Properties

Label 539.2.q
Level 539
Weight 2
Character orbit q
Rep. character \(\chi_{539}(214,\cdot)\)
Character field \(\Q(\zeta_{15})\)
Dimension 288
Newform subspaces 9
Sturm bound 112
Trace bound 3

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Defining parameters

Level: \( N \) = \( 539 = 7^{2} \cdot 11 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 539.q (of order \(15\) and degree \(8\))
Character conductor: \(\operatorname{cond}(\chi)\) = \( 77 \)
Character field: \(\Q(\zeta_{15})\)
Newform subspaces: \( 9 \)
Sturm bound: \(112\)
Trace bound: \(3\)
Distinguishing \(T_p\): \(2\), \(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(539, [\chi])\).

Total New Old
Modular forms 512 352 160
Cusp forms 384 288 96
Eisenstein series 128 64 64

Trace form

\( 288q + 7q^{2} + 3q^{3} + 39q^{4} + q^{5} + 12q^{6} + 19q^{9} + O(q^{10}) \) \( 288q + 7q^{2} + 3q^{3} + 39q^{4} + q^{5} + 12q^{6} + 19q^{9} - 4q^{10} + 9q^{11} + 16q^{12} + 4q^{13} + 27q^{16} + 11q^{17} - 57q^{18} + 7q^{19} + 40q^{20} - 108q^{22} + 2q^{23} + 7q^{24} - 15q^{25} + 15q^{26} + 6q^{27} - 84q^{29} + 41q^{30} + 9q^{31} - 8q^{32} + 26q^{33} - 72q^{34} - 134q^{36} - 15q^{37} - 3q^{38} - 53q^{39} - 5q^{40} - 62q^{41} - 60q^{43} - 45q^{44} + 16q^{45} + 17q^{46} - 19q^{47} - 146q^{48} - 70q^{50} - 55q^{51} + 3q^{52} - 33q^{53} - 44q^{54} - 4q^{55} - 72q^{57} + 57q^{58} - 3q^{59} + 137q^{60} - 18q^{61} + 80q^{62} - 204q^{64} + 22q^{65} + 47q^{66} + 104q^{67} + 21q^{68} + 126q^{69} - 36q^{71} - 31q^{72} - 26q^{73} + 47q^{74} + 11q^{75} + 144q^{76} + 348q^{78} - 48q^{79} + 3q^{80} + 65q^{81} - 13q^{82} + 28q^{83} + 118q^{85} + 145q^{86} - 66q^{87} - 101q^{88} + 34q^{89} + 18q^{90} - 168q^{92} - 68q^{93} - 45q^{94} - 98q^{95} + 55q^{96} + 68q^{97} + 104q^{99} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(539, [\chi])\) into newform subspaces

Label Dim. \(A\) Field CM Traces $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\)
539.2.q.a \(8\) \(4.304\) \(\Q(\zeta_{15})\) None \(-2\) \(-1\) \(5\) \(0\) \(q+(-1+\zeta_{15}-\zeta_{15}^{5}+\zeta_{15}^{7})q^{2}+\cdots\)
539.2.q.b \(16\) \(4.304\) \(\mathbb{Q}[x]/(x^{16} - \cdots)\) None \(1\) \(-4\) \(3\) \(0\) \(q+(\beta _{2}+\beta _{4}+\beta _{11}-\beta _{12})q^{2}+(\beta _{5}-\beta _{7}+\cdots)q^{3}+\cdots\)
539.2.q.c \(16\) \(4.304\) \(\mathbb{Q}[x]/(x^{16} - \cdots)\) None \(1\) \(4\) \(-3\) \(0\) \(q+(\beta _{2}+\beta _{4}+\beta _{11}-\beta _{12})q^{2}+(-\beta _{5}+\cdots)q^{3}+\cdots\)
539.2.q.d \(16\) \(4.304\) 16.0.\(\cdots\).1 \(\Q(\sqrt{-7}) \) \(2\) \(0\) \(0\) \(0\) \(q+(\beta _{1}-\beta _{4}-\beta _{5}+\beta _{6}+\beta _{7}-\beta _{9}-\beta _{10}+\cdots)q^{2}+\cdots\)
539.2.q.e \(32\) \(4.304\) None \(2\) \(0\) \(0\) \(0\)
539.2.q.f \(32\) \(4.304\) None \(3\) \(-2\) \(-5\) \(0\)
539.2.q.g \(32\) \(4.304\) None \(3\) \(2\) \(5\) \(0\)
539.2.q.h \(40\) \(4.304\) None \(-3\) \(4\) \(-4\) \(0\)
539.2.q.i \(96\) \(4.304\) None \(0\) \(0\) \(0\) \(0\)

Decomposition of \(S_{2}^{\mathrm{old}}(539, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(539, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(77, [\chi])\)\(^{\oplus 2}\)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ (\( 1 + 2 T + 2 T^{2} - 6 T^{3} - 17 T^{4} - 24 T^{5} - T^{6} + 47 T^{7} + 103 T^{8} + 94 T^{9} - 4 T^{10} - 192 T^{11} - 272 T^{12} - 192 T^{13} + 128 T^{14} + 256 T^{15} + 256 T^{16} \))(\( 1 - T + 4 T^{2} - 7 T^{3} + 14 T^{4} - 34 T^{5} + 48 T^{6} - 115 T^{7} + 169 T^{8} - 299 T^{9} + 523 T^{10} - 758 T^{11} + 1321 T^{12} - 1850 T^{13} + 2863 T^{14} - 4328 T^{15} + 5781 T^{16} - 8656 T^{17} + 11452 T^{18} - 14800 T^{19} + 21136 T^{20} - 24256 T^{21} + 33472 T^{22} - 38272 T^{23} + 43264 T^{24} - 58880 T^{25} + 49152 T^{26} - 69632 T^{27} + 57344 T^{28} - 57344 T^{29} + 65536 T^{30} - 32768 T^{31} + 65536 T^{32} \))(\( 1 - T + 4 T^{2} - 7 T^{3} + 14 T^{4} - 34 T^{5} + 48 T^{6} - 115 T^{7} + 169 T^{8} - 299 T^{9} + 523 T^{10} - 758 T^{11} + 1321 T^{12} - 1850 T^{13} + 2863 T^{14} - 4328 T^{15} + 5781 T^{16} - 8656 T^{17} + 11452 T^{18} - 14800 T^{19} + 21136 T^{20} - 24256 T^{21} + 33472 T^{22} - 38272 T^{23} + 43264 T^{24} - 58880 T^{25} + 49152 T^{26} - 69632 T^{27} + 57344 T^{28} - 57344 T^{29} + 65536 T^{30} - 32768 T^{31} + 65536 T^{32} \))(\( ( 1 - T + 2 T^{2} - 5 T^{3} + 5 T^{4} + T^{5} + 6 T^{6} + 5 T^{7} - 21 T^{8} + 10 T^{9} + 24 T^{10} + 8 T^{11} + 80 T^{12} - 160 T^{13} + 128 T^{14} - 128 T^{15} + 256 T^{16} )^{2} \))
$3$ (\( 1 + T + 3 T^{2} + 8 T^{3} + 8 T^{4} - 7 T^{5} + 6 T^{6} - 56 T^{7} - 137 T^{8} - 168 T^{9} + 54 T^{10} - 189 T^{11} + 648 T^{12} + 1944 T^{13} + 2187 T^{14} + 2187 T^{15} + 6561 T^{16} \))(\( ( 1 + 2 T + 3 T^{2} - 10 T^{3} - 30 T^{4} - 52 T^{5} - T^{6} + 150 T^{7} + 379 T^{8} + 450 T^{9} - 9 T^{10} - 1404 T^{11} - 2430 T^{12} - 2430 T^{13} + 2187 T^{14} + 4374 T^{15} + 6561 T^{16} )^{2} \))(\( ( 1 - 2 T + 3 T^{2} + 10 T^{3} - 30 T^{4} + 52 T^{5} - T^{6} - 150 T^{7} + 379 T^{8} - 450 T^{9} - 9 T^{10} + 1404 T^{11} - 2430 T^{12} + 2430 T^{13} + 2187 T^{14} - 4374 T^{15} + 6561 T^{16} )^{2} \))(\( ( 1 - 3 T + 6 T^{2} - 9 T^{3} + 9 T^{4} - 27 T^{5} + 54 T^{6} - 81 T^{7} + 81 T^{8} )^{2}( 1 + 3 T + 6 T^{2} + 9 T^{3} + 9 T^{4} + 27 T^{5} + 54 T^{6} + 81 T^{7} + 81 T^{8} )^{2} \))
$5$ (\( ( 1 - 5 T + 15 T^{2} - 25 T^{3} + 25 T^{4} )^{2}( 1 + 5 T + 10 T^{2} + 25 T^{3} + 75 T^{4} + 125 T^{5} + 250 T^{6} + 625 T^{7} + 625 T^{8} ) \))(\( 1 - 3 T + 12 T^{2} - 51 T^{3} + 138 T^{4} - 474 T^{5} + 1189 T^{6} - 3123 T^{7} + 8088 T^{8} - 16326 T^{9} + 38041 T^{10} - 69267 T^{11} + 129643 T^{12} - 221502 T^{13} + 294093 T^{14} - 572844 T^{15} + 907531 T^{16} - 2864220 T^{17} + 7352325 T^{18} - 27687750 T^{19} + 81026875 T^{20} - 216459375 T^{21} + 594390625 T^{22} - 1275468750 T^{23} + 3159375000 T^{24} - 6099609375 T^{25} + 11611328125 T^{26} - 23144531250 T^{27} + 33691406250 T^{28} - 62255859375 T^{29} + 73242187500 T^{30} - 91552734375 T^{31} + 152587890625 T^{32} \))(\( 1 + 3 T + 12 T^{2} + 51 T^{3} + 138 T^{4} + 474 T^{5} + 1189 T^{6} + 3123 T^{7} + 8088 T^{8} + 16326 T^{9} + 38041 T^{10} + 69267 T^{11} + 129643 T^{12} + 221502 T^{13} + 294093 T^{14} + 572844 T^{15} + 907531 T^{16} + 2864220 T^{17} + 7352325 T^{18} + 27687750 T^{19} + 81026875 T^{20} + 216459375 T^{21} + 594390625 T^{22} + 1275468750 T^{23} + 3159375000 T^{24} + 6099609375 T^{25} + 11611328125 T^{26} + 23144531250 T^{27} + 33691406250 T^{28} + 62255859375 T^{29} + 73242187500 T^{30} + 91552734375 T^{31} + 152587890625 T^{32} \))(\( ( 1 + 5 T^{2} - 125 T^{6} - 625 T^{8} - 3125 T^{10} + 78125 T^{14} + 390625 T^{16} )^{2} \))
$7$ 1
$11$ (\( 1 - 4 T + 10 T^{2} + 64 T^{3} - 261 T^{4} + 704 T^{5} + 1210 T^{6} - 5324 T^{7} + 14641 T^{8} \))(\( 1 + 5 T - T^{2} - 20 T^{3} + 50 T^{4} + 265 T^{5} + 1266 T^{6} - 1690 T^{7} - 28761 T^{8} - 18590 T^{9} + 153186 T^{10} + 352715 T^{11} + 732050 T^{12} - 3221020 T^{13} - 1771561 T^{14} + 97435855 T^{15} + 214358881 T^{16} \))(\( 1 + 5 T - T^{2} - 20 T^{3} + 50 T^{4} + 265 T^{5} + 1266 T^{6} - 1690 T^{7} - 28761 T^{8} - 18590 T^{9} + 153186 T^{10} + 352715 T^{11} + 732050 T^{12} - 3221020 T^{13} - 1771561 T^{14} + 97435855 T^{15} + 214358881 T^{16} \))(\( 1 - 4 T + 11 T^{2} - 68 T^{3} + 272 T^{4} - 824 T^{5} + 3597 T^{6} - 14008 T^{7} + 41391 T^{8} - 154088 T^{9} + 435237 T^{10} - 1096744 T^{11} + 3982352 T^{12} - 10951468 T^{13} + 19487171 T^{14} - 77948684 T^{15} + 214358881 T^{16} \))
$13$ (\( ( 1 - 5 T + 27 T^{2} - 115 T^{3} + 584 T^{4} - 1495 T^{5} + 4563 T^{6} - 10985 T^{7} + 28561 T^{8} )^{2} \))(\( ( 1 + 5 T + 35 T^{2} + 90 T^{3} + 611 T^{4} + 670 T^{5} + 4380 T^{6} - 8865 T^{7} + 38151 T^{8} - 115245 T^{9} + 740220 T^{10} + 1471990 T^{11} + 17450771 T^{12} + 33416370 T^{13} + 168938315 T^{14} + 313742585 T^{15} + 815730721 T^{16} )^{2} \))(\( ( 1 - 5 T + 35 T^{2} - 90 T^{3} + 611 T^{4} - 670 T^{5} + 4380 T^{6} + 8865 T^{7} + 38151 T^{8} + 115245 T^{9} + 740220 T^{10} - 1471990 T^{11} + 17450771 T^{12} - 33416370 T^{13} + 168938315 T^{14} - 313742585 T^{15} + 815730721 T^{16} )^{2} \))(\( ( 1 - 13 T^{2} + 169 T^{4} - 2197 T^{6} + 28561 T^{8} )^{4} \))
$17$ (\( 1 - 4 T + 17 T^{2} + 120 T^{3} - 740 T^{4} + 3144 T^{5} - 649 T^{6} - 49990 T^{7} + 275299 T^{8} - 849830 T^{9} - 187561 T^{10} + 15446472 T^{11} - 61805540 T^{12} + 170382840 T^{13} + 410338673 T^{14} - 1641354692 T^{15} + 6975757441 T^{16} \))(\( 1 + 11 T + 82 T^{2} + 151 T^{3} - 1257 T^{4} - 14573 T^{5} - 43518 T^{6} + 93859 T^{7} + 1568504 T^{8} + 5547395 T^{9} - 2761766 T^{10} - 97790693 T^{11} - 344680251 T^{12} + 129416362 T^{13} + 3449705386 T^{14} + 9240382008 T^{15} + 336684367 T^{16} + 157086494136 T^{17} + 996964856554 T^{18} + 635822586506 T^{19} - 28788039243771 T^{20} - 138848799990901 T^{21} - 66662317386854 T^{22} + 2276310702906835 T^{23} + 10941503449238264 T^{24} + 11130539500131923 T^{25} - 87732022559739582 T^{26} - 499444344891135709 T^{27} - 732356152197809577 T^{28} + 1495591282968796487 T^{29} + 13806981777870876178 T^{30} + 31486653566607973723 T^{31} + 48661191875666868481 T^{32} \))(\( 1 - 11 T + 82 T^{2} - 151 T^{3} - 1257 T^{4} + 14573 T^{5} - 43518 T^{6} - 93859 T^{7} + 1568504 T^{8} - 5547395 T^{9} - 2761766 T^{10} + 97790693 T^{11} - 344680251 T^{12} - 129416362 T^{13} + 3449705386 T^{14} - 9240382008 T^{15} + 336684367 T^{16} - 157086494136 T^{17} + 996964856554 T^{18} - 635822586506 T^{19} - 28788039243771 T^{20} + 138848799990901 T^{21} - 66662317386854 T^{22} - 2276310702906835 T^{23} + 10941503449238264 T^{24} - 11130539500131923 T^{25} - 87732022559739582 T^{26} + 499444344891135709 T^{27} - 732356152197809577 T^{28} - 1495591282968796487 T^{29} + 13806981777870876178 T^{30} - 31486653566607973723 T^{31} + 48661191875666868481 T^{32} \))(\( ( 1 + 17 T^{2} - 4913 T^{6} - 83521 T^{8} - 1419857 T^{10} + 410338673 T^{14} + 6975757441 T^{16} )^{2} \))
$19$ (\( 1 - 3 T - 26 T^{2} + 21 T^{3} + 252 T^{4} + 1302 T^{5} - 1792 T^{6} - 24084 T^{7} + 90161 T^{8} - 457596 T^{9} - 646912 T^{10} + 8930418 T^{11} + 32840892 T^{12} + 51998079 T^{13} - 1223192906 T^{14} - 2681615217 T^{15} + 16983563041 T^{16} \))(\( 1 + 9 T + 38 T^{2} + 45 T^{3} - 713 T^{4} - 9387 T^{5} - 55694 T^{6} - 172071 T^{7} + 11894 T^{8} + 3422331 T^{9} + 28508642 T^{10} + 136939077 T^{11} + 373421069 T^{12} + 123703272 T^{13} - 4687437236 T^{14} - 41957891316 T^{15} - 224298429341 T^{16} - 797199935004 T^{17} - 1692164842196 T^{18} + 848480742648 T^{19} + 48664607133149 T^{20} + 339074711620623 T^{21} + 1341214179003602 T^{22} + 3059124962403609 T^{23} + 202002498809654 T^{24} - 55525194844530309 T^{25} - 341463604161968894 T^{26} - 1093494060277581753 T^{27} - 1578093537294172793 T^{28} + 1892384255801567655 T^{29} + 30362254059749596598 T^{30} + \)\(13\!\cdots\!91\)\( T^{31} + \)\(28\!\cdots\!81\)\( T^{32} \))(\( 1 - 9 T + 38 T^{2} - 45 T^{3} - 713 T^{4} + 9387 T^{5} - 55694 T^{6} + 172071 T^{7} + 11894 T^{8} - 3422331 T^{9} + 28508642 T^{10} - 136939077 T^{11} + 373421069 T^{12} - 123703272 T^{13} - 4687437236 T^{14} + 41957891316 T^{15} - 224298429341 T^{16} + 797199935004 T^{17} - 1692164842196 T^{18} - 848480742648 T^{19} + 48664607133149 T^{20} - 339074711620623 T^{21} + 1341214179003602 T^{22} - 3059124962403609 T^{23} + 202002498809654 T^{24} + 55525194844530309 T^{25} - 341463604161968894 T^{26} + 1093494060277581753 T^{27} - 1578093537294172793 T^{28} - 1892384255801567655 T^{29} + 30362254059749596598 T^{30} - \)\(13\!\cdots\!91\)\( T^{31} + \)\(28\!\cdots\!81\)\( T^{32} \))(\( ( 1 + 19 T^{2} - 6859 T^{6} - 130321 T^{8} - 2476099 T^{10} + 893871739 T^{14} + 16983563041 T^{16} )^{2} \))
$23$ (\( ( 1 - 8 T + 7 T^{2} - 88 T^{3} + 1248 T^{4} - 2024 T^{5} + 3703 T^{6} - 97336 T^{7} + 279841 T^{8} )^{2} \))(\( ( 1 - 8 T - 19 T^{2} + 140 T^{3} + 1118 T^{4} - 1732 T^{5} - 36615 T^{6} + 59134 T^{7} + 444387 T^{8} + 1360082 T^{9} - 19369335 T^{10} - 21073244 T^{11} + 312862238 T^{12} + 901088020 T^{13} - 2812681891 T^{14} - 27238603576 T^{15} + 78310985281 T^{16} )^{2} \))(\( ( 1 - 8 T - 19 T^{2} + 140 T^{3} + 1118 T^{4} - 1732 T^{5} - 36615 T^{6} + 59134 T^{7} + 444387 T^{8} + 1360082 T^{9} - 19369335 T^{10} - 21073244 T^{11} + 312862238 T^{12} + 901088020 T^{13} - 2812681891 T^{14} - 27238603576 T^{15} + 78310985281 T^{16} )^{2} \))(\( ( 1 - 8 T + 23 T^{2} - 40 T^{3} + 320 T^{4} - 5872 T^{5} + 29049 T^{6} - 29360 T^{7} - 44961 T^{8} - 675280 T^{9} + 15366921 T^{10} - 71444624 T^{11} + 89549120 T^{12} - 257453720 T^{13} + 3404825447 T^{14} - 27238603576 T^{15} + 78310985281 T^{16} )^{2} \))
$29$ (\( ( 1 - 12 T + 25 T^{2} + 288 T^{3} - 2471 T^{4} + 8352 T^{5} + 21025 T^{6} - 292668 T^{7} + 707281 T^{8} )^{2} \))(\( ( 1 + 9 T - 22 T^{2} - 429 T^{3} - 762 T^{4} + 11754 T^{5} + 81940 T^{6} - 191700 T^{7} - 3727369 T^{8} - 5559300 T^{9} + 68911540 T^{10} + 286668306 T^{11} - 538948122 T^{12} - 8799282921 T^{13} - 13086113062 T^{14} + 155248886781 T^{15} + 500246412961 T^{16} )^{2} \))(\( ( 1 + 9 T - 22 T^{2} - 429 T^{3} - 762 T^{4} + 11754 T^{5} + 81940 T^{6} - 191700 T^{7} - 3727369 T^{8} - 5559300 T^{9} + 68911540 T^{10} + 286668306 T^{11} - 538948122 T^{12} - 8799282921 T^{13} - 13086113062 T^{14} + 155248886781 T^{15} + 500246412961 T^{16} )^{2} \))(\( ( 1 + 2 T - 25 T^{2} - 108 T^{3} + 509 T^{4} - 3132 T^{5} - 21025 T^{6} + 48778 T^{7} + 707281 T^{8} )^{4} \))
$31$ (\( ( 1 - 11 T + 60 T^{2} + 11 T^{3} - 991 T^{4} + 341 T^{5} + 57660 T^{6} - 327701 T^{7} + 923521 T^{8} )( 1 + 19 T + 210 T^{2} + 1691 T^{3} + 10649 T^{4} + 52421 T^{5} + 201810 T^{6} + 566029 T^{7} + 923521 T^{8} ) \))(\( 1 + 11 T + 87 T^{2} + 8 T^{3} - 4067 T^{4} - 51562 T^{5} - 279707 T^{6} - 605448 T^{7} + 7071411 T^{8} + 79682271 T^{9} + 453595654 T^{10} + 966291599 T^{11} - 4382753906 T^{12} - 56567005227 T^{13} - 210031083446 T^{14} - 225648686906 T^{15} + 2581204556074 T^{16} - 6995109294086 T^{17} - 201839871191606 T^{18} - 1685187652717557 T^{19} - 4047565270023026 T^{20} + 27664108097802449 T^{21} + 402567812610602374 T^{22} + 2192267573511126081 T^{23} + 6031143063961699251 T^{24} - 16007816357933935608 T^{25} - \)\(22\!\cdots\!07\)\( T^{26} - \)\(13\!\cdots\!22\)\( T^{27} - \)\(32\!\cdots\!87\)\( T^{28} + \)\(19\!\cdots\!28\)\( T^{29} + \)\(65\!\cdots\!27\)\( T^{30} + \)\(25\!\cdots\!61\)\( T^{31} + \)\(72\!\cdots\!81\)\( T^{32} \))(\( 1 - 11 T + 87 T^{2} - 8 T^{3} - 4067 T^{4} + 51562 T^{5} - 279707 T^{6} + 605448 T^{7} + 7071411 T^{8} - 79682271 T^{9} + 453595654 T^{10} - 966291599 T^{11} - 4382753906 T^{12} + 56567005227 T^{13} - 210031083446 T^{14} + 225648686906 T^{15} + 2581204556074 T^{16} + 6995109294086 T^{17} - 201839871191606 T^{18} + 1685187652717557 T^{19} - 4047565270023026 T^{20} - 27664108097802449 T^{21} + 402567812610602374 T^{22} - 2192267573511126081 T^{23} + 6031143063961699251 T^{24} + 16007816357933935608 T^{25} - \)\(22\!\cdots\!07\)\( T^{26} + \)\(13\!\cdots\!22\)\( T^{27} - \)\(32\!\cdots\!87\)\( T^{28} - \)\(19\!\cdots\!28\)\( T^{29} + \)\(65\!\cdots\!27\)\( T^{30} - \)\(25\!\cdots\!61\)\( T^{31} + \)\(72\!\cdots\!81\)\( T^{32} \))(\( ( 1 + 31 T^{2} - 29791 T^{6} - 923521 T^{8} - 28629151 T^{10} + 27512614111 T^{14} + 852891037441 T^{16} )^{2} \))
$37$ (\( 1 + 13 T + 112 T^{2} + 71 T^{3} - 4102 T^{4} - 46406 T^{5} - 63616 T^{6} + 1263548 T^{7} + 15622363 T^{8} + 46751276 T^{9} - 87090304 T^{10} - 2350603118 T^{11} - 7687808422 T^{12} + 4923420947 T^{13} + 287361357808 T^{14} + 1234114402729 T^{15} + 3512479453921 T^{16} \))(\( 1 + 6 T + 99 T^{2} + 662 T^{3} + 5739 T^{4} + 29514 T^{5} + 237308 T^{6} + 716560 T^{7} + 4809854 T^{8} + 17329244 T^{9} + 22995413 T^{10} + 480402038 T^{11} + 2695419596 T^{12} + 38023386470 T^{13} + 235277510708 T^{14} + 3056978579838 T^{15} + 9263636009781 T^{16} + 113108207454006 T^{17} + 322094912159252 T^{18} + 1925998594864910 T^{19} + 5051650285458956 T^{20} + 33312978265784366 T^{21} + 58999938419961917 T^{22} + 1645097662215777452 T^{23} + 16894513351359737534 T^{24} + 93125384267560375120 T^{25} + \)\(11\!\cdots\!92\)\( T^{26} + \)\(52\!\cdots\!82\)\( T^{27} + \)\(37\!\cdots\!59\)\( T^{28} + \)\(16\!\cdots\!14\)\( T^{29} + \)\(89\!\cdots\!11\)\( T^{30} + \)\(20\!\cdots\!58\)\( T^{31} + \)\(12\!\cdots\!41\)\( T^{32} \))(\( 1 + 6 T + 99 T^{2} + 662 T^{3} + 5739 T^{4} + 29514 T^{5} + 237308 T^{6} + 716560 T^{7} + 4809854 T^{8} + 17329244 T^{9} + 22995413 T^{10} + 480402038 T^{11} + 2695419596 T^{12} + 38023386470 T^{13} + 235277510708 T^{14} + 3056978579838 T^{15} + 9263636009781 T^{16} + 113108207454006 T^{17} + 322094912159252 T^{18} + 1925998594864910 T^{19} + 5051650285458956 T^{20} + 33312978265784366 T^{21} + 58999938419961917 T^{22} + 1645097662215777452 T^{23} + 16894513351359737534 T^{24} + 93125384267560375120 T^{25} + \)\(11\!\cdots\!92\)\( T^{26} + \)\(52\!\cdots\!82\)\( T^{27} + \)\(37\!\cdots\!59\)\( T^{28} + \)\(16\!\cdots\!14\)\( T^{29} + \)\(89\!\cdots\!11\)\( T^{30} + \)\(20\!\cdots\!58\)\( T^{31} + \)\(12\!\cdots\!41\)\( T^{32} \))(\( ( 1 - 6 T - T^{2} - 222 T^{3} + 1369 T^{4} )^{4}( 1 + 6 T + 37 T^{2} + 450 T^{3} + 2700 T^{4} + 7764 T^{5} + 98531 T^{6} + 582300 T^{7} + 1619639 T^{8} + 21545100 T^{9} + 134888939 T^{10} + 393269892 T^{11} + 5060234700 T^{12} + 31204780650 T^{13} + 94931877133 T^{14} + 569591262798 T^{15} + 3512479453921 T^{16} ) \))
$41$ (\( ( 1 + T - 25 T^{2} - 221 T^{3} + 1064 T^{4} - 9061 T^{5} - 42025 T^{6} + 68921 T^{7} + 2825761 T^{8} )^{2} \))(\( ( 1 - 22 T + 160 T^{2} - 414 T^{3} + 2641 T^{4} - 46342 T^{5} + 334288 T^{6} - 1062556 T^{7} + 2274033 T^{8} - 43564796 T^{9} + 561938128 T^{10} - 3193936982 T^{11} + 7462834801 T^{12} - 47964467214 T^{13} + 760016678560 T^{14} - 4284594025382 T^{15} + 7984925229121 T^{16} )^{2} \))(\( ( 1 + 22 T + 160 T^{2} + 414 T^{3} + 2641 T^{4} + 46342 T^{5} + 334288 T^{6} + 1062556 T^{7} + 2274033 T^{8} + 43564796 T^{9} + 561938128 T^{10} + 3193936982 T^{11} + 7462834801 T^{12} + 47964467214 T^{13} + 760016678560 T^{14} + 4284594025382 T^{15} + 7984925229121 T^{16} )^{2} \))(\( ( 1 - 41 T^{2} + 1681 T^{4} - 68921 T^{6} + 2825761 T^{8} )^{4} \))
$43$ (\( ( 1 - 7 T + 87 T^{2} - 301 T^{3} + 1849 T^{4} )^{4} \))(\( ( 1 - 4 T + 45 T^{2} - 172 T^{3} + 1849 T^{4} )^{8} \))(\( ( 1 - 4 T + 45 T^{2} - 172 T^{3} + 1849 T^{4} )^{8} \))(\( ( 1 - 12 T + 101 T^{2} - 696 T^{3} + 4009 T^{4} - 29928 T^{5} + 186749 T^{6} - 954084 T^{7} + 3418801 T^{8} )^{4} \))
$47$ (\( 1 + 6 T + 7 T^{2} - 690 T^{3} - 6420 T^{4} - 23196 T^{5} + 57521 T^{6} + 1846680 T^{7} + 13766039 T^{8} + 86793960 T^{9} + 127063889 T^{10} - 2408278308 T^{11} - 31327552020 T^{12} - 158248054830 T^{13} + 75454507303 T^{14} + 3039738722778 T^{15} + 23811286661761 T^{16} \))(\( 1 - 7 T + 149 T^{2} - 2128 T^{3} + 19612 T^{4} - 240919 T^{5} + 2462607 T^{6} - 21813848 T^{7} + 211617499 T^{8} - 1862688919 T^{9} + 15071552909 T^{10} - 126763363055 T^{11} + 1005269592991 T^{12} - 7378260675254 T^{13} + 56266129109578 T^{14} - 404849575321212 T^{15} + 2719197771655735 T^{16} - 19027930040096964 T^{17} + 124291879203057802 T^{18} - 766033158086896042 T^{19} + 4905394932795915871 T^{20} - 29072544387192516385 T^{21} + \)\(16\!\cdots\!61\)\( T^{22} - \)\(94\!\cdots\!97\)\( T^{23} + \)\(50\!\cdots\!39\)\( T^{24} - \)\(24\!\cdots\!16\)\( T^{25} + \)\(12\!\cdots\!43\)\( T^{26} - \)\(59\!\cdots\!57\)\( T^{27} + \)\(22\!\cdots\!92\)\( T^{28} - \)\(11\!\cdots\!56\)\( T^{29} + \)\(38\!\cdots\!81\)\( T^{30} - \)\(84\!\cdots\!01\)\( T^{31} + \)\(56\!\cdots\!21\)\( T^{32} \))(\( 1 + 7 T + 149 T^{2} + 2128 T^{3} + 19612 T^{4} + 240919 T^{5} + 2462607 T^{6} + 21813848 T^{7} + 211617499 T^{8} + 1862688919 T^{9} + 15071552909 T^{10} + 126763363055 T^{11} + 1005269592991 T^{12} + 7378260675254 T^{13} + 56266129109578 T^{14} + 404849575321212 T^{15} + 2719197771655735 T^{16} + 19027930040096964 T^{17} + 124291879203057802 T^{18} + 766033158086896042 T^{19} + 4905394932795915871 T^{20} + 29072544387192516385 T^{21} + \)\(16\!\cdots\!61\)\( T^{22} + \)\(94\!\cdots\!97\)\( T^{23} + \)\(50\!\cdots\!39\)\( T^{24} + \)\(24\!\cdots\!16\)\( T^{25} + \)\(12\!\cdots\!43\)\( T^{26} + \)\(59\!\cdots\!57\)\( T^{27} + \)\(22\!\cdots\!92\)\( T^{28} + \)\(11\!\cdots\!56\)\( T^{29} + \)\(38\!\cdots\!81\)\( T^{30} + \)\(84\!\cdots\!01\)\( T^{31} + \)\(56\!\cdots\!21\)\( T^{32} \))(\( ( 1 + 47 T^{2} - 103823 T^{6} - 4879681 T^{8} - 229345007 T^{10} + 506623120463 T^{14} + 23811286661761 T^{16} )^{2} \))
$53$ (\( 1 + 12 T + 103 T^{2} - 420 T^{3} - 9840 T^{4} - 102312 T^{5} - 99751 T^{6} + 3945120 T^{7} + 55833959 T^{8} + 209091360 T^{9} - 280200559 T^{10} - 15231903624 T^{11} - 77642333040 T^{12} - 175642107060 T^{13} + 2282929196287 T^{14} + 14096533678044 T^{15} + 62259690411361 T^{16} \))(\( 1 + 2 T + 177 T^{2} + 1582 T^{3} + 21921 T^{4} + 231770 T^{5} + 2684050 T^{6} + 23678026 T^{7} + 239791912 T^{8} + 2114692878 T^{9} + 17637992799 T^{10} + 151508122160 T^{11} + 1184498641232 T^{12} + 9380769650632 T^{13} + 69924170534704 T^{14} + 544239771304768 T^{15} + 3774658249521589 T^{16} + 28844707879152704 T^{17} + 196416995031983536 T^{18} + 1396580843277140264 T^{19} + 9346264023166912592 T^{20} + 63360013840205424880 T^{21} + \)\(39\!\cdots\!71\)\( T^{22} + \)\(24\!\cdots\!86\)\( T^{23} + \)\(14\!\cdots\!32\)\( T^{24} + \)\(78\!\cdots\!58\)\( T^{25} + \)\(46\!\cdots\!50\)\( T^{26} + \)\(21\!\cdots\!90\)\( T^{27} + \)\(10\!\cdots\!61\)\( T^{28} + \)\(41\!\cdots\!86\)\( T^{29} + \)\(24\!\cdots\!13\)\( T^{30} + \)\(14\!\cdots\!14\)\( T^{31} + \)\(38\!\cdots\!21\)\( T^{32} \))(\( 1 + 2 T + 177 T^{2} + 1582 T^{3} + 21921 T^{4} + 231770 T^{5} + 2684050 T^{6} + 23678026 T^{7} + 239791912 T^{8} + 2114692878 T^{9} + 17637992799 T^{10} + 151508122160 T^{11} + 1184498641232 T^{12} + 9380769650632 T^{13} + 69924170534704 T^{14} + 544239771304768 T^{15} + 3774658249521589 T^{16} + 28844707879152704 T^{17} + 196416995031983536 T^{18} + 1396580843277140264 T^{19} + 9346264023166912592 T^{20} + 63360013840205424880 T^{21} + \)\(39\!\cdots\!71\)\( T^{22} + \)\(24\!\cdots\!86\)\( T^{23} + \)\(14\!\cdots\!32\)\( T^{24} + \)\(78\!\cdots\!58\)\( T^{25} + \)\(46\!\cdots\!50\)\( T^{26} + \)\(21\!\cdots\!90\)\( T^{27} + \)\(10\!\cdots\!61\)\( T^{28} + \)\(41\!\cdots\!86\)\( T^{29} + \)\(24\!\cdots\!13\)\( T^{30} + \)\(14\!\cdots\!14\)\( T^{31} + \)\(38\!\cdots\!21\)\( T^{32} \))(\( ( 1 - 10 T + 47 T^{2} - 530 T^{3} + 2809 T^{4} )^{4}( 1 + 10 T + 53 T^{2} + 590 T^{3} + 5900 T^{4} + 55820 T^{5} + 444723 T^{6} + 3293380 T^{7} + 25043319 T^{8} + 174549140 T^{9} + 1249226907 T^{10} + 8310314140 T^{11} + 46553837900 T^{12} + 246735340870 T^{13} + 1174711139837 T^{14} + 11747111398370 T^{15} + 62259690411361 T^{16} ) \))
$59$ (\( 1 - 18 T + 239 T^{2} - 1374 T^{3} + 5292 T^{4} + 7812 T^{5} + 470053 T^{6} - 8281674 T^{7} + 95738891 T^{8} - 488618766 T^{9} + 1636254493 T^{10} + 1604420748 T^{11} + 64125074412 T^{12} - 982305986826 T^{13} + 10081147540199 T^{14} - 44795726726742 T^{15} + 146830437604321 T^{16} \))(\( 1 - 25 T + 419 T^{2} - 5200 T^{3} + 58281 T^{4} - 598550 T^{5} + 6209893 T^{6} - 62948850 T^{7} + 627371027 T^{8} - 5955201125 T^{9} + 54113866884 T^{10} - 468835214275 T^{11} + 3967819915452 T^{12} - 33125257031325 T^{13} + 272714366831568 T^{14} - 2199975935884200 T^{15} + 17198372786595766 T^{16} - 129798580217167800 T^{17} + 949318710940688208 T^{18} - 6803232163836497175 T^{19} + 48079506298521362172 T^{20} - \)\(33\!\cdots\!25\)\( T^{21} + \)\(22\!\cdots\!44\)\( T^{22} - \)\(14\!\cdots\!75\)\( T^{23} + \)\(92\!\cdots\!67\)\( T^{24} - \)\(54\!\cdots\!50\)\( T^{25} + \)\(31\!\cdots\!93\)\( T^{26} - \)\(18\!\cdots\!50\)\( T^{27} + \)\(10\!\cdots\!61\)\( T^{28} - \)\(54\!\cdots\!00\)\( T^{29} + \)\(25\!\cdots\!59\)\( T^{30} - \)\(91\!\cdots\!75\)\( T^{31} + \)\(21\!\cdots\!41\)\( T^{32} \))(\( 1 + 25 T + 419 T^{2} + 5200 T^{3} + 58281 T^{4} + 598550 T^{5} + 6209893 T^{6} + 62948850 T^{7} + 627371027 T^{8} + 5955201125 T^{9} + 54113866884 T^{10} + 468835214275 T^{11} + 3967819915452 T^{12} + 33125257031325 T^{13} + 272714366831568 T^{14} + 2199975935884200 T^{15} + 17198372786595766 T^{16} + 129798580217167800 T^{17} + 949318710940688208 T^{18} + 6803232163836497175 T^{19} + 48079506298521362172 T^{20} + \)\(33\!\cdots\!25\)\( T^{21} + \)\(22\!\cdots\!44\)\( T^{22} + \)\(14\!\cdots\!75\)\( T^{23} + \)\(92\!\cdots\!67\)\( T^{24} + \)\(54\!\cdots\!50\)\( T^{25} + \)\(31\!\cdots\!93\)\( T^{26} + \)\(18\!\cdots\!50\)\( T^{27} + \)\(10\!\cdots\!61\)\( T^{28} + \)\(54\!\cdots\!00\)\( T^{29} + \)\(25\!\cdots\!59\)\( T^{30} + \)\(91\!\cdots\!75\)\( T^{31} + \)\(21\!\cdots\!41\)\( T^{32} \))(\( ( 1 + 59 T^{2} - 205379 T^{6} - 12117361 T^{8} - 714924299 T^{10} + 2488651484819 T^{14} + 146830437604321 T^{16} )^{2} \))
$61$ (\( 1 + 18 T + 141 T^{2} - 538 T^{3} - 19524 T^{4} - 199396 T^{5} - 460901 T^{6} + 9059916 T^{7} + 127866487 T^{8} + 552654876 T^{9} - 1715012621 T^{10} - 45259103476 T^{11} - 270326199684 T^{12} - 454392809938 T^{13} + 7264372784901 T^{14} + 56569371048378 T^{15} + 191707312997281 T^{16} \))(\( 1 - 7 T + 157 T^{2} - 2810 T^{3} + 23827 T^{4} - 352454 T^{5} + 4248359 T^{6} - 37300018 T^{7} + 438042179 T^{8} - 4457193753 T^{9} + 37214201488 T^{10} - 377860830849 T^{11} + 3428826064944 T^{12} - 26601329050551 T^{13} + 245019222253856 T^{14} - 2011514545995138 T^{15} + 14453308043412074 T^{16} - 122702387305703418 T^{17} + 911716526006598176 T^{18} - 6037996269223116531 T^{19} + 47474980511870297904 T^{20} - \)\(31\!\cdots\!49\)\( T^{21} + \)\(19\!\cdots\!68\)\( T^{22} - \)\(14\!\cdots\!13\)\( T^{23} + \)\(83\!\cdots\!99\)\( T^{24} - \)\(43\!\cdots\!38\)\( T^{25} + \)\(30\!\cdots\!59\)\( T^{26} - \)\(15\!\cdots\!94\)\( T^{27} + \)\(63\!\cdots\!67\)\( T^{28} - \)\(45\!\cdots\!10\)\( T^{29} + \)\(15\!\cdots\!37\)\( T^{30} - \)\(42\!\cdots\!07\)\( T^{31} + \)\(36\!\cdots\!61\)\( T^{32} \))(\( 1 + 7 T + 157 T^{2} + 2810 T^{3} + 23827 T^{4} + 352454 T^{5} + 4248359 T^{6} + 37300018 T^{7} + 438042179 T^{8} + 4457193753 T^{9} + 37214201488 T^{10} + 377860830849 T^{11} + 3428826064944 T^{12} + 26601329050551 T^{13} + 245019222253856 T^{14} + 2011514545995138 T^{15} + 14453308043412074 T^{16} + 122702387305703418 T^{17} + 911716526006598176 T^{18} + 6037996269223116531 T^{19} + 47474980511870297904 T^{20} + \)\(31\!\cdots\!49\)\( T^{21} + \)\(19\!\cdots\!68\)\( T^{22} + \)\(14\!\cdots\!13\)\( T^{23} + \)\(83\!\cdots\!99\)\( T^{24} + \)\(43\!\cdots\!38\)\( T^{25} + \)\(30\!\cdots\!59\)\( T^{26} + \)\(15\!\cdots\!94\)\( T^{27} + \)\(63\!\cdots\!67\)\( T^{28} + \)\(45\!\cdots\!10\)\( T^{29} + \)\(15\!\cdots\!37\)\( T^{30} + \)\(42\!\cdots\!07\)\( T^{31} + \)\(36\!\cdots\!61\)\( T^{32} \))(\( ( 1 + 61 T^{2} - 226981 T^{6} - 13845841 T^{8} - 844596301 T^{10} + 3142742836021 T^{14} + 191707312997281 T^{16} )^{2} \))
$67$ (\( ( 1 + 19 T + 138 T^{2} + 1691 T^{3} + 21053 T^{4} + 113297 T^{5} + 619482 T^{6} + 5714497 T^{7} + 20151121 T^{8} )^{2} \))(\( ( 1 - 15 T - 110 T^{2} + 1095 T^{3} + 30612 T^{4} - 158730 T^{5} - 2667925 T^{6} + 1012950 T^{7} + 259471373 T^{8} + 67867650 T^{9} - 11976315325 T^{10} - 47740110990 T^{11} + 616866116052 T^{12} + 1478386992165 T^{13} - 9950422038590 T^{14} - 90910674079845 T^{15} + 406067677556641 T^{16} )^{2} \))(\( ( 1 - 15 T - 110 T^{2} + 1095 T^{3} + 30612 T^{4} - 158730 T^{5} - 2667925 T^{6} + 1012950 T^{7} + 259471373 T^{8} + 67867650 T^{9} - 11976315325 T^{10} - 47740110990 T^{11} + 616866116052 T^{12} + 1478386992165 T^{13} - 9950422038590 T^{14} - 90910674079845 T^{15} + 406067677556641 T^{16} )^{2} \))(\( ( 1 - 4 T + 67 T^{2} - 740 T^{3} + 2960 T^{4} + 19784 T^{5} - 30619 T^{6} + 3660040 T^{7} - 34791281 T^{8} + 245222680 T^{9} - 137448691 T^{10} + 5950295192 T^{11} + 59647318160 T^{12} - 999092579180 T^{13} + 6060711605323 T^{14} - 24242846421292 T^{15} + 406067677556641 T^{16} )^{2} \))
$71$ (\( ( 1 + 8 T - 47 T^{2} - 434 T^{3} + 1365 T^{4} - 30814 T^{5} - 236927 T^{6} + 2863288 T^{7} + 25411681 T^{8} )^{2} \))(\( ( 1 + 14 T - 9 T^{2} - 147 T^{3} + 6927 T^{4} + 52059 T^{5} + 728307 T^{6} + 4047848 T^{7} - 14238189 T^{8} + 287397208 T^{9} + 3671395587 T^{10} + 18632488749 T^{11} + 176026714287 T^{12} - 265221714597 T^{13} - 1152902555289 T^{14} + 127331682217474 T^{15} + 645753531245761 T^{16} )^{2} \))(\( ( 1 + 14 T - 9 T^{2} - 147 T^{3} + 6927 T^{4} + 52059 T^{5} + 728307 T^{6} + 4047848 T^{7} - 14238189 T^{8} + 287397208 T^{9} + 3671395587 T^{10} + 18632488749 T^{11} + 176026714287 T^{12} - 265221714597 T^{13} - 1152902555289 T^{14} + 127331682217474 T^{15} + 645753531245761 T^{16} )^{2} \))(\( ( 1 - 16 T + 71 T^{2} )^{8}( 1 + 16 T + 185 T^{2} + 1824 T^{3} + 16049 T^{4} + 129504 T^{5} + 932585 T^{6} + 5726576 T^{7} + 25411681 T^{8} )^{2} \))
$73$ (\( 1 + 15 T + 208 T^{2} + 2445 T^{3} + 29070 T^{4} + 272130 T^{5} + 2789288 T^{6} + 23441760 T^{7} + 210261239 T^{8} + 1711248480 T^{9} + 14864115752 T^{10} + 105863196210 T^{11} + 825536865870 T^{12} + 5068660044885 T^{13} + 31477519068112 T^{14} + 165710977786455 T^{15} + 806460091894081 T^{16} \))(\( 1 - 3 T + 11 T^{2} + 696 T^{3} - 1251 T^{4} - 98282 T^{5} + 1060537 T^{6} - 2326410 T^{7} - 65290511 T^{8} + 308702063 T^{9} + 6492765102 T^{10} - 92451983031 T^{11} + 281336636896 T^{12} + 4375330012525 T^{13} - 27106436766918 T^{14} - 268331027179726 T^{15} + 5635646597034826 T^{16} - 19588164984119998 T^{17} - 144450201530906022 T^{18} + 1702077755482437925 T^{19} + 7989465616702099936 T^{20} - \)\(19\!\cdots\!83\)\( T^{21} + \)\(98\!\cdots\!78\)\( T^{22} + \)\(34\!\cdots\!11\)\( T^{23} - \)\(52\!\cdots\!91\)\( T^{24} - \)\(13\!\cdots\!30\)\( T^{25} + \)\(45\!\cdots\!13\)\( T^{26} - \)\(30\!\cdots\!14\)\( T^{27} - \)\(28\!\cdots\!71\)\( T^{28} + \)\(11\!\cdots\!68\)\( T^{29} + \)\(13\!\cdots\!99\)\( T^{30} - \)\(26\!\cdots\!71\)\( T^{31} + \)\(65\!\cdots\!61\)\( T^{32} \))(\( 1 + 3 T + 11 T^{2} - 696 T^{3} - 1251 T^{4} + 98282 T^{5} + 1060537 T^{6} + 2326410 T^{7} - 65290511 T^{8} - 308702063 T^{9} + 6492765102 T^{10} + 92451983031 T^{11} + 281336636896 T^{12} - 4375330012525 T^{13} - 27106436766918 T^{14} + 268331027179726 T^{15} + 5635646597034826 T^{16} + 19588164984119998 T^{17} - 144450201530906022 T^{18} - 1702077755482437925 T^{19} + 7989465616702099936 T^{20} + \)\(19\!\cdots\!83\)\( T^{21} + \)\(98\!\cdots\!78\)\( T^{22} - \)\(34\!\cdots\!11\)\( T^{23} - \)\(52\!\cdots\!91\)\( T^{24} + \)\(13\!\cdots\!30\)\( T^{25} + \)\(45\!\cdots\!13\)\( T^{26} + \)\(30\!\cdots\!14\)\( T^{27} - \)\(28\!\cdots\!71\)\( T^{28} - \)\(11\!\cdots\!68\)\( T^{29} + \)\(13\!\cdots\!99\)\( T^{30} + \)\(26\!\cdots\!71\)\( T^{31} + \)\(65\!\cdots\!61\)\( T^{32} \))(\( ( 1 + 73 T^{2} - 389017 T^{6} - 28398241 T^{8} - 2073071593 T^{10} + 11047398519097 T^{14} + 806460091894081 T^{16} )^{2} \))
$79$ (\( 1 - 9 T - 11 T^{2} + 1446 T^{3} - 17334 T^{4} + 85743 T^{5} + 200726 T^{6} - 10774872 T^{7} + 128238887 T^{8} - 851214888 T^{9} + 1252730966 T^{10} + 42274642977 T^{11} - 675160704054 T^{12} + 4449423552954 T^{13} - 2673962010731 T^{14} - 172835180875431 T^{15} + 1517108809906561 T^{16} \))(\( 1 - 9 T + 156 T^{2} - 861 T^{3} + 2506 T^{4} + 98394 T^{5} - 1985283 T^{6} + 19471377 T^{7} - 143824296 T^{8} + 713258136 T^{9} + 7748031243 T^{10} - 94344490293 T^{11} + 1366072992785 T^{12} - 5625357609390 T^{13} + 11704956821451 T^{14} + 387894213236796 T^{15} - 6232535914916185 T^{16} + 30643642845706884 T^{17} + 73050635522675691 T^{18} - 2773520690376036210 T^{19} + 53208653720888165585 T^{20} - \)\(29\!\cdots\!07\)\( T^{21} + \)\(18\!\cdots\!03\)\( T^{22} + \)\(13\!\cdots\!24\)\( T^{23} - \)\(21\!\cdots\!56\)\( T^{24} + \)\(23\!\cdots\!63\)\( T^{25} - \)\(18\!\cdots\!83\)\( T^{26} + \)\(73\!\cdots\!26\)\( T^{27} + \)\(14\!\cdots\!46\)\( T^{28} - \)\(40\!\cdots\!79\)\( T^{29} + \)\(57\!\cdots\!36\)\( T^{30} - \)\(26\!\cdots\!91\)\( T^{31} + \)\(23\!\cdots\!21\)\( T^{32} \))(\( 1 - 9 T + 156 T^{2} - 861 T^{3} + 2506 T^{4} + 98394 T^{5} - 1985283 T^{6} + 19471377 T^{7} - 143824296 T^{8} + 713258136 T^{9} + 7748031243 T^{10} - 94344490293 T^{11} + 1366072992785 T^{12} - 5625357609390 T^{13} + 11704956821451 T^{14} + 387894213236796 T^{15} - 6232535914916185 T^{16} + 30643642845706884 T^{17} + 73050635522675691 T^{18} - 2773520690376036210 T^{19} + 53208653720888165585 T^{20} - \)\(29\!\cdots\!07\)\( T^{21} + \)\(18\!\cdots\!03\)\( T^{22} + \)\(13\!\cdots\!24\)\( T^{23} - \)\(21\!\cdots\!56\)\( T^{24} + \)\(23\!\cdots\!63\)\( T^{25} - \)\(18\!\cdots\!83\)\( T^{26} + \)\(73\!\cdots\!26\)\( T^{27} + \)\(14\!\cdots\!46\)\( T^{28} - \)\(40\!\cdots\!79\)\( T^{29} + \)\(57\!\cdots\!36\)\( T^{30} - \)\(26\!\cdots\!91\)\( T^{31} + \)\(23\!\cdots\!21\)\( T^{32} \))(\( ( 1 + 8 T - 15 T^{2} + 632 T^{3} + 6241 T^{4} )^{4}( 1 - 8 T + 79 T^{2} - 1384 T^{3} + 11072 T^{4} - 29168 T^{5} + 781073 T^{6} - 5046064 T^{7} + 1418431 T^{8} - 398639056 T^{9} + 4874676593 T^{10} - 14380961552 T^{11} + 431255296832 T^{12} - 4258646056216 T^{13} + 19203908986159 T^{14} - 153631271889272 T^{15} + 1517108809906561 T^{16} ) \))
$83$ (\( ( 1 + 9 T - 2 T^{2} - 765 T^{3} - 6719 T^{4} - 63495 T^{5} - 13778 T^{6} + 5146083 T^{7} + 47458321 T^{8} )^{2} \))(\( ( 1 + 23 T + 98 T^{2} - 1745 T^{3} - 15470 T^{4} + 98632 T^{5} + 1750974 T^{6} - 1504700 T^{7} - 137136651 T^{8} - 124890100 T^{9} + 12062459886 T^{10} + 56396495384 T^{11} - 734180225870 T^{12} - 6873625922035 T^{13} + 32040156590162 T^{14} + 624129172761421 T^{15} + 2252292232139041 T^{16} )^{2} \))(\( ( 1 - 23 T + 98 T^{2} + 1745 T^{3} - 15470 T^{4} - 98632 T^{5} + 1750974 T^{6} + 1504700 T^{7} - 137136651 T^{8} + 124890100 T^{9} + 12062459886 T^{10} - 56396495384 T^{11} - 734180225870 T^{12} + 6873625922035 T^{13} + 32040156590162 T^{14} - 624129172761421 T^{15} + 2252292232139041 T^{16} )^{2} \))(\( ( 1 - 83 T^{2} + 6889 T^{4} - 571787 T^{6} + 47458321 T^{8} )^{4} \))
$89$ (\( ( 1 - 12 T - 50 T^{2} - 192 T^{3} + 16899 T^{4} - 17088 T^{5} - 396050 T^{6} - 8459628 T^{7} + 62742241 T^{8} )^{2} \))(\( ( 1 + 17 T - 23 T^{2} - 1534 T^{3} + 282 T^{4} + 47093 T^{5} - 689650 T^{6} - 2747686 T^{7} + 49542809 T^{8} - 244544054 T^{9} - 5462717650 T^{10} + 33199105117 T^{11} + 17693311962 T^{12} - 8565947194766 T^{13} - 11430569692103 T^{14} + 751932693223993 T^{15} + 3936588805702081 T^{16} )^{2} \))(\( ( 1 - 17 T - 23 T^{2} + 1534 T^{3} + 282 T^{4} - 47093 T^{5} - 689650 T^{6} + 2747686 T^{7} + 49542809 T^{8} + 244544054 T^{9} - 5462717650 T^{10} - 33199105117 T^{11} + 17693311962 T^{12} + 8565947194766 T^{13} - 11430569692103 T^{14} - 751932693223993 T^{15} + 3936588805702081 T^{16} )^{2} \))(\( ( 1 - 89 T^{2} + 7921 T^{4} )^{8} \))
$97$ (\( ( 1 - 7 T - 63 T^{2} - 185 T^{3} + 11276 T^{4} - 17945 T^{5} - 592767 T^{6} - 6388711 T^{7} + 88529281 T^{8} )^{2} \))(\( ( 1 + 30 T + 361 T^{2} + 2160 T^{3} + 10062 T^{4} + 83370 T^{5} - 577417 T^{6} - 32754150 T^{7} - 458148745 T^{8} - 3177152550 T^{9} - 5432916553 T^{10} + 76089548010 T^{11} + 890781625422 T^{12} + 18548654955120 T^{13} + 300702893779369 T^{14} + 2423948534343390 T^{15} + 7837433594376961 T^{16} )^{2} \))(\( ( 1 - 30 T + 361 T^{2} - 2160 T^{3} + 10062 T^{4} - 83370 T^{5} - 577417 T^{6} + 32754150 T^{7} - 458148745 T^{8} + 3177152550 T^{9} - 5432916553 T^{10} - 76089548010 T^{11} + 890781625422 T^{12} - 18548654955120 T^{13} + 300702893779369 T^{14} - 2423948534343390 T^{15} + 7837433594376961 T^{16} )^{2} \))(\( ( 1 - 97 T^{2} + 9409 T^{4} - 912673 T^{6} + 88529281 T^{8} )^{4} \))
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