Properties

Label 847.2.l
Level 847
Weight 2
Character orbit l
Rep. character \(\chi_{847}(118,\cdot)\)
Character field \(\Q(\zeta_{10})\)
Dimension 256
Newform subspaces 13
Sturm bound 176
Trace bound 8

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Defining parameters

Level: \( N \) = \( 847 = 7 \cdot 11^{2} \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 847.l (of order \(10\) and degree \(4\))
Character conductor: \(\operatorname{cond}(\chi)\) = \( 77 \)
Character field: \(\Q(\zeta_{10})\)
Newform subspaces: \( 13 \)
Sturm bound: \(176\)
Trace bound: \(8\)
Distinguishing \(T_p\): \(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(847, [\chi])\).

Total New Old
Modular forms 400 320 80
Cusp forms 304 256 48
Eisenstein series 96 64 32

Trace form

\( 256q + 10q^{2} + 60q^{4} + 10q^{7} + 48q^{9} + O(q^{10}) \) \( 256q + 10q^{2} + 60q^{4} + 10q^{7} + 48q^{9} + 17q^{14} - 26q^{16} + 35q^{18} - 88q^{23} + 70q^{25} - 25q^{28} - 10q^{29} + 40q^{30} - 40q^{35} - 90q^{36} + 6q^{37} - 30q^{39} - 86q^{42} - 5q^{46} - 28q^{49} + 60q^{50} + 10q^{51} - 26q^{53} - 18q^{56} + 90q^{57} - 13q^{58} - 52q^{60} + 20q^{63} + 44q^{64} - 132q^{67} - 8q^{70} + 120q^{71} + 45q^{72} - 90q^{74} + 4q^{78} - 10q^{79} + 52q^{81} + 70q^{84} - 80q^{85} - 29q^{86} - 6q^{91} - 105q^{92} - 40q^{93} + 60q^{95} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(847, [\chi])\) into newform subspaces

Label Dim. \(A\) Field CM Traces $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\)
847.2.l.a \(8\) \(6.763\) 8.0.37515625.1 \(\Q(\sqrt{-7}) \) \(-5\) \(0\) \(0\) \(0\) \(q+(-\beta _{1}-\beta _{2}+\beta _{3}+\beta _{4}+\beta _{5}-\beta _{7})q^{2}+\cdots\)
847.2.l.b \(8\) \(6.763\) 8.0.37515625.1 \(\Q(\sqrt{-7}) \) \(0\) \(0\) \(0\) \(0\) \(q-\beta _{4}q^{2}-5\beta _{1}q^{4}+\beta _{5}q^{7}-3\beta _{7}q^{8}+\cdots\)
847.2.l.c \(8\) \(6.763\) 8.0.37515625.1 \(\Q(\sqrt{-7}) \) \(0\) \(0\) \(0\) \(0\) \(q+(-1+\beta _{1}-\beta _{3}-\beta _{4}-\beta _{5}+\beta _{6}+\cdots)q^{2}+\cdots\)
847.2.l.d \(8\) \(6.763\) 8.0.37515625.1 \(\Q(\sqrt{-7}) \) \(5\) \(0\) \(0\) \(0\) \(q+(\beta _{1}+\beta _{2}-\beta _{3}-\beta _{4}-\beta _{5}+\beta _{7})q^{2}+\cdots\)
847.2.l.e \(16\) \(6.763\) \(\mathbb{Q}[x]/(x^{16} + \cdots)\) None \(-10\) \(0\) \(0\) \(10\) \(q+(-1+\beta _{4}+\beta _{5}+\beta _{8}-\beta _{9}-\beta _{14}+\cdots)q^{2}+\cdots\)
847.2.l.f \(16\) \(6.763\) \(\mathbb{Q}[x]/(x^{16} + \cdots)\) None \(0\) \(0\) \(0\) \(0\) \(q+(-\beta _{8}-\beta _{15})q^{2}+(-\beta _{2}-\beta _{6})q^{3}+\cdots\)
847.2.l.g \(16\) \(6.763\) \(\Q(\zeta_{40})\) None \(0\) \(0\) \(0\) \(0\) \(q-\zeta_{40}^{6}q^{2}-\zeta_{40}^{5}q^{3}+\zeta_{40}^{13}q^{5}+\cdots\)
847.2.l.h \(16\) \(6.763\) \(\mathbb{Q}[x]/(x^{16} + \cdots)\) None \(0\) \(0\) \(0\) \(0\) \(q+(\beta _{8}-\beta _{10}+\beta _{15})q^{2}+(-\beta _{2}+\beta _{7}+\cdots)q^{3}+\cdots\)
847.2.l.i \(16\) \(6.763\) \(\mathbb{Q}[x]/(x^{16} + \cdots)\) None \(10\) \(0\) \(0\) \(10\) \(q+(-\beta _{8}-\beta _{9}-\beta _{11}+\beta _{12})q^{2}+(-\beta _{1}+\cdots)q^{3}+\cdots\)
847.2.l.j \(16\) \(6.763\) \(\mathbb{Q}[x]/(x^{16} + \cdots)\) None \(10\) \(0\) \(0\) \(-10\) \(q+(1-\beta _{4}-\beta _{5}-\beta _{8}+\beta _{9}+\beta _{14})q^{2}+\cdots\)
847.2.l.k \(32\) \(6.763\) None \(0\) \(0\) \(0\) \(0\)
847.2.l.l \(32\) \(6.763\) None \(0\) \(0\) \(0\) \(0\)
847.2.l.m \(64\) \(6.763\) None \(0\) \(0\) \(0\) \(0\)

Decomposition of \(S_{2}^{\mathrm{old}}(847, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(847, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(77, [\chi])\)\(^{\oplus 2}\)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ (\( ( 1 + T + 2 T^{2} )^{4}( 1 + T - T^{2} - 3 T^{3} - T^{4} - 6 T^{5} - 4 T^{6} + 8 T^{7} + 16 T^{8} ) \))(\( ( 1 - T - T^{2} + 3 T^{3} - T^{4} + 6 T^{5} - 4 T^{6} - 8 T^{7} + 16 T^{8} )( 1 + T - T^{2} - 3 T^{3} - T^{4} - 6 T^{5} - 4 T^{6} + 8 T^{7} + 16 T^{8} ) \))(\( ( 1 - T - T^{2} + 3 T^{3} - T^{4} + 6 T^{5} - 4 T^{6} - 8 T^{7} + 16 T^{8} )( 1 + T - T^{2} - 3 T^{3} - T^{4} - 6 T^{5} - 4 T^{6} + 8 T^{7} + 16 T^{8} ) \))(\( ( 1 - T + 2 T^{2} )^{4}( 1 - T - T^{2} + 3 T^{3} - T^{4} + 6 T^{5} - 4 T^{6} - 8 T^{7} + 16 T^{8} ) \))(\( ( 1 + T + 4 T^{2} + 2 T^{3} + 9 T^{4} + 4 T^{5} + 16 T^{6} + 8 T^{7} + 16 T^{8} )^{2}( 1 + 4 T + 4 T^{2} - 7 T^{3} - 21 T^{4} - 14 T^{5} + 16 T^{6} + 32 T^{7} + 16 T^{8} )^{2} \))(\( ( 1 + 2 T^{2} )^{8}( 1 - 2 T^{2} + 4 T^{4} - 8 T^{6} + 16 T^{8} )^{2} \))(\( ( 1 + 2 T^{2} - 8 T^{6} - 16 T^{8} - 32 T^{10} + 128 T^{14} + 256 T^{16} )^{2} \))(\( ( 1 - 2 T^{2} + 4 T^{4} - 8 T^{6} + 16 T^{8} )^{4} \))(\( ( 1 - 3 T + 5 T^{2} - 6 T^{3} + 4 T^{4} )^{4}( 1 + T + 4 T^{2} + 2 T^{3} + 9 T^{4} + 4 T^{5} + 16 T^{6} + 8 T^{7} + 16 T^{8} )^{2} \))(\( ( 1 - 4 T + 4 T^{2} + 7 T^{3} - 21 T^{4} + 14 T^{5} + 16 T^{6} - 32 T^{7} + 16 T^{8} )^{2}( 1 - T + 4 T^{2} - 2 T^{3} + 9 T^{4} - 4 T^{5} + 16 T^{6} - 8 T^{7} + 16 T^{8} )^{2} \))
$3$ (\( ( 1 + 3 T^{2} + 9 T^{4} + 27 T^{6} + 81 T^{8} )^{2} \))(\( ( 1 + 3 T^{2} + 9 T^{4} + 27 T^{6} + 81 T^{8} )^{2} \))(\( ( 1 + 3 T^{2} + 9 T^{4} + 27 T^{6} + 81 T^{8} )^{2} \))(\( ( 1 + 3 T^{2} + 9 T^{4} + 27 T^{6} + 81 T^{8} )^{2} \))(\( 1 + 2 T^{2} - 15 T^{4} - 50 T^{6} + 5 T^{8} + 766 T^{10} + 2187 T^{12} - 3790 T^{14} - 28100 T^{16} - 34110 T^{18} + 177147 T^{20} + 558414 T^{22} + 32805 T^{24} - 2952450 T^{26} - 7971615 T^{28} + 9565938 T^{30} + 43046721 T^{32} \))(\( ( 1 - 3 T^{2} + 10 T^{4} - 43 T^{6} + 199 T^{8} - 387 T^{10} + 810 T^{12} - 2187 T^{14} + 6561 T^{16} )^{2} \))(\( ( 1 + T^{2} - 8 T^{4} - 17 T^{6} + 55 T^{8} - 153 T^{10} - 648 T^{12} + 729 T^{14} + 6561 T^{16} )^{2} \))(\( ( 1 + 7 T^{2} + 10 T^{4} - 33 T^{6} - 161 T^{8} - 297 T^{10} + 810 T^{12} + 5103 T^{14} + 6561 T^{16} )^{2} \))(\( 1 + 2 T^{2} + 10 T^{6} + 65 T^{8} + 466 T^{10} + 912 T^{12} + 1220 T^{14} + 6865 T^{16} + 10980 T^{18} + 73872 T^{20} + 339714 T^{22} + 426465 T^{24} + 590490 T^{26} + 9565938 T^{30} + 43046721 T^{32} \))(\( 1 + 2 T^{2} - 15 T^{4} - 50 T^{6} + 5 T^{8} + 766 T^{10} + 2187 T^{12} - 3790 T^{14} - 28100 T^{16} - 34110 T^{18} + 177147 T^{20} + 558414 T^{22} + 32805 T^{24} - 2952450 T^{26} - 7971615 T^{28} + 9565938 T^{30} + 43046721 T^{32} \))
$5$ (\( ( 1 - 5 T + 15 T^{2} - 25 T^{3} + 25 T^{4} )^{2}( 1 + 5 T + 15 T^{2} + 25 T^{3} + 25 T^{4} )^{2} \))(\( ( 1 - 5 T + 15 T^{2} - 25 T^{3} + 25 T^{4} )^{2}( 1 + 5 T + 15 T^{2} + 25 T^{3} + 25 T^{4} )^{2} \))(\( ( 1 - 5 T + 15 T^{2} - 25 T^{3} + 25 T^{4} )^{2}( 1 + 5 T + 15 T^{2} + 25 T^{3} + 25 T^{4} )^{2} \))(\( ( 1 - 5 T + 15 T^{2} - 25 T^{3} + 25 T^{4} )^{2}( 1 + 5 T + 15 T^{2} + 25 T^{3} + 25 T^{4} )^{2} \))(\( 1 - 15 T^{2} + 150 T^{4} - 1220 T^{6} + 9090 T^{8} - 60275 T^{10} + 359050 T^{12} - 1999650 T^{14} + 10337275 T^{16} - 49991250 T^{18} + 224406250 T^{20} - 941796875 T^{22} + 3550781250 T^{24} - 11914062500 T^{26} + 36621093750 T^{28} - 91552734375 T^{30} + 152587890625 T^{32} \))(\( ( 1 + 11 T^{2} + 26 T^{4} - 269 T^{6} - 2489 T^{8} - 6725 T^{10} + 16250 T^{12} + 171875 T^{14} + 390625 T^{16} )^{2} \))(\( ( 1 + 5 T^{2} - 125 T^{6} - 625 T^{8} - 3125 T^{10} + 78125 T^{14} + 390625 T^{16} )^{2} \))(\( ( 1 + T^{2} + 26 T^{4} + 41 T^{6} + 511 T^{8} + 1025 T^{10} + 16250 T^{12} + 15625 T^{14} + 390625 T^{16} )^{2} \))(\( 1 + 25 T^{2} + 235 T^{4} + 830 T^{6} - 985 T^{8} - 8750 T^{10} + 76100 T^{12} + 846725 T^{14} + 4674775 T^{16} + 21168125 T^{18} + 47562500 T^{20} - 136718750 T^{22} - 384765625 T^{24} + 8105468750 T^{26} + 57373046875 T^{28} + 152587890625 T^{30} + 152587890625 T^{32} \))(\( 1 - 15 T^{2} + 150 T^{4} - 1220 T^{6} + 9090 T^{8} - 60275 T^{10} + 359050 T^{12} - 1999650 T^{14} + 10337275 T^{16} - 49991250 T^{18} + 224406250 T^{20} - 941796875 T^{22} + 3550781250 T^{24} - 11914062500 T^{26} + 36621093750 T^{28} - 91552734375 T^{30} + 152587890625 T^{32} \))
$7$ (\( 1 - 7 T^{2} + 49 T^{4} - 343 T^{6} + 2401 T^{8} \))(\( 1 - 7 T^{2} + 49 T^{4} - 343 T^{6} + 2401 T^{8} \))(\( 1 - 7 T^{2} + 49 T^{4} - 343 T^{6} + 2401 T^{8} \))(\( 1 - 7 T^{2} + 49 T^{4} - 343 T^{6} + 2401 T^{8} \))(\( 1 - 10 T + 66 T^{2} - 330 T^{3} + 1387 T^{4} - 5090 T^{5} + 17028 T^{6} - 51660 T^{7} + 143385 T^{8} - 361620 T^{9} + 834372 T^{10} - 1745870 T^{11} + 3330187 T^{12} - 5546310 T^{13} + 7764834 T^{14} - 8235430 T^{15} + 5764801 T^{16} \))(\( 1 - 26 T^{2} + 327 T^{4} - 2878 T^{6} + 21305 T^{8} - 141022 T^{10} + 785127 T^{12} - 3058874 T^{14} + 5764801 T^{16} \))(\( 1 - 4 T^{2} - 33 T^{4} + 328 T^{6} + 305 T^{8} + 16072 T^{10} - 79233 T^{12} - 470596 T^{14} + 5764801 T^{16} \))(\( 1 + 24 T^{2} + 327 T^{4} + 3322 T^{6} + 26205 T^{8} + 162778 T^{10} + 785127 T^{12} + 2823576 T^{14} + 5764801 T^{16} \))(\( 1 - 10 T + 46 T^{2} - 110 T^{3} + 67 T^{4} + 430 T^{5} - 1612 T^{6} + 3140 T^{7} - 6135 T^{8} + 21980 T^{9} - 78988 T^{10} + 147490 T^{11} + 160867 T^{12} - 1848770 T^{13} + 5411854 T^{14} - 8235430 T^{15} + 5764801 T^{16} \))(\( 1 + 10 T + 66 T^{2} + 330 T^{3} + 1387 T^{4} + 5090 T^{5} + 17028 T^{6} + 51660 T^{7} + 143385 T^{8} + 361620 T^{9} + 834372 T^{10} + 1745870 T^{11} + 3330187 T^{12} + 5546310 T^{13} + 7764834 T^{14} + 8235430 T^{15} + 5764801 T^{16} \))
$11$ (\( \))(\( \))(\( \))(\( \))(\( \))(\( \))(\( \))(\( \))(\( \))(\( \))
$13$ (\( ( 1 - 13 T^{2} + 169 T^{4} - 2197 T^{6} + 28561 T^{8} )^{2} \))(\( ( 1 - 13 T^{2} + 169 T^{4} - 2197 T^{6} + 28561 T^{8} )^{2} \))(\( ( 1 - 13 T^{2} + 169 T^{4} - 2197 T^{6} + 28561 T^{8} )^{2} \))(\( ( 1 - 13 T^{2} + 169 T^{4} - 2197 T^{6} + 28561 T^{8} )^{2} \))(\( 1 - 47 T^{2} + 1100 T^{4} - 14690 T^{6} + 87190 T^{8} - 72371 T^{10} + 14716072 T^{12} - 637390320 T^{14} + 11513489835 T^{16} - 107718964080 T^{18} + 420305732392 T^{20} - 349320994139 T^{22} + 71123561563990 T^{24} - 2025141245261810 T^{26} + 25627893634729100 T^{28} - 185056690127866583 T^{30} + 665416609183179841 T^{32} \))(\( ( 1 + 22 T^{2} + 75 T^{4} + 692 T^{6} + 32309 T^{8} + 116948 T^{10} + 2142075 T^{12} + 106189798 T^{14} + 815730721 T^{16} )^{2} \))(\( ( 1 + 14 T^{2} + 27 T^{4} - 1988 T^{6} - 32395 T^{8} - 335972 T^{10} + 771147 T^{12} + 67575326 T^{14} + 815730721 T^{16} )^{2} \))(\( ( 1 - 18 T^{2} + 75 T^{4} - 2348 T^{6} + 59349 T^{8} - 396812 T^{10} + 2142075 T^{12} - 86882562 T^{14} + 815730721 T^{16} )^{2} \))(\( 1 - 37 T^{2} + 905 T^{4} - 17590 T^{6} + 252535 T^{8} - 3649636 T^{10} + 49961932 T^{12} - 672618165 T^{14} + 9550335135 T^{16} - 113672469885 T^{18} + 1426962739852 T^{20} - 17616095891524 T^{22} + 206000557627735 T^{24} - 2424930871623910 T^{26} + 21084767035845305 T^{28} - 145682926270873693 T^{30} + 665416609183179841 T^{32} \))(\( 1 - 47 T^{2} + 1100 T^{4} - 14690 T^{6} + 87190 T^{8} - 72371 T^{10} + 14716072 T^{12} - 637390320 T^{14} + 11513489835 T^{16} - 107718964080 T^{18} + 420305732392 T^{20} - 349320994139 T^{22} + 71123561563990 T^{24} - 2025141245261810 T^{26} + 25627893634729100 T^{28} - 185056690127866583 T^{30} + 665416609183179841 T^{32} \))
$17$ (\( ( 1 - 17 T^{2} + 289 T^{4} - 4913 T^{6} + 83521 T^{8} )^{2} \))(\( ( 1 - 17 T^{2} + 289 T^{4} - 4913 T^{6} + 83521 T^{8} )^{2} \))(\( ( 1 - 17 T^{2} + 289 T^{4} - 4913 T^{6} + 83521 T^{8} )^{2} \))(\( ( 1 - 17 T^{2} + 289 T^{4} - 4913 T^{6} + 83521 T^{8} )^{2} \))(\( 1 - 28 T^{2} + 635 T^{4} - 13075 T^{6} + 139165 T^{8} + 1031651 T^{10} - 40976873 T^{12} + 1379790840 T^{14} - 30384779175 T^{16} + 398759552760 T^{18} - 3422429409833 T^{20} + 24901547196419 T^{22} + 970781284276765 T^{24} - 26359120248370675 T^{26} + 369965120640898235 T^{28} - 4714579143663226012 T^{30} + 48661191875666868481 T^{32} \))(\( ( 1 - 44 T^{2} + 1247 T^{4} - 29452 T^{6} + 585505 T^{8} - 8511628 T^{10} + 104150687 T^{12} - 1062053036 T^{14} + 6975757441 T^{16} )^{2} \))(\( ( 1 - 17 T^{2} + 289 T^{4} - 4913 T^{6} + 83521 T^{8} )^{4} \))(\( ( 1 + 56 T^{2} + 1247 T^{4} + 16948 T^{6} + 238705 T^{8} + 4897972 T^{10} + 104150687 T^{12} + 1351703864 T^{14} + 6975757441 T^{16} )^{2} \))(\( 1 - 53 T^{2} + 935 T^{4} + 10075 T^{6} - 695885 T^{8} + 10961626 T^{10} - 1176848 T^{12} - 2883558960 T^{14} + 66310473675 T^{16} - 833348539440 T^{18} - 98291521808 T^{20} + 264587003927194 T^{22} - 4854324966830285 T^{24} + 20311138547023675 T^{26} + 544751791809826535 T^{28} - 8924024807648249237 T^{30} + 48661191875666868481 T^{32} \))(\( 1 - 28 T^{2} + 635 T^{4} - 13075 T^{6} + 139165 T^{8} + 1031651 T^{10} - 40976873 T^{12} + 1379790840 T^{14} - 30384779175 T^{16} + 398759552760 T^{18} - 3422429409833 T^{20} + 24901547196419 T^{22} + 970781284276765 T^{24} - 26359120248370675 T^{26} + 369965120640898235 T^{28} - 4714579143663226012 T^{30} + 48661191875666868481 T^{32} \))
$19$ (\( ( 1 - 19 T^{2} + 361 T^{4} - 6859 T^{6} + 130321 T^{8} )^{2} \))(\( ( 1 - 19 T^{2} + 361 T^{4} - 6859 T^{6} + 130321 T^{8} )^{2} \))(\( ( 1 - 19 T^{2} + 361 T^{4} - 6859 T^{6} + 130321 T^{8} )^{2} \))(\( ( 1 - 19 T^{2} + 361 T^{4} - 6859 T^{6} + 130321 T^{8} )^{2} \))(\( 1 + 74 T^{2} + 3935 T^{4} + 142985 T^{6} + 4147525 T^{8} + 97507007 T^{10} + 1951099453 T^{12} + 36276904890 T^{14} + 663576736605 T^{16} + 13095962665290 T^{18} + 254269231814413 T^{20} + 4587303047988167 T^{22} + 70439752301623525 T^{24} + 876650508871675985 T^{26} + 8709394206525343535 T^{28} + 59126494747933424954 T^{30} + \)\(28\!\cdots\!81\)\( T^{32} \))(\( ( 1 - 36 T^{2} + 615 T^{4} - 2474 T^{6} - 106611 T^{8} - 893114 T^{10} + 80147415 T^{12} - 1693651716 T^{14} + 16983563041 T^{16} )^{2} \))(\( ( 1 - 28 T^{2} + 423 T^{4} - 1736 T^{6} - 104095 T^{8} - 626696 T^{10} + 55125783 T^{12} - 1317284668 T^{14} + 16983563041 T^{16} )^{2} \))(\( ( 1 - 26 T^{2} + 615 T^{4} - 12034 T^{6} + 117209 T^{8} - 4344274 T^{10} + 80147415 T^{12} - 1223192906 T^{14} + 16983563041 T^{16} )^{2} \))(\( 1 - 131 T^{2} + 7835 T^{4} - 269555 T^{6} + 5344225 T^{8} - 36384548 T^{10} - 1284840662 T^{12} + 54300385290 T^{14} - 1231352034765 T^{16} + 19602439089690 T^{18} - 167441719912502 T^{20} - 1711743115446788 T^{22} + 90763982192788225 T^{24} - 1652659565121548555 T^{26} + 17341322390883371435 T^{28} - \)\(10\!\cdots\!51\)\( T^{30} + \)\(28\!\cdots\!81\)\( T^{32} \))(\( 1 + 74 T^{2} + 3935 T^{4} + 142985 T^{6} + 4147525 T^{8} + 97507007 T^{10} + 1951099453 T^{12} + 36276904890 T^{14} + 663576736605 T^{16} + 13095962665290 T^{18} + 254269231814413 T^{20} + 4587303047988167 T^{22} + 70439752301623525 T^{24} + 876650508871675985 T^{26} + 8709394206525343535 T^{28} + 59126494747933424954 T^{30} + \)\(28\!\cdots\!81\)\( T^{32} \))
$23$ (\( ( 1 - 8 T + 41 T^{2} - 144 T^{3} + 209 T^{4} - 3312 T^{5} + 21689 T^{6} - 97336 T^{7} + 279841 T^{8} )^{2} \))(\( ( 1 + 8 T + 23 T^{2} )^{8} \))(\( ( 1 - 8 T + 41 T^{2} - 144 T^{3} + 209 T^{4} - 3312 T^{5} + 21689 T^{6} - 97336 T^{7} + 279841 T^{8} )^{2} \))(\( ( 1 - 8 T + 41 T^{2} - 144 T^{3} + 209 T^{4} - 3312 T^{5} + 21689 T^{6} - 97336 T^{7} + 279841 T^{8} )^{2} \))(\( ( 1 + T + 35 T^{2} + 23 T^{3} + 529 T^{4} )^{8} \))(\( ( 1 + 2 T + 27 T^{2} + 46 T^{3} + 529 T^{4} )^{8} \))(\( ( 1 + 3 T + 23 T^{2} )^{16} \))(\( ( 1 + 2 T + 27 T^{2} + 46 T^{3} + 529 T^{4} )^{8} \))(\( ( 1 + T + 35 T^{2} + 23 T^{3} + 529 T^{4} )^{8} \))(\( ( 1 + T + 35 T^{2} + 23 T^{3} + 529 T^{4} )^{8} \))
$29$ (\( ( 1 - 2 T + 29 T^{2} )^{4}( 1 - 2 T - 25 T^{2} + 108 T^{3} + 509 T^{4} + 3132 T^{5} - 21025 T^{6} - 48778 T^{7} + 707281 T^{8} ) \))(\( ( 1 - 2 T - 25 T^{2} + 108 T^{3} + 509 T^{4} + 3132 T^{5} - 21025 T^{6} - 48778 T^{7} + 707281 T^{8} )( 1 + 2 T - 25 T^{2} - 108 T^{3} + 509 T^{4} - 3132 T^{5} - 21025 T^{6} + 48778 T^{7} + 707281 T^{8} ) \))(\( ( 1 - 2 T - 25 T^{2} + 108 T^{3} + 509 T^{4} + 3132 T^{5} - 21025 T^{6} - 48778 T^{7} + 707281 T^{8} )( 1 + 2 T - 25 T^{2} - 108 T^{3} + 509 T^{4} - 3132 T^{5} - 21025 T^{6} + 48778 T^{7} + 707281 T^{8} ) \))(\( ( 1 + 2 T + 29 T^{2} )^{4}( 1 + 2 T - 25 T^{2} - 108 T^{3} + 509 T^{4} - 3132 T^{5} - 21025 T^{6} + 48778 T^{7} + 707281 T^{8} ) \))(\( ( 1 - 10 T + 101 T^{2} - 540 T^{3} + 3490 T^{4} - 14430 T^{5} + 92159 T^{6} - 454810 T^{7} + 3087199 T^{8} - 13189490 T^{9} + 77505719 T^{10} - 351933270 T^{11} + 2468410690 T^{12} - 11076020460 T^{13} + 60077155421 T^{14} - 172498763090 T^{15} + 500246412961 T^{16} )^{2} \))(\( ( 1 + 76 T^{2} + 1335 T^{4} - 59926 T^{6} - 3413971 T^{8} - 50397766 T^{10} + 944220135 T^{12} + 45206572396 T^{14} + 500246412961 T^{16} )^{2} \))(\( ( 1 + 56 T^{2} + 2295 T^{4} + 81424 T^{6} + 2629649 T^{8} + 68477584 T^{10} + 1623209895 T^{12} + 33310105976 T^{14} + 500246412961 T^{16} )^{2} \))(\( ( 1 - 14 T^{2} + 1335 T^{4} - 9886 T^{6} + 1278809 T^{8} - 8314126 T^{10} + 944220135 T^{12} - 8327526494 T^{14} + 500246412961 T^{16} )^{2} \))(\( ( 1 + 5 T + 66 T^{2} + 555 T^{3} + 3920 T^{4} + 28200 T^{5} + 189134 T^{6} + 1026380 T^{7} + 6343899 T^{8} + 29765020 T^{9} + 159061694 T^{10} + 687769800 T^{11} + 2772541520 T^{12} + 11383687695 T^{13} + 39258339186 T^{14} + 86249381545 T^{15} + 500246412961 T^{16} )^{2} \))(\( ( 1 + 10 T + 101 T^{2} + 540 T^{3} + 3490 T^{4} + 14430 T^{5} + 92159 T^{6} + 454810 T^{7} + 3087199 T^{8} + 13189490 T^{9} + 77505719 T^{10} + 351933270 T^{11} + 2468410690 T^{12} + 11076020460 T^{13} + 60077155421 T^{14} + 172498763090 T^{15} + 500246412961 T^{16} )^{2} \))
$31$ (\( ( 1 + 31 T^{2} + 961 T^{4} + 29791 T^{6} + 923521 T^{8} )^{2} \))(\( ( 1 + 31 T^{2} + 961 T^{4} + 29791 T^{6} + 923521 T^{8} )^{2} \))(\( ( 1 + 31 T^{2} + 961 T^{4} + 29791 T^{6} + 923521 T^{8} )^{2} \))(\( ( 1 + 31 T^{2} + 961 T^{4} + 29791 T^{6} + 923521 T^{8} )^{2} \))(\( 1 + 144 T^{2} + 8455 T^{4} + 160530 T^{6} - 8385030 T^{8} - 615070998 T^{10} - 11789511487 T^{12} + 326129343750 T^{14} + 22130035940015 T^{16} + 313410299343750 T^{18} - 10887861437985727 T^{20} - 545877774801343638 T^{22} - 7151516935673908230 T^{24} + \)\(13\!\cdots\!30\)\( T^{26} + \)\(66\!\cdots\!55\)\( T^{28} + \)\(10\!\cdots\!24\)\( T^{30} + \)\(72\!\cdots\!81\)\( T^{32} \))(\( ( 1 + 11 T^{2} + 1890 T^{4} + 63139 T^{6} + 1653599 T^{8} + 60676579 T^{10} + 1745454690 T^{12} + 9762540491 T^{14} + 852891037441 T^{16} )^{2} \))(\( ( 1 + 17 T^{2} - 672 T^{4} - 27761 T^{6} + 173855 T^{8} - 26678321 T^{10} - 620606112 T^{12} + 15087562577 T^{14} + 852891037441 T^{16} )^{2} \))(\( ( 1 + 81 T^{2} + 1890 T^{4} - 67831 T^{6} - 4535241 T^{8} - 65185591 T^{10} + 1745454690 T^{12} + 71887798161 T^{14} + 852891037441 T^{16} )^{2} \))(\( 1 + 39 T^{2} + 3430 T^{4} + 163065 T^{6} + 6773220 T^{8} + 296986872 T^{10} + 10144743998 T^{12} + 345739802400 T^{14} + 11501584169495 T^{16} + 332255950106400 T^{18} + 9368884121776958 T^{20} + 263576942108675832 T^{22} + 5776818632616130020 T^{24} + \)\(13\!\cdots\!65\)\( T^{26} + \)\(27\!\cdots\!30\)\( T^{28} + \)\(29\!\cdots\!19\)\( T^{30} + \)\(72\!\cdots\!81\)\( T^{32} \))(\( 1 + 144 T^{2} + 8455 T^{4} + 160530 T^{6} - 8385030 T^{8} - 615070998 T^{10} - 11789511487 T^{12} + 326129343750 T^{14} + 22130035940015 T^{16} + 313410299343750 T^{18} - 10887861437985727 T^{20} - 545877774801343638 T^{22} - 7151516935673908230 T^{24} + \)\(13\!\cdots\!30\)\( T^{26} + \)\(66\!\cdots\!55\)\( T^{28} + \)\(10\!\cdots\!24\)\( T^{30} + \)\(72\!\cdots\!81\)\( T^{32} \))
$37$ (\( ( 1 - 6 T - T^{2} + 228 T^{3} - 1331 T^{4} + 8436 T^{5} - 1369 T^{6} - 303918 T^{7} + 1874161 T^{8} )^{2} \))(\( ( 1 - 6 T - T^{2} + 228 T^{3} - 1331 T^{4} + 8436 T^{5} - 1369 T^{6} - 303918 T^{7} + 1874161 T^{8} )^{2} \))(\( ( 1 + 6 T + 37 T^{2} )^{4}( 1 - 6 T - T^{2} + 228 T^{3} - 1331 T^{4} + 8436 T^{5} - 1369 T^{6} - 303918 T^{7} + 1874161 T^{8} ) \))(\( ( 1 - 6 T - T^{2} + 228 T^{3} - 1331 T^{4} + 8436 T^{5} - 1369 T^{6} - 303918 T^{7} + 1874161 T^{8} )^{2} \))(\( ( 1 - 7 T - 16 T^{2} + 233 T^{3} - 272 T^{4} - 20242 T^{5} + 119506 T^{6} + 231534 T^{7} - 3495357 T^{8} + 8566758 T^{9} + 163603714 T^{10} - 1025318026 T^{11} - 509771792 T^{12} + 16157141981 T^{13} - 41051622544 T^{14} - 664523139931 T^{15} + 3512479453921 T^{16} )^{2} \))(\( ( 1 - 3 T - 28 T^{2} + 195 T^{3} + 451 T^{4} + 7215 T^{5} - 38332 T^{6} - 151959 T^{7} + 1874161 T^{8} )^{4} \))(\( ( 1 - T - 36 T^{2} + 73 T^{3} + 1259 T^{4} + 2701 T^{5} - 49284 T^{6} - 50653 T^{7} + 1874161 T^{8} )^{4} \))(\( ( 1 - 3 T - 28 T^{2} + 195 T^{3} + 451 T^{4} + 7215 T^{5} - 38332 T^{6} - 151959 T^{7} + 1874161 T^{8} )^{4} \))(\( ( 1 - 2 T - 31 T^{2} + 88 T^{3} - 122 T^{4} - 18302 T^{5} + 81031 T^{6} + 289434 T^{7} - 1619457 T^{8} + 10709058 T^{9} + 110931439 T^{10} - 927051206 T^{11} - 228647642 T^{12} + 6102268216 T^{13} - 79537518679 T^{14} - 189863754266 T^{15} + 3512479453921 T^{16} )^{2} \))(\( ( 1 - 7 T - 16 T^{2} + 233 T^{3} - 272 T^{4} - 20242 T^{5} + 119506 T^{6} + 231534 T^{7} - 3495357 T^{8} + 8566758 T^{9} + 163603714 T^{10} - 1025318026 T^{11} - 509771792 T^{12} + 16157141981 T^{13} - 41051622544 T^{14} - 664523139931 T^{15} + 3512479453921 T^{16} )^{2} \))
$41$ (\( ( 1 - 41 T^{2} + 1681 T^{4} - 68921 T^{6} + 2825761 T^{8} )^{2} \))(\( ( 1 - 41 T^{2} + 1681 T^{4} - 68921 T^{6} + 2825761 T^{8} )^{2} \))(\( ( 1 - 41 T^{2} + 1681 T^{4} - 68921 T^{6} + 2825761 T^{8} )^{2} \))(\( ( 1 - 41 T^{2} + 1681 T^{4} - 68921 T^{6} + 2825761 T^{8} )^{2} \))(\( 1 - 14 T^{2} - 4055 T^{4} + 148070 T^{6} + 6691085 T^{8} - 424370282 T^{10} + 977382083 T^{12} + 383909674130 T^{14} - 14906091923060 T^{16} + 645352162212530 T^{18} + 2761848172240163 T^{20} - 2015803076282565962 T^{22} + 53427813426693086285 T^{24} + \)\(19\!\cdots\!70\)\( T^{26} - \)\(91\!\cdots\!55\)\( T^{28} - \)\(53\!\cdots\!54\)\( T^{30} + \)\(63\!\cdots\!41\)\( T^{32} \))(\( ( 1 - 80 T^{2} + 3959 T^{4} - 141250 T^{6} + 4790941 T^{8} - 237441250 T^{10} + 11187187799 T^{12} - 380008339280 T^{14} + 7984925229121 T^{16} )^{2} \))(\( ( 1 + 8 T^{2} - 1617 T^{4} - 26384 T^{6} + 2507105 T^{8} - 44351504 T^{10} - 4569255537 T^{12} + 38000833928 T^{14} + 7984925229121 T^{16} )^{2} \))(\( ( 1 - 70 T^{2} + 3959 T^{4} - 197450 T^{6} + 7312441 T^{8} - 331913450 T^{10} + 11187187799 T^{12} - 332507296870 T^{14} + 7984925229121 T^{16} )^{2} \))(\( 1 - 54 T^{2} + 5160 T^{4} - 368270 T^{6} + 20358825 T^{8} - 1151359622 T^{10} + 57899888128 T^{12} - 2567003976540 T^{14} + 113908617753985 T^{16} - 4315133684563740 T^{18} + 163611245776465408 T^{20} - 5469078223378356902 T^{22} + \)\(16\!\cdots\!25\)\( T^{24} - \)\(49\!\cdots\!70\)\( T^{26} + \)\(11\!\cdots\!60\)\( T^{28} - \)\(20\!\cdots\!94\)\( T^{30} + \)\(63\!\cdots\!41\)\( T^{32} \))(\( 1 - 14 T^{2} - 4055 T^{4} + 148070 T^{6} + 6691085 T^{8} - 424370282 T^{10} + 977382083 T^{12} + 383909674130 T^{14} - 14906091923060 T^{16} + 645352162212530 T^{18} + 2761848172240163 T^{20} - 2015803076282565962 T^{22} + 53427813426693086285 T^{24} + \)\(19\!\cdots\!70\)\( T^{26} - \)\(91\!\cdots\!55\)\( T^{28} - \)\(53\!\cdots\!54\)\( T^{30} + \)\(63\!\cdots\!41\)\( T^{32} \))
$43$ (\( ( 1 - 12 T + 101 T^{2} - 696 T^{3} + 4009 T^{4} - 29928 T^{5} + 186749 T^{6} - 954084 T^{7} + 3418801 T^{8} )( 1 + 12 T + 101 T^{2} + 696 T^{3} + 4009 T^{4} + 29928 T^{5} + 186749 T^{6} + 954084 T^{7} + 3418801 T^{8} ) \))(\( ( 1 - 12 T + 43 T^{2} )^{4}( 1 + 12 T + 43 T^{2} )^{4} \))(\( ( 1 - 12 T + 101 T^{2} - 696 T^{3} + 4009 T^{4} - 29928 T^{5} + 186749 T^{6} - 954084 T^{7} + 3418801 T^{8} )( 1 + 12 T + 101 T^{2} + 696 T^{3} + 4009 T^{4} + 29928 T^{5} + 186749 T^{6} + 954084 T^{7} + 3418801 T^{8} ) \))(\( ( 1 - 12 T + 101 T^{2} - 696 T^{3} + 4009 T^{4} - 29928 T^{5} + 186749 T^{6} - 954084 T^{7} + 3418801 T^{8} )( 1 + 12 T + 101 T^{2} + 696 T^{3} + 4009 T^{4} + 29928 T^{5} + 186749 T^{6} + 954084 T^{7} + 3418801 T^{8} ) \))(\( ( 1 - 13 T + 43 T^{2} )^{8}( 1 + 13 T + 43 T^{2} )^{8} \))(\( ( 1 + 2 T^{2} + 2854 T^{4} + 3698 T^{6} + 3418801 T^{8} )^{4} \))(\( ( 1 - 68 T^{2} + 1849 T^{4} )^{8} \))(\( ( 1 + 2 T^{2} + 2854 T^{4} + 3698 T^{6} + 3418801 T^{8} )^{4} \))(\( ( 1 - 13 T + 43 T^{2} )^{8}( 1 + 13 T + 43 T^{2} )^{8} \))(\( ( 1 - 13 T + 43 T^{2} )^{8}( 1 + 13 T + 43 T^{2} )^{8} \))
$47$ (\( ( 1 + 47 T^{2} + 2209 T^{4} + 103823 T^{6} + 4879681 T^{8} )^{2} \))(\( ( 1 + 47 T^{2} + 2209 T^{4} + 103823 T^{6} + 4879681 T^{8} )^{2} \))(\( ( 1 + 47 T^{2} + 2209 T^{4} + 103823 T^{6} + 4879681 T^{8} )^{2} \))(\( ( 1 + 47 T^{2} + 2209 T^{4} + 103823 T^{6} + 4879681 T^{8} )^{2} \))(\( 1 + 53 T^{2} + 3175 T^{4} + 66910 T^{6} + 7037915 T^{8} + 53044514 T^{10} - 3576233308 T^{12} - 622916000195 T^{14} - 13560550353665 T^{16} - 1376021444430755 T^{18} - 17450877724614748 T^{20} + 571778238428155106 T^{22} + \)\(16\!\cdots\!15\)\( T^{24} + \)\(35\!\cdots\!90\)\( T^{26} + \)\(36\!\cdots\!75\)\( T^{28} + \)\(13\!\cdots\!57\)\( T^{30} + \)\(56\!\cdots\!21\)\( T^{32} \))(\( ( 1 + 74 T^{2} - 133 T^{4} - 108008 T^{6} - 3128795 T^{8} - 238589672 T^{10} - 648997573 T^{12} + 797661934346 T^{14} + 23811286661761 T^{16} )^{2} \))(\( ( 1 + 74 T^{2} + 3267 T^{4} + 78292 T^{6} - 1423195 T^{8} + 172947028 T^{10} + 15941917827 T^{12} + 797661934346 T^{14} + 23811286661761 T^{16} )^{2} \))(\( ( 1 - 26 T^{2} - 133 T^{4} - 100408 T^{6} + 7474405 T^{8} - 221801272 T^{10} - 648997573 T^{12} - 280259598554 T^{14} + 23811286661761 T^{16} )^{2} \))(\( 1 + 173 T^{2} + 13870 T^{4} + 773020 T^{6} + 37466570 T^{8} + 1409514809 T^{10} + 35630259662 T^{12} + 1022803179550 T^{14} + 50909696806195 T^{16} + 2259372223625950 T^{18} + 173864301097727822 T^{20} + 15193463635625307161 T^{22} + \)\(89\!\cdots\!70\)\( T^{24} + \)\(40\!\cdots\!80\)\( T^{26} + \)\(16\!\cdots\!70\)\( T^{28} + \)\(44\!\cdots\!37\)\( T^{30} + \)\(56\!\cdots\!21\)\( T^{32} \))(\( 1 + 53 T^{2} + 3175 T^{4} + 66910 T^{6} + 7037915 T^{8} + 53044514 T^{10} - 3576233308 T^{12} - 622916000195 T^{14} - 13560550353665 T^{16} - 1376021444430755 T^{18} - 17450877724614748 T^{20} + 571778238428155106 T^{22} + \)\(16\!\cdots\!15\)\( T^{24} + \)\(35\!\cdots\!90\)\( T^{26} + \)\(36\!\cdots\!75\)\( T^{28} + \)\(13\!\cdots\!57\)\( T^{30} + \)\(56\!\cdots\!21\)\( T^{32} \))
$53$ (\( ( 1 + 10 T + 47 T^{2} - 60 T^{3} - 3091 T^{4} - 3180 T^{5} + 132023 T^{6} + 1488770 T^{7} + 7890481 T^{8} )^{2} \))(\( ( 1 + 10 T + 47 T^{2} - 60 T^{3} - 3091 T^{4} - 3180 T^{5} + 132023 T^{6} + 1488770 T^{7} + 7890481 T^{8} )^{2} \))(\( ( 1 - 10 T + 53 T^{2} )^{4}( 1 + 10 T + 47 T^{2} - 60 T^{3} - 3091 T^{4} - 3180 T^{5} + 132023 T^{6} + 1488770 T^{7} + 7890481 T^{8} ) \))(\( ( 1 + 10 T + 47 T^{2} - 60 T^{3} - 3091 T^{4} - 3180 T^{5} + 132023 T^{6} + 1488770 T^{7} + 7890481 T^{8} )^{2} \))(\( ( 1 + 5 T - 106 T^{2} - 615 T^{3} + 5032 T^{4} + 26970 T^{5} - 90598 T^{6} - 412150 T^{7} + 2297575 T^{8} - 21843950 T^{9} - 254489782 T^{10} + 4015212690 T^{11} + 39704900392 T^{12} - 257190228195 T^{13} - 2349422279674 T^{14} + 5873555699185 T^{15} + 62259690411361 T^{16} )^{2} \))(\( ( 1 - 2 T - 49 T^{2} + 204 T^{3} + 2189 T^{4} + 10812 T^{5} - 137641 T^{6} - 297754 T^{7} + 7890481 T^{8} )^{4} \))(\( ( 1 - 53 T^{2} + 2809 T^{4} - 148877 T^{6} + 7890481 T^{8} )^{4} \))(\( ( 1 - 2 T - 49 T^{2} + 204 T^{3} + 2189 T^{4} + 10812 T^{5} - 137641 T^{6} - 297754 T^{7} + 7890481 T^{8} )^{4} \))(\( ( 1 - 71 T^{2} - 150 T^{3} + 6432 T^{4} + 1950 T^{5} - 488983 T^{6} + 192300 T^{7} + 26002355 T^{8} + 10191900 T^{9} - 1373553247 T^{10} + 290310150 T^{11} + 50751573792 T^{12} - 62729323950 T^{13} - 1573669640159 T^{14} + 62259690411361 T^{16} )^{2} \))(\( ( 1 + 5 T - 106 T^{2} - 615 T^{3} + 5032 T^{4} + 26970 T^{5} - 90598 T^{6} - 412150 T^{7} + 2297575 T^{8} - 21843950 T^{9} - 254489782 T^{10} + 4015212690 T^{11} + 39704900392 T^{12} - 257190228195 T^{13} - 2349422279674 T^{14} + 5873555699185 T^{15} + 62259690411361 T^{16} )^{2} \))
$59$ (\( ( 1 + 59 T^{2} + 3481 T^{4} + 205379 T^{6} + 12117361 T^{8} )^{2} \))(\( ( 1 + 59 T^{2} + 3481 T^{4} + 205379 T^{6} + 12117361 T^{8} )^{2} \))(\( ( 1 + 59 T^{2} + 3481 T^{4} + 205379 T^{6} + 12117361 T^{8} )^{2} \))(\( ( 1 + 59 T^{2} + 3481 T^{4} + 205379 T^{6} + 12117361 T^{8} )^{2} \))(\( 1 + 66 T^{2} + 3735 T^{4} + 138340 T^{6} + 4049100 T^{8} - 4975802 T^{10} - 47292660557 T^{12} - 4443691805340 T^{14} - 377534465906645 T^{16} - 15468491174388540 T^{18} - 573062240619630077 T^{20} - 209881983651955082 T^{22} + \)\(59\!\cdots\!00\)\( T^{24} + \)\(70\!\cdots\!40\)\( T^{26} + \)\(66\!\cdots\!35\)\( T^{28} + \)\(40\!\cdots\!26\)\( T^{30} + \)\(21\!\cdots\!41\)\( T^{32} \))(\( ( 1 + 29 T^{2} - 190 T^{4} - 155459 T^{6} - 710921 T^{8} - 541152779 T^{10} - 2302298590 T^{12} + 1223235475589 T^{14} + 146830437604321 T^{16} )^{2} \))(\( ( 1 + 113 T^{2} + 9288 T^{4} + 656191 T^{6} + 41818055 T^{8} + 2284200871 T^{10} + 112546048968 T^{12} + 4766400301433 T^{14} + 146830437604321 T^{16} )^{2} \))(\( ( 1 + 79 T^{2} - 190 T^{4} - 295009 T^{6} - 19508321 T^{8} - 1026926329 T^{10} - 2302298590 T^{12} + 3332262157639 T^{14} + 146830437604321 T^{16} )^{2} \))(\( 1 + 181 T^{2} + 12520 T^{4} + 229955 T^{6} - 15999940 T^{8} - 623745812 T^{10} + 48156426998 T^{12} + 5123867326280 T^{14} + 307569246051235 T^{16} + 17836182162780680 T^{18} + 583528810404912278 T^{20} - 26309931206498861492 T^{22} - \)\(23\!\cdots\!40\)\( T^{24} + \)\(11\!\cdots\!55\)\( T^{26} + \)\(22\!\cdots\!20\)\( T^{28} + \)\(11\!\cdots\!41\)\( T^{30} + \)\(21\!\cdots\!41\)\( T^{32} \))(\( 1 + 66 T^{2} + 3735 T^{4} + 138340 T^{6} + 4049100 T^{8} - 4975802 T^{10} - 47292660557 T^{12} - 4443691805340 T^{14} - 377534465906645 T^{16} - 15468491174388540 T^{18} - 573062240619630077 T^{20} - 209881983651955082 T^{22} + \)\(59\!\cdots\!00\)\( T^{24} + \)\(70\!\cdots\!40\)\( T^{26} + \)\(66\!\cdots\!35\)\( T^{28} + \)\(40\!\cdots\!26\)\( T^{30} + \)\(21\!\cdots\!41\)\( T^{32} \))
$61$ (\( ( 1 - 61 T^{2} + 3721 T^{4} - 226981 T^{6} + 13845841 T^{8} )^{2} \))(\( ( 1 - 61 T^{2} + 3721 T^{4} - 226981 T^{6} + 13845841 T^{8} )^{2} \))(\( ( 1 - 61 T^{2} + 3721 T^{4} - 226981 T^{6} + 13845841 T^{8} )^{2} \))(\( ( 1 - 61 T^{2} + 3721 T^{4} - 226981 T^{6} + 13845841 T^{8} )^{2} \))(\( 1 + 26 T^{2} - 3215 T^{4} + 359620 T^{6} + 6106250 T^{8} - 2354446462 T^{10} + 79985426783 T^{12} + 3226358950810 T^{14} - 711263352045185 T^{16} + 12005281655964010 T^{18} + 1107465501554559503 T^{20} - \)\(12\!\cdots\!82\)\( T^{22} + \)\(11\!\cdots\!50\)\( T^{24} + \)\(25\!\cdots\!20\)\( T^{26} - \)\(85\!\cdots\!15\)\( T^{28} + \)\(25\!\cdots\!66\)\( T^{30} + \)\(36\!\cdots\!61\)\( T^{32} \))(\( ( 1 - 110 T^{2} + 9519 T^{4} - 730570 T^{6} + 45281321 T^{8} - 2718450970 T^{10} + 131798560479 T^{12} - 5667241179710 T^{14} + 191707312997281 T^{16} )^{2} \))(\( ( 1 - 112 T^{2} + 8823 T^{4} - 571424 T^{6} + 31169105 T^{8} - 2126268704 T^{10} + 122161855143 T^{12} - 5770281928432 T^{14} + 191707312997281 T^{16} )^{2} \))(\( ( 1 - 120 T^{2} + 9519 T^{4} - 598370 T^{6} + 36723021 T^{8} - 2226534770 T^{10} + 131798560479 T^{12} - 6182444923320 T^{14} + 191707312997281 T^{16} )^{2} \))(\( 1 - 189 T^{2} + 16680 T^{4} - 1152085 T^{6} + 74808120 T^{8} - 4782904102 T^{10} + 347814490078 T^{12} - 24590631974850 T^{14} + 1546838296855255 T^{16} - 91501741578416850 T^{18} + 4815784127116065598 T^{20} - \)\(24\!\cdots\!22\)\( T^{22} + \)\(14\!\cdots\!20\)\( T^{24} - \)\(82\!\cdots\!85\)\( T^{26} + \)\(44\!\cdots\!80\)\( T^{28} - \)\(18\!\cdots\!49\)\( T^{30} + \)\(36\!\cdots\!61\)\( T^{32} \))(\( 1 + 26 T^{2} - 3215 T^{4} + 359620 T^{6} + 6106250 T^{8} - 2354446462 T^{10} + 79985426783 T^{12} + 3226358950810 T^{14} - 711263352045185 T^{16} + 12005281655964010 T^{18} + 1107465501554559503 T^{20} - \)\(12\!\cdots\!82\)\( T^{22} + \)\(11\!\cdots\!50\)\( T^{24} + \)\(25\!\cdots\!20\)\( T^{26} - \)\(85\!\cdots\!15\)\( T^{28} + \)\(25\!\cdots\!66\)\( T^{30} + \)\(36\!\cdots\!61\)\( T^{32} \))
$67$ (\( ( 1 - 4 T - 51 T^{2} + 472 T^{3} + 1529 T^{4} + 31624 T^{5} - 228939 T^{6} - 1203052 T^{7} + 20151121 T^{8} )^{2} \))(\( ( 1 + 4 T + 67 T^{2} )^{8} \))(\( ( 1 - 4 T - 51 T^{2} + 472 T^{3} + 1529 T^{4} + 31624 T^{5} - 228939 T^{6} - 1203052 T^{7} + 20151121 T^{8} )^{2} \))(\( ( 1 - 4 T - 51 T^{2} + 472 T^{3} + 1529 T^{4} + 31624 T^{5} - 228939 T^{6} - 1203052 T^{7} + 20151121 T^{8} )^{2} \))(\( ( 1 + T + 159 T^{2} - 238 T^{3} + 12569 T^{4} - 15946 T^{5} + 713751 T^{6} + 300763 T^{7} + 20151121 T^{8} )^{4} \))(\( ( 1 + 6 T + 63 T^{2} + 402 T^{3} + 4489 T^{4} )^{8} \))(\( ( 1 - 11 T + 67 T^{2} )^{16} \))(\( ( 1 + 6 T + 63 T^{2} + 402 T^{3} + 4489 T^{4} )^{8} \))(\( ( 1 + T + 159 T^{2} - 238 T^{3} + 12569 T^{4} - 15946 T^{5} + 713751 T^{6} + 300763 T^{7} + 20151121 T^{8} )^{4} \))(\( ( 1 + T + 159 T^{2} - 238 T^{3} + 12569 T^{4} - 15946 T^{5} + 713751 T^{6} + 300763 T^{7} + 20151121 T^{8} )^{4} \))
$71$ (\( ( 1 - 16 T + 185 T^{2} - 1824 T^{3} + 16049 T^{4} - 129504 T^{5} + 932585 T^{6} - 5726576 T^{7} + 25411681 T^{8} )^{2} \))(\( ( 1 - 16 T + 185 T^{2} - 1824 T^{3} + 16049 T^{4} - 129504 T^{5} + 932585 T^{6} - 5726576 T^{7} + 25411681 T^{8} )^{2} \))(\( ( 1 + 16 T + 71 T^{2} )^{4}( 1 - 16 T + 185 T^{2} - 1824 T^{3} + 16049 T^{4} - 129504 T^{5} + 932585 T^{6} - 5726576 T^{7} + 25411681 T^{8} ) \))(\( ( 1 - 16 T + 185 T^{2} - 1824 T^{3} + 16049 T^{4} - 129504 T^{5} + 932585 T^{6} - 5726576 T^{7} + 25411681 T^{8} )^{2} \))(\( ( 1 - 17 T + 71 T^{2} - 733 T^{3} + 14111 T^{4} - 123166 T^{5} + 1421694 T^{6} - 11515034 T^{7} + 60440697 T^{8} - 817567414 T^{9} + 7166759454 T^{10} - 44082466226 T^{11} + 358584230591 T^{12} - 1322500114283 T^{13} + 9095120158391 T^{14} - 154617042692647 T^{15} + 645753531245761 T^{16} )^{2} \))(\( ( 1 + T - 70 T^{2} - 141 T^{3} + 4829 T^{4} - 10011 T^{5} - 352870 T^{6} + 357911 T^{7} + 25411681 T^{8} )^{4} \))(\( ( 1 + 9 T + 10 T^{2} - 549 T^{3} - 5651 T^{4} - 38979 T^{5} + 50410 T^{6} + 3221199 T^{7} + 25411681 T^{8} )^{4} \))(\( ( 1 + T - 70 T^{2} - 141 T^{3} + 4829 T^{4} - 10011 T^{5} - 352870 T^{6} + 357911 T^{7} + 25411681 T^{8} )^{4} \))(\( ( 1 + 28 T + 311 T^{2} + 1907 T^{3} + 10031 T^{4} - 8251 T^{5} - 1509411 T^{6} - 19087514 T^{7} - 158934393 T^{8} - 1355213494 T^{9} - 7608940851 T^{10} - 2953123661 T^{11} + 254904572111 T^{12} + 3440665372357 T^{13} + 39839188299431 T^{14} + 254663364434948 T^{15} + 645753531245761 T^{16} )^{2} \))(\( ( 1 - 17 T + 71 T^{2} - 733 T^{3} + 14111 T^{4} - 123166 T^{5} + 1421694 T^{6} - 11515034 T^{7} + 60440697 T^{8} - 817567414 T^{9} + 7166759454 T^{10} - 44082466226 T^{11} + 358584230591 T^{12} - 1322500114283 T^{13} + 9095120158391 T^{14} - 154617042692647 T^{15} + 645753531245761 T^{16} )^{2} \))
$73$ (\( ( 1 - 73 T^{2} + 5329 T^{4} - 389017 T^{6} + 28398241 T^{8} )^{2} \))(\( ( 1 - 73 T^{2} + 5329 T^{4} - 389017 T^{6} + 28398241 T^{8} )^{2} \))(\( ( 1 - 73 T^{2} + 5329 T^{4} - 389017 T^{6} + 28398241 T^{8} )^{2} \))(\( ( 1 - 73 T^{2} + 5329 T^{4} - 389017 T^{6} + 28398241 T^{8} )^{2} \))(\( 1 - 372 T^{2} + 55545 T^{4} - 3172630 T^{6} - 181947780 T^{8} + 40780939994 T^{10} - 1644589386923 T^{12} - 183485755284000 T^{14} + 25871064101490595 T^{16} - 977795589908436000 T^{18} - 46703445755881602443 T^{20} + \)\(61\!\cdots\!66\)\( T^{22} - \)\(14\!\cdots\!80\)\( T^{24} - \)\(13\!\cdots\!70\)\( T^{26} + \)\(12\!\cdots\!45\)\( T^{28} - \)\(45\!\cdots\!48\)\( T^{30} + \)\(65\!\cdots\!61\)\( T^{32} \))(\( ( 1 - 136 T^{2} + 7367 T^{4} + 134182 T^{6} - 48744995 T^{8} + 715055878 T^{10} + 209209841447 T^{12} - 20581454775304 T^{14} + 806460091894081 T^{16} )^{2} \))(\( ( 1 - 136 T^{2} + 13167 T^{4} - 1065968 T^{6} + 74804705 T^{8} - 5680543472 T^{10} + 373919639247 T^{12} - 20581454775304 T^{14} + 806460091894081 T^{16} )^{2} \))(\( ( 1 - 86 T^{2} + 7367 T^{4} - 475618 T^{6} + 10406905 T^{8} - 2534568322 T^{10} + 209209841447 T^{12} - 13014743460854 T^{14} + 806460091894081 T^{16} )^{2} \))(\( 1 + 23 T^{2} + 14710 T^{4} + 27475 T^{6} + 102964580 T^{8} - 4851528886 T^{10} + 427407600722 T^{12} - 53265322210310 T^{14} + 1562285407061275 T^{16} - 283850902058741990 T^{18} + 12137624050535130002 T^{20} - \)\(73\!\cdots\!54\)\( T^{22} + \)\(83\!\cdots\!80\)\( T^{24} + \)\(11\!\cdots\!75\)\( T^{26} + \)\(33\!\cdots\!10\)\( T^{28} + \)\(28\!\cdots\!07\)\( T^{30} + \)\(65\!\cdots\!61\)\( T^{32} \))(\( 1 - 372 T^{2} + 55545 T^{4} - 3172630 T^{6} - 181947780 T^{8} + 40780939994 T^{10} - 1644589386923 T^{12} - 183485755284000 T^{14} + 25871064101490595 T^{16} - 977795589908436000 T^{18} - 46703445755881602443 T^{20} + \)\(61\!\cdots\!66\)\( T^{22} - \)\(14\!\cdots\!80\)\( T^{24} - \)\(13\!\cdots\!70\)\( T^{26} + \)\(12\!\cdots\!45\)\( T^{28} - \)\(45\!\cdots\!48\)\( T^{30} + \)\(65\!\cdots\!61\)\( T^{32} \))
$79$ (\( ( 1 - 8 T - 15 T^{2} + 752 T^{3} - 4831 T^{4} + 59408 T^{5} - 93615 T^{6} - 3944312 T^{7} + 38950081 T^{8} )( 1 + 8 T - 15 T^{2} - 752 T^{3} - 4831 T^{4} - 59408 T^{5} - 93615 T^{6} + 3944312 T^{7} + 38950081 T^{8} ) \))(\( ( 1 - 8 T - 15 T^{2} + 752 T^{3} - 4831 T^{4} + 59408 T^{5} - 93615 T^{6} - 3944312 T^{7} + 38950081 T^{8} )( 1 + 8 T - 15 T^{2} - 752 T^{3} - 4831 T^{4} - 59408 T^{5} - 93615 T^{6} + 3944312 T^{7} + 38950081 T^{8} ) \))(\( ( 1 - 8 T + 79 T^{2} )^{4}( 1 - 8 T - 15 T^{2} + 752 T^{3} - 4831 T^{4} + 59408 T^{5} - 93615 T^{6} - 3944312 T^{7} + 38950081 T^{8} ) \))(\( ( 1 - 8 T - 15 T^{2} + 752 T^{3} - 4831 T^{4} + 59408 T^{5} - 93615 T^{6} - 3944312 T^{7} + 38950081 T^{8} )( 1 + 8 T - 15 T^{2} - 752 T^{3} - 4831 T^{4} - 59408 T^{5} - 93615 T^{6} + 3944312 T^{7} + 38950081 T^{8} ) \))(\( ( 1 - 25 T + 346 T^{2} - 2300 T^{3} + 6340 T^{4} + 22975 T^{5} + 585524 T^{6} - 13891600 T^{7} + 176078799 T^{8} - 1097436400 T^{9} + 3654255284 T^{10} + 11327571025 T^{11} + 246943513540 T^{12} - 7077229717700 T^{13} + 84108259610266 T^{14} - 480097724653975 T^{15} + 1517108809906561 T^{16} )^{2} \))(\( ( 1 + 140 T^{2} + 13359 T^{4} + 996520 T^{6} + 56139281 T^{8} + 6219281320 T^{10} + 520334132079 T^{12} + 34032243772940 T^{14} + 1517108809906561 T^{16} )^{2} \))(\( ( 1 + 86 T^{2} + 1155 T^{4} - 437396 T^{6} - 44824411 T^{8} - 2729788436 T^{10} + 44987343555 T^{12} + 20905521174806 T^{14} + 1517108809906561 T^{16} )^{2} \))(\( ( 1 + 140 T^{2} + 13359 T^{4} + 996520 T^{6} + 56139281 T^{8} + 6219281320 T^{10} + 520334132079 T^{12} + 34032243772940 T^{14} + 1517108809906561 T^{16} )^{2} \))(\( ( 1 + 25 T + 501 T^{2} + 8100 T^{3} + 108715 T^{4} + 1313450 T^{5} + 14241744 T^{6} + 141629175 T^{7} + 1311396489 T^{8} + 11188704825 T^{9} + 88882724304 T^{10} + 647582074550 T^{11} + 4234458055915 T^{12} + 24924156831900 T^{13} + 121786815216021 T^{14} + 480097724653975 T^{15} + 1517108809906561 T^{16} )^{2} \))(\( ( 1 + 25 T + 346 T^{2} + 2300 T^{3} + 6340 T^{4} - 22975 T^{5} + 585524 T^{6} + 13891600 T^{7} + 176078799 T^{8} + 1097436400 T^{9} + 3654255284 T^{10} - 11327571025 T^{11} + 246943513540 T^{12} + 7077229717700 T^{13} + 84108259610266 T^{14} + 480097724653975 T^{15} + 1517108809906561 T^{16} )^{2} \))
$83$ (\( ( 1 - 83 T^{2} + 6889 T^{4} - 571787 T^{6} + 47458321 T^{8} )^{2} \))(\( ( 1 - 83 T^{2} + 6889 T^{4} - 571787 T^{6} + 47458321 T^{8} )^{2} \))(\( ( 1 - 83 T^{2} + 6889 T^{4} - 571787 T^{6} + 47458321 T^{8} )^{2} \))(\( ( 1 - 83 T^{2} + 6889 T^{4} - 571787 T^{6} + 47458321 T^{8} )^{2} \))(\( 1 - 332 T^{2} + 41195 T^{4} - 388130 T^{6} - 520282430 T^{8} + 65781382474 T^{10} - 2171848089803 T^{12} - 327622469474250 T^{14} + 48330517524443895 T^{16} - 2256991192208108250 T^{18} - \)\(10\!\cdots\!63\)\( T^{20} + \)\(21\!\cdots\!06\)\( T^{22} - \)\(11\!\cdots\!30\)\( T^{24} - \)\(60\!\cdots\!70\)\( T^{26} + \)\(44\!\cdots\!95\)\( T^{28} - \)\(24\!\cdots\!28\)\( T^{30} + \)\(50\!\cdots\!81\)\( T^{32} \))(\( ( 1 - 88 T^{2} + 1775 T^{4} + 434242 T^{6} - 50130611 T^{8} + 2991493138 T^{10} + 84238519775 T^{12} - 28770752856472 T^{14} + 2252292232139041 T^{16} )^{2} \))(\( ( 1 - 76 T^{2} - 1113 T^{4} + 608152 T^{6} - 38552095 T^{8} + 4189559128 T^{10} - 52821111273 T^{12} - 24847468376044 T^{14} + 2252292232139041 T^{16} )^{2} \))(\( ( 1 - 98 T^{2} + 1775 T^{4} + 520682 T^{6} - 62944151 T^{8} + 3586978298 T^{10} + 84238519775 T^{12} - 32040156590162 T^{14} + 2252292232139041 T^{16} )^{2} \))(\( 1 - 97 T^{2} + 16850 T^{4} - 1726555 T^{6} + 145607140 T^{8} - 9460226116 T^{10} + 616698856582 T^{12} - 21019950912420 T^{14} + 1372062609692295 T^{16} - 144806441835661380 T^{18} + 29267492296001518822 T^{20} - \)\(30\!\cdots\!04\)\( T^{22} + \)\(32\!\cdots\!40\)\( T^{24} - \)\(26\!\cdots\!95\)\( T^{26} + \)\(18\!\cdots\!50\)\( T^{28} - \)\(71\!\cdots\!13\)\( T^{30} + \)\(50\!\cdots\!81\)\( T^{32} \))(\( 1 - 332 T^{2} + 41195 T^{4} - 388130 T^{6} - 520282430 T^{8} + 65781382474 T^{10} - 2171848089803 T^{12} - 327622469474250 T^{14} + 48330517524443895 T^{16} - 2256991192208108250 T^{18} - \)\(10\!\cdots\!63\)\( T^{20} + \)\(21\!\cdots\!06\)\( T^{22} - \)\(11\!\cdots\!30\)\( T^{24} - \)\(60\!\cdots\!70\)\( T^{26} + \)\(44\!\cdots\!95\)\( T^{28} - \)\(24\!\cdots\!28\)\( T^{30} + \)\(50\!\cdots\!81\)\( T^{32} \))
$89$ (\( ( 1 - 89 T^{2} )^{8} \))(\( ( 1 - 89 T^{2} )^{8} \))(\( ( 1 - 89 T^{2} )^{8} \))(\( ( 1 - 89 T^{2} )^{8} \))(\( ( 1 - 547 T^{2} + 136828 T^{4} - 20978449 T^{6} + 2212310245 T^{8} - 166170294529 T^{10} + 8584895351548 T^{12} - 271848766155667 T^{14} + 3936588805702081 T^{16} )^{2} \))(\( ( 1 + 84 T^{2} + 2481 T^{4} + 665364 T^{6} + 62742241 T^{8} )^{4} \))(\( ( 1 - 173 T^{2} + 7921 T^{4} )^{8} \))(\( ( 1 + 84 T^{2} + 2481 T^{4} + 665364 T^{6} + 62742241 T^{8} )^{4} \))(\( ( 1 - 547 T^{2} + 136828 T^{4} - 20978449 T^{6} + 2212310245 T^{8} - 166170294529 T^{10} + 8584895351548 T^{12} - 271848766155667 T^{14} + 3936588805702081 T^{16} )^{2} \))(\( ( 1 - 547 T^{2} + 136828 T^{4} - 20978449 T^{6} + 2212310245 T^{8} - 166170294529 T^{10} + 8584895351548 T^{12} - 271848766155667 T^{14} + 3936588805702081 T^{16} )^{2} \))
$97$ (\( ( 1 + 97 T^{2} + 9409 T^{4} + 912673 T^{6} + 88529281 T^{8} )^{2} \))(\( ( 1 + 97 T^{2} + 9409 T^{4} + 912673 T^{6} + 88529281 T^{8} )^{2} \))(\( ( 1 + 97 T^{2} + 9409 T^{4} + 912673 T^{6} + 88529281 T^{8} )^{2} \))(\( ( 1 + 97 T^{2} + 9409 T^{4} + 912673 T^{6} + 88529281 T^{8} )^{2} \))(\( 1 - 87 T^{2} + 11800 T^{4} - 1772925 T^{6} + 303958140 T^{8} - 33126817776 T^{10} + 3022618026302 T^{12} - 377313722967420 T^{14} + 40714509281804075 T^{16} - 3550144819400454780 T^{18} + \)\(26\!\cdots\!62\)\( T^{20} - \)\(27\!\cdots\!04\)\( T^{22} + \)\(23\!\cdots\!40\)\( T^{24} - \)\(13\!\cdots\!25\)\( T^{26} + \)\(81\!\cdots\!00\)\( T^{28} - \)\(56\!\cdots\!03\)\( T^{30} + \)\(61\!\cdots\!21\)\( T^{32} \))(\( ( 1 + 143 T^{2} + 23010 T^{4} + 3346903 T^{6} + 302137079 T^{8} + 31491010327 T^{10} + 2037058755810 T^{12} + 119114996704847 T^{14} + 7837433594376961 T^{16} )^{2} \))(\( ( 1 + 149 T^{2} + 12792 T^{4} + 504067 T^{6} - 45253945 T^{8} + 4742766403 T^{10} + 1132466562552 T^{12} + 124112828734421 T^{14} + 7837433594376961 T^{16} )^{2} \))(\( ( 1 + 213 T^{2} + 23010 T^{4} + 1146173 T^{6} + 67664799 T^{8} + 10784341757 T^{10} + 2037058755810 T^{12} + 177423037049877 T^{14} + 7837433594376961 T^{16} )^{2} \))(\( 1 + 378 T^{2} + 43195 T^{4} - 1829100 T^{6} - 951404880 T^{8} - 97555724586 T^{10} - 1788063513133 T^{12} + 794596480873200 T^{14} + 120216740714426075 T^{16} + 7476358288535938800 T^{18} - \)\(15\!\cdots\!73\)\( T^{20} - \)\(81\!\cdots\!94\)\( T^{22} - \)\(74\!\cdots\!80\)\( T^{24} - \)\(13\!\cdots\!00\)\( T^{26} + \)\(29\!\cdots\!95\)\( T^{28} + \)\(24\!\cdots\!82\)\( T^{30} + \)\(61\!\cdots\!21\)\( T^{32} \))(\( 1 - 87 T^{2} + 11800 T^{4} - 1772925 T^{6} + 303958140 T^{8} - 33126817776 T^{10} + 3022618026302 T^{12} - 377313722967420 T^{14} + 40714509281804075 T^{16} - 3550144819400454780 T^{18} + \)\(26\!\cdots\!62\)\( T^{20} - \)\(27\!\cdots\!04\)\( T^{22} + \)\(23\!\cdots\!40\)\( T^{24} - \)\(13\!\cdots\!25\)\( T^{26} + \)\(81\!\cdots\!00\)\( T^{28} - \)\(56\!\cdots\!03\)\( T^{30} + \)\(61\!\cdots\!21\)\( T^{32} \))
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