Properties

Label 56.2
Level 56
Weight 2
Dimension 42
Nonzero newspaces 6
Newform subspaces 10
Sturm bound 384
Trace bound 2

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Defining parameters

Level: \( N \) = \( 56 = 2^{3} \cdot 7 \)
Weight: \( k \) = \( 2 \)
Nonzero newspaces: \( 6 \)
Newform subspaces: \( 10 \)
Sturm bound: \(384\)
Trace bound: \(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_1(56))\).

Total New Old
Modular forms 132 62 70
Cusp forms 61 42 19
Eisenstein series 71 20 51

Trace form

\( 42q - 6q^{2} - 6q^{3} - 6q^{4} - 6q^{6} - 6q^{7} - 12q^{8} - 18q^{9} + O(q^{10}) \) \( 42q - 6q^{2} - 6q^{3} - 6q^{4} - 6q^{6} - 6q^{7} - 12q^{8} - 18q^{9} - 6q^{10} - 12q^{11} - 6q^{12} - 6q^{13} - 6q^{14} - 24q^{15} - 6q^{16} - 18q^{17} + 12q^{18} - 6q^{19} + 12q^{20} + 18q^{22} + 12q^{23} + 30q^{24} + 6q^{25} + 24q^{26} + 24q^{27} + 42q^{28} + 42q^{30} + 12q^{31} + 24q^{32} - 6q^{33} + 30q^{34} - 6q^{35} + 18q^{36} - 6q^{37} + 12q^{38} + 24q^{40} - 36q^{41} - 42q^{42} - 12q^{43} - 30q^{44} - 6q^{45} - 36q^{46} + 12q^{47} - 66q^{48} - 6q^{49} - 54q^{50} + 36q^{51} - 60q^{52} + 6q^{53} - 78q^{54} + 24q^{55} - 48q^{56} - 36q^{58} + 30q^{59} - 72q^{60} - 60q^{62} + 30q^{63} - 54q^{64} - 12q^{65} - 30q^{66} + 12q^{67} - 6q^{68} + 12q^{69} + 18q^{70} + 24q^{71} + 12q^{72} - 30q^{73} + 78q^{74} + 54q^{75} + 66q^{76} + 18q^{77} + 72q^{78} + 12q^{79} + 108q^{80} + 12q^{81} + 66q^{82} + 30q^{83} + 102q^{84} + 102q^{86} + 36q^{87} + 66q^{88} - 6q^{89} + 72q^{90} - 18q^{91} + 66q^{92} + 6q^{93} + 30q^{94} - 12q^{95} + 18q^{96} + 36q^{97} + 18q^{98} - 84q^{99} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_1(56))\)

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
56.2.a \(\chi_{56}(1, \cdot)\) 56.2.a.a 1 1
56.2.a.b 1
56.2.b \(\chi_{56}(29, \cdot)\) 56.2.b.a 2 1
56.2.b.b 4
56.2.e \(\chi_{56}(27, \cdot)\) 56.2.e.a 2 1
56.2.e.b 4
56.2.f \(\chi_{56}(55, \cdot)\) None 0 1
56.2.i \(\chi_{56}(9, \cdot)\) 56.2.i.a 2 2
56.2.i.b 2
56.2.l \(\chi_{56}(31, \cdot)\) None 0 2
56.2.m \(\chi_{56}(3, \cdot)\) 56.2.m.a 12 2
56.2.p \(\chi_{56}(37, \cdot)\) 56.2.p.a 12 2

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_1(56))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_1(56)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_1(14))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(28))\)\(^{\oplus 2}\)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ (\( \))(\( \))(\( 1 + 2 T^{2} \))(\( 1 + T + 2 T^{3} + 4 T^{4} \))(\( 1 - T + 2 T^{2} \))(\( ( 1 + 2 T + 2 T^{2} )^{2} \))(\( \))(\( \))(\( ( 1 + 2 T^{3} + 8 T^{6} )^{2} \))(\( 1 + 2 T + 2 T^{2} - 4 T^{4} - 4 T^{5} - 4 T^{6} - 8 T^{7} - 16 T^{8} + 32 T^{10} + 64 T^{11} + 64 T^{12} \))
$3$ (\( 1 + 3 T^{2} \))(\( 1 - 2 T + 3 T^{2} \))(\( 1 - 4 T^{2} + 9 T^{4} \))(\( 1 - 2 T^{2} + 2 T^{4} - 18 T^{6} + 81 T^{8} \))(\( ( 1 - 3 T^{2} )^{2} \))(\( ( 1 + 9 T^{4} )^{2} \))(\( ( 1 + 3 T^{2} )( 1 + 3 T + 3 T^{2} ) \))(\( 1 - T - 2 T^{2} - 3 T^{3} + 9 T^{4} \))(\( ( 1 + 3 T + 9 T^{2} + 18 T^{3} + 33 T^{4} + 63 T^{5} + 102 T^{6} + 189 T^{7} + 297 T^{8} + 486 T^{9} + 729 T^{10} + 729 T^{11} + 729 T^{12} )^{2} \))(\( 1 + 9 T^{2} + 35 T^{4} + 92 T^{6} + 253 T^{8} + 683 T^{10} + 1774 T^{12} + 6147 T^{14} + 20493 T^{16} + 67068 T^{18} + 229635 T^{20} + 531441 T^{22} + 531441 T^{24} \))
$5$ (\( 1 - 2 T + 5 T^{2} \))(\( 1 + 4 T + 5 T^{2} \))(\( 1 - 8 T^{2} + 25 T^{4} \))(\( 1 - 6 T^{2} + 42 T^{4} - 150 T^{6} + 625 T^{8} \))(\( ( 1 + 5 T^{2} )^{2} \))(\( ( 1 + 4 T^{2} + 25 T^{4} )^{2} \))(\( 1 - T - 4 T^{2} - 5 T^{3} + 25 T^{4} \))(\( 1 - T - 4 T^{2} - 5 T^{3} + 25 T^{4} \))(\( 1 - 15 T^{2} + 99 T^{4} - 412 T^{6} + 1641 T^{8} - 7989 T^{10} + 39846 T^{12} - 199725 T^{14} + 1025625 T^{16} - 6437500 T^{18} + 38671875 T^{20} - 146484375 T^{22} + 244140625 T^{24} \))(\( 1 + 17 T^{2} + 147 T^{4} + 788 T^{6} + 2793 T^{8} + 5371 T^{10} + 7494 T^{12} + 134275 T^{14} + 1745625 T^{16} + 12312500 T^{18} + 57421875 T^{20} + 166015625 T^{22} + 244140625 T^{24} \))
$7$ (\( 1 + T \))(\( 1 - T \))(\( ( 1 - T )^{2} \))(\( ( 1 + T )^{4} \))(\( 1 + 7 T^{2} \))(\( 1 - 10 T^{2} + 49 T^{4} \))(\( 1 - 4 T + 7 T^{2} \))(\( 1 + 4 T + 7 T^{2} \))(\( 1 + 6 T^{2} - 33 T^{4} - 700 T^{6} - 1617 T^{8} + 14406 T^{10} + 117649 T^{12} \))(\( ( 1 + 2 T + 5 T^{2} + 32 T^{3} + 35 T^{4} + 98 T^{5} + 343 T^{6} )^{2} \))
$11$ (\( 1 + 4 T + 11 T^{2} \))(\( 1 + 11 T^{2} \))(\( ( 1 - 6 T + 11 T^{2} )( 1 + 6 T + 11 T^{2} ) \))(\( 1 - 24 T^{2} + 318 T^{4} - 2904 T^{6} + 14641 T^{8} \))(\( ( 1 + 4 T + 11 T^{2} )^{2} \))(\( ( 1 - 2 T + 11 T^{2} )^{4} \))(\( 1 - T - 10 T^{2} - 11 T^{3} + 121 T^{4} \))(\( 1 + 3 T - 2 T^{2} + 33 T^{3} + 121 T^{4} \))(\( ( 1 + 3 T - 21 T^{2} - 28 T^{3} + 393 T^{4} + 153 T^{5} - 4846 T^{6} + 1683 T^{7} + 47553 T^{8} - 37268 T^{9} - 307461 T^{10} + 483153 T^{11} + 1771561 T^{12} )^{2} \))(\( 1 + 29 T^{2} + 327 T^{4} + 2696 T^{6} + 24957 T^{8} - 44693 T^{10} - 3466650 T^{12} - 5407853 T^{14} + 365395437 T^{16} + 4776128456 T^{18} + 70095354087 T^{20} + 752185313429 T^{22} + 3138428376721 T^{24} \))
$13$ (\( 1 - 2 T + 13 T^{2} \))(\( 1 + 13 T^{2} \))(\( 1 - 8 T^{2} + 169 T^{4} \))(\( 1 - 38 T^{2} + 682 T^{4} - 6422 T^{6} + 28561 T^{8} \))(\( ( 1 + 13 T^{2} )^{2} \))(\( ( 1 + 20 T^{2} + 169 T^{4} )^{2} \))(\( ( 1 - 2 T + 13 T^{2} )^{2} \))(\( ( 1 + 6 T + 13 T^{2} )^{2} \))(\( ( 1 + 42 T^{2} + 903 T^{4} + 13340 T^{6} + 152607 T^{8} + 1199562 T^{10} + 4826809 T^{12} )^{2} \))(\( ( 1 - 46 T^{2} + 1191 T^{4} - 18804 T^{6} + 201279 T^{8} - 1313806 T^{10} + 4826809 T^{12} )^{2} \))
$17$ (\( 1 + 6 T + 17 T^{2} \))(\( 1 + 2 T + 17 T^{2} \))(\( ( 1 + 6 T + 17 T^{2} )^{2} \))(\( ( 1 - 2 T + 17 T^{2} )^{4} \))(\( ( 1 - 17 T^{2} )^{2} \))(\( ( 1 - 10 T^{2} + 289 T^{4} )^{2} \))(\( 1 + 3 T - 8 T^{2} + 51 T^{3} + 289 T^{4} \))(\( 1 - 5 T + 8 T^{2} - 85 T^{3} + 289 T^{4} \))(\( ( 1 + 3 T + 39 T^{2} + 108 T^{3} + 729 T^{4} + 753 T^{5} + 12014 T^{6} + 12801 T^{7} + 210681 T^{8} + 530604 T^{9} + 3257319 T^{10} + 4259571 T^{11} + 24137569 T^{12} )^{2} \))(\( ( 1 + T - 33 T^{2} + 8 T^{3} + 565 T^{4} - 425 T^{5} - 9946 T^{6} - 7225 T^{7} + 163285 T^{8} + 39304 T^{9} - 2756193 T^{10} + 1419857 T^{11} + 24137569 T^{12} )^{2} \))
$19$ (\( 1 - 8 T + 19 T^{2} \))(\( 1 + 2 T + 19 T^{2} \))(\( 1 - 20 T^{2} + 361 T^{4} \))(\( 1 - 66 T^{2} + 1794 T^{4} - 23826 T^{6} + 130321 T^{8} \))(\( ( 1 - 19 T^{2} )^{2} \))(\( ( 1 - 32 T^{2} + 361 T^{4} )^{2} \))(\( 1 + 5 T + 6 T^{2} + 95 T^{3} + 361 T^{4} \))(\( ( 1 - 7 T + 19 T^{2} )( 1 + 8 T + 19 T^{2} ) \))(\( ( 1 + 3 T + 39 T^{2} + 108 T^{3} + 705 T^{4} + 2265 T^{5} + 12706 T^{6} + 43035 T^{7} + 254505 T^{8} + 740772 T^{9} + 5082519 T^{10} + 7428297 T^{11} + 47045881 T^{12} )^{2} \))(\( 1 + 53 T^{2} + 1215 T^{4} + 18608 T^{6} + 219741 T^{8} - 1587413 T^{10} - 104375754 T^{12} - 573056093 T^{14} + 28636866861 T^{16} + 875429753648 T^{18} + 20635029094815 T^{20} + 324946511663453 T^{22} + 2213314919066161 T^{24} \))
$23$ (\( 1 + 23 T^{2} \))(\( 1 - 8 T + 23 T^{2} \))(\( ( 1 + 6 T + 23 T^{2} )^{2} \))(\( ( 1 - 2 T + 30 T^{2} - 46 T^{3} + 529 T^{4} )^{2} \))(\( ( 1 - 8 T + 23 T^{2} )( 1 + 8 T + 23 T^{2} ) \))(\( ( 1 - 30 T^{2} + 529 T^{4} )^{2} \))(\( 1 - 3 T - 14 T^{2} - 69 T^{3} + 529 T^{4} \))(\( 1 - 7 T + 26 T^{2} - 161 T^{3} + 529 T^{4} \))(\( 1 + 69 T^{2} + 1863 T^{4} + 44136 T^{6} + 1489365 T^{8} + 33541107 T^{10} + 591428630 T^{12} + 17743245603 T^{14} + 416785390965 T^{16} + 6533711996904 T^{18} + 145893365578503 T^{20} + 2858429273741781 T^{22} + 21914624432020321 T^{24} \))(\( ( 1 - T - 61 T^{2} + 12 T^{3} + 2381 T^{4} + 29 T^{5} - 63238 T^{6} + 667 T^{7} + 1259549 T^{8} + 146004 T^{9} - 17070301 T^{10} - 6436343 T^{11} + 148035889 T^{12} )^{2} \))
$29$ (\( 1 - 6 T + 29 T^{2} \))(\( 1 - 2 T + 29 T^{2} \))(\( 1 - 50 T^{2} + 841 T^{4} \))(\( 1 - 76 T^{2} + 2854 T^{4} - 63916 T^{6} + 707281 T^{8} \))(\( ( 1 - 2 T + 29 T^{2} )( 1 + 2 T + 29 T^{2} ) \))(\( ( 1 - 10 T + 29 T^{2} )^{2}( 1 + 10 T + 29 T^{2} )^{2} \))(\( ( 1 + 6 T + 29 T^{2} )^{2} \))(\( ( 1 - 2 T + 29 T^{2} )^{2} \))(\( ( 1 - 126 T^{2} + 7623 T^{4} - 278868 T^{6} + 6410943 T^{8} - 89117406 T^{10} + 594823321 T^{12} )^{2} \))(\( ( 1 - 102 T^{2} + 4567 T^{4} - 141988 T^{6} + 3840847 T^{8} - 72142662 T^{10} + 594823321 T^{12} )^{2} \))
$31$ (\( 1 - 8 T + 31 T^{2} \))(\( 1 - 4 T + 31 T^{2} \))(\( ( 1 + 4 T + 31 T^{2} )^{2} \))(\( ( 1 + 4 T - 2 T^{2} + 124 T^{3} + 961 T^{4} )^{2} \))(\( ( 1 + 31 T^{2} )^{2} \))(\( ( 1 + 38 T^{2} + 961 T^{4} )^{2} \))(\( 1 - T - 30 T^{2} - 31 T^{3} + 961 T^{4} \))(\( 1 - 5 T - 6 T^{2} - 155 T^{3} + 961 T^{4} \))(\( 1 - 87 T^{2} + 3867 T^{4} - 78332 T^{6} - 455595 T^{8} + 87513459 T^{10} - 3532899090 T^{12} + 84100434099 T^{14} - 420751549995 T^{16} - 69519938340092 T^{18} + 3298129641784347 T^{20} - 71307660967329687 T^{22} + 787662783788549761 T^{24} \))(\( ( 1 - 5 T - 71 T^{2} + 126 T^{3} + 4725 T^{4} - 4133 T^{5} - 158330 T^{6} - 128123 T^{7} + 4540725 T^{8} + 3753666 T^{9} - 65569991 T^{10} - 143145755 T^{11} + 887503681 T^{12} )^{2} \))
$37$ (\( 1 + 2 T + 37 T^{2} \))(\( 1 + 6 T + 37 T^{2} \))(\( 1 - 2 T^{2} + 1369 T^{4} \))(\( 1 - 108 T^{2} + 5382 T^{4} - 147852 T^{6} + 1874161 T^{8} \))(\( ( 1 - 6 T + 37 T^{2} )( 1 + 6 T + 37 T^{2} ) \))(\( ( 1 - 10 T^{2} + 1369 T^{4} )^{2} \))(\( 1 - 5 T - 12 T^{2} - 185 T^{3} + 1369 T^{4} \))(\( 1 + 3 T - 28 T^{2} + 111 T^{3} + 1369 T^{4} \))(\( 1 + 105 T^{2} + 3339 T^{4} + 152764 T^{6} + 14203497 T^{8} + 501483003 T^{10} + 10611705558 T^{12} + 686530231107 T^{14} + 26619640141017 T^{16} + 391950629144476 T^{18} + 11728168896642219 T^{20} + 504901359103874145 T^{22} + 6582952005840035281 T^{24} \))(\( 1 + 161 T^{2} + 13515 T^{4} + 840428 T^{6} + 43791513 T^{8} + 2009438227 T^{10} + 80476513014 T^{12} + 2750920932763 T^{14} + 82072345795593 T^{16} + 2156308314463052 T^{18} + 47471159819742315 T^{20} + 774182083959273689 T^{22} + 6582952005840035281 T^{24} \))
$41$ (\( 1 - 2 T + 41 T^{2} \))(\( 1 + 2 T + 41 T^{2} \))(\( ( 1 - 6 T + 41 T^{2} )^{2} \))(\( ( 1 + 8 T + 30 T^{2} + 328 T^{3} + 1681 T^{4} )^{2} \))(\( ( 1 - 41 T^{2} )^{2} \))(\( ( 1 - 41 T^{2} )^{4} \))(\( ( 1 + 10 T + 41 T^{2} )^{2} \))(\( ( 1 + 2 T + 41 T^{2} )^{2} \))(\( ( 1 - 102 T^{2} + 6783 T^{4} - 298852 T^{6} + 11402223 T^{8} - 288227622 T^{10} + 4750104241 T^{12} )^{2} \))(\( ( 1 + 2 T + 83 T^{2} + 80 T^{3} + 3403 T^{4} + 3362 T^{5} + 68921 T^{6} )^{4} \))
$43$ (\( 1 + 4 T + 43 T^{2} \))(\( 1 - 8 T + 43 T^{2} \))(\( ( 1 - 10 T + 43 T^{2} )( 1 + 10 T + 43 T^{2} ) \))(\( 1 - 152 T^{2} + 9406 T^{4} - 281048 T^{6} + 3418801 T^{8} \))(\( ( 1 - 12 T + 43 T^{2} )^{2} \))(\( ( 1 + 6 T + 43 T^{2} )^{4} \))(\( ( 1 + 4 T + 43 T^{2} )^{2} \))(\( ( 1 + 4 T + 43 T^{2} )^{2} \))(\( ( 1 + 43 T^{2} )^{12} \))(\( ( 1 - 94 T^{2} + 3543 T^{4} - 112068 T^{6} + 6551007 T^{8} - 321367294 T^{10} + 6321363049 T^{12} )^{2} \))
$47$ (\( 1 + 8 T + 47 T^{2} \))(\( 1 + 4 T + 47 T^{2} \))(\( ( 1 + 47 T^{2} )^{2} \))(\( ( 1 + 47 T^{2} )^{4} \))(\( ( 1 + 47 T^{2} )^{2} \))(\( ( 1 + 70 T^{2} + 2209 T^{4} )^{2} \))(\( 1 + T - 46 T^{2} + 47 T^{3} + 2209 T^{4} \))(\( 1 + 5 T - 22 T^{2} + 235 T^{3} + 2209 T^{4} \))(\( 1 - 159 T^{2} + 13131 T^{4} - 642364 T^{6} + 18269013 T^{8} - 74583429 T^{10} - 12209351154 T^{12} - 164754794661 T^{14} + 89146955624853 T^{16} - 6924179875597756 T^{18} + 312666005155583691 T^{20} - 8363262025496977791 T^{22} + \)\(11\!\cdots\!41\)\( T^{24} \))(\( ( 1 - 15 T + 57 T^{2} - 78 T^{3} + 2013 T^{4} - 1383 T^{5} - 125066 T^{6} - 65001 T^{7} + 4446717 T^{8} - 8098194 T^{9} + 278141817 T^{10} - 3440175105 T^{11} + 10779215329 T^{12} )^{2} \))
$53$ (\( 1 - 6 T + 53 T^{2} \))(\( 1 + 10 T + 53 T^{2} \))(\( 1 - 74 T^{2} + 2809 T^{4} \))(\( 1 - 132 T^{2} + 8886 T^{4} - 370788 T^{6} + 7890481 T^{8} \))(\( ( 1 - 10 T + 53 T^{2} )( 1 + 10 T + 53 T^{2} ) \))(\( ( 1 - 14 T + 53 T^{2} )^{2}( 1 + 14 T + 53 T^{2} )^{2} \))(\( 1 - 9 T + 28 T^{2} - 477 T^{3} + 2809 T^{4} \))(\( 1 - T - 52 T^{2} - 53 T^{3} + 2809 T^{4} \))(\( 1 + 105 T^{2} - 693 T^{4} - 10308 T^{6} + 39259017 T^{8} + 1185419067 T^{10} - 43539019882 T^{12} + 3329842159203 T^{14} + 309772527717177 T^{16} - 228470234517732 T^{18} - 43145965455073173 T^{20} + 18363184388378870145 T^{22} + \)\(49\!\cdots\!41\)\( T^{24} \))(\( 1 + 161 T^{2} + 12171 T^{4} + 564332 T^{6} + 17660601 T^{8} + 173268691 T^{10} - 11132063562 T^{12} + 486711753019 T^{14} + 139350636639081 T^{16} + 12508058244650828 T^{18} + 757762691996674731 T^{20} + 28156882728847600889 T^{22} + \)\(49\!\cdots\!41\)\( T^{24} \))
$59$ (\( 1 + 59 T^{2} \))(\( 1 - 6 T + 59 T^{2} \))(\( 1 - 116 T^{2} + 3481 T^{4} \))(\( 1 - 178 T^{2} + 14050 T^{4} - 619618 T^{6} + 12117361 T^{8} \))(\( ( 1 - 59 T^{2} )^{2} \))(\( ( 1 - 112 T^{2} + 3481 T^{4} )^{2} \))(\( 1 + 3 T - 50 T^{2} + 177 T^{3} + 3481 T^{4} \))(\( 1 + 15 T + 166 T^{2} + 885 T^{3} + 3481 T^{4} \))(\( ( 1 - 21 T + 309 T^{2} - 3402 T^{3} + 29601 T^{4} - 225789 T^{5} + 1760606 T^{6} - 13321551 T^{7} + 103041081 T^{8} - 698699358 T^{9} + 3744264549 T^{10} - 15013410279 T^{11} + 42180533641 T^{12} )^{2} \))(\( 1 + 177 T^{2} + 16739 T^{4} + 834316 T^{6} + 7372221 T^{8} - 3081613133 T^{10} - 272281952338 T^{12} - 10727095315973 T^{14} + 89331863228781 T^{16} + 35191894105224556 T^{18} + 2457794695058729219 T^{20} + 90467665334213527977 T^{22} + \)\(17\!\cdots\!81\)\( T^{24} \))
$61$ (\( 1 + 6 T + 61 T^{2} \))(\( 1 - 4 T + 61 T^{2} \))(\( 1 + 40 T^{2} + 3721 T^{4} \))(\( 1 - 230 T^{2} + 20650 T^{4} - 855830 T^{6} + 13845841 T^{8} \))(\( ( 1 + 61 T^{2} )^{2} \))(\( ( 1 + 68 T^{2} + 3721 T^{4} )^{2} \))(\( 1 + 3 T - 52 T^{2} + 183 T^{3} + 3721 T^{4} \))(\( 1 - 5 T - 36 T^{2} - 305 T^{3} + 3721 T^{4} \))(\( 1 - 159 T^{2} + 7419 T^{4} - 352004 T^{6} + 54786249 T^{8} - 3286587357 T^{10} + 113418284406 T^{12} - 12229391555397 T^{14} + 758561692640409 T^{16} - 18135377856569444 T^{18} + 1422276555126827739 T^{20} - \)\(11\!\cdots\!59\)\( T^{22} + \)\(26\!\cdots\!21\)\( T^{24} \))(\( 1 + 73 T^{2} - 3941 T^{4} - 121236 T^{6} + 22314201 T^{8} - 326849717 T^{10} - 131320604138 T^{12} - 1216207796957 T^{14} + 308958879088041 T^{16} - 6246124106030196 T^{18} - 755518520522284421 T^{20} + 52074032551390429873 T^{22} + \)\(26\!\cdots\!21\)\( T^{24} \))
$67$ (\( 1 + 4 T + 67 T^{2} \))(\( 1 + 12 T + 67 T^{2} \))(\( ( 1 - 67 T^{2} )^{2} \))(\( 1 + 112 T^{2} + 8782 T^{4} + 502768 T^{6} + 20151121 T^{8} \))(\( ( 1 - 4 T + 67 T^{2} )^{2} \))(\( ( 1 + 2 T + 67 T^{2} )^{4} \))(\( ( 1 - 5 T + 67 T^{2} )( 1 + 16 T + 67 T^{2} ) \))(\( 1 - 9 T + 14 T^{2} - 603 T^{3} + 4489 T^{4} \))(\( ( 1 - 15 T - 3 T^{2} + 142 T^{3} + 9993 T^{4} - 15123 T^{5} - 719466 T^{6} - 1013241 T^{7} + 44858577 T^{8} + 42708346 T^{9} - 60453363 T^{10} - 20251876605 T^{11} + 90458382169 T^{12} )^{2} \))(\( 1 + 361 T^{2} + 73723 T^{4} + 10542804 T^{6} + 1164287325 T^{8} + 103688059123 T^{10} + 7623484120510 T^{12} + 465455697403147 T^{14} + 23461694764841325 T^{16} + 953684993364861876 T^{18} + 29936527392508244443 T^{20} + \)\(65\!\cdots\!89\)\( T^{22} + \)\(81\!\cdots\!61\)\( T^{24} \))
$71$ (\( 1 + 8 T + 71 T^{2} \))(\( 1 + 71 T^{2} \))(\( ( 1 + 71 T^{2} )^{2} \))(\( ( 1 + 8 T + 71 T^{2} )^{4} \))(\( ( 1 - 16 T + 71 T^{2} )( 1 + 16 T + 71 T^{2} ) \))(\( ( 1 - 42 T^{2} + 5041 T^{4} )^{2} \))(\( ( 1 - 16 T + 71 T^{2} )^{2} \))(\( ( 1 + 71 T^{2} )^{2} \))(\( ( 1 - 16 T + 71 T^{2} )^{6}( 1 + 16 T + 71 T^{2} )^{6} \))(\( ( 1 - 8 T + 157 T^{2} - 704 T^{3} + 11147 T^{4} - 40328 T^{5} + 357911 T^{6} )^{4} \))
$73$ (\( 1 - 10 T + 73 T^{2} \))(\( 1 + 14 T + 73 T^{2} \))(\( ( 1 - 2 T + 73 T^{2} )^{2} \))(\( ( 1 + 6 T + 73 T^{2} )^{4} \))(\( ( 1 - 73 T^{2} )^{2} \))(\( ( 1 + 70 T^{2} + 5329 T^{4} )^{2} \))(\( ( 1 - 10 T + 73 T^{2} )( 1 + 17 T + 73 T^{2} ) \))(\( ( 1 - 10 T + 73 T^{2} )( 1 + 17 T + 73 T^{2} ) \))(\( ( 1 - 9 T + 183 T^{2} - 1404 T^{3} + 16701 T^{4} - 147195 T^{5} + 1376278 T^{6} - 10745235 T^{7} + 88999629 T^{8} - 546179868 T^{9} + 5196878103 T^{10} - 18657644337 T^{11} + 151334226289 T^{12} )^{2} \))(\( ( 1 + 5 T - 101 T^{2} + 52 T^{3} + 5233 T^{4} - 28921 T^{5} - 430618 T^{6} - 2111233 T^{7} + 27886657 T^{8} + 20228884 T^{9} - 2868222341 T^{10} + 10365357965 T^{11} + 151334226289 T^{12} )^{2} \))
$79$ (\( 1 - 16 T + 79 T^{2} \))(\( 1 + 8 T + 79 T^{2} \))(\( ( 1 - 8 T + 79 T^{2} )^{2} \))(\( ( 1 + 79 T^{2} )^{4} \))(\( ( 1 - 8 T + 79 T^{2} )( 1 + 8 T + 79 T^{2} ) \))(\( ( 1 - 122 T^{2} + 6241 T^{4} )^{2} \))(\( 1 - 11 T + 42 T^{2} - 869 T^{3} + 6241 T^{4} \))(\( 1 + T - 78 T^{2} + 79 T^{3} + 6241 T^{4} \))(\( 1 + 261 T^{2} + 30519 T^{4} + 2829272 T^{6} + 267713397 T^{8} + 18988265187 T^{10} + 1205673992502 T^{12} + 118505763032067 T^{14} + 10427458497935157 T^{16} + 687760531456810712 T^{18} + 46300643769538335159 T^{20} + \)\(24\!\cdots\!61\)\( T^{22} + \)\(59\!\cdots\!41\)\( T^{24} \))(\( ( 1 + 11 T - 137 T^{2} - 656 T^{3} + 26965 T^{4} + 70973 T^{5} - 1975630 T^{6} + 5606867 T^{7} + 168288565 T^{8} - 323433584 T^{9} - 5336161097 T^{10} + 33847620389 T^{11} + 243087455521 T^{12} )^{2} \))
$83$ (\( 1 - 8 T + 83 T^{2} \))(\( 1 - 6 T + 83 T^{2} \))(\( 1 + 76 T^{2} + 6889 T^{4} \))(\( 1 - 210 T^{2} + 23970 T^{4} - 1446690 T^{6} + 47458321 T^{8} \))(\( ( 1 - 83 T^{2} )^{2} \))(\( ( 1 - 160 T^{2} + 6889 T^{4} )^{2} \))(\( ( 1 + 4 T + 83 T^{2} )^{2} \))(\( ( 1 - 12 T + 83 T^{2} )^{2} \))(\( ( 1 - 270 T^{2} + 28455 T^{4} - 2146852 T^{6} + 196026495 T^{8} - 12813746670 T^{10} + 326940373369 T^{12} )^{2} \))(\( ( 1 - 446 T^{2} + 86503 T^{4} - 9357636 T^{6} + 595919167 T^{8} - 21166411166 T^{10} + 326940373369 T^{12} )^{2} \))
$89$ (\( 1 + 6 T + 89 T^{2} \))(\( 1 - 10 T + 89 T^{2} \))(\( ( 1 - 6 T + 89 T^{2} )^{2} \))(\( ( 1 + 16 T + 174 T^{2} + 1424 T^{3} + 7921 T^{4} )^{2} \))(\( ( 1 - 89 T^{2} )^{2} \))(\( ( 1 + 38 T^{2} + 7921 T^{4} )^{2} \))(\( 1 - 9 T - 8 T^{2} - 801 T^{3} + 7921 T^{4} \))(\( 1 + 7 T - 40 T^{2} + 623 T^{3} + 7921 T^{4} \))(\( ( 1 - 3 T + 92 T^{2} - 267 T^{3} + 7921 T^{4} )^{6} \))(\( ( 1 + 5 T - 133 T^{2} - 1452 T^{3} + 5297 T^{4} + 76103 T^{5} + 246758 T^{6} + 6773167 T^{7} + 41957537 T^{8} - 1023614988 T^{9} - 8344718053 T^{10} + 27920297245 T^{11} + 496981290961 T^{12} )^{2} \))
$97$ (\( 1 + 6 T + 97 T^{2} \))(\( 1 + 2 T + 97 T^{2} \))(\( ( 1 + 10 T + 97 T^{2} )^{2} \))(\( ( 1 - 8 T + 142 T^{2} - 776 T^{3} + 9409 T^{4} )^{2} \))(\( ( 1 - 97 T^{2} )^{2} \))(\( ( 1 - 170 T^{2} + 9409 T^{4} )^{2} \))(\( ( 1 - 6 T + 97 T^{2} )^{2} \))(\( ( 1 + 2 T + 97 T^{2} )^{2} \))(\( ( 1 - 222 T^{2} + 20703 T^{4} - 1529300 T^{6} + 194794527 T^{8} - 19653500382 T^{10} + 832972004929 T^{12} )^{2} \))(\( ( 1 - 10 T + 255 T^{2} - 1968 T^{3} + 24735 T^{4} - 94090 T^{5} + 912673 T^{6} )^{4} \))
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