Properties

Label 84.2
Level 84
Weight 2
Dimension 72
Nonzero newspaces 8
Newform subspaces 13
Sturm bound 768
Trace bound 5

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Defining parameters

Level: \( N \) = \( 84\( 84 = 2^{2} \cdot 3 \cdot 7 \) \)
Weight: \( k \) = \( 2 \)
Nonzero newspaces: \( 8 \)
Newform subspaces: \( 13 \)
Sturm bound: \(768\)
Trace bound: \(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_1(84))\).

Total New Old
Modular forms 252 88 164
Cusp forms 133 72 61
Eisenstein series 119 16 103

Trace form

\( 72q + q^{3} - 6q^{4} + 6q^{5} - 6q^{6} + 8q^{7} - 6q^{8} - 11q^{9} + O(q^{10}) \) \( 72q + q^{3} - 6q^{4} + 6q^{5} - 6q^{6} + 8q^{7} - 6q^{8} - 11q^{9} - 24q^{10} - 6q^{11} - 18q^{12} - 34q^{13} - 24q^{14} - 18q^{15} - 30q^{16} - 12q^{17} - 18q^{18} - 16q^{19} - 35q^{21} + 12q^{22} - 12q^{23} + 6q^{24} - 42q^{25} + 30q^{26} - 2q^{27} + 42q^{28} - 36q^{29} + 30q^{30} - 4q^{31} + 30q^{32} + 3q^{33} + 24q^{34} + 6q^{35} + 30q^{36} - 14q^{37} + 18q^{38} + 38q^{39} + 24q^{40} + 24q^{41} + 48q^{42} + 40q^{43} + 24q^{44} + 39q^{45} + 24q^{46} + 18q^{47} + 54q^{48} + 18q^{49} + 42q^{50} + 21q^{51} + 24q^{52} + 24q^{53} + 48q^{54} + 18q^{56} - 10q^{57} - 12q^{59} + 12q^{60} - 40q^{61} - 17q^{63} - 42q^{64} - 6q^{65} - 18q^{66} - 60q^{67} - 36q^{68} - 12q^{69} - 72q^{70} - 30q^{72} - 58q^{73} - 66q^{74} + 10q^{75} - 84q^{76} - 120q^{78} - 12q^{79} - 72q^{80} + 73q^{81} - 84q^{82} - 12q^{83} - 114q^{84} - 36q^{85} - 66q^{86} + 12q^{87} - 120q^{88} + 12q^{89} - 120q^{90} + 20q^{91} - 84q^{92} + 5q^{93} - 24q^{94} - 18q^{95} - 66q^{96} + 56q^{97} - 42q^{98} - 54q^{99} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_1(84))\)

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
84.2.a \(\chi_{84}(1, \cdot)\) 84.2.a.a 1 1
84.2.a.b 1
84.2.b \(\chi_{84}(55, \cdot)\) 84.2.b.a 4 1
84.2.b.b 4
84.2.e \(\chi_{84}(71, \cdot)\) 84.2.e.a 12 1
84.2.f \(\chi_{84}(41, \cdot)\) 84.2.f.a 2 1
84.2.i \(\chi_{84}(25, \cdot)\) 84.2.i.a 2 2
84.2.k \(\chi_{84}(5, \cdot)\) 84.2.k.a 2 2
84.2.k.b 2
84.2.k.c 2
84.2.n \(\chi_{84}(11, \cdot)\) 84.2.n.a 24 2
84.2.o \(\chi_{84}(19, \cdot)\) 84.2.o.a 8 2
84.2.o.b 8

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_1(84))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_1(84)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_1(14))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(21))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(28))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(42))\)\(^{\oplus 2}\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ (\( 1 + T + 2 T^{3} + 4 T^{4} \))(\( 1 + T + 2 T^{3} + 4 T^{4} \))(\( 1 + 2 T^{2} + T^{4} - 4 T^{6} + 4 T^{8} + 32 T^{10} + 64 T^{12} \))(\( 1 - T + T^{2} - 2 T^{3} + 6 T^{4} - 4 T^{5} + 4 T^{6} - 8 T^{7} + 16 T^{8} \))(\( 1 - T + T^{2} - 2 T^{3} - 4 T^{5} + 4 T^{6} - 8 T^{7} + 16 T^{8} \))
$3$ (\( 1 + T \))(\( 1 - T \))(\( ( 1 + T )^{4} \))(\( ( 1 - T )^{4} \))(\( 1 + 2 T^{2} - 5 T^{4} - 44 T^{6} - 45 T^{8} + 162 T^{10} + 729 T^{12} \))(\( 1 + 3 T^{2} \))(\( 1 - T + T^{2} \))(\( 1 + 3 T + 3 T^{2} \))(\( 1 + 3 T^{2} \))(\( 1 - 3 T + 3 T^{2} \))(\( ( 1 + T + T^{2} )^{4} \))(\( ( 1 - T + T^{2} )^{4} \))
$5$ (\( 1 - 4 T + 5 T^{2} \))(\( 1 + 5 T^{2} \))(\( 1 - 6 T^{2} + 42 T^{4} - 150 T^{6} + 625 T^{8} \))(\( 1 - 6 T^{2} + 42 T^{4} - 150 T^{6} + 625 T^{8} \))(\( ( 1 - 12 T^{2} + 83 T^{4} - 472 T^{6} + 2075 T^{8} - 7500 T^{10} + 15625 T^{12} )^{2} \))(\( ( 1 + 5 T^{2} )^{2} \))(\( 1 - 2 T - T^{2} - 10 T^{3} + 25 T^{4} \))(\( 1 - 3 T + 4 T^{2} - 15 T^{3} + 25 T^{4} \))(\( 1 + 3 T + 4 T^{2} + 15 T^{3} + 25 T^{4} \))(\( 1 - 5 T^{2} + 25 T^{4} \))(\( 1 + 9 T^{2} + 45 T^{4} + 96 T^{5} + 66 T^{6} + 864 T^{7} - 394 T^{8} + 4320 T^{9} + 1650 T^{10} + 12000 T^{11} + 28125 T^{12} + 140625 T^{14} + 390625 T^{16} \))(\( 1 + 9 T^{2} + 45 T^{4} + 96 T^{5} + 66 T^{6} + 864 T^{7} - 394 T^{8} + 4320 T^{9} + 1650 T^{10} + 12000 T^{11} + 28125 T^{12} + 140625 T^{14} + 390625 T^{16} \))
$7$ (\( 1 + T \))(\( 1 - T \))(\( 1 - 2 T - 2 T^{2} - 14 T^{3} + 49 T^{4} \))(\( 1 + 2 T - 2 T^{2} + 14 T^{3} + 49 T^{4} \))(\( ( 1 + T^{2} )^{6} \))(\( 1 - 4 T + 7 T^{2} \))(\( 1 - T + 7 T^{2} \))(\( 1 - 4 T + 7 T^{2} \))(\( 1 - 4 T + 7 T^{2} \))(\( 1 + 5 T + 7 T^{2} \))(\( 1 - 2 T - 16 T^{3} + 65 T^{4} - 112 T^{5} - 686 T^{7} + 2401 T^{8} \))(\( 1 + 2 T + 16 T^{3} + 65 T^{4} + 112 T^{5} + 686 T^{7} + 2401 T^{8} \))
$11$ (\( 1 - 2 T + 11 T^{2} \))(\( 1 + 6 T + 11 T^{2} \))(\( 1 - 34 T^{2} + 514 T^{4} - 4114 T^{6} + 14641 T^{8} \))(\( 1 - 34 T^{2} + 514 T^{4} - 4114 T^{6} + 14641 T^{8} \))(\( ( 1 + 32 T^{2} + 607 T^{4} + 8240 T^{6} + 73447 T^{8} + 468512 T^{10} + 1771561 T^{12} )^{2} \))(\( ( 1 - 11 T^{2} )^{2} \))(\( 1 + 2 T - 7 T^{2} + 22 T^{3} + 121 T^{4} \))(\( 1 + 9 T + 38 T^{2} + 99 T^{3} + 121 T^{4} \))(\( 1 - 9 T + 38 T^{2} - 99 T^{3} + 121 T^{4} \))(\( 1 + 11 T^{2} + 121 T^{4} \))(\( ( 1 - 8 T + 34 T^{2} - 112 T^{3} + 339 T^{4} - 1232 T^{5} + 4114 T^{6} - 10648 T^{7} + 14641 T^{8} )( 1 + 2 T + 25 T^{2} + 58 T^{3} + 372 T^{4} + 638 T^{5} + 3025 T^{6} + 2662 T^{7} + 14641 T^{8} ) \))(\( ( 1 - 2 T + 25 T^{2} - 58 T^{3} + 372 T^{4} - 638 T^{5} + 3025 T^{6} - 2662 T^{7} + 14641 T^{8} )( 1 + 8 T + 34 T^{2} + 112 T^{3} + 339 T^{4} + 1232 T^{5} + 4114 T^{6} + 10648 T^{7} + 14641 T^{8} ) \))
$13$ (\( 1 + 6 T + 13 T^{2} \))(\( 1 - 2 T + 13 T^{2} \))(\( 1 - 12 T^{2} + 102 T^{4} - 2028 T^{6} + 28561 T^{8} \))(\( 1 - 12 T^{2} + 102 T^{4} - 2028 T^{6} + 28561 T^{8} \))(\( ( 1 + 29 T^{2} - 4 T^{3} + 377 T^{4} + 2197 T^{6} )^{4} \))(\( ( 1 - 2 T + 13 T^{2} )( 1 + 2 T + 13 T^{2} ) \))(\( ( 1 + 3 T + 13 T^{2} )^{2} \))(\( ( 1 - 13 T^{2} )^{2} \))(\( ( 1 - 13 T^{2} )^{2} \))(\( ( 1 - 5 T + 13 T^{2} )( 1 + 5 T + 13 T^{2} ) \))(\( 1 - 66 T^{2} + 2241 T^{4} - 49362 T^{6} + 759092 T^{8} - 8342178 T^{10} + 64005201 T^{12} - 318569394 T^{14} + 815730721 T^{16} \))(\( 1 - 66 T^{2} + 2241 T^{4} - 49362 T^{6} + 759092 T^{8} - 8342178 T^{10} + 64005201 T^{12} - 318569394 T^{14} + 815730721 T^{16} \))
$17$ (\( 1 + 4 T + 17 T^{2} \))(\( 1 + 17 T^{2} \))(\( 1 - 22 T^{2} + 682 T^{4} - 6358 T^{6} + 83521 T^{8} \))(\( 1 - 22 T^{2} + 682 T^{4} - 6358 T^{6} + 83521 T^{8} \))(\( ( 1 - 68 T^{2} + 2311 T^{4} - 49064 T^{6} + 667879 T^{8} - 5679428 T^{10} + 24137569 T^{12} )^{2} \))(\( ( 1 + 17 T^{2} )^{2} \))(\( 1 + 8 T + 47 T^{2} + 136 T^{3} + 289 T^{4} \))(\( 1 - 3 T - 8 T^{2} - 51 T^{3} + 289 T^{4} \))(\( 1 + 3 T - 8 T^{2} + 51 T^{3} + 289 T^{4} \))(\( 1 - 17 T^{2} + 289 T^{4} \))(\( 1 + 40 T^{2} + 786 T^{4} + 1104 T^{5} + 10208 T^{6} + 39072 T^{7} + 129731 T^{8} + 664224 T^{9} + 2950112 T^{10} + 5423952 T^{11} + 65647506 T^{12} + 965502760 T^{14} + 6975757441 T^{16} \))(\( 1 + 40 T^{2} + 786 T^{4} + 1104 T^{5} + 10208 T^{6} + 39072 T^{7} + 129731 T^{8} + 664224 T^{9} + 2950112 T^{10} + 5423952 T^{11} + 65647506 T^{12} + 965502760 T^{14} + 6975757441 T^{16} \))
$19$ (\( 1 + 4 T + 19 T^{2} \))(\( 1 + 4 T + 19 T^{2} \))(\( ( 1 - 6 T + 30 T^{2} - 114 T^{3} + 361 T^{4} )^{2} \))(\( ( 1 + 6 T + 30 T^{2} + 114 T^{3} + 361 T^{4} )^{2} \))(\( ( 1 - 50 T^{2} + 1675 T^{4} - 38132 T^{6} + 604675 T^{8} - 6516050 T^{10} + 47045881 T^{12} )^{2} \))(\( ( 1 - 8 T + 19 T^{2} )( 1 + 8 T + 19 T^{2} ) \))(\( ( 1 - 8 T + 19 T^{2} )( 1 + 7 T + 19 T^{2} ) \))(\( 1 - 3 T + 22 T^{2} - 57 T^{3} + 361 T^{4} \))(\( 1 - 3 T + 22 T^{2} - 57 T^{3} + 361 T^{4} \))(\( ( 1 + 7 T + 19 T^{2} )( 1 + 8 T + 19 T^{2} ) \))(\( 1 + 6 T - 33 T^{2} - 150 T^{3} + 1165 T^{4} + 1968 T^{5} - 34182 T^{6} - 11868 T^{7} + 759066 T^{8} - 225492 T^{9} - 12339702 T^{10} + 13498512 T^{11} + 151823965 T^{12} - 371414850 T^{13} - 1552514073 T^{14} + 5363230434 T^{15} + 16983563041 T^{16} \))(\( 1 - 6 T - 33 T^{2} + 150 T^{3} + 1165 T^{4} - 1968 T^{5} - 34182 T^{6} + 11868 T^{7} + 759066 T^{8} + 225492 T^{9} - 12339702 T^{10} - 13498512 T^{11} + 151823965 T^{12} + 371414850 T^{13} - 1552514073 T^{14} - 5363230434 T^{15} + 16983563041 T^{16} \))
$23$ (\( 1 - 2 T + 23 T^{2} \))(\( 1 + 6 T + 23 T^{2} \))(\( 1 - 82 T^{2} + 2722 T^{4} - 43378 T^{6} + 279841 T^{8} \))(\( 1 - 82 T^{2} + 2722 T^{4} - 43378 T^{6} + 279841 T^{8} \))(\( ( 1 + 112 T^{2} + 5671 T^{4} + 166576 T^{6} + 2999959 T^{8} + 31342192 T^{10} + 148035889 T^{12} )^{2} \))(\( ( 1 - 23 T^{2} )^{2} \))(\( 1 + 8 T + 41 T^{2} + 184 T^{3} + 529 T^{4} \))(\( 1 - 9 T + 50 T^{2} - 207 T^{3} + 529 T^{4} \))(\( 1 + 9 T + 50 T^{2} + 207 T^{3} + 529 T^{4} \))(\( 1 + 23 T^{2} + 529 T^{4} \))(\( 1 + 52 T^{2} + 1242 T^{4} + 2784 T^{5} + 24080 T^{6} + 140352 T^{7} + 497843 T^{8} + 3228096 T^{9} + 12738320 T^{10} + 33872928 T^{11} + 347562522 T^{12} + 7697866228 T^{14} + 78310985281 T^{16} \))(\( 1 + 52 T^{2} + 1242 T^{4} - 2784 T^{5} + 24080 T^{6} - 140352 T^{7} + 497843 T^{8} - 3228096 T^{9} + 12738320 T^{10} - 33872928 T^{11} + 347562522 T^{12} + 7697866228 T^{14} + 78310985281 T^{16} \))
$29$ (\( 1 + 2 T + 29 T^{2} \))(\( 1 - 6 T + 29 T^{2} \))(\( ( 1 + 2 T + 29 T^{2} )^{4} \))(\( ( 1 + 2 T + 29 T^{2} )^{4} \))(\( ( 1 - 30 T^{2} + 1047 T^{4} - 12836 T^{6} + 880527 T^{8} - 21218430 T^{10} + 594823321 T^{12} )^{2} \))(\( ( 1 - 29 T^{2} )^{2} \))(\( ( 1 - 4 T + 29 T^{2} )^{2} \))(\( ( 1 - 29 T^{2} )^{2} \))(\( ( 1 - 29 T^{2} )^{2} \))(\( ( 1 - 29 T^{2} )^{2} \))(\( ( 1 + 8 T + 71 T^{2} + 344 T^{3} + 1924 T^{4} + 9976 T^{5} + 59711 T^{6} + 195112 T^{7} + 707281 T^{8} )^{2} \))(\( ( 1 + 8 T + 71 T^{2} + 344 T^{3} + 1924 T^{4} + 9976 T^{5} + 59711 T^{6} + 195112 T^{7} + 707281 T^{8} )^{2} \))
$31$ (\( 1 + 31 T^{2} \))(\( 1 - 8 T + 31 T^{2} \))(\( ( 1 + 31 T^{2} )^{4} \))(\( ( 1 + 31 T^{2} )^{4} \))(\( ( 1 - 118 T^{2} + 7327 T^{4} - 280660 T^{6} + 7041247 T^{8} - 108975478 T^{10} + 887503681 T^{12} )^{2} \))(\( ( 1 - 4 T + 31 T^{2} )( 1 + 4 T + 31 T^{2} ) \))(\( 1 + 3 T - 22 T^{2} + 93 T^{3} + 961 T^{4} \))(\( ( 1 - 4 T + 31 T^{2} )( 1 + 7 T + 31 T^{2} ) \))(\( ( 1 - 4 T + 31 T^{2} )( 1 + 7 T + 31 T^{2} ) \))(\( ( 1 - 4 T + 31 T^{2} )( 1 + 7 T + 31 T^{2} ) \))(\( 1 - 6 T - 4 T^{2} + 336 T^{3} - 2729 T^{4} + 10764 T^{5} + 2216 T^{6} - 444234 T^{7} + 3877768 T^{8} - 13771254 T^{9} + 2129576 T^{10} + 320670324 T^{11} - 2520288809 T^{12} + 9619394736 T^{13} - 3550014724 T^{14} - 165075684666 T^{15} + 852891037441 T^{16} \))(\( 1 + 6 T - 4 T^{2} - 336 T^{3} - 2729 T^{4} - 10764 T^{5} + 2216 T^{6} + 444234 T^{7} + 3877768 T^{8} + 13771254 T^{9} + 2129576 T^{10} - 320670324 T^{11} - 2520288809 T^{12} - 9619394736 T^{13} - 3550014724 T^{14} + 165075684666 T^{15} + 852891037441 T^{16} \))
$37$ (\( 1 - 2 T + 37 T^{2} \))(\( 1 - 2 T + 37 T^{2} \))(\( ( 1 + 6 T + 66 T^{2} + 222 T^{3} + 1369 T^{4} )^{2} \))(\( ( 1 + 6 T + 66 T^{2} + 222 T^{3} + 1369 T^{4} )^{2} \))(\( ( 1 + 4 T + 83 T^{2} + 312 T^{3} + 3071 T^{4} + 5476 T^{5} + 50653 T^{6} )^{4} \))(\( ( 1 - 10 T + 37 T^{2} )^{2} \))(\( ( 1 - 11 T + 37 T^{2} )( 1 + 10 T + 37 T^{2} ) \))(\( 1 + 7 T + 12 T^{2} + 259 T^{3} + 1369 T^{4} \))(\( 1 + 7 T + 12 T^{2} + 259 T^{3} + 1369 T^{4} \))(\( ( 1 - 10 T + 37 T^{2} )( 1 - T + 37 T^{2} ) \))(\( 1 - 6 T - 69 T^{2} - 18 T^{3} + 4753 T^{4} + 9780 T^{5} - 152586 T^{6} - 146184 T^{7} + 2893194 T^{8} - 5408808 T^{9} - 208890234 T^{10} + 495386340 T^{11} + 8907887233 T^{12} - 1248191226 T^{13} - 177035122221 T^{14} - 569591262798 T^{15} + 3512479453921 T^{16} \))(\( 1 - 6 T - 69 T^{2} - 18 T^{3} + 4753 T^{4} + 9780 T^{5} - 152586 T^{6} - 146184 T^{7} + 2893194 T^{8} - 5408808 T^{9} - 208890234 T^{10} + 495386340 T^{11} + 8907887233 T^{12} - 1248191226 T^{13} - 177035122221 T^{14} - 569591262798 T^{15} + 3512479453921 T^{16} \))
$41$ (\( 1 + 41 T^{2} \))(\( 1 - 12 T + 41 T^{2} \))(\( 1 - 102 T^{2} + 5130 T^{4} - 171462 T^{6} + 2825761 T^{8} \))(\( 1 - 102 T^{2} + 5130 T^{4} - 171462 T^{6} + 2825761 T^{8} \))(\( ( 1 - 180 T^{2} + 14903 T^{4} - 753928 T^{6} + 25051943 T^{8} - 508636980 T^{10} + 4750104241 T^{12} )^{2} \))(\( ( 1 + 41 T^{2} )^{2} \))(\( ( 1 - 6 T + 41 T^{2} )^{2} \))(\( ( 1 + 6 T + 41 T^{2} )^{2} \))(\( ( 1 - 6 T + 41 T^{2} )^{2} \))(\( ( 1 + 41 T^{2} )^{2} \))(\( 1 - 120 T^{2} + 9948 T^{4} - 607176 T^{6} + 27583238 T^{8} - 1020662856 T^{10} + 28110670428 T^{12} - 570012508920 T^{14} + 7984925229121 T^{16} \))(\( 1 - 120 T^{2} + 9948 T^{4} - 607176 T^{6} + 27583238 T^{8} - 1020662856 T^{10} + 28110670428 T^{12} - 570012508920 T^{14} + 7984925229121 T^{16} \))
$43$ (\( 1 + 4 T + 43 T^{2} \))(\( 1 + 4 T + 43 T^{2} \))(\( 1 - 24 T^{2} + 3774 T^{4} - 44376 T^{6} + 3418801 T^{8} \))(\( 1 - 24 T^{2} + 3774 T^{4} - 44376 T^{6} + 3418801 T^{8} \))(\( ( 1 - 74 T^{2} + 4039 T^{4} - 189452 T^{6} + 7468111 T^{8} - 252991274 T^{10} + 6321363049 T^{12} )^{2} \))(\( ( 1 + 8 T + 43 T^{2} )^{2} \))(\( ( 1 - 11 T + 43 T^{2} )^{2} \))(\( ( 1 - 4 T + 43 T^{2} )^{2} \))(\( ( 1 - 4 T + 43 T^{2} )^{2} \))(\( ( 1 - 13 T + 43 T^{2} )^{2} \))(\( 1 - 210 T^{2} + 23793 T^{4} - 1729746 T^{6} + 88400276 T^{8} - 3198300354 T^{10} + 81343532193 T^{12} - 1327486240290 T^{14} + 11688200277601 T^{16} \))(\( 1 - 210 T^{2} + 23793 T^{4} - 1729746 T^{6} + 88400276 T^{8} - 3198300354 T^{10} + 81343532193 T^{12} - 1327486240290 T^{14} + 11688200277601 T^{16} \))
$47$ (\( 1 - 12 T + 47 T^{2} \))(\( 1 - 12 T + 47 T^{2} \))(\( ( 1 + 4 T + 30 T^{2} + 188 T^{3} + 2209 T^{4} )^{2} \))(\( ( 1 - 4 T + 30 T^{2} - 188 T^{3} + 2209 T^{4} )^{2} \))(\( ( 1 + 178 T^{2} + 15631 T^{4} + 882364 T^{6} + 34528879 T^{8} + 868583218 T^{10} + 10779215329 T^{12} )^{2} \))(\( ( 1 + 47 T^{2} )^{2} \))(\( 1 + 6 T - 11 T^{2} + 282 T^{3} + 2209 T^{4} \))(\( 1 - 3 T - 38 T^{2} - 141 T^{3} + 2209 T^{4} \))(\( 1 + 3 T - 38 T^{2} + 141 T^{3} + 2209 T^{4} \))(\( 1 - 47 T^{2} + 2209 T^{4} \))(\( 1 - 4 T - 144 T^{2} + 456 T^{3} + 12722 T^{4} - 27948 T^{5} - 805600 T^{6} + 556940 T^{7} + 41968563 T^{8} + 26176180 T^{9} - 1779570400 T^{10} - 2901645204 T^{11} + 62079301682 T^{12} + 104581323192 T^{13} - 1552207007376 T^{14} - 2026492481852 T^{15} + 23811286661761 T^{16} \))(\( 1 + 4 T - 144 T^{2} - 456 T^{3} + 12722 T^{4} + 27948 T^{5} - 805600 T^{6} - 556940 T^{7} + 41968563 T^{8} - 26176180 T^{9} - 1779570400 T^{10} + 2901645204 T^{11} + 62079301682 T^{12} - 104581323192 T^{13} - 1552207007376 T^{14} + 2026492481852 T^{15} + 23811286661761 T^{16} \))
$53$ (\( 1 + 6 T + 53 T^{2} \))(\( 1 + 6 T + 53 T^{2} \))(\( ( 1 - 8 T + 54 T^{2} - 424 T^{3} + 2809 T^{4} )^{2} \))(\( ( 1 - 8 T + 54 T^{2} - 424 T^{3} + 2809 T^{4} )^{2} \))(\( ( 1 - 262 T^{2} + 31111 T^{4} - 2121556 T^{6} + 87390799 T^{8} - 2067306022 T^{10} + 22164361129 T^{12} )^{2} \))(\( ( 1 - 53 T^{2} )^{2} \))(\( 1 - 12 T + 91 T^{2} - 636 T^{3} + 2809 T^{4} \))(\( 1 - 9 T + 80 T^{2} - 477 T^{3} + 2809 T^{4} \))(\( 1 + 9 T + 80 T^{2} + 477 T^{3} + 2809 T^{4} \))(\( 1 + 53 T^{2} + 2809 T^{4} \))(\( 1 + 4 T - 135 T^{2} - 900 T^{3} + 9413 T^{4} + 70368 T^{5} - 312982 T^{6} - 1938608 T^{7} + 10598262 T^{8} - 102746224 T^{9} - 879166438 T^{10} + 10476176736 T^{11} + 74273097653 T^{12} - 376375943700 T^{13} - 2992188752415 T^{14} + 4698844559348 T^{15} + 62259690411361 T^{16} \))(\( 1 + 4 T - 135 T^{2} - 900 T^{3} + 9413 T^{4} + 70368 T^{5} - 312982 T^{6} - 1938608 T^{7} + 10598262 T^{8} - 102746224 T^{9} - 879166438 T^{10} + 10476176736 T^{11} + 74273097653 T^{12} - 376375943700 T^{13} - 2992188752415 T^{14} + 4698844559348 T^{15} + 62259690411361 T^{16} \))
$59$ (\( 1 + 8 T + 59 T^{2} \))(\( 1 + 59 T^{2} \))(\( ( 1 - 4 T + 59 T^{2} )^{4} \))(\( ( 1 + 4 T + 59 T^{2} )^{4} \))(\( ( 1 + 74 T^{2} + 3563 T^{4} + 111204 T^{6} + 12402803 T^{8} + 896684714 T^{10} + 42180533641 T^{12} )^{2} \))(\( ( 1 + 59 T^{2} )^{2} \))(\( 1 + 4 T - 43 T^{2} + 236 T^{3} + 3481 T^{4} \))(\( 1 + 3 T - 50 T^{2} + 177 T^{3} + 3481 T^{4} \))(\( 1 - 3 T - 50 T^{2} - 177 T^{3} + 3481 T^{4} \))(\( 1 - 59 T^{2} + 3481 T^{4} \))(\( 1 + 14 T - 13 T^{2} - 1110 T^{3} - 3463 T^{4} + 11848 T^{5} - 87914 T^{6} + 1416852 T^{7} + 32978194 T^{8} + 83594268 T^{9} - 306028634 T^{10} + 2433330392 T^{11} - 41962421143 T^{12} - 793565971890 T^{13} - 548346937333 T^{14} + 34841120787466 T^{15} + 146830437604321 T^{16} \))(\( 1 - 14 T - 13 T^{2} + 1110 T^{3} - 3463 T^{4} - 11848 T^{5} - 87914 T^{6} - 1416852 T^{7} + 32978194 T^{8} - 83594268 T^{9} - 306028634 T^{10} - 2433330392 T^{11} - 41962421143 T^{12} + 793565971890 T^{13} - 548346937333 T^{14} - 34841120787466 T^{15} + 146830437604321 T^{16} \))
$61$ (\( 1 - 6 T + 61 T^{2} \))(\( 1 + 10 T + 61 T^{2} \))(\( 1 - 132 T^{2} + 10710 T^{4} - 491172 T^{6} + 13845841 T^{8} \))(\( 1 - 132 T^{2} + 10710 T^{4} - 491172 T^{6} + 13845841 T^{8} \))(\( ( 1 + 4 T + 37 T^{2} + 436 T^{3} + 2257 T^{4} + 14884 T^{5} + 226981 T^{6} )^{4} \))(\( ( 1 - 14 T + 61 T^{2} )( 1 + 14 T + 61 T^{2} ) \))(\( 1 - 6 T - 25 T^{2} - 366 T^{3} + 3721 T^{4} \))(\( 1 + 21 T + 208 T^{2} + 1281 T^{3} + 3721 T^{4} \))(\( 1 + 21 T + 208 T^{2} + 1281 T^{3} + 3721 T^{4} \))(\( ( 1 - T + 61 T^{2} )( 1 + 13 T + 61 T^{2} ) \))(\( 1 - 12 T + 180 T^{2} - 1584 T^{3} + 12426 T^{4} - 46860 T^{5} + 10032 T^{6} + 3576756 T^{7} - 33274477 T^{8} + 218182116 T^{9} + 37329072 T^{10} - 10636329660 T^{11} + 172048420266 T^{12} - 1337840540784 T^{13} + 9273667384980 T^{14} - 37712914032252 T^{15} + 191707312997281 T^{16} \))(\( 1 - 12 T + 180 T^{2} - 1584 T^{3} + 12426 T^{4} - 46860 T^{5} + 10032 T^{6} + 3576756 T^{7} - 33274477 T^{8} + 218182116 T^{9} + 37329072 T^{10} - 10636329660 T^{11} + 172048420266 T^{12} - 1337840540784 T^{13} + 9273667384980 T^{14} - 37712914032252 T^{15} + 191707312997281 T^{16} \))
$67$ (\( 1 + 8 T + 67 T^{2} \))(\( 1 - 8 T + 67 T^{2} \))(\( 1 - 144 T^{2} + 10830 T^{4} - 646416 T^{6} + 20151121 T^{8} \))(\( 1 - 144 T^{2} + 10830 T^{4} - 646416 T^{6} + 20151121 T^{8} \))(\( ( 1 - 330 T^{2} + 48695 T^{4} - 4163660 T^{6} + 218591855 T^{8} - 6649869930 T^{10} + 90458382169 T^{12} )^{2} \))(\( ( 1 + 16 T + 67 T^{2} )^{2} \))(\( 1 + 13 T + 102 T^{2} + 871 T^{3} + 4489 T^{4} \))(\( ( 1 - 11 T + 67 T^{2} )( 1 + 16 T + 67 T^{2} ) \))(\( ( 1 - 11 T + 67 T^{2} )( 1 + 16 T + 67 T^{2} ) \))(\( ( 1 - 11 T + 67 T^{2} )( 1 + 16 T + 67 T^{2} ) \))(\( 1 - 42 T + 1023 T^{2} - 18270 T^{3} + 261141 T^{4} - 3133152 T^{5} + 32837970 T^{6} - 307895844 T^{7} + 2631022010 T^{8} - 20629021548 T^{9} + 147409647330 T^{10} - 942336194976 T^{11} + 5262283889061 T^{12} - 24666785704890 T^{13} + 92538924958887 T^{14} - 254549887423566 T^{15} + 406067677556641 T^{16} \))(\( 1 + 42 T + 1023 T^{2} + 18270 T^{3} + 261141 T^{4} + 3133152 T^{5} + 32837970 T^{6} + 307895844 T^{7} + 2631022010 T^{8} + 20629021548 T^{9} + 147409647330 T^{10} + 942336194976 T^{11} + 5262283889061 T^{12} + 24666785704890 T^{13} + 92538924958887 T^{14} + 254549887423566 T^{15} + 406067677556641 T^{16} \))
$71$ (\( 1 - 14 T + 71 T^{2} \))(\( 1 - 6 T + 71 T^{2} \))(\( 1 - 114 T^{2} + 8418 T^{4} - 574674 T^{6} + 25411681 T^{8} \))(\( 1 - 114 T^{2} + 8418 T^{4} - 574674 T^{6} + 25411681 T^{8} \))(\( ( 1 + 152 T^{2} + 19111 T^{4} + 1530752 T^{6} + 96338551 T^{8} + 3862575512 T^{10} + 128100283921 T^{12} )^{2} \))(\( ( 1 - 71 T^{2} )^{2} \))(\( ( 1 + 10 T + 71 T^{2} )^{2} \))(\( 1 - 34 T^{2} + 5041 T^{4} \))(\( 1 - 34 T^{2} + 5041 T^{4} \))(\( ( 1 - 71 T^{2} )^{2} \))(\( 1 - 288 T^{2} + 41724 T^{4} - 4336608 T^{6} + 350671046 T^{8} - 21860840928 T^{10} + 1060276978044 T^{12} - 36892881769248 T^{14} + 645753531245761 T^{16} \))(\( 1 - 288 T^{2} + 41724 T^{4} - 4336608 T^{6} + 350671046 T^{8} - 21860840928 T^{10} + 1060276978044 T^{12} - 36892881769248 T^{14} + 645753531245761 T^{16} \))
$73$ (\( 1 + 2 T + 73 T^{2} \))(\( 1 + 10 T + 73 T^{2} \))(\( 1 - 236 T^{2} + 24310 T^{4} - 1257644 T^{6} + 28398241 T^{8} \))(\( 1 - 236 T^{2} + 24310 T^{4} - 1257644 T^{6} + 28398241 T^{8} \))(\( ( 1 - 6 T + 191 T^{2} - 772 T^{3} + 13943 T^{4} - 31974 T^{5} + 389017 T^{6} )^{4} \))(\( ( 1 - 10 T + 73 T^{2} )( 1 + 10 T + 73 T^{2} ) \))(\( 1 - 11 T + 48 T^{2} - 803 T^{3} + 5329 T^{4} \))(\( 1 + 21 T + 220 T^{2} + 1533 T^{3} + 5329 T^{4} \))(\( 1 + 21 T + 220 T^{2} + 1533 T^{3} + 5329 T^{4} \))(\( ( 1 - 7 T + 73 T^{2} )( 1 + 10 T + 73 T^{2} ) \))(\( 1 + 18 T + 347 T^{2} + 4302 T^{3} + 50601 T^{4} + 463140 T^{5} + 4384558 T^{6} + 34967736 T^{7} + 313616978 T^{8} + 2552644728 T^{9} + 23365309582 T^{10} + 180169333380 T^{11} + 1436979392841 T^{12} + 8918353993086 T^{13} + 52512976522283 T^{14} + 198853173343746 T^{15} + 806460091894081 T^{16} \))(\( 1 + 18 T + 347 T^{2} + 4302 T^{3} + 50601 T^{4} + 463140 T^{5} + 4384558 T^{6} + 34967736 T^{7} + 313616978 T^{8} + 2552644728 T^{9} + 23365309582 T^{10} + 180169333380 T^{11} + 1436979392841 T^{12} + 8918353993086 T^{13} + 52512976522283 T^{14} + 198853173343746 T^{15} + 806460091894081 T^{16} \))
$79$ (\( 1 - 12 T + 79 T^{2} \))(\( 1 + 4 T + 79 T^{2} \))(\( 1 - 288 T^{2} + 33150 T^{4} - 1797408 T^{6} + 38950081 T^{8} \))(\( 1 - 288 T^{2} + 33150 T^{4} - 1797408 T^{6} + 38950081 T^{8} \))(\( ( 1 - 142 T^{2} + 6241 T^{4} )^{6} \))(\( ( 1 + 4 T + 79 T^{2} )^{2} \))(\( 1 - 3 T - 70 T^{2} - 237 T^{3} + 6241 T^{4} \))(\( 1 - T - 78 T^{2} - 79 T^{3} + 6241 T^{4} \))(\( 1 - T - 78 T^{2} - 79 T^{3} + 6241 T^{4} \))(\( ( 1 + 4 T + 79 T^{2} )( 1 + 13 T + 79 T^{2} ) \))(\( 1 + 6 T + 132 T^{2} + 720 T^{3} + 9999 T^{4} + 52644 T^{5} - 314880 T^{6} + 1496442 T^{7} - 48671848 T^{8} + 118218918 T^{9} - 1965166080 T^{10} + 25955545116 T^{11} + 389461859919 T^{12} + 2215480607280 T^{13} + 32087544128772 T^{14} + 115223453916954 T^{15} + 1517108809906561 T^{16} \))(\( 1 - 6 T + 132 T^{2} - 720 T^{3} + 9999 T^{4} - 52644 T^{5} - 314880 T^{6} - 1496442 T^{7} - 48671848 T^{8} - 118218918 T^{9} - 1965166080 T^{10} - 25955545116 T^{11} + 389461859919 T^{12} - 2215480607280 T^{13} + 32087544128772 T^{14} - 115223453916954 T^{15} + 1517108809906561 T^{16} \))
$83$ (\( 1 + 4 T + 83 T^{2} \))(\( 1 + 12 T + 83 T^{2} \))(\( ( 1 + 4 T + 102 T^{2} + 332 T^{3} + 6889 T^{4} )^{2} \))(\( ( 1 - 4 T + 102 T^{2} - 332 T^{3} + 6889 T^{4} )^{2} \))(\( ( 1 + 370 T^{2} + 64699 T^{4} + 6784180 T^{6} + 445711411 T^{8} + 17559578770 T^{10} + 326940373369 T^{12} )^{2} \))(\( ( 1 + 83 T^{2} )^{2} \))(\( ( 1 - 2 T + 83 T^{2} )^{2} \))(\( ( 1 - 12 T + 83 T^{2} )^{2} \))(\( ( 1 + 12 T + 83 T^{2} )^{2} \))(\( ( 1 + 83 T^{2} )^{2} \))(\( ( 1 - 2 T + 229 T^{2} - 802 T^{3} + 24432 T^{4} - 66566 T^{5} + 1577581 T^{6} - 1143574 T^{7} + 47458321 T^{8} )^{2} \))(\( ( 1 + 2 T + 229 T^{2} + 802 T^{3} + 24432 T^{4} + 66566 T^{5} + 1577581 T^{6} + 1143574 T^{7} + 47458321 T^{8} )^{2} \))
$89$ (\( 1 + 89 T^{2} \))(\( 1 - 12 T + 89 T^{2} \))(\( 1 - 294 T^{2} + 36618 T^{4} - 2328774 T^{6} + 62742241 T^{8} \))(\( 1 - 294 T^{2} + 36618 T^{4} - 2328774 T^{6} + 62742241 T^{8} \))(\( ( 1 - 212 T^{2} + 30551 T^{4} - 3163080 T^{6} + 241994471 T^{8} - 13301355092 T^{10} + 496981290961 T^{12} )^{2} \))(\( ( 1 + 89 T^{2} )^{2} \))(\( 1 - 89 T^{2} + 7921 T^{4} \))(\( 1 + 9 T - 8 T^{2} + 801 T^{3} + 7921 T^{4} \))(\( 1 - 9 T - 8 T^{2} - 801 T^{3} + 7921 T^{4} \))(\( 1 - 89 T^{2} + 7921 T^{4} \))(\( 1 + 264 T^{2} + 38610 T^{4} + 50400 T^{5} + 4052064 T^{6} + 8952768 T^{7} + 353995811 T^{8} + 796796352 T^{9} + 32096398944 T^{10} + 35530437600 T^{11} + 2422477925010 T^{12} + 131203060813704 T^{14} + 3936588805702081 T^{16} \))(\( 1 + 264 T^{2} + 38610 T^{4} + 50400 T^{5} + 4052064 T^{6} + 8952768 T^{7} + 353995811 T^{8} + 796796352 T^{9} + 32096398944 T^{10} + 35530437600 T^{11} + 2422477925010 T^{12} + 131203060813704 T^{14} + 3936588805702081 T^{16} \))
$97$ (\( 1 + 2 T + 97 T^{2} \))(\( 1 + 10 T + 97 T^{2} \))(\( 1 - 204 T^{2} + 28950 T^{4} - 1919436 T^{6} + 88529281 T^{8} \))(\( 1 - 204 T^{2} + 28950 T^{4} - 1919436 T^{6} + 88529281 T^{8} \))(\( ( 1 + 2 T + 97 T^{2} )^{12} \))(\( ( 1 - 14 T + 97 T^{2} )( 1 + 14 T + 97 T^{2} ) \))(\( ( 1 - 10 T + 97 T^{2} )^{2} \))(\( 1 - 146 T^{2} + 9409 T^{4} \))(\( 1 - 146 T^{2} + 9409 T^{4} \))(\( ( 1 - 14 T + 97 T^{2} )( 1 + 14 T + 97 T^{2} ) \))(\( 1 - 594 T^{2} + 165777 T^{4} - 28537554 T^{6} + 3324136868 T^{8} - 268509845586 T^{10} + 14676118616337 T^{12} - 494785370927826 T^{14} + 7837433594376961 T^{16} \))(\( 1 - 594 T^{2} + 165777 T^{4} - 28537554 T^{6} + 3324136868 T^{8} - 268509845586 T^{10} + 14676118616337 T^{12} - 494785370927826 T^{14} + 7837433594376961 T^{16} \))
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