Properties

Label 147.2
Level 147
Weight 2
Dimension 517
Nonzero newspaces 8
Newform subspaces 21
Sturm bound 3136
Trace bound 1

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Defining parameters

Level: \( N \) = \( 147\( 147 = 3 \cdot 7^{2} \) \)
Weight: \( k \) = \( 2 \)
Nonzero newspaces: \( 8 \)
Newform subspaces: \( 21 \)
Sturm bound: \(3136\)
Trace bound: \(1\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_1(147))\).

Total New Old
Modular forms 904 613 291
Cusp forms 665 517 148
Eisenstein series 239 96 143

Trace form

\( 517q + 3q^{2} - 16q^{3} - 39q^{4} - 6q^{5} - 30q^{6} - 44q^{7} - 21q^{8} - 28q^{9} + O(q^{10}) \) \( 517q + 3q^{2} - 16q^{3} - 39q^{4} - 6q^{5} - 30q^{6} - 44q^{7} - 21q^{8} - 28q^{9} - 72q^{10} - 24q^{11} - 34q^{12} - 56q^{13} - 24q^{14} - 33q^{15} - 71q^{16} - 6q^{17} - 18q^{18} - 50q^{19} - 30q^{20} - 25q^{21} - 90q^{22} - 24q^{23} - 42q^{24} - 83q^{25} - 42q^{26} - 28q^{27} - 108q^{28} - 42q^{29} - 51q^{30} - 86q^{31} - 69q^{32} - 45q^{33} - 108q^{34} - 42q^{35} - 28q^{36} - 28q^{37} + 36q^{38} + 14q^{39} + 132q^{40} + 78q^{41} + 51q^{42} - 30q^{43} + 168q^{44} + 15q^{45} + 138q^{46} + 24q^{47} + 141q^{48} + 78q^{49} + 81q^{50} + 57q^{51} + 180q^{52} + 42q^{53} - 18q^{54} + 96q^{55} + 150q^{56} - 65q^{57} + 48q^{58} - 24q^{59} + 51q^{60} - 6q^{61} - 84q^{62} - 36q^{63} - 75q^{64} - 96q^{65} - 57q^{66} - 106q^{67} - 138q^{68} - 45q^{69} - 150q^{70} - 48q^{71} - 42q^{72} - 56q^{73} - 90q^{74} - 22q^{75} - 182q^{76} - 72q^{77} + 15q^{78} - 118q^{79} - 48q^{80} + 80q^{81} + 6q^{82} + 84q^{83} + 164q^{84} - 42q^{85} + 174q^{86} + 129q^{87} + 216q^{88} + 138q^{89} + 372q^{90} + 22q^{91} + 126q^{92} + 247q^{93} + 168q^{94} + 228q^{95} + 225q^{96} + 58q^{97} + 438q^{98} + 96q^{99} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_1(147))\)

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
147.2.a \(\chi_{147}(1, \cdot)\) 147.2.a.a 1 1
147.2.a.b 1
147.2.a.c 1
147.2.a.d 2
147.2.a.e 2
147.2.c \(\chi_{147}(146, \cdot)\) 147.2.c.a 2 1
147.2.c.b 8
147.2.e \(\chi_{147}(67, \cdot)\) 147.2.e.a 2 2
147.2.e.b 2
147.2.e.c 2
147.2.e.d 4
147.2.e.e 4
147.2.g \(\chi_{147}(68, \cdot)\) 147.2.g.a 2 2
147.2.g.b 16
147.2.i \(\chi_{147}(22, \cdot)\) 147.2.i.a 24 6
147.2.i.b 36
147.2.k \(\chi_{147}(20, \cdot)\) 147.2.k.a 96 6
147.2.m \(\chi_{147}(4, \cdot)\) 147.2.m.a 48 12
147.2.m.b 60
147.2.o \(\chi_{147}(5, \cdot)\) 147.2.o.a 12 12
147.2.o.b 192

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_1(147))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_1(147)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_1(21))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(49))\)\(^{\oplus 2}\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ (\( 1 + T + 2 T^{2} \))(\( 1 - 2 T + 2 T^{2} \))(\( 1 - 2 T + 2 T^{2} \))(\( 1 + 2 T + 3 T^{2} + 4 T^{3} + 4 T^{4} \))(\( 1 + 2 T + 3 T^{2} + 4 T^{3} + 4 T^{4} \))(\( ( 1 - 2 T^{2} )^{2} \))(\( ( 1 - 2 T^{2} + 7 T^{4} - 8 T^{6} + 16 T^{8} )^{2} \))(\( 1 + 2 T + 2 T^{2} + 4 T^{3} + 4 T^{4} \))(\( 1 - T - T^{2} - 2 T^{3} + 4 T^{4} \))(\( 1 - T - T^{2} - 2 T^{3} + 4 T^{4} \))(\( 1 - 2 T + T^{2} + 2 T^{3} - 3 T^{4} + 4 T^{5} + 4 T^{6} - 16 T^{7} + 16 T^{8} \))(\( 1 - 2 T + T^{2} + 2 T^{3} - 3 T^{4} + 4 T^{5} + 4 T^{6} - 16 T^{7} + 16 T^{8} \))(\( 1 + 2 T^{2} + 4 T^{4} \))(\( ( 1 + 2 T^{2} - 3 T^{4} - 2 T^{6} + 17 T^{8} - 8 T^{10} - 48 T^{12} + 128 T^{14} + 256 T^{16} )^{2} \))(\( 1 - 2 T^{2} + 8 T^{6} - 16 T^{8} + 64 T^{12} - 256 T^{16} + 512 T^{18} - 2048 T^{22} + 4096 T^{24} \))
$3$ (\( 1 + T \))(\( 1 + T \))(\( 1 - T \))(\( ( 1 + T )^{2} \))(\( ( 1 - T )^{2} \))(\( 1 + 3 T^{2} \))(\( 1 - 4 T^{2} + 14 T^{4} - 36 T^{6} + 81 T^{8} \))(\( 1 - T + T^{2} \))(\( 1 + T + T^{2} \))(\( 1 - T + T^{2} \))(\( ( 1 + T + T^{2} )^{2} \))(\( ( 1 - T + T^{2} )^{2} \))(\( 1 - 3 T + 3 T^{2} \))(\( 1 + 4 T^{2} + 2 T^{4} - 16 T^{6} - 29 T^{8} - 144 T^{10} + 162 T^{12} + 2916 T^{14} + 6561 T^{16} \))(\( 1 - 3 T + 6 T^{2} - 9 T^{3} + 9 T^{4} - 27 T^{6} + 81 T^{8} - 243 T^{9} + 486 T^{10} - 729 T^{11} + 729 T^{12} \))
$5$ (\( 1 - 2 T + 5 T^{2} \))(\( 1 - 2 T + 5 T^{2} \))(\( 1 + 2 T + 5 T^{2} \))(\( 1 + 4 T + 12 T^{2} + 20 T^{3} + 25 T^{4} \))(\( 1 - 4 T + 12 T^{2} - 20 T^{3} + 25 T^{4} \))(\( ( 1 + 5 T^{2} )^{2} \))(\( ( 1 + 12 T^{2} + 84 T^{4} + 300 T^{6} + 625 T^{8} )^{2} \))(\( 1 + 2 T - T^{2} + 10 T^{3} + 25 T^{4} \))(\( 1 - 2 T - T^{2} - 10 T^{3} + 25 T^{4} \))(\( 1 + 2 T - T^{2} + 10 T^{3} + 25 T^{4} \))(\( 1 + 4 T + 4 T^{2} + 8 T^{3} + 39 T^{4} + 40 T^{5} + 100 T^{6} + 500 T^{7} + 625 T^{8} \))(\( 1 - 4 T + 4 T^{2} - 8 T^{3} + 39 T^{4} - 40 T^{5} + 100 T^{6} - 500 T^{7} + 625 T^{8} \))(\( 1 - 5 T^{2} + 25 T^{4} \))(\( ( 1 - 12 T^{2} + 60 T^{4} - 408 T^{6} + 2831 T^{8} - 10200 T^{10} + 37500 T^{12} - 187500 T^{14} + 390625 T^{16} )^{2} \))(\( 1 + 5 T^{2} - 125 T^{6} - 625 T^{8} + 15625 T^{12} - 390625 T^{16} - 1953125 T^{18} + 48828125 T^{22} + 244140625 T^{24} \))
$7$ (\( 1 + T - 6 T^{2} - 13 T^{3} + 29 T^{4} + 120 T^{5} - 83 T^{6} + 840 T^{7} + 1421 T^{8} - 4459 T^{9} - 14406 T^{10} + 16807 T^{11} + 117649 T^{12} \))
$11$ (\( 1 - 4 T + 11 T^{2} \))(\( 1 + 2 T + 11 T^{2} \))(\( 1 + 2 T + 11 T^{2} \))(\( ( 1 + 2 T + 11 T^{2} )^{2} \))(\( ( 1 + 2 T + 11 T^{2} )^{2} \))(\( ( 1 - 11 T^{2} )^{2} \))(\( ( 1 - 32 T^{2} + 490 T^{4} - 3872 T^{6} + 14641 T^{8} )^{2} \))(\( 1 - 2 T - 7 T^{2} - 22 T^{3} + 121 T^{4} \))(\( 1 + 4 T + 5 T^{2} + 44 T^{3} + 121 T^{4} \))(\( 1 + 4 T + 5 T^{2} + 44 T^{3} + 121 T^{4} \))(\( ( 1 - 2 T - 7 T^{2} - 22 T^{3} + 121 T^{4} )^{2} \))(\( ( 1 - 2 T - 7 T^{2} - 22 T^{3} + 121 T^{4} )^{2} \))(\( 1 + 11 T^{2} + 121 T^{4} \))(\( ( 1 + 32 T^{2} + 534 T^{4} + 7936 T^{6} + 101555 T^{8} + 960256 T^{10} + 7818294 T^{12} + 56689952 T^{14} + 214358881 T^{16} )^{2} \))(\( 1 - 11 T^{2} + 1331 T^{6} - 14641 T^{8} + 1771561 T^{12} - 214358881 T^{16} + 2357947691 T^{18} - 285311670611 T^{22} + 3138428376721 T^{24} \))
$13$ (\( 1 - 2 T + 13 T^{2} \))(\( 1 + T + 13 T^{2} \))(\( 1 - T + 13 T^{2} \))(\( 1 + 8 T + 40 T^{2} + 104 T^{3} + 169 T^{4} \))(\( 1 - 8 T + 40 T^{2} - 104 T^{3} + 169 T^{4} \))(\( ( 1 - 7 T + 13 T^{2} )( 1 + 7 T + 13 T^{2} ) \))(\( ( 1 - 32 T^{2} + 496 T^{4} - 5408 T^{6} + 28561 T^{8} )^{2} \))(\( ( 1 + T + 13 T^{2} )^{2} \))(\( ( 1 + 2 T + 13 T^{2} )^{2} \))(\( ( 1 - 2 T + 13 T^{2} )^{2} \))(\( ( 1 - 8 T + 40 T^{2} - 104 T^{3} + 169 T^{4} )^{2} \))(\( ( 1 + 8 T + 40 T^{2} + 104 T^{3} + 169 T^{4} )^{2} \))(\( ( 1 - 7 T + 13 T^{2} )( 1 + 7 T + 13 T^{2} ) \))(\( ( 1 - 32 T^{2} + 496 T^{4} - 5408 T^{6} + 28561 T^{8} )^{4} \))(\( ( 1 - 7 T + 36 T^{2} - 161 T^{3} + 659 T^{4} - 2520 T^{5} + 9073 T^{6} - 32760 T^{7} + 111371 T^{8} - 353717 T^{9} + 1028196 T^{10} - 2599051 T^{11} + 4826809 T^{12} )( 1 + 7 T + 36 T^{2} + 161 T^{3} + 659 T^{4} + 2520 T^{5} + 9073 T^{6} + 32760 T^{7} + 111371 T^{8} + 353717 T^{9} + 1028196 T^{10} + 2599051 T^{11} + 4826809 T^{12} ) \))
$17$ (\( 1 - 6 T + 17 T^{2} \))(\( 1 + 17 T^{2} \))(\( 1 + 17 T^{2} \))(\( 1 + 4 T + 20 T^{2} + 68 T^{3} + 289 T^{4} \))(\( 1 - 4 T + 20 T^{2} - 68 T^{3} + 289 T^{4} \))(\( ( 1 + 17 T^{2} )^{2} \))(\( ( 1 + 52 T^{2} + 1204 T^{4} + 15028 T^{6} + 83521 T^{8} )^{2} \))(\( 1 - 17 T^{2} + 289 T^{4} \))(\( 1 - 6 T + 19 T^{2} - 102 T^{3} + 289 T^{4} \))(\( 1 + 6 T + 19 T^{2} + 102 T^{3} + 289 T^{4} \))(\( 1 + 4 T - 4 T^{2} - 56 T^{3} - 161 T^{4} - 952 T^{5} - 1156 T^{6} + 19652 T^{7} + 83521 T^{8} \))(\( 1 - 4 T - 4 T^{2} + 56 T^{3} - 161 T^{4} + 952 T^{5} - 1156 T^{6} - 19652 T^{7} + 83521 T^{8} \))(\( 1 - 17 T^{2} + 289 T^{4} \))(\( ( 1 - 52 T^{2} + 1500 T^{4} - 32552 T^{6} + 584639 T^{8} - 9407528 T^{10} + 125281500 T^{12} - 1255153588 T^{14} + 6975757441 T^{16} )^{2} \))(\( 1 + 17 T^{2} - 4913 T^{6} - 83521 T^{8} + 24137569 T^{12} - 6975757441 T^{16} - 118587876497 T^{18} + 34271896307633 T^{22} + 582622237229761 T^{24} \))
$19$ (\( 1 + 4 T + 19 T^{2} \))(\( 1 + T + 19 T^{2} \))(\( 1 - T + 19 T^{2} \))(\( 1 + 30 T^{2} + 361 T^{4} \))(\( 1 + 30 T^{2} + 361 T^{4} \))(\( ( 1 - 7 T + 19 T^{2} )( 1 + 7 T + 19 T^{2} ) \))(\( ( 1 - 44 T^{2} + 1078 T^{4} - 15884 T^{6} + 130321 T^{8} )^{2} \))(\( ( 1 - 8 T + 19 T^{2} )( 1 + 7 T + 19 T^{2} ) \))(\( 1 + 4 T - 3 T^{2} + 76 T^{3} + 361 T^{4} \))(\( 1 - 4 T - 3 T^{2} - 76 T^{3} + 361 T^{4} \))(\( 1 - 30 T^{2} + 539 T^{4} - 10830 T^{6} + 130321 T^{8} \))(\( 1 - 30 T^{2} + 539 T^{4} - 10830 T^{6} + 130321 T^{8} \))(\( ( 1 - 8 T + 19 T^{2} )( 1 - T + 19 T^{2} ) \))(\( ( 1 + 44 T^{2} + 858 T^{4} + 15664 T^{6} + 332867 T^{8} + 5654704 T^{10} + 111815418 T^{12} + 2070018764 T^{14} + 16983563041 T^{16} )^{2} \))(\( ( 1 - 8 T + 45 T^{2} - 208 T^{3} + 809 T^{4} - 2520 T^{5} + 4789 T^{6} - 47880 T^{7} + 292049 T^{8} - 1426672 T^{9} + 5864445 T^{10} - 19808792 T^{11} + 47045881 T^{12} )( 1 - T - 18 T^{2} + 37 T^{3} + 305 T^{4} - 1008 T^{5} - 4787 T^{6} - 19152 T^{7} + 110105 T^{8} + 253783 T^{9} - 2345778 T^{10} - 2476099 T^{11} + 47045881 T^{12} ) \))
$23$ (\( 1 + 23 T^{2} \))(\( 1 + 23 T^{2} \))(\( 1 + 23 T^{2} \))(\( 1 + 4 T + 18 T^{2} + 92 T^{3} + 529 T^{4} \))(\( 1 + 4 T + 18 T^{2} + 92 T^{3} + 529 T^{4} \))(\( ( 1 - 23 T^{2} )^{2} \))(\( ( 1 - 72 T^{2} + 2282 T^{4} - 38088 T^{6} + 279841 T^{8} )^{2} \))(\( 1 - 23 T^{2} + 529 T^{4} \))(\( 1 - 23 T^{2} + 529 T^{4} \))(\( 1 - 23 T^{2} + 529 T^{4} \))(\( 1 - 4 T - 2 T^{2} + 112 T^{3} - 573 T^{4} + 2576 T^{5} - 1058 T^{6} - 48668 T^{7} + 279841 T^{8} \))(\( 1 - 4 T - 2 T^{2} + 112 T^{3} - 573 T^{4} + 2576 T^{5} - 1058 T^{6} - 48668 T^{7} + 279841 T^{8} \))(\( 1 + 23 T^{2} + 529 T^{4} \))(\( ( 1 + 72 T^{2} + 2902 T^{4} + 88128 T^{6} + 2185347 T^{8} + 46619712 T^{10} + 812098582 T^{12} + 10658584008 T^{14} + 78310985281 T^{16} )^{2} \))(\( 1 - 23 T^{2} + 12167 T^{6} - 279841 T^{8} + 148035889 T^{12} - 78310985281 T^{16} + 1801152661463 T^{18} - 952809757913927 T^{22} + 21914624432020321 T^{24} \))
$29$ (\( 1 + 2 T + 29 T^{2} \))(\( 1 - 4 T + 29 T^{2} \))(\( 1 - 4 T + 29 T^{2} \))(\( 1 + 8 T + 66 T^{2} + 232 T^{3} + 841 T^{4} \))(\( 1 + 8 T + 66 T^{2} + 232 T^{3} + 841 T^{4} \))(\( ( 1 - 29 T^{2} )^{2} \))(\( ( 1 - 32 T^{2} + 1546 T^{4} - 26912 T^{6} + 707281 T^{8} )^{2} \))(\( ( 1 - 4 T + 29 T^{2} )^{2} \))(\( ( 1 + 2 T + 29 T^{2} )^{2} \))(\( ( 1 + 2 T + 29 T^{2} )^{2} \))(\( ( 1 + 8 T + 66 T^{2} + 232 T^{3} + 841 T^{4} )^{2} \))(\( ( 1 + 8 T + 66 T^{2} + 232 T^{3} + 841 T^{4} )^{2} \))(\( ( 1 - 29 T^{2} )^{2} \))(\( ( 1 - 32 T^{2} + 1546 T^{4} - 26912 T^{6} + 707281 T^{8} )^{4} \))(\( ( 1 + 29 T^{2} + 841 T^{4} + 24389 T^{6} + 707281 T^{8} + 20511149 T^{10} + 594823321 T^{12} )^{2} \))
$31$ (\( 1 + 31 T^{2} \))(\( 1 + 9 T + 31 T^{2} \))(\( 1 - 9 T + 31 T^{2} \))(\( 1 - 8 T + 70 T^{2} - 248 T^{3} + 961 T^{4} \))(\( 1 + 8 T + 70 T^{2} + 248 T^{3} + 961 T^{4} \))(\( ( 1 - 7 T + 31 T^{2} )( 1 + 7 T + 31 T^{2} ) \))(\( ( 1 - 116 T^{2} + 5278 T^{4} - 111476 T^{6} + 923521 T^{8} )^{2} \))(\( 1 - 9 T + 50 T^{2} - 279 T^{3} + 961 T^{4} \))(\( 1 - 31 T^{2} + 961 T^{4} \))(\( 1 - 31 T^{2} + 961 T^{4} \))(\( 1 - 8 T - 6 T^{2} - 64 T^{3} + 1955 T^{4} - 1984 T^{5} - 5766 T^{6} - 238328 T^{7} + 923521 T^{8} \))(\( 1 + 8 T - 6 T^{2} + 64 T^{3} + 1955 T^{4} + 1984 T^{5} - 5766 T^{6} + 238328 T^{7} + 923521 T^{8} \))(\( ( 1 + 4 T + 31 T^{2} )( 1 + 11 T + 31 T^{2} ) \))(\( ( 1 + 116 T^{2} + 8178 T^{4} + 389296 T^{6} + 14002547 T^{8} + 374113456 T^{10} + 7552554738 T^{12} + 102950426996 T^{14} + 852891037441 T^{16} )^{2} \))(\( ( 1 + 4 T - 15 T^{2} - 184 T^{3} - 271 T^{4} + 4620 T^{5} + 26881 T^{6} + 143220 T^{7} - 260431 T^{8} - 5481544 T^{9} - 13852815 T^{10} + 114516604 T^{11} + 887503681 T^{12} )( 1 + 11 T + 90 T^{2} + 649 T^{3} + 4349 T^{4} + 27720 T^{5} + 170101 T^{6} + 859320 T^{7} + 4179389 T^{8} + 19334359 T^{9} + 83116890 T^{10} + 314920661 T^{11} + 887503681 T^{12} ) \))
$37$ (\( 1 - 6 T + 37 T^{2} \))(\( 1 - 3 T + 37 T^{2} \))(\( 1 - 3 T + 37 T^{2} \))(\( ( 1 + 4 T + 37 T^{2} )^{2} \))(\( ( 1 + 4 T + 37 T^{2} )^{2} \))(\( ( 1 - T + 37 T^{2} )^{2} \))(\( ( 1 - 4 T + 28 T^{2} - 148 T^{3} + 1369 T^{4} )^{4} \))(\( 1 + 3 T - 28 T^{2} + 111 T^{3} + 1369 T^{4} \))(\( 1 + 6 T - T^{2} + 222 T^{3} + 1369 T^{4} \))(\( 1 + 6 T - T^{2} + 222 T^{3} + 1369 T^{4} \))(\( ( 1 - 4 T - 21 T^{2} - 148 T^{3} + 1369 T^{4} )^{2} \))(\( ( 1 - 4 T - 21 T^{2} - 148 T^{3} + 1369 T^{4} )^{2} \))(\( ( 1 - 10 T + 37 T^{2} )( 1 + 11 T + 37 T^{2} ) \))(\( ( 1 + 4 T - 12 T^{2} - 184 T^{3} - 1177 T^{4} - 6808 T^{5} - 16428 T^{6} + 202612 T^{7} + 1874161 T^{8} )^{4} \))(\( ( 1 + 11 T + 37 T^{2} )^{6}( 1 + 10 T + 63 T^{2} + 260 T^{3} + 269 T^{4} - 6930 T^{5} - 79253 T^{6} - 256410 T^{7} + 368261 T^{8} + 13169780 T^{9} + 118072143 T^{10} + 693439570 T^{11} + 2565726409 T^{12} ) \))
$41$ (\( 1 + 2 T + 41 T^{2} \))(\( 1 - 10 T + 41 T^{2} \))(\( 1 + 10 T + 41 T^{2} \))(\( 1 + 4 T + 68 T^{2} + 164 T^{3} + 1681 T^{4} \))(\( 1 - 4 T + 68 T^{2} - 164 T^{3} + 1681 T^{4} \))(\( ( 1 + 41 T^{2} )^{2} \))(\( ( 1 + 108 T^{2} + 6180 T^{4} + 181548 T^{6} + 2825761 T^{8} )^{2} \))(\( ( 1 - 10 T + 41 T^{2} )^{2} \))(\( ( 1 - 2 T + 41 T^{2} )^{2} \))(\( ( 1 + 2 T + 41 T^{2} )^{2} \))(\( ( 1 - 4 T + 68 T^{2} - 164 T^{3} + 1681 T^{4} )^{2} \))(\( ( 1 + 4 T + 68 T^{2} + 164 T^{3} + 1681 T^{4} )^{2} \))(\( ( 1 + 41 T^{2} )^{2} \))(\( ( 1 + 108 T^{2} + 6180 T^{4} + 181548 T^{6} + 2825761 T^{8} )^{4} \))(\( ( 1 - 41 T^{2} + 1681 T^{4} - 68921 T^{6} + 2825761 T^{8} - 115856201 T^{10} + 4750104241 T^{12} )^{2} \))
$43$ (\( 1 + 4 T + 43 T^{2} \))(\( 1 - 5 T + 43 T^{2} \))(\( 1 - 5 T + 43 T^{2} \))(\( 1 + 54 T^{2} + 1849 T^{4} \))(\( 1 + 54 T^{2} + 1849 T^{4} \))(\( ( 1 + 5 T + 43 T^{2} )^{2} \))(\( ( 1 - 2 T + 43 T^{2} )^{8} \))(\( ( 1 - 5 T + 43 T^{2} )^{2} \))(\( ( 1 + 4 T + 43 T^{2} )^{2} \))(\( ( 1 + 4 T + 43 T^{2} )^{2} \))(\( ( 1 + 54 T^{2} + 1849 T^{4} )^{2} \))(\( ( 1 + 54 T^{2} + 1849 T^{4} )^{2} \))(\( ( 1 + 5 T + 43 T^{2} )^{2} \))(\( ( 1 - 2 T + 43 T^{2} )^{16} \))(\( ( 1 - 5 T - 18 T^{2} + 305 T^{3} - 751 T^{4} - 9360 T^{5} + 79093 T^{6} - 402480 T^{7} - 1388599 T^{8} + 24249635 T^{9} - 61538418 T^{10} - 735042215 T^{11} + 6321363049 T^{12} )^{2} \))
$47$ (\( 1 + 47 T^{2} \))(\( 1 - 6 T + 47 T^{2} \))(\( 1 + 6 T + 47 T^{2} \))(\( 1 + 86 T^{2} + 2209 T^{4} \))(\( 1 + 86 T^{2} + 2209 T^{4} \))(\( ( 1 + 47 T^{2} )^{2} \))(\( ( 1 + 12 T^{2} - 3234 T^{4} + 26508 T^{6} + 4879681 T^{8} )^{2} \))(\( 1 + 6 T - 11 T^{2} + 282 T^{3} + 2209 T^{4} \))(\( 1 - 47 T^{2} + 2209 T^{4} \))(\( 1 - 47 T^{2} + 2209 T^{4} \))(\( 1 - 86 T^{2} + 5187 T^{4} - 189974 T^{6} + 4879681 T^{8} \))(\( 1 - 86 T^{2} + 5187 T^{4} - 189974 T^{6} + 4879681 T^{8} \))(\( 1 - 47 T^{2} + 2209 T^{4} \))(\( ( 1 - 12 T^{2} + 3378 T^{4} + 91824 T^{6} + 5260979 T^{8} + 202839216 T^{10} + 16483562418 T^{12} - 129350583948 T^{14} + 23811286661761 T^{16} )^{2} \))(\( 1 + 47 T^{2} - 103823 T^{6} - 4879681 T^{8} + 10779215329 T^{12} - 23811286661761 T^{16} - 1119130473102767 T^{18} + 2472159215084012303 T^{22} + \)\(11\!\cdots\!41\)\( T^{24} \))
$53$ (\( 1 - 6 T + 53 T^{2} \))(\( 1 - 12 T + 53 T^{2} \))(\( 1 - 12 T + 53 T^{2} \))(\( ( 1 + 2 T + 53 T^{2} )^{2} \))(\( ( 1 + 2 T + 53 T^{2} )^{2} \))(\( ( 1 - 53 T^{2} )^{2} \))(\( ( 1 - 172 T^{2} + 12726 T^{4} - 483148 T^{6} + 7890481 T^{8} )^{2} \))(\( 1 + 12 T + 91 T^{2} + 636 T^{3} + 2809 T^{4} \))(\( 1 + 6 T - 17 T^{2} + 318 T^{3} + 2809 T^{4} \))(\( 1 + 6 T - 17 T^{2} + 318 T^{3} + 2809 T^{4} \))(\( ( 1 - 2 T - 49 T^{2} - 106 T^{3} + 2809 T^{4} )^{2} \))(\( ( 1 - 2 T - 49 T^{2} - 106 T^{3} + 2809 T^{4} )^{2} \))(\( 1 + 53 T^{2} + 2809 T^{4} \))(\( ( 1 + 172 T^{2} + 16858 T^{4} + 1222576 T^{6} + 70959139 T^{8} + 3434215984 T^{10} + 133017728698 T^{12} + 3812270114188 T^{14} + 62259690411361 T^{16} )^{2} \))(\( 1 - 53 T^{2} + 148877 T^{6} - 7890481 T^{8} + 22164361129 T^{12} - 62259690411361 T^{16} + 3299763591802133 T^{18} - 9269035929372191597 T^{22} + \)\(49\!\cdots\!41\)\( T^{24} \))
$59$ (\( 1 + 12 T + 59 T^{2} \))(\( 1 - 12 T + 59 T^{2} \))(\( 1 + 12 T + 59 T^{2} \))(\( 1 - 8 T + 126 T^{2} - 472 T^{3} + 3481 T^{4} \))(\( 1 + 8 T + 126 T^{2} + 472 T^{3} + 3481 T^{4} \))(\( ( 1 + 59 T^{2} )^{2} \))(\( ( 1 + 108 T^{2} + 9366 T^{4} + 375948 T^{6} + 12117361 T^{8} )^{2} \))(\( 1 + 12 T + 85 T^{2} + 708 T^{3} + 3481 T^{4} \))(\( 1 + 12 T + 85 T^{2} + 708 T^{3} + 3481 T^{4} \))(\( 1 - 12 T + 85 T^{2} - 708 T^{3} + 3481 T^{4} \))(\( 1 - 8 T - 62 T^{2} - 64 T^{3} + 8619 T^{4} - 3776 T^{5} - 215822 T^{6} - 1643032 T^{7} + 12117361 T^{8} \))(\( 1 + 8 T - 62 T^{2} + 64 T^{3} + 8619 T^{4} + 3776 T^{5} - 215822 T^{6} + 1643032 T^{7} + 12117361 T^{8} \))(\( 1 - 59 T^{2} + 3481 T^{4} \))(\( ( 1 - 108 T^{2} + 2298 T^{4} - 259632 T^{6} + 35002211 T^{8} - 903778992 T^{10} + 27845695578 T^{12} - 4555497633228 T^{14} + 146830437604321 T^{16} )^{2} \))(\( 1 + 59 T^{2} - 205379 T^{6} - 12117361 T^{8} + 42180533641 T^{12} - 146830437604321 T^{16} - 8662995818654939 T^{18} + 30155888444737842659 T^{22} + \)\(17\!\cdots\!81\)\( T^{24} \))
$61$ (\( 1 - 2 T + 61 T^{2} \))(\( 1 + 10 T + 61 T^{2} \))(\( 1 - 10 T + 61 T^{2} \))(\( 1 + 16 T + 168 T^{2} + 976 T^{3} + 3721 T^{4} \))(\( 1 - 16 T + 168 T^{2} - 976 T^{3} + 3721 T^{4} \))(\( ( 1 - 14 T + 61 T^{2} )( 1 + 14 T + 61 T^{2} ) \))(\( ( 1 - 128 T^{2} + 11536 T^{4} - 476288 T^{6} + 13845841 T^{8} )^{2} \))(\( 1 - 10 T + 39 T^{2} - 610 T^{3} + 3721 T^{4} \))(\( 1 - 2 T - 57 T^{2} - 122 T^{3} + 3721 T^{4} \))(\( 1 + 2 T - 57 T^{2} + 122 T^{3} + 3721 T^{4} \))(\( 1 + 16 T + 88 T^{2} + 736 T^{3} + 8887 T^{4} + 44896 T^{5} + 327448 T^{6} + 3631696 T^{7} + 13845841 T^{8} \))(\( 1 - 16 T + 88 T^{2} - 736 T^{3} + 8887 T^{4} - 44896 T^{5} + 327448 T^{6} - 3631696 T^{7} + 13845841 T^{8} \))(\( ( 1 - T + 61 T^{2} )( 1 + 13 T + 61 T^{2} ) \))(\( ( 1 + 128 T^{2} + 4848 T^{4} + 524032 T^{6} + 58268591 T^{8} + 1949923072 T^{10} + 67124637168 T^{12} + 6594607918208 T^{14} + 191707312997281 T^{16} )^{2} \))(\( ( 1 - 13 T + 61 T^{2} )^{6}( 1 - T - 60 T^{2} + 121 T^{3} + 3539 T^{4} - 10920 T^{5} - 204959 T^{6} - 666120 T^{7} + 13168619 T^{8} + 27464701 T^{9} - 830750460 T^{10} - 844596301 T^{11} + 51520374361 T^{12} ) \))
$67$ (\( 1 - 4 T + 67 T^{2} \))(\( 1 + 5 T + 67 T^{2} \))(\( 1 + 5 T + 67 T^{2} \))(\( 1 + 102 T^{2} + 4489 T^{4} \))(\( 1 + 102 T^{2} + 4489 T^{4} \))(\( ( 1 - 11 T + 67 T^{2} )^{2} \))(\( ( 1 + 12 T + 98 T^{2} + 804 T^{3} + 4489 T^{4} )^{4} \))(\( ( 1 - 16 T + 67 T^{2} )( 1 + 11 T + 67 T^{2} ) \))(\( 1 + 4 T - 51 T^{2} + 268 T^{3} + 4489 T^{4} \))(\( 1 + 4 T - 51 T^{2} + 268 T^{3} + 4489 T^{4} \))(\( 1 - 102 T^{2} + 5915 T^{4} - 457878 T^{6} + 20151121 T^{8} \))(\( 1 - 102 T^{2} + 5915 T^{4} - 457878 T^{6} + 20151121 T^{8} \))(\( ( 1 - 5 T + 67 T^{2} )( 1 + 16 T + 67 T^{2} ) \))(\( ( 1 - 12 T + 46 T^{2} + 432 T^{3} - 4533 T^{4} + 28944 T^{5} + 206494 T^{6} - 3609156 T^{7} + 20151121 T^{8} )^{4} \))(\( ( 1 - 16 T + 189 T^{2} - 1952 T^{3} + 18569 T^{4} - 166320 T^{5} + 1416997 T^{6} - 11143440 T^{7} + 83356241 T^{8} - 587089376 T^{9} + 3808561869 T^{10} - 21602001712 T^{11} + 90458382169 T^{12} )( 1 + 5 T - 42 T^{2} - 545 T^{3} + 89 T^{4} + 36960 T^{5} + 178837 T^{6} + 2476320 T^{7} + 399521 T^{8} - 163915835 T^{9} - 846347082 T^{10} + 6750625535 T^{11} + 90458382169 T^{12} ) \))
$71$ (\( 1 + 71 T^{2} \))(\( 1 + 6 T + 71 T^{2} \))(\( 1 + 6 T + 71 T^{2} \))(\( 1 + 4 T + 18 T^{2} + 284 T^{3} + 5041 T^{4} \))(\( 1 + 4 T + 18 T^{2} + 284 T^{3} + 5041 T^{4} \))(\( ( 1 - 71 T^{2} )^{2} \))(\( ( 1 - 200 T^{2} + 19690 T^{4} - 1008200 T^{6} + 25411681 T^{8} )^{2} \))(\( ( 1 + 6 T + 71 T^{2} )^{2} \))(\( ( 1 + 71 T^{2} )^{2} \))(\( ( 1 + 71 T^{2} )^{2} \))(\( ( 1 + 4 T + 18 T^{2} + 284 T^{3} + 5041 T^{4} )^{2} \))(\( ( 1 + 4 T + 18 T^{2} + 284 T^{3} + 5041 T^{4} )^{2} \))(\( ( 1 - 71 T^{2} )^{2} \))(\( ( 1 - 200 T^{2} + 19690 T^{4} - 1008200 T^{6} + 25411681 T^{8} )^{4} \))(\( ( 1 + 71 T^{2} + 5041 T^{4} + 357911 T^{6} + 25411681 T^{8} + 1804229351 T^{10} + 128100283921 T^{12} )^{2} \))
$73$ (\( 1 - 6 T + 73 T^{2} \))(\( 1 - 3 T + 73 T^{2} \))(\( 1 + 3 T + 73 T^{2} \))(\( 1 + 8 T + 64 T^{2} + 584 T^{3} + 5329 T^{4} \))(\( 1 - 8 T + 64 T^{2} - 584 T^{3} + 5329 T^{4} \))(\( ( 1 - 7 T + 73 T^{2} )( 1 + 7 T + 73 T^{2} ) \))(\( ( 1 - 144 T^{2} + 14784 T^{4} - 767376 T^{6} + 28398241 T^{8} )^{2} \))(\( 1 + 3 T - 64 T^{2} + 219 T^{3} + 5329 T^{4} \))(\( 1 - 6 T - 37 T^{2} - 438 T^{3} + 5329 T^{4} \))(\( 1 + 6 T - 37 T^{2} + 438 T^{3} + 5329 T^{4} \))(\( 1 + 8 T - 656 T^{3} - 5905 T^{4} - 47888 T^{5} + 3112136 T^{7} + 28398241 T^{8} \))(\( 1 - 8 T + 656 T^{3} - 5905 T^{4} + 47888 T^{5} - 3112136 T^{7} + 28398241 T^{8} \))(\( ( 1 - 17 T + 73 T^{2} )( 1 - 10 T + 73 T^{2} ) \))(\( ( 1 + 144 T^{2} + 5952 T^{4} + 594144 T^{6} + 79666271 T^{8} + 3166193376 T^{10} + 169026330432 T^{12} + 21792128585616 T^{14} + 806460091894081 T^{16} )^{2} \))(\( ( 1 - 17 T + 216 T^{2} - 2431 T^{3} + 25559 T^{4} - 257040 T^{5} + 2503873 T^{6} - 18763920 T^{7} + 136203911 T^{8} - 945700327 T^{9} + 6134020056 T^{10} - 35242217081 T^{11} + 151334226289 T^{12} )( 1 - 10 T + 27 T^{2} + 460 T^{3} - 6571 T^{4} + 32130 T^{5} + 158383 T^{6} + 2345490 T^{7} - 35016859 T^{8} + 178947820 T^{9} + 766752507 T^{10} - 20730715930 T^{11} + 151334226289 T^{12} ) \))
$79$ (\( 1 + 16 T + 79 T^{2} \))(\( 1 + T + 79 T^{2} \))(\( 1 + T + 79 T^{2} \))(\( 1 - 16 T + 190 T^{2} - 1264 T^{3} + 6241 T^{4} \))(\( 1 - 16 T + 190 T^{2} - 1264 T^{3} + 6241 T^{4} \))(\( ( 1 + 13 T + 79 T^{2} )^{2} \))(\( ( 1 - 4 T + 154 T^{2} - 316 T^{3} + 6241 T^{4} )^{4} \))(\( 1 - T - 78 T^{2} - 79 T^{3} + 6241 T^{4} \))(\( 1 - 16 T + 177 T^{2} - 1264 T^{3} + 6241 T^{4} \))(\( 1 - 16 T + 177 T^{2} - 1264 T^{3} + 6241 T^{4} \))(\( 1 + 16 T + 66 T^{2} + 512 T^{3} + 9635 T^{4} + 40448 T^{5} + 411906 T^{6} + 7888624 T^{7} + 38950081 T^{8} \))(\( 1 + 16 T + 66 T^{2} + 512 T^{3} + 9635 T^{4} + 40448 T^{5} + 411906 T^{6} + 7888624 T^{7} + 38950081 T^{8} \))(\( ( 1 - 17 T + 79 T^{2} )( 1 + 4 T + 79 T^{2} ) \))(\( ( 1 + 4 T - 138 T^{2} - 16 T^{3} + 16211 T^{4} - 1264 T^{5} - 861258 T^{6} + 1972156 T^{7} + 38950081 T^{8} )^{4} \))(\( ( 1 - 4 T - 63 T^{2} + 568 T^{3} + 2705 T^{4} - 55692 T^{5} + 9073 T^{6} - 4399668 T^{7} + 16881905 T^{8} + 280046152 T^{9} - 2453855103 T^{10} - 12308225596 T^{11} + 243087455521 T^{12} )( 1 + 17 T + 210 T^{2} + 2227 T^{3} + 21269 T^{4} + 185640 T^{5} + 1475629 T^{6} + 14665560 T^{7} + 132739829 T^{8} + 1097997853 T^{9} + 8179517010 T^{10} + 52309958783 T^{11} + 243087455521 T^{12} ) \))
$83$ (\( 1 - 12 T + 83 T^{2} \))(\( 1 + 6 T + 83 T^{2} \))(\( 1 - 6 T + 83 T^{2} \))(\( 1 - 8 T + 54 T^{2} - 664 T^{3} + 6889 T^{4} \))(\( 1 + 8 T + 54 T^{2} + 664 T^{3} + 6889 T^{4} \))(\( ( 1 + 83 T^{2} )^{2} \))(\( ( 1 + 108 T^{2} + 6894 T^{4} + 744012 T^{6} + 47458321 T^{8} )^{2} \))(\( ( 1 + 6 T + 83 T^{2} )^{2} \))(\( ( 1 + 12 T + 83 T^{2} )^{2} \))(\( ( 1 - 12 T + 83 T^{2} )^{2} \))(\( ( 1 + 8 T + 54 T^{2} + 664 T^{3} + 6889 T^{4} )^{2} \))(\( ( 1 - 8 T + 54 T^{2} - 664 T^{3} + 6889 T^{4} )^{2} \))(\( ( 1 + 83 T^{2} )^{2} \))(\( ( 1 + 108 T^{2} + 6894 T^{4} + 744012 T^{6} + 47458321 T^{8} )^{4} \))(\( ( 1 - 83 T^{2} + 6889 T^{4} - 571787 T^{6} + 47458321 T^{8} - 3939040643 T^{10} + 326940373369 T^{12} )^{2} \))
$89$ (\( 1 - 14 T + 89 T^{2} \))(\( 1 + 16 T + 89 T^{2} \))(\( 1 - 16 T + 89 T^{2} \))(\( 1 - 20 T + 260 T^{2} - 1780 T^{3} + 7921 T^{4} \))(\( 1 + 20 T + 260 T^{2} + 1780 T^{3} + 7921 T^{4} \))(\( ( 1 + 89 T^{2} )^{2} \))(\( ( 1 + 180 T^{2} + 23604 T^{4} + 1425780 T^{6} + 62742241 T^{8} )^{2} \))(\( 1 - 16 T + 167 T^{2} - 1424 T^{3} + 7921 T^{4} \))(\( 1 - 14 T + 107 T^{2} - 1246 T^{3} + 7921 T^{4} \))(\( 1 + 14 T + 107 T^{2} + 1246 T^{3} + 7921 T^{4} \))(\( 1 - 20 T + 140 T^{2} - 1640 T^{3} + 24079 T^{4} - 145960 T^{5} + 1108940 T^{6} - 14099380 T^{7} + 62742241 T^{8} \))(\( 1 + 20 T + 140 T^{2} + 1640 T^{3} + 24079 T^{4} + 145960 T^{5} + 1108940 T^{6} + 14099380 T^{7} + 62742241 T^{8} \))(\( 1 - 89 T^{2} + 7921 T^{4} \))(\( ( 1 - 180 T^{2} + 8796 T^{4} - 1397160 T^{6} + 237766175 T^{8} - 11066904360 T^{10} + 551880751836 T^{12} - 89456632372980 T^{14} + 3936588805702081 T^{16} )^{2} \))(\( 1 + 89 T^{2} - 704969 T^{6} - 62742241 T^{8} + 496981290961 T^{12} - 3936588805702081 T^{16} - 350356403707485209 T^{18} + \)\(27\!\cdots\!89\)\( T^{22} + \)\(24\!\cdots\!21\)\( T^{24} \))
$97$ (\( 1 + 18 T + 97 T^{2} \))(\( 1 - 6 T + 97 T^{2} \))(\( 1 + 6 T + 97 T^{2} \))(\( 1 + 8 T + 208 T^{2} + 776 T^{3} + 9409 T^{4} \))(\( 1 - 8 T + 208 T^{2} - 776 T^{3} + 9409 T^{4} \))(\( ( 1 - 14 T + 97 T^{2} )( 1 + 14 T + 97 T^{2} ) \))(\( ( 1 - 144 T^{2} + 19200 T^{4} - 1354896 T^{6} + 88529281 T^{8} )^{2} \))(\( ( 1 - 6 T + 97 T^{2} )^{2} \))(\( ( 1 - 18 T + 97 T^{2} )^{2} \))(\( ( 1 + 18 T + 97 T^{2} )^{2} \))(\( ( 1 - 8 T + 208 T^{2} - 776 T^{3} + 9409 T^{4} )^{2} \))(\( ( 1 + 8 T + 208 T^{2} + 776 T^{3} + 9409 T^{4} )^{2} \))(\( ( 1 - 14 T + 97 T^{2} )( 1 + 14 T + 97 T^{2} ) \))(\( ( 1 - 144 T^{2} + 19200 T^{4} - 1354896 T^{6} + 88529281 T^{8} )^{4} \))(\( ( 1 - 14 T + 99 T^{2} - 28 T^{3} - 9211 T^{4} + 131670 T^{5} - 949913 T^{6} + 12771990 T^{7} - 86666299 T^{8} - 25554844 T^{9} + 8764398819 T^{10} - 120222763598 T^{11} + 832972004929 T^{12} )( 1 + 14 T + 99 T^{2} + 28 T^{3} - 9211 T^{4} - 131670 T^{5} - 949913 T^{6} - 12771990 T^{7} - 86666299 T^{8} + 25554844 T^{9} + 8764398819 T^{10} + 120222763598 T^{11} + 832972004929 T^{12} ) \))
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