + 8748 T^{15} + 6561 T^{16} \))(\( 1 - 4 T + 8 T^{2} - 12 T^{3} + 16 T^{4} - 12 T^{5} + 24 T^{6} - 68 T^{7} + 142 T^{8} - 204 T^{9} + 216 T^{10} - 324 T^{11} + 1296 T^{12} - 2916 T^{13} + 5832 T^{14} - 8748 T^{15} + 6561 T^{16} \))
$5$ (\( 1 + 5 T^{2} \))(\( 1 + 4 T + 5 T^{2} \))(\( 1 - 4 T + 5 T^{2} \))(\( 1 + 5 T^{2} \))(\( ( 1 + 5 T^{2} )^{2} \))(\( ( 1 - 4 T + 5 T^{2} )( 1 + 4 T + 5 T^{2} ) \))(\( ( 1 - 4 T + 5 T^{2} )( 1 + 4 T + 5 T^{2} ) \))(\( ( 1 - 4 T + 5 T^{2} )( 1 + 4 T + 5 T^{2} ) \))(\( ( 1 - 34 T^{4} + 625 T^{8} )^{2} \))(\( ( 1 - 34 T^{4} + 625 T^{8} )^{2} \))(\( ( 1 - 2 T + 5 T^{2} )^{2}( 1 - 8 T^{2} + 25 T^{4} ) \))(\( ( 1 - 2 T + 5 T^{2} )^{2}( 1 - 8 T^{2} + 25 T^{4} ) \))(\( ( 1 + 2 T^{2} - 16 T^{3} + 2 T^{4} - 80 T^{5} + 50 T^{6} + 625 T^{8} )^{2} \))(\( ( 1 + 2 T^{2} - 16 T^{3} + 2 T^{4} - 80 T^{5} + 50 T^{6} + 625 T^{8} )^{2} \))
$7$ (\( 1 + 7 T^{2} \))(\( 1 + 7 T^{2} \))(\( 1 + 7 T^{2} \))(\( 1 + 7 T^{2} \))(\( ( 1 + 7 T^{2} )^{2} \))(\( ( 1 + 4 T + 7 T^{2} )^{2} \))(\( ( 1 + 7 T^{2} )^{2} \))(\( ( 1 - 4 T + 7 T^{2} )^{2} \))(\( ( 1 - 12 T^{2} + 86 T^{4} - 588 T^{6} + 2401 T^{8} )^{2} \))(\( ( 1 - 12 T^{2} + 86 T^{4} - 588 T^{6} + 2401 T^{8} )^{2} \))(\( ( 1 + 2 T + 2 T^{2} + 14 T^{3} + 49 T^{4} )^{2} \))(\( ( 1 - 2 T + 2 T^{2} - 14 T^{3} + 49 T^{4} )^{2} \))(\( 1 - 8 T + 32 T^{2} - 104 T^{3} + 252 T^{4} - 392 T^{5} + 480 T^{6} - 168 T^{7} - 1274 T^{8} - 1176 T^{9} + 23520 T^{10} - 134456 T^{11} + 605052 T^{12} - 1747928 T^{13} + 3764768 T^{14} - 6588344 T^{15} + 5764801 T^{16} \))(\( 1 + 8 T + 32 T^{2} + 104 T^{3} + 252 T^{4} + 392 T^{5} + 480 T^{6} + 168 T^{7} - 1274 T^{8} + 1176 T^{9} + 23520 T^{10} + 134456 T^{11} + 605052 T^{12} + 1747928 T^{13} + 3764768 T^{14} + 6588344 T^{15} + 5764801 T^{16} \))
$11$ (\( 1 + 6 T + 11 T^{2} \))(\( 1 + 11 T^{2} \))(\( 1 + 11 T^{2} \))(\( 1 - 6 T + 11 T^{2} \))(\( 1 + 14 T^{2} + 121 T^{4} \))(\( 1 - 18 T^{2} + 121 T^{4} \))(\( ( 1 - 11 T^{2} )^{2} \))(\( 1 - 18 T^{2} + 121 T^{4} \))(\( 1 - 68 T^{4} - 12762 T^{8} - 995588 T^{12} + 214358881 T^{16} \))(\( 1 - 68 T^{4} - 12762 T^{8} - 995588 T^{12} + 214358881 T^{16} \))(\( ( 1 + 6 T + 11 T^{2} )^{2}( 1 - 4 T + 8 T^{2} - 44 T^{3} + 121 T^{4} ) \))(\( ( 1 - 6 T + 11 T^{2} )^{2}( 1 + 4 T + 8 T^{2} + 44 T^{3} + 121 T^{4} ) \))(\( 1 - 4 T + 8 T^{2} + 68 T^{3} - 304 T^{4} + 804 T^{5} + 1304 T^{6} - 7396 T^{7} + 43982 T^{8} - 81356 T^{9} + 157784 T^{10} + 1070124 T^{11} - 4450864 T^{12} + 10951468 T^{13} + 14172488 T^{14} - 77948684 T^{15} + 214358881 T^{16} \))(\( 1 + 4 T + 8 T^{2} - 68 T^{3} - 304 T^{4} - 804 T^{5} + 1304 T^{6} + 7396 T^{7} + 43982 T^{8} + 81356 T^{9} + 157784 T^{10} - 1070124 T^{11} - 4450864 T^{12} - 10951468 T^{13} + 14172488 T^{14} + 77948684 T^{15} + 214358881 T^{16} \))
$13$ (\( 1 + 13 T^{2} \))(\( 1 + 4 T + 13 T^{2} \))(\( 1 - 4 T + 13 T^{2} \))(\( 1 + 13 T^{2} \))(\( ( 1 + 13 T^{2} )^{2} \))(\( 1 - 22 T^{2} + 169 T^{4} \))(\( ( 1 - 4 T + 13 T^{2} )( 1 + 4 T + 13 T^{2} ) \))(\( 1 - 22 T^{2} + 169 T^{4} \))(\( ( 1 + 8 T + 32 T^{2} + 120 T^{3} + 446 T^{4} + 1560 T^{5} + 5408 T^{6} + 17576 T^{7} + 28561 T^{8} )^{2} \))(\( ( 1 - 8 T + 32 T^{2} - 120 T^{3} + 446 T^{4} - 1560 T^{5} + 5408 T^{6} - 17576 T^{7} + 28561 T^{8} )^{2} \))(\( 1 + 4 T + 6 T^{2} + 4 T^{3} + 2 T^{4} + 52 T^{5} + 1014 T^{6} + 8788 T^{7} + 28561 T^{8} \))(\( 1 + 4 T + 6 T^{2} + 4 T^{3} + 2 T^{4} + 52 T^{5} + 1014 T^{6} + 8788 T^{7} + 28561 T^{8} \))(\( 1 - 8 T + 36 T^{2} - 104 T^{3} + 200 T^{4} + 72 T^{5} - 4084 T^{6} + 34216 T^{7} - 153618 T^{8} + 444808 T^{9} - 690196 T^{10} + 158184 T^{11} + 5712200 T^{12} - 38614472 T^{13} + 173765124 T^{14} - 501988136 T^{15} + 815730721 T^{16} \))(\( 1 - 8 T + 36 T^{2} - 104 T^{3} + 200 T^{4} + 72 T^{5} - 4084 T^{6} + 34216 T^{7} - 153618 T^{8} + 444808 T^{9} - 690196 T^{10} + 158184 T^{11} + 5712200 T^{12} - 38614472 T^{13} + 173765124 T^{14} - 501988136 T^{15} + 815730721 T^{16} \))
$17$ (\( 1 + 6 T + 17 T^{2} \))(\( 1 + 2 T + 17 T^{2} \))(\( 1 + 2 T + 17 T^{2} \))(\( 1 + 6 T + 17 T^{2} \))(\( ( 1 - 6 T + 17 T^{2} )^{2} \))(\( ( 1 + 2 T + 17 T^{2} )^{2} \))(\( ( 1 - 2 T + 17 T^{2} )^{2} \))(\( ( 1 + 2 T + 17 T^{2} )^{2} \))(\( ( 1 + 22 T^{2} + 289 T^{4} )^{4} \))(\( ( 1 + 22 T^{2} + 289 T^{4} )^{4} \))(\( ( 1 - 26 T^{2} + 289 T^{4} )^{2} \))(\( ( 1 - 26 T^{2} + 289 T^{4} )^{2} \))(\( 1 - 72 T^{2} + 2620 T^{4} - 64376 T^{6} + 1215110 T^{8} - 18604664 T^{10} + 218825020 T^{12} - 1737904968 T^{14} + 6975757441 T^{16} \))(\( 1 - 72 T^{2} + 2620 T^{4} - 64376 T^{6} + 1215110 T^{8} - 18604664 T^{10} + 218825020 T^{12} - 1737904968 T^{14} + 6975757441 T^{16} \))
$19$ (\( 1 + 2 T + 19 T^{2} \))(\( 1 + 19 T^{2} \))(\( 1 + 19 T^{2} \))(\( 1 - 2 T + 19 T^{2} \))(\( 1 - 34 T^{2} + 361 T^{4} \))(\( 1 - 34 T^{2} + 361 T^{4} \))(\( ( 1 - 19 T^{2} )^{2} \))(\( 1 - 34 T^{2} + 361 T^{4} \))(\( 1 + 124 T^{4} - 27546 T^{8} + 16159804 T^{12} + 16983563041 T^{16} \))(\( 1 + 124 T^{4} - 27546 T^{8} + 16159804 T^{12} + 16983563041 T^{16} \))(\( 1 + 8 T + 18 T^{2} - 160 T^{3} - 1246 T^{4} - 3040 T^{5} + 6498 T^{6} + 54872 T^{7} + 130321 T^{8} \))(\( 1 - 8 T + 18 T^{2} + 160 T^{3} - 1246 T^{4} + 3040 T^{5} + 6498 T^{6} - 54872 T^{7} + 130321 T^{8} \))(\( 1 - 4 T - 8 T^{2} - 28 T^{3} + 336 T^{4} - 1308 T^{5} + 14056 T^{6} - 28228 T^{7} - 115058 T^{8} - 536332 T^{9} + 5074216 T^{10} - 8971572 T^{11} + 43787856 T^{12} - 69330772 T^{13} - 376367048 T^{14} - 3575486956 T^{15} + 16983563041 T^{16} \))(\( 1 + 4 T - 8 T^{2} + 28 T^{3} + 336 T^{4} + 1308 T^{5} + 14056 T^{6} + 28228 T^{7} - 115058 T^{8} + 536332 T^{9} + 5074216 T^{10} + 8971572 T^{11} + 43787856 T^{12} + 69330772 T^{13} - 376367048 T^{14} + 3575486956 T^{15} + 16983563041 T^{16} \))
$23$ (\( 1 + 23 T^{2} \))(\( 1 + 23 T^{2} \))(\( 1 + 23 T^{2} \))(\( 1 + 23 T^{2} \))(\( ( 1 + 23 T^{2} )^{2} \))(\( ( 1 - 4 T + 23 T^{2} )^{2} \))(\( ( 1 + 23 T^{2} )^{2} \))(\( ( 1 + 4 T + 23 T^{2} )^{2} \))(\( ( 1 - 44 T^{2} + 1110 T^{4} - 23276 T^{6} + 279841 T^{8} )^{2} \))(\( ( 1 - 44 T^{2} + 1110 T^{4} - 23276 T^{6} + 279841 T^{8} )^{2} \))(\( 1 + 12 T + 72 T^{2} + 300 T^{3} + 1246 T^{4} + 6900 T^{5} + 38088 T^{6} + 146004 T^{7} + 279841 T^{8} \))(\( 1 - 12 T + 72 T^{2} - 300 T^{3} + 1246 T^{4} - 6900 T^{5} + 38088 T^{6} - 146004 T^{7} + 279841 T^{8} \))(\( 1 - 8 T + 32 T^{2} - 200 T^{3} + 2108 T^{4} - 11560 T^{5} + 45024 T^{6} - 277224 T^{7} + 1704454 T^{8} - 6376152 T^{9} + 23817696 T^{10} - 140650520 T^{11} + 589904828 T^{12} - 1287268600 T^{13} + 4737148448 T^{14} - 27238603576 T^{15} + 78310985281 T^{16} \))(\( 1 + 8 T + 32 T^{2} + 200 T^{3} + 2108 T^{4} + 11560 T^{5} + 45024 T^{6} + 277224 T^{7} + 1704454 T^{8} + 6376152 T^{9} + 23817696 T^{10} + 140650520 T^{11} + 589904828 T^{12} + 1287268600 T^{13} + 4737148448 T^{14} + 27238603576 T^{15} + 78310985281 T^{16} \))
$29$ (\( 1 + 29 T^{2} \))(\( 1 + 4 T + 29 T^{2} \))(\( 1 - 4 T + 29 T^{2} \))(\( 1 + 29 T^{2} \))(\( ( 1 + 29 T^{2} )^{2} \))(\( 1 - 22 T^{2} + 841 T^{4} \))(\( ( 1 - 4 T + 29 T^{2} )( 1 + 4 T + 29 T^{2} ) \))(\( 1 - 22 T^{2} + 841 T^{4} \))(\( ( 1 + 1022 T^{4} + 707281 T^{8} )^{2} \))(\( ( 1 + 1022 T^{4} + 707281 T^{8} )^{2} \))(\( 1 - 4 T + 6 T^{2} + 204 T^{3} - 830 T^{4} + 5916 T^{5} + 5046 T^{6} - 97556 T^{7} + 707281 T^{8} \))(\( 1 - 4 T + 6 T^{2} + 204 T^{3} - 830 T^{4} + 5916 T^{5} + 5046 T^{6} - 97556 T^{7} + 707281 T^{8} \))(\( 1 - 12 T^{2} + 64 T^{3} + 72 T^{4} - 4416 T^{5} + 14140 T^{6} + 57344 T^{7} - 693650 T^{8} + 1662976 T^{9} + 11891740 T^{10} - 107701824 T^{11} + 50924232 T^{12} + 1312713536 T^{13} - 7137879852 T^{14} + 500246412961 T^{16} \))(\( 1 - 12 T^{2} + 64 T^{3} + 72 T^{4} - 4416 T^{5} + 14140 T^{6} + 57344 T^{7} - 693650 T^{8} + 1662976 T^{9} + 11891740 T^{10} - 107701824 T^{11} + 50924232 T^{12} + 1312713536 T^{13} - 7137879852 T^{14} + 500246412961 T^{16} \))
$31$ (\( 1 + 31 T^{2} \))(\( 1 + 31 T^{2} \))(\( 1 + 31 T^{2} \))(\( 1 + 31 T^{2} \))(\( ( 1 + 31 T^{2} )^{2} \))(\( ( 1 + 31 T^{2} )^{2} \))(\( ( 1 + 31 T^{2} )^{2} \))(\( ( 1 + 31 T^{2} )^{2} \))(\( ( 1 + 30 T^{2} + 961 T^{4} )^{4} \))(\( ( 1 + 30 T^{2} + 961 T^{4} )^{4} \))(\( ( 1 - 4 T + 31 T^{2} )^{4} \))(\( ( 1 + 4 T + 31 T^{2} )^{4} \))(\( ( 1 + 8 T + 70 T^{2} + 248 T^{3} + 961 T^{4} )^{4} \))(\( ( 1 - 8 T + 70 T^{2} - 248 T^{3} + 961 T^{4} )^{4} \))
$37$ (\( 1 + 37 T^{2} \))(\( 1 - 12 T + 37 T^{2} \))(\( 1 + 12 T + 37 T^{2} \))(\( 1 + 37 T^{2} \))(\( ( 1 + 37 T^{2} )^{2} \))(\( 1 + 26 T^{2} + 1369 T^{4} \))(\( ( 1 - 12 T + 37 T^{2} )( 1 + 12 T + 37 T^{2} ) \))(\( 1 + 26 T^{2} + 1369 T^{4} \))(\( ( 1 - 8 T + 32 T^{2} - 312 T^{3} + 3038 T^{4} - 11544 T^{5} + 43808 T^{6} - 405224 T^{7} + 1874161 T^{8} )^{2} \))(\( ( 1 + 8 T + 32 T^{2} + 312 T^{3} + 3038 T^{4} + 11544 T^{5} + 43808 T^{6} + 405224 T^{7} + 1874161 T^{8} )^{2} \))(\( 1 + 4 T + 6 T^{2} + 4 T^{3} + 2 T^{4} + 148 T^{5} + 8214 T^{6} + 202612 T^{7} + 1874161 T^{8} \))(\( 1 + 4 T + 6 T^{2} + 4 T^{3} + 2 T^{4} + 148 T^{5} + 8214 T^{6} + 202612 T^{7} + 1874161 T^{8} \))(\( 1 - 8 T - 44 T^{2} + 760 T^{3} - 1272 T^{4} - 22872 T^{5} + 112156 T^{6} + 219496 T^{7} - 4000082 T^{8} + 8121352 T^{9} + 153541564 T^{10} - 1158535416 T^{11} - 2383932792 T^{12} + 52701407320 T^{13} - 112891961996 T^{14} - 759455017064 T^{15} + 3512479453921 T^{16} \))(\( 1 - 8 T - 44 T^{2} + 760 T^{3} - 1272 T^{4} - 22872 T^{5} + 112156 T^{6} + 219496 T^{7} - 4000082 T^{8} + 8121352 T^{9} + 153541564 T^{10} - 1158535416 T^{11} - 2383932792 T^{12} + 52701407320 T^{13} - 112891961996 T^{14} - 759455017064 T^{15} + 3512479453921 T^{16} \))
$41$ (\( 1 - 6 T + 41 T^{2} \))(\( 1 + 10 T + 41 T^{2} \))(\( 1 + 10 T + 41 T^{2} \))(\( 1 - 6 T + 41 T^{2} \))(\( ( 1 - 6 T + 41 T^{2} )^{2} \))(\( ( 1 - 6 T + 41 T^{2} )^{2} \))(\( ( 1 + 10 T + 41 T^{2} )^{2} \))(\( ( 1 - 6 T + 41 T^{2} )^{2} \))(\( ( 1 - 34 T^{2} + 1681 T^{4} )^{4} \))(\( ( 1 - 34 T^{2} + 1681 T^{4} )^{4} \))(\( 1 + 12 T + 72 T^{2} + 516 T^{3} + 3694 T^{4} + 21156 T^{5} + 121032 T^{6} + 827052 T^{7} + 2825761 T^{8} \))(\( 1 + 12 T + 72 T^{2} + 516 T^{3} + 3694 T^{4} + 21156 T^{5} + 121032 T^{6} + 827052 T^{7} + 2825761 T^{8} \))(\( 1 - 8 T + 32 T^{2} - 88 T^{3} + 1788 T^{4} - 23048 T^{5} + 131040 T^{6} - 748824 T^{7} + 3829318 T^{8} - 30701784 T^{9} + 220278240 T^{10} - 1588491208 T^{11} + 5052460668 T^{12} - 10195345688 T^{13} + 152003335712 T^{14} - 1558034191048 T^{15} + 7984925229121 T^{16} \))(\( 1 - 8 T + 32 T^{2} - 88 T^{3} + 1788 T^{4} - 23048 T^{5} + 131040 T^{6} - 748824 T^{7} + 3829318 T^{8} - 30701784 T^{9} + 220278240 T^{10} - 1588491208 T^{11} + 5052460668 T^{12} - 10195345688 T^{13} + 152003335712 T^{14} - 1558034191048 T^{15} + 7984925229121 T^{16} \))
$43$ (\( 1 - 10 T + 43 T^{2} \))(\( 1 + 43 T^{2} \))(\( 1 + 43 T^{2} \))(\( 1 + 10 T + 43 T^{2} \))(\( 1 + 14 T^{2} + 1849 T^{4} \))(\( 1 - 50 T^{2} + 1849 T^{4} \))(\( ( 1 - 43 T^{2} )^{2} \))(\( 1 - 50 T^{2} + 1849 T^{4} \))(\( 1 - 452 T^{4} - 2831322 T^{8} - 1545298052 T^{12} + 11688200277601 T^{16} \))(\( 1 - 452 T^{4} - 2831322 T^{8} - 1545298052 T^{12} + 11688200277601 T^{16} \))(\( 1 - 16 T + 162 T^{2} - 1384 T^{3} + 10178 T^{4} - 59512 T^{5} + 299538 T^{6} - 1272112 T^{7} + 3418801 T^{8} \))(\( 1 + 16 T + 162 T^{2} + 1384 T^{3} + 10178 T^{4} + 59512 T^{5} + 299538 T^{6} + 1272112 T^{7} + 3418801 T^{8} \))(\( 1 + 12 T + 56 T^{2} - 604 T^{3} - 8560 T^{4} - 59644 T^{5} - 32984 T^{6} + 2234540 T^{7} + 25213774 T^{8} + 96085220 T^{9} - 60987416 T^{10} - 4742115508 T^{11} - 29264936560 T^{12} - 88793099572 T^{13} + 353996330744 T^{14} + 3261823333284 T^{15} + 11688200277601 T^{16} \))(\( 1 - 12 T + 56 T^{2} + 604 T^{3} - 8560 T^{4} + 59644 T^{5} - 32984 T^{6} - 2234540 T^{7} + 25213774 T^{8} - 96085220 T^{9} - 60987416 T^{10} + 4742115508 T^{11} - 29264936560 T^{12} + 88793099572 T^{13} + 353996330744 T^{14} - 3261823333284 T^{15} + 11688200277601 T^{16} \))
$47$ (\( 1 + 47 T^{2} \))(\( 1 + 47 T^{2} \))(\( 1 + 47 T^{2} \))(\( 1 + 47 T^{2} \))(\( ( 1 + 47 T^{2} )^{2} \))(\( ( 1 - 8 T + 47 T^{2} )^{2} \))(\( ( 1 + 47 T^{2} )^{2} \))(\( ( 1 + 8 T + 47 T^{2} )^{2} \))(\( ( 1 - 2 T^{2} + 2209 T^{4} )^{4} \))(\( ( 1 - 2 T^{2} + 2209 T^{4} )^{4} \))(\( 1 - 52 T^{2} + 486 T^{4} - 114868 T^{6} + 4879681 T^{8} \))(\( 1 - 52 T^{2} + 486 T^{4} - 114868 T^{6} + 4879681 T^{8} \))(\( 1 - 312 T^{2} + 44348 T^{4} - 3794696 T^{6} + 215798406 T^{8} - 8382483464 T^{10} + 216404092988 T^{12} - 3363115182648 T^{14} + 23811286661761 T^{16} \))(\( 1 - 312 T^{2} + 44348 T^{4} - 3794696 T^{6} + 215798406 T^{8} - 8382483464 T^{10} + 216404092988 T^{12} - 3363115182648 T^{14} + 23811286661761 T^{16} \))
$53$ (\( 1 + 53 T^{2} \))(\( 1 + 4 T + 53 T^{2} \))(\( 1 - 4 T + 53 T^{2} \))(\( 1 + 53 T^{2} \))(\( ( 1 + 53 T^{2} )^{2} \))(\( 1 - 70 T^{2} + 2809 T^{4} \))(\( ( 1 - 4 T + 53 T^{2} )( 1 + 4 T + 53 T^{2} ) \))(\( 1 - 70 T^{2} + 2809 T^{4} \))(\( ( 1 - 24 T + 288 T^{2} - 2856 T^{3} + 23966 T^{4} - 151368 T^{5} + 808992 T^{6} - 3573048 T^{7} + 7890481 T^{8} )^{2} \))(\( ( 1 + 24 T + 288 T^{2} + 2856 T^{3} + 23966 T^{4} + 151368 T^{5} + 808992 T^{6} + 3573048 T^{7} + 7890481 T^{8} )^{2} \))(\( 1 + 4 T + 54 T^{2} + 708 T^{3} + 3490 T^{4} + 37524 T^{5} + 151686 T^{6} + 595508 T^{7} + 7890481 T^{8} \))(\( 1 + 4 T + 54 T^{2} + 708 T^{3} + 3490 T^{4} + 37524 T^{5} + 151686 T^{6} + 595508 T^{7} + 7890481 T^{8} \))(\( 1 + 8 T + 100 T^{2} + 424 T^{3} + 3016 T^{4} + 19768 T^{5} + 118540 T^{6} + 2296472 T^{7} + 11779502 T^{8} + 121713016 T^{9} + 332978860 T^{10} + 2943000536 T^{11} + 23797690696 T^{12} + 177314889032 T^{13} + 2216436112900 T^{14} + 9397689118696 T^{15} + 62259690411361 T^{16} \))(\( 1 + 8 T + 100 T^{2} + 424 T^{3} + 3016 T^{4} + 19768 T^{5} + 118540 T^{6} + 2296472 T^{7} + 11779502 T^{8} + 121713016 T^{9} + 332978860 T^{10} + 2943000536 T^{11} + 23797690696 T^{12} + 177314889032 T^{13} + 2216436112900 T^{14} + 9397689118696 T^{15} + 62259690411361 T^{16} \))
$59$ (\( 1 + 6 T + 59 T^{2} \))(\( 1 + 59 T^{2} \))(\( 1 + 59 T^{2} \))(\( 1 - 6 T + 59 T^{2} \))(\( 1 - 82 T^{2} + 3481 T^{4} \))(\( 1 + 78 T^{2} + 3481 T^{4} \))(\( ( 1 - 59 T^{2} )^{2} \))(\( 1 + 78 T^{2} + 3481 T^{4} \))(\( 1 - 9668 T^{4} + 43107750 T^{8} - 117150646148 T^{12} + 146830437604321 T^{16} \))(\( 1 - 9668 T^{4} + 43107750 T^{8} - 117150646148 T^{12} + 146830437604321 T^{16} \))(\( 1 + 16 T + 114 T^{2} + 696 T^{3} + 4834 T^{4} + 41064 T^{5} + 396834 T^{6} + 3286064 T^{7} + 12117361 T^{8} \))(\( 1 - 16 T + 114 T^{2} - 696 T^{3} + 4834 T^{4} - 41064 T^{5} + 396834 T^{6} - 3286064 T^{7} + 12117361 T^{8} \))(\( 1 + 20 T + 136 T^{2} - 1124 T^{3} - 27632 T^{4} - 217412 T^{5} - 194408 T^{6} + 12195572 T^{7} + 142178894 T^{8} + 719538748 T^{9} - 676734248 T^{10} - 44651859148 T^{11} - 334826919152 T^{12} - 803574912076 T^{13} + 5736552575176 T^{14} + 49773029696380 T^{15} + 146830437604321 T^{16} \))(\( 1 - 20 T + 136 T^{2} + 1124 T^{3} - 27632 T^{4} + 217412 T^{5} - 194408 T^{6} - 12195572 T^{7} + 142178894 T^{8} - 719538748 T^{9} - 676734248 T^{10} + 44651859148 T^{11} - 334826919152 T^{12} + 803574912076 T^{13} + 5736552575176 T^{14} - 49773029696380 T^{15} + 146830437604321 T^{16} \))
$61$ (\( 1 + 61 T^{2} \))(\( 1 - 12 T + 61 T^{2} \))(\( 1 + 12 T + 61 T^{2} \))(\( 1 + 61 T^{2} \))(\( ( 1 + 61 T^{2} )^{2} \))(\( 1 - 118 T^{2} + 3721 T^{4} \))(\( ( 1 - 12 T + 61 T^{2} )( 1 + 12 T + 61 T^{2} ) \))(\( 1 - 118 T^{2} + 3721 T^{4} \))(\( ( 1 - 8 T + 32 T^{2} - 504 T^{3} + 7934 T^{4} - 30744 T^{5} + 119072 T^{6} - 1815848 T^{7} + 13845841 T^{8} )^{2} \))(\( ( 1 + 8 T + 32 T^{2} + 504 T^{3} + 7934 T^{4} + 30744 T^{5} + 119072 T^{6} + 1815848 T^{7} + 13845841 T^{8} )^{2} \))(\( 1 + 4 T + 6 T^{2} + 4 T^{3} + 2 T^{4} + 244 T^{5} + 22326 T^{6} + 907924 T^{7} + 13845841 T^{8} \))(\( 1 + 4 T + 6 T^{2} + 4 T^{3} + 2 T^{4} + 244 T^{5} + 22326 T^{6} + 907924 T^{7} + 13845841 T^{8} \))(\( 1 + 24 T + 132 T^{2} - 648 T^{3} + 72 T^{4} + 138600 T^{5} + 1067820 T^{6} + 2497416 T^{7} + 88430 T^{8} + 152342376 T^{9} + 3973358220 T^{10} + 31459566600 T^{11} + 996900552 T^{12} - 547298403048 T^{13} + 6800689415652 T^{14} + 75425828064504 T^{15} + 191707312997281 T^{16} \))(\( 1 + 24 T + 132 T^{2} - 648 T^{3} + 72 T^{4} + 138600 T^{5} + 1067820 T^{6} + 2497416 T^{7} + 88430 T^{8} + 152342376 T^{9} + 3973358220 T^{10} + 31459566600 T^{11} + 996900552 T^{12} - 547298403048 T^{13} + 6800689415652 T^{14} + 75425828064504 T^{15} + 191707312997281 T^{16} \))
$67$ (\( 1 - 14 T + 67 T^{2} \))(\( 1 + 67 T^{2} \))(\( 1 + 67 T^{2} \))(\( 1 + 14 T + 67 T^{2} \))(\( 1 + 62 T^{2} + 4489 T^{4} \))(\( 1 - 34 T^{2} + 4489 T^{4} \))(\( ( 1 - 67 T^{2} )^{2} \))(\( 1 - 34 T^{2} + 4489 T^{4} \))(\( 1 + 4732 T^{4} + 41046246 T^{8} + 95355104572 T^{12} + 406067677556641 T^{16} \))(\( 1 + 4732 T^{4} + 41046246 T^{8} + 95355104572 T^{12} + 406067677556641 T^{16} \))(\( 1 + 8 T + 18 T^{2} - 736 T^{3} - 5854 T^{4} - 49312 T^{5} + 80802 T^{6} + 2406104 T^{7} + 20151121 T^{8} \))(\( 1 - 8 T + 18 T^{2} + 736 T^{3} - 5854 T^{4} + 49312 T^{5} + 80802 T^{6} - 2406104 T^{7} + 20151121 T^{8} \))(\( 1 + 36 T + 504 T^{2} + 2652 T^{3} - 10800 T^{4} - 217380 T^{5} - 1006488 T^{6} - 637596 T^{7} + 11635982 T^{8} - 42718932 T^{9} - 4518124632 T^{10} - 65379860940 T^{11} - 217632106800 T^{12} + 3580531783764 T^{13} + 45591024613176 T^{14} + 218185617791628 T^{15} + 406067677556641 T^{16} \))(\( 1 - 36 T + 504 T^{2} - 2652 T^{3} - 10800 T^{4} + 217380 T^{5} - 1006488 T^{6} + 637596 T^{7} + 11635982 T^{8} + 42718932 T^{9} - 4518124632 T^{10} + 65379860940 T^{11} - 217632106800 T^{12} - 3580531783764 T^{13} + 45591024613176 T^{14} - 218185617791628 T^{15} + 406067677556641 T^{16} \))
$71$ (\( 1 + 71 T^{2} \))(\( 1 + 71 T^{2} \))(\( 1 + 71 T^{2} \))(\( 1 + 71 T^{2} \))(\( ( 1 + 71 T^{2} )^{2} \))(\( ( 1 - 12 T + 71 T^{2} )^{2} \))(\( ( 1 + 71 T^{2} )^{2} \))(\( ( 1 + 12 T + 71 T^{2} )^{2} \))(\( ( 1 - 236 T^{2} + 23574 T^{4} - 1189676 T^{6} + 25411681 T^{8} )^{2} \))(\( ( 1 - 236 T^{2} + 23574 T^{4} - 1189676 T^{6} + 25411681 T^{8} )^{2} \))(\( 1 - 12 T + 72 T^{2} - 876 T^{3} + 10654 T^{4} - 62196 T^{5} + 362952 T^{6} - 4294932 T^{7} + 25411681 T^{8} \))(\( 1 + 12 T + 72 T^{2} + 876 T^{3} + 10654 T^{4} + 62196 T^{5} + 362952 T^{6} + 4294932 T^{7} + 25411681 T^{8} \))(\( 1 - 24 T + 288 T^{2} - 2904 T^{3} + 21308 T^{4} - 62712 T^{5} - 415008 T^{6} + 11818440 T^{7} - 144298362 T^{8} + 839109240 T^{9} - 2092055328 T^{10} - 22445314632 T^{11} + 541472098748 T^{12} - 5239482035304 T^{13} + 36892881769248 T^{14} - 218282883801384 T^{15} + 645753531245761 T^{16} \))(\( 1 + 24 T + 288 T^{2} + 2904 T^{3} + 21308 T^{4} + 62712 T^{5} - 415008 T^{6} - 11818440 T^{7} - 144298362 T^{8} - 839109240 T^{9} - 2092055328 T^{10} + 22445314632 T^{11} + 541472098748 T^{12} + 5239482035304 T^{13} + 36892881769248 T^{14} + 218282883801384 T^{15} + 645753531245761 T^{16} \))
$73$ (\( 1 + 2 T + 73 T^{2} \))(\( 1 + 6 T + 73 T^{2} \))(\( 1 + 6 T + 73 T^{2} \))(\( 1 + 2 T + 73 T^{2} \))(\( ( 1 - 2 T + 73 T^{2} )^{2} \))(\( ( 1 + 14 T + 73 T^{2} )^{2} \))(\( ( 1 - 6 T + 73 T^{2} )^{2} \))(\( ( 1 + 14 T + 73 T^{2} )^{2} \))(\( ( 1 - 196 T^{2} + 18534 T^{4} - 1044484 T^{6} + 28398241 T^{8} )^{2} \))(\( ( 1 - 196 T^{2} + 18534 T^{4} - 1044484 T^{6} + 28398241 T^{8} )^{2} \))(\( ( 1 - 14 T + 98 T^{2} - 1022 T^{3} + 5329 T^{4} )^{2} \))(\( ( 1 - 14 T + 98 T^{2} - 1022 T^{3} + 5329 T^{4} )^{2} \))(\( 1 + 32 T + 512 T^{2} + 7008 T^{3} + 95708 T^{4} + 1109024 T^{5} + 11042304 T^{6} + 107800160 T^{7} + 989395590 T^{8} + 7869411680 T^{9} + 58844438016 T^{10} + 431429189408 T^{11} + 2717938849628 T^{12} + 14528085723744 T^{13} + 77483123859968 T^{14} + 353516752611104 T^{15} + 806460091894081 T^{16} \))(\( 1 + 32 T + 512 T^{2} + 7008 T^{3} + 95708 T^{4} + 1109024 T^{5} + 11042304 T^{6} + 107800160 T^{7} + 989395590 T^{8} + 7869411680 T^{9} + 58844438016 T^{10} + 431429189408 T^{11} + 2717938849628 T^{12} + 14528085723744 T^{13} + 77483123859968 T^{14} + 353516752611104 T^{15} + 806460091894081 T^{16} \))
$79$ (\( 1 + 79 T^{2} \))(\( 1 + 79 T^{2} \))(\( 1 + 79 T^{2} \))(\( 1 + 79 T^{2} \))(\( ( 1 + 79 T^{2} )^{2} \))(\( ( 1 - 8 T + 79 T^{2} )^{2} \))(\( ( 1 + 79 T^{2} )^{2} \))(\( ( 1 + 8 T + 79 T^{2} )^{2} \))(\( ( 1 + 60 T^{2} + 1094 T^{4} + 374460 T^{6} + 38950081 T^{8} )^{2} \))(\( ( 1 + 60 T^{2} + 1094 T^{4} + 374460 T^{6} + 38950081 T^{8} )^{2} \))(\( ( 1 - 122 T^{2} + 6241 T^{4} )^{2} \))(\( ( 1 - 122 T^{2} + 6241 T^{4} )^{2} \))(\( 1 - 120 T^{2} + 10940 T^{4} + 170552 T^{6} - 23710074 T^{8} + 1064415032 T^{10} + 426113886140 T^{12} - 29170494662520 T^{14} + 1517108809906561 T^{16} \))(\( 1 - 120 T^{2} + 10940 T^{4} + 170552 T^{6} - 23710074 T^{8} + 1064415032 T^{10} + 426113886140 T^{12} - 29170494662520 T^{14} + 1517108809906561 T^{16} \))
$83$ (\( 1 + 18 T + 83 T^{2} \))(\( 1 + 83 T^{2} \))(\( 1 + 83 T^{2} \))(\( 1 - 18 T + 83 T^{2} \))(\( 1 + 158 T^{2} + 6889 T^{4} \))(\( 1 - 130 T^{2} + 6889 T^{4} \))(\( ( 1 - 83 T^{2} )^{2} \))(\( 1 - 130 T^{2} + 6889 T^{4} \))(\( 1 + 8188 T^{4} + 45970278 T^{8} + 388588732348 T^{12} + 2252292232139041 T^{16} \))(\( 1 + 8188 T^{4} + 45970278 T^{8} + 388588732348 T^{12} + 2252292232139041 T^{16} \))(\( 1 - 16 T + 114 T^{2} - 792 T^{3} + 6370 T^{4} - 65736 T^{5} + 785346 T^{6} - 9148592 T^{7} + 47458321 T^{8} \))(\( 1 + 16 T + 114 T^{2} + 792 T^{3} + 6370 T^{4} + 65736 T^{5} + 785346 T^{6} + 9148592 T^{7} + 47458321 T^{8} \))(\( 1 - 20 T + 184 T^{2} - 2684 T^{3} + 37008 T^{4} - 337916 T^{5} + 3624872 T^{6} - 38211284 T^{7} + 335969678 T^{8} - 3171536572 T^{9} + 24971743208 T^{10} - 193215975892 T^{11} + 1756337543568 T^{12} - 10572385085812 T^{13} + 60157028699896 T^{14} - 542721019792540 T^{15} + 2252292232139041 T^{16} \))(\( 1 + 20 T + 184 T^{2} + 2684 T^{3} + 37008 T^{4} + 337916 T^{5} + 3624872 T^{6} + 38211284 T^{7} + 335969678 T^{8} + 3171536572 T^{9} + 24971743208 T^{10} + 193215975892 T^{11} + 1756337543568 T^{12} + 10572385085812 T^{13} + 60157028699896 T^{14} + 542721019792540 T^{15} + 2252292232139041 T^{16} \))
$89$ (\( 1 + 18 T + 89 T^{2} \))(\( 1 - 10 T + 89 T^{2} \))(\( 1 - 10 T + 89 T^{2} \))(\( 1 + 18 T + 89 T^{2} \))(\( ( 1 - 18 T + 89 T^{2} )^{2} \))(\( ( 1 - 2 T + 89 T^{2} )^{2} \))(\( ( 1 + 10 T + 89 T^{2} )^{2} \))(\( ( 1 - 2 T + 89 T^{2} )^{2} \))(\( ( 1 - 260 T^{2} + 31014 T^{4} - 2059460 T^{6} + 62742241 T^{8} )^{2} \))(\( ( 1 - 260 T^{2} + 31014 T^{4} - 2059460 T^{6} + 62742241 T^{8} )^{2} \))(\( 1 - 12 T + 72 T^{2} - 516 T^{3} + 1582 T^{4} - 45924 T^{5} + 570312 T^{6} - 8459628 T^{7} + 62742241 T^{8} \))(\( 1 - 12 T + 72 T^{2} - 516 T^{3} + 1582 T^{4} - 45924 T^{5} + 570312 T^{6} - 8459628 T^{7} + 62742241 T^{8} \))(\( 1 + 16 T + 128 T^{2} + 2096 T^{3} + 18652 T^{4} - 22640 T^{5} - 553088 T^{6} - 12396240 T^{7} - 224072442 T^{8} - 1103265360 T^{9} - 4381010048 T^{10} - 15960498160 T^{11} + 1170268279132 T^{12} + 11704188605104 T^{13} + 63613605243008 T^{14} + 707701358328464 T^{15} + 3936588805702081 T^{16} \))(\( 1 + 16 T + 128 T^{2} + 2096 T^{3} + 18652 T^{4} - 22640 T^{5} - 553088 T^{6} - 12396240 T^{7} - 224072442 T^{8} - 1103265360 T^{9} - 4381010048 T^{10} - 15960498160 T^{11} + 1170268279132 T^{12} + 11704188605104 T^{13} + 63613605243008 T^{14} + 707701358328464 T^{15} + 3936588805702081 T^{16} \))
$97$ (\( 1 - 10 T + 97 T^{2} \))(\( 1 + 18 T + 97 T^{2} \))(\( 1 + 18 T + 97 T^{2} \))(\( 1 - 10 T + 97 T^{2} \))(\( ( 1 + 10 T + 97 T^{2} )^{2} \))(\( ( 1 + 2 T + 97 T^{2} )^{2} \))(\( ( 1 - 18 T + 97 T^{2} )^{2} \))(\( ( 1 + 2 T + 97 T^{2} )^{2} \))(\( ( 1 + 182 T^{2} + 9409 T^{4} )^{4} \))(\( ( 1 + 182 T^{2} + 9409 T^{4} )^{4} \))(\( ( 1 + 20 T + 222 T^{2} + 1940 T^{3} + 9409 T^{4} )^{2} \))(\( ( 1 + 20 T + 222 T^{2} + 1940 T^{3} + 9409 T^{4} )^{2} \))(\( ( 1 - 16 T + 428 T^{2} - 4368 T^{3} + 63222 T^{4} - 423696 T^{5} + 4027052 T^{6} - 14602768 T^{7} + 88529281 T^{8} )^{2} \))(\( ( 1 - 16 T + 428 T^{2} - 4368 T^{3} + 63222 T^{4} - 423696 T^{5} + 4027052 T^{6} - 14602768 T^{7} + 88529281 T^{8} )^{2} \))
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