Properties

Label 722.2.c
Level 722
Weight 2
Character orbit c
Rep. character \(\chi_{722}(429,\cdot)\)
Character field \(\Q(\zeta_{3})\)
Dimension 54
Newform subspaces 14
Sturm bound 190
Trace bound 7

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Defining parameters

Level: \( N \) = \( 722 = 2 \cdot 19^{2} \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 722.c (of order \(3\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) = \( 19 \)
Character field: \(\Q(\zeta_{3})\)
Newform subspaces: \( 14 \)
Sturm bound: \(190\)
Trace bound: \(7\)
Distinguishing \(T_p\): \(3\), \(5\), \(7\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(722, [\chi])\).

Total New Old
Modular forms 230 54 176
Cusp forms 150 54 96
Eisenstein series 80 0 80

Trace form

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

Decomposition of \(S_{2}^{\mathrm{new}}(722, [\chi])\) into newform subspaces

Label Dim. \(A\) Field CM Traces $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\)
722.2.c.a \(2\) \(5.765\) \(\Q(\sqrt{-3}) \) None \(-1\) \(-3\) \(-2\) \(-6\) \(q+(-1+\zeta_{6})q^{2}+(-3+3\zeta_{6})q^{3}-\zeta_{6}q^{4}+\cdots\)
722.2.c.b \(2\) \(5.765\) \(\Q(\sqrt{-3}) \) None \(-1\) \(1\) \(0\) \(-8\) \(q+(-1+\zeta_{6})q^{2}+(1-\zeta_{6})q^{3}-\zeta_{6}q^{4}+\cdots\)
722.2.c.c \(2\) \(5.765\) \(\Q(\sqrt{-3}) \) None \(-1\) \(1\) \(0\) \(-2\) \(q+(-1+\zeta_{6})q^{2}+(1-\zeta_{6})q^{3}-\zeta_{6}q^{4}+\cdots\)
722.2.c.d \(2\) \(5.765\) \(\Q(\sqrt{-3}) \) None \(-1\) \(1\) \(4\) \(6\) \(q+(-1+\zeta_{6})q^{2}+(1-\zeta_{6})q^{3}-\zeta_{6}q^{4}+\cdots\)
722.2.c.e \(2\) \(5.765\) \(\Q(\sqrt{-3}) \) None \(1\) \(-1\) \(0\) \(-2\) \(q+(1-\zeta_{6})q^{2}+(-1+\zeta_{6})q^{3}-\zeta_{6}q^{4}+\cdots\)
722.2.c.f \(2\) \(5.765\) \(\Q(\sqrt{-3}) \) None \(1\) \(-1\) \(4\) \(6\) \(q+(1-\zeta_{6})q^{2}+(-1+\zeta_{6})q^{3}-\zeta_{6}q^{4}+\cdots\)
722.2.c.g \(2\) \(5.765\) \(\Q(\sqrt{-3}) \) None \(1\) \(3\) \(-2\) \(-6\) \(q+(1-\zeta_{6})q^{2}+(3-3\zeta_{6})q^{3}-\zeta_{6}q^{4}+\cdots\)
722.2.c.h \(4\) \(5.765\) \(\Q(\sqrt{-3}, \sqrt{5})\) None \(-2\) \(2\) \(5\) \(-4\) \(q+(-1-\beta _{3})q^{2}+2\beta _{1}q^{3}+\beta _{3}q^{4}+\cdots\)
722.2.c.i \(4\) \(5.765\) \(\Q(\sqrt{-3}, \sqrt{5})\) None \(2\) \(-2\) \(5\) \(-4\) \(q+(1+\beta _{3})q^{2}-2\beta _{1}q^{3}+\beta _{3}q^{4}+(3+\cdots)q^{5}+\cdots\)
722.2.c.j \(4\) \(5.765\) \(\Q(\sqrt{-3}, \sqrt{7})\) None \(2\) \(0\) \(-2\) \(4\) \(q+(1+\beta _{2})q^{2}-\beta _{1}q^{3}+\beta _{2}q^{4}+(-1+\cdots)q^{5}+\cdots\)
722.2.c.k \(6\) \(5.765\) \(\Q(\zeta_{18})\) None \(-3\) \(0\) \(-6\) \(12\) \(q+(-1+\zeta_{18})q^{2}+(\zeta_{18}^{2}-\zeta_{18}^{3})q^{3}+\cdots\)
722.2.c.l \(6\) \(5.765\) \(\Q(\zeta_{18})\) None \(3\) \(0\) \(-6\) \(12\) \(q+\zeta_{18}q^{2}+(-\zeta_{18}^{2}+\zeta_{18}^{3}+\zeta_{18}^{4}+\cdots)q^{3}+\cdots\)
722.2.c.m \(8\) \(5.765\) 8.0.324000000.2 None \(-4\) \(-2\) \(2\) \(-4\) \(q+(-1-\beta _{4})q^{2}+(\beta _{3}+\beta _{6})q^{3}+\beta _{4}q^{4}+\cdots\)
722.2.c.n \(8\) \(5.765\) 8.0.324000000.2 None \(4\) \(2\) \(2\) \(-4\) \(q+(1+\beta _{4})q^{2}+(-\beta _{3}-\beta _{6})q^{3}+\beta _{4}q^{4}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(722, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(722, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(38, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(361, [\chi])\)\(^{\oplus 2}\)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ (\( 1 + T + T^{2} \))(\( 1 + T + T^{2} \))(\( 1 + T + T^{2} \))(\( 1 + T + T^{2} \))(\( 1 - T + T^{2} \))(\( 1 - T + T^{2} \))(\( 1 - T + T^{2} \))(\( ( 1 + T + T^{2} )^{2} \))(\( ( 1 - T + T^{2} )^{2} \))(\( ( 1 - T + T^{2} )^{2} \))(\( ( 1 + T + T^{2} )^{3} \))(\( ( 1 - T + T^{2} )^{3} \))(\( ( 1 + T + T^{2} )^{4} \))(\( ( 1 - T + T^{2} )^{4} \))
$3$ (\( ( 1 + 3 T^{2} )( 1 + 3 T + 3 T^{2} ) \))(\( 1 - T - 2 T^{2} - 3 T^{3} + 9 T^{4} \))(\( 1 - T - 2 T^{2} - 3 T^{3} + 9 T^{4} \))(\( 1 - T - 2 T^{2} - 3 T^{3} + 9 T^{4} \))(\( 1 + T - 2 T^{2} + 3 T^{3} + 9 T^{4} \))(\( 1 + T - 2 T^{2} + 3 T^{3} + 9 T^{4} \))(\( ( 1 - 3 T + 3 T^{2} )( 1 + 3 T^{2} ) \))(\( 1 - 2 T + 2 T^{2} + 8 T^{3} - 17 T^{4} + 24 T^{5} + 18 T^{6} - 54 T^{7} + 81 T^{8} \))(\( 1 + 2 T + 2 T^{2} - 8 T^{3} - 17 T^{4} - 24 T^{5} + 18 T^{6} + 54 T^{7} + 81 T^{8} \))(\( 1 + T^{2} - 8 T^{4} + 9 T^{6} + 81 T^{8} \))(\( 1 - 6 T^{2} + 2 T^{3} + 18 T^{4} - 6 T^{5} - 53 T^{6} - 18 T^{7} + 162 T^{8} + 54 T^{9} - 486 T^{10} + 729 T^{12} \))(\( 1 - 6 T^{2} - 2 T^{3} + 18 T^{4} + 6 T^{5} - 53 T^{6} + 18 T^{7} + 162 T^{8} - 54 T^{9} - 486 T^{10} + 729 T^{12} \))(\( 1 + 2 T - 2 T^{2} + 10 T^{4} - 2 T^{5} + 24 T^{6} + 30 T^{7} - 101 T^{8} + 90 T^{9} + 216 T^{10} - 54 T^{11} + 810 T^{12} - 1458 T^{14} + 4374 T^{15} + 6561 T^{16} \))(\( 1 - 2 T - 2 T^{2} + 10 T^{4} + 2 T^{5} + 24 T^{6} - 30 T^{7} - 101 T^{8} - 90 T^{9} + 216 T^{10} + 54 T^{11} + 810 T^{12} - 1458 T^{14} - 4374 T^{15} + 6561 T^{16} \))
$5$ (\( 1 + 2 T - T^{2} + 10 T^{3} + 25 T^{4} \))(\( 1 - 5 T^{2} + 25 T^{4} \))(\( 1 - 5 T^{2} + 25 T^{4} \))(\( 1 - 4 T + 11 T^{2} - 20 T^{3} + 25 T^{4} \))(\( 1 - 5 T^{2} + 25 T^{4} \))(\( 1 - 4 T + 11 T^{2} - 20 T^{3} + 25 T^{4} \))(\( 1 + 2 T - T^{2} + 10 T^{3} + 25 T^{4} \))(\( 1 - 5 T + 10 T^{2} - 25 T^{3} + 75 T^{4} - 125 T^{5} + 250 T^{6} - 625 T^{7} + 625 T^{8} \))(\( 1 - 5 T + 10 T^{2} - 25 T^{3} + 75 T^{4} - 125 T^{5} + 250 T^{6} - 625 T^{7} + 625 T^{8} \))(\( 1 + 2 T - 12 T^{3} - 29 T^{4} - 60 T^{5} + 250 T^{7} + 625 T^{8} \))(\( ( 1 + 2 T - T^{2} + 10 T^{3} + 25 T^{4} )^{3} \))(\( ( 1 + 2 T - T^{2} + 10 T^{3} + 25 T^{4} )^{3} \))(\( 1 - 2 T - 5 T^{2} - 22 T^{3} + 64 T^{4} + 112 T^{5} + 285 T^{6} - 838 T^{7} - 1229 T^{8} - 4190 T^{9} + 7125 T^{10} + 14000 T^{11} + 40000 T^{12} - 68750 T^{13} - 78125 T^{14} - 156250 T^{15} + 390625 T^{16} \))(\( 1 - 2 T - 5 T^{2} - 22 T^{3} + 64 T^{4} + 112 T^{5} + 285 T^{6} - 838 T^{7} - 1229 T^{8} - 4190 T^{9} + 7125 T^{10} + 14000 T^{11} + 40000 T^{12} - 68750 T^{13} - 78125 T^{14} - 156250 T^{15} + 390625 T^{16} \))
$7$ (\( ( 1 + 3 T + 7 T^{2} )^{2} \))(\( ( 1 + 4 T + 7 T^{2} )^{2} \))(\( ( 1 + T + 7 T^{2} )^{2} \))(\( ( 1 - 3 T + 7 T^{2} )^{2} \))(\( ( 1 + T + 7 T^{2} )^{2} \))(\( ( 1 - 3 T + 7 T^{2} )^{2} \))(\( ( 1 + 3 T + 7 T^{2} )^{2} \))(\( ( 1 + 2 T + 10 T^{2} + 14 T^{3} + 49 T^{4} )^{2} \))(\( ( 1 + 2 T + 10 T^{2} + 14 T^{3} + 49 T^{4} )^{2} \))(\( ( 1 - 2 T + 8 T^{2} - 14 T^{3} + 49 T^{4} )^{2} \))(\( ( 1 - 6 T + 21 T^{2} - 60 T^{3} + 147 T^{4} - 294 T^{5} + 343 T^{6} )^{2} \))(\( ( 1 - 6 T + 21 T^{2} - 60 T^{3} + 147 T^{4} - 294 T^{5} + 343 T^{6} )^{2} \))(\( ( 1 + 2 T + 22 T^{2} + 30 T^{3} + 206 T^{4} + 210 T^{5} + 1078 T^{6} + 686 T^{7} + 2401 T^{8} )^{2} \))(\( ( 1 + 2 T + 22 T^{2} + 30 T^{3} + 206 T^{4} + 210 T^{5} + 1078 T^{6} + 686 T^{7} + 2401 T^{8} )^{2} \))
$11$ (\( ( 1 + 2 T + 11 T^{2} )^{2} \))(\( ( 1 - 3 T + 11 T^{2} )^{2} \))(\( ( 1 + 6 T + 11 T^{2} )^{2} \))(\( ( 1 - 2 T + 11 T^{2} )^{2} \))(\( ( 1 + 6 T + 11 T^{2} )^{2} \))(\( ( 1 - 2 T + 11 T^{2} )^{2} \))(\( ( 1 + 2 T + 11 T^{2} )^{2} \))(\( ( 1 + 2 T + 18 T^{2} + 22 T^{3} + 121 T^{4} )^{2} \))(\( ( 1 + 2 T + 18 T^{2} + 22 T^{3} + 121 T^{4} )^{2} \))(\( ( 1 + 4 T + 19 T^{2} + 44 T^{3} + 121 T^{4} )^{2} \))(\( ( 1 - 6 T + 36 T^{2} - 113 T^{3} + 396 T^{4} - 726 T^{5} + 1331 T^{6} )^{2} \))(\( ( 1 - 6 T + 36 T^{2} - 113 T^{3} + 396 T^{4} - 726 T^{5} + 1331 T^{6} )^{2} \))(\( ( 1 - 2 T + 18 T^{2} - 54 T^{3} + 230 T^{4} - 594 T^{5} + 2178 T^{6} - 2662 T^{7} + 14641 T^{8} )^{2} \))(\( ( 1 - 2 T + 18 T^{2} - 54 T^{3} + 230 T^{4} - 594 T^{5} + 2178 T^{6} - 2662 T^{7} + 14641 T^{8} )^{2} \))
$13$ (\( 1 - 3 T - 4 T^{2} - 39 T^{3} + 169 T^{4} \))(\( ( 1 - 7 T + 13 T^{2} )( 1 + 5 T + 13 T^{2} ) \))(\( ( 1 - 7 T + 13 T^{2} )( 1 + 2 T + 13 T^{2} ) \))(\( 1 - T - 12 T^{2} - 13 T^{3} + 169 T^{4} \))(\( ( 1 - 2 T + 13 T^{2} )( 1 + 7 T + 13 T^{2} ) \))(\( 1 + T - 12 T^{2} + 13 T^{3} + 169 T^{4} \))(\( 1 + 3 T - 4 T^{2} + 39 T^{3} + 169 T^{4} \))(\( 1 - 5 T - 6 T^{2} - 25 T^{3} + 467 T^{4} - 325 T^{5} - 1014 T^{6} - 10985 T^{7} + 28561 T^{8} \))(\( 1 + 5 T - 6 T^{2} + 25 T^{3} + 467 T^{4} + 325 T^{5} - 1014 T^{6} + 10985 T^{7} + 28561 T^{8} \))(\( ( 1 - 7 T + 13 T^{2} )^{2}( 1 + 5 T + 13 T^{2} )^{2} \))(\( 1 - 6 T - 3 T^{2} + 62 T^{3} + 126 T^{4} - 702 T^{5} + 441 T^{6} - 9126 T^{7} + 21294 T^{8} + 136214 T^{9} - 85683 T^{10} - 2227758 T^{11} + 4826809 T^{12} \))(\( 1 + 6 T - 3 T^{2} - 62 T^{3} + 126 T^{4} + 702 T^{5} + 441 T^{6} + 9126 T^{7} + 21294 T^{8} - 136214 T^{9} - 85683 T^{10} + 2227758 T^{11} + 4826809 T^{12} \))(\( 1 + 18 T + 163 T^{2} + 1030 T^{3} + 5200 T^{4} + 21792 T^{5} + 78869 T^{6} + 270590 T^{7} + 954099 T^{8} + 3517670 T^{9} + 13328861 T^{10} + 47877024 T^{11} + 148517200 T^{12} + 382431790 T^{13} + 786769867 T^{14} + 1129473306 T^{15} + 815730721 T^{16} \))(\( 1 - 18 T + 163 T^{2} - 1030 T^{3} + 5200 T^{4} - 21792 T^{5} + 78869 T^{6} - 270590 T^{7} + 954099 T^{8} - 3517670 T^{9} + 13328861 T^{10} - 47877024 T^{11} + 148517200 T^{12} - 382431790 T^{13} + 786769867 T^{14} - 1129473306 T^{15} + 815730721 T^{16} \))
$17$ (\( 1 - T - 16 T^{2} - 17 T^{3} + 289 T^{4} \))(\( 1 - 6 T + 19 T^{2} - 102 T^{3} + 289 T^{4} \))(\( 1 + 3 T - 8 T^{2} + 51 T^{3} + 289 T^{4} \))(\( 1 + 3 T - 8 T^{2} + 51 T^{3} + 289 T^{4} \))(\( 1 + 3 T - 8 T^{2} + 51 T^{3} + 289 T^{4} \))(\( 1 + 3 T - 8 T^{2} + 51 T^{3} + 289 T^{4} \))(\( 1 - T - 16 T^{2} - 17 T^{3} + 289 T^{4} \))(\( 1 - 9 T + 28 T^{2} - 171 T^{3} + 1143 T^{4} - 2907 T^{5} + 8092 T^{6} - 44217 T^{7} + 83521 T^{8} \))(\( 1 - 9 T + 28 T^{2} - 171 T^{3} + 1143 T^{4} - 2907 T^{5} + 8092 T^{6} - 44217 T^{7} + 83521 T^{8} \))(\( ( 1 - 17 T^{2} + 289 T^{4} )^{2} \))(\( 1 + 3 T + 3 T^{2} + 36 T^{3} - 39 T^{4} + 201 T^{5} + 6694 T^{6} + 3417 T^{7} - 11271 T^{8} + 176868 T^{9} + 250563 T^{10} + 4259571 T^{11} + 24137569 T^{12} \))(\( 1 + 3 T + 3 T^{2} + 36 T^{3} - 39 T^{4} + 201 T^{5} + 6694 T^{6} + 3417 T^{7} - 11271 T^{8} + 176868 T^{9} + 250563 T^{10} + 4259571 T^{11} + 24137569 T^{12} \))(\( 1 + 6 T - 23 T^{2} - 110 T^{3} + 680 T^{4} + 724 T^{5} - 18869 T^{6} - 6720 T^{7} + 334479 T^{8} - 114240 T^{9} - 5453141 T^{10} + 3557012 T^{11} + 56794280 T^{12} - 156184270 T^{13} - 555164087 T^{14} + 2462032038 T^{15} + 6975757441 T^{16} \))(\( 1 + 6 T - 23 T^{2} - 110 T^{3} + 680 T^{4} + 724 T^{5} - 18869 T^{6} - 6720 T^{7} + 334479 T^{8} - 114240 T^{9} - 5453141 T^{10} + 3557012 T^{11} + 56794280 T^{12} - 156184270 T^{13} - 555164087 T^{14} + 2462032038 T^{15} + 6975757441 T^{16} \))
$19$ 1
$23$ (\( 1 + 5 T + 2 T^{2} + 115 T^{3} + 529 T^{4} \))(\( 1 - 6 T + 13 T^{2} - 138 T^{3} + 529 T^{4} \))(\( 1 + 3 T - 14 T^{2} + 69 T^{3} + 529 T^{4} \))(\( 1 - T - 22 T^{2} - 23 T^{3} + 529 T^{4} \))(\( 1 + 3 T - 14 T^{2} + 69 T^{3} + 529 T^{4} \))(\( 1 - T - 22 T^{2} - 23 T^{3} + 529 T^{4} \))(\( 1 + 5 T + 2 T^{2} + 115 T^{3} + 529 T^{4} \))(\( 1 + 6 T - 14 T^{2} + 24 T^{3} + 1143 T^{4} + 552 T^{5} - 7406 T^{6} + 73002 T^{7} + 279841 T^{8} \))(\( 1 + 6 T - 14 T^{2} + 24 T^{3} + 1143 T^{4} + 552 T^{5} - 7406 T^{6} + 73002 T^{7} + 279841 T^{8} \))(\( 1 + 2 T - 36 T^{2} - 12 T^{3} + 979 T^{4} - 276 T^{5} - 19044 T^{6} + 24334 T^{7} + 279841 T^{8} \))(\( 1 - 57 T^{2} - 16 T^{3} + 1938 T^{4} + 456 T^{5} - 50329 T^{6} + 10488 T^{7} + 1025202 T^{8} - 194672 T^{9} - 15950937 T^{10} + 148035889 T^{12} \))(\( 1 - 57 T^{2} - 16 T^{3} + 1938 T^{4} + 456 T^{5} - 50329 T^{6} + 10488 T^{7} + 1025202 T^{8} - 194672 T^{9} - 15950937 T^{10} + 148035889 T^{12} \))(\( 1 - 10 T + 58 T^{2} - 360 T^{3} + 1810 T^{4} - 8870 T^{5} + 52968 T^{6} - 275510 T^{7} + 1319419 T^{8} - 6336730 T^{9} + 28020072 T^{10} - 107921290 T^{11} + 506512210 T^{12} - 2317083480 T^{13} + 8586081562 T^{14} - 34048254470 T^{15} + 78310985281 T^{16} \))(\( 1 - 10 T + 58 T^{2} - 360 T^{3} + 1810 T^{4} - 8870 T^{5} + 52968 T^{6} - 275510 T^{7} + 1319419 T^{8} - 6336730 T^{9} + 28020072 T^{10} - 107921290 T^{11} + 506512210 T^{12} - 2317083480 T^{13} + 8586081562 T^{14} - 34048254470 T^{15} + 78310985281 T^{16} \))
$29$ (\( 1 - 3 T - 20 T^{2} - 87 T^{3} + 841 T^{4} \))(\( 1 - 29 T^{2} + 841 T^{4} \))(\( 1 - 9 T + 52 T^{2} - 261 T^{3} + 841 T^{4} \))(\( 1 - 5 T - 4 T^{2} - 145 T^{3} + 841 T^{4} \))(\( 1 + 9 T + 52 T^{2} + 261 T^{3} + 841 T^{4} \))(\( 1 + 5 T - 4 T^{2} + 145 T^{3} + 841 T^{4} \))(\( 1 + 3 T - 20 T^{2} + 87 T^{3} + 841 T^{4} \))(\( 1 + 7 T + 10 T^{2} - 133 T^{3} - 741 T^{4} - 3857 T^{5} + 8410 T^{6} + 170723 T^{7} + 707281 T^{8} \))(\( 1 - 7 T + 10 T^{2} + 133 T^{3} - 741 T^{4} + 3857 T^{5} + 8410 T^{6} - 170723 T^{7} + 707281 T^{8} \))(\( 1 + 2 T - 48 T^{2} - 12 T^{3} + 1747 T^{4} - 348 T^{5} - 40368 T^{6} + 48778 T^{7} + 707281 T^{8} \))(\( 1 + 12 T + 45 T^{2} + 100 T^{3} + 402 T^{4} - 4956 T^{5} - 60619 T^{6} - 143724 T^{7} + 338082 T^{8} + 2438900 T^{9} + 31827645 T^{10} + 246133788 T^{11} + 594823321 T^{12} \))(\( 1 - 12 T + 45 T^{2} - 100 T^{3} + 402 T^{4} + 4956 T^{5} - 60619 T^{6} + 143724 T^{7} + 338082 T^{8} - 2438900 T^{9} + 31827645 T^{10} - 246133788 T^{11} + 594823321 T^{12} \))(\( 1 - 2 T - 81 T^{2} - 6 T^{3} + 3832 T^{4} + 3568 T^{5} - 129527 T^{6} - 52266 T^{7} + 3643051 T^{8} - 1515714 T^{9} - 108932207 T^{10} + 87019952 T^{11} + 2710300792 T^{12} - 123066894 T^{13} - 48180689001 T^{14} - 34499752618 T^{15} + 500246412961 T^{16} \))(\( 1 + 2 T - 81 T^{2} + 6 T^{3} + 3832 T^{4} - 3568 T^{5} - 129527 T^{6} + 52266 T^{7} + 3643051 T^{8} + 1515714 T^{9} - 108932207 T^{10} - 87019952 T^{11} + 2710300792 T^{12} + 123066894 T^{13} - 48180689001 T^{14} + 34499752618 T^{15} + 500246412961 T^{16} \))
$31$ (\( ( 1 + 6 T + 31 T^{2} )^{2} \))(\( ( 1 + 2 T + 31 T^{2} )^{2} \))(\( ( 1 - 4 T + 31 T^{2} )^{2} \))(\( ( 1 + 8 T + 31 T^{2} )^{2} \))(\( ( 1 + 4 T + 31 T^{2} )^{2} \))(\( ( 1 - 8 T + 31 T^{2} )^{2} \))(\( ( 1 - 6 T + 31 T^{2} )^{2} \))(\( ( 1 + 2 T + 58 T^{2} + 62 T^{3} + 961 T^{4} )^{2} \))(\( ( 1 - 2 T + 58 T^{2} - 62 T^{3} + 961 T^{4} )^{2} \))(\( ( 1 - 6 T + 64 T^{2} - 186 T^{3} + 961 T^{4} )^{2} \))(\( ( 1 + 6 T + 69 T^{2} + 380 T^{3} + 2139 T^{4} + 5766 T^{5} + 29791 T^{6} )^{2} \))(\( ( 1 - 6 T + 69 T^{2} - 380 T^{3} + 2139 T^{4} - 5766 T^{5} + 29791 T^{6} )^{2} \))(\( ( 1 - 26 T + 370 T^{2} - 3414 T^{3} + 22454 T^{4} - 105834 T^{5} + 355570 T^{6} - 774566 T^{7} + 923521 T^{8} )^{2} \))(\( ( 1 + 26 T + 370 T^{2} + 3414 T^{3} + 22454 T^{4} + 105834 T^{5} + 355570 T^{6} + 774566 T^{7} + 923521 T^{8} )^{2} \))
$37$ (\( ( 1 - 6 T + 37 T^{2} )^{2} \))(\( ( 1 - 10 T + 37 T^{2} )^{2} \))(\( ( 1 + 2 T + 37 T^{2} )^{2} \))(\( ( 1 + 2 T + 37 T^{2} )^{2} \))(\( ( 1 - 2 T + 37 T^{2} )^{2} \))(\( ( 1 - 2 T + 37 T^{2} )^{2} \))(\( ( 1 + 6 T + 37 T^{2} )^{2} \))(\( ( 1 + 19 T + 163 T^{2} + 703 T^{3} + 1369 T^{4} )^{2} \))(\( ( 1 - 19 T + 163 T^{2} - 703 T^{3} + 1369 T^{4} )^{2} \))(\( ( 1 + 6 T + 76 T^{2} + 222 T^{3} + 1369 T^{4} )^{2} \))(\( ( 1 + 6 T + 87 T^{2} + 308 T^{3} + 3219 T^{4} + 8214 T^{5} + 50653 T^{6} )^{2} \))(\( ( 1 - 6 T + 87 T^{2} - 308 T^{3} + 3219 T^{4} - 8214 T^{5} + 50653 T^{6} )^{2} \))(\( ( 1 - 4 T + 129 T^{2} - 398 T^{3} + 6789 T^{4} - 14726 T^{5} + 176601 T^{6} - 202612 T^{7} + 1874161 T^{8} )^{2} \))(\( ( 1 + 4 T + 129 T^{2} + 398 T^{3} + 6789 T^{4} + 14726 T^{5} + 176601 T^{6} + 202612 T^{7} + 1874161 T^{8} )^{2} \))
$41$ (\( 1 + 12 T + 103 T^{2} + 492 T^{3} + 1681 T^{4} \))(\( 1 - 9 T + 40 T^{2} - 369 T^{3} + 1681 T^{4} \))(\( 1 - 41 T^{2} + 1681 T^{4} \))(\( 1 - 8 T + 23 T^{2} - 328 T^{3} + 1681 T^{4} \))(\( 1 - 41 T^{2} + 1681 T^{4} \))(\( 1 + 8 T + 23 T^{2} + 328 T^{3} + 1681 T^{4} \))(\( 1 - 12 T + 103 T^{2} - 492 T^{3} + 1681 T^{4} \))(\( 1 + 5 T - 52 T^{2} - 25 T^{3} + 3223 T^{4} - 1025 T^{5} - 87412 T^{6} + 344605 T^{7} + 2825761 T^{8} \))(\( 1 - 5 T - 52 T^{2} + 25 T^{3} + 3223 T^{4} + 1025 T^{5} - 87412 T^{6} - 344605 T^{7} + 2825761 T^{8} \))(\( 1 - 10 T + 21 T^{2} + 30 T^{3} + 460 T^{4} + 1230 T^{5} + 35301 T^{6} - 689210 T^{7} + 2825761 T^{8} \))(\( 1 - 120 T^{2} - 2 T^{3} + 9480 T^{4} + 120 T^{5} - 452557 T^{6} + 4920 T^{7} + 15935880 T^{8} - 137842 T^{9} - 339091320 T^{10} + 4750104241 T^{12} \))(\( 1 - 120 T^{2} + 2 T^{3} + 9480 T^{4} - 120 T^{5} - 452557 T^{6} - 4920 T^{7} + 15935880 T^{8} + 137842 T^{9} - 339091320 T^{10} + 4750104241 T^{12} \))(\( 1 - 12 T - 39 T^{2} + 452 T^{3} + 6316 T^{4} - 25736 T^{5} - 348341 T^{6} + 408076 T^{7} + 16207687 T^{8} + 16731116 T^{9} - 585561221 T^{10} - 1773750856 T^{11} + 17847506476 T^{12} + 52367002852 T^{13} - 185254065399 T^{14} - 2337051286572 T^{15} + 7984925229121 T^{16} \))(\( 1 + 12 T - 39 T^{2} - 452 T^{3} + 6316 T^{4} + 25736 T^{5} - 348341 T^{6} - 408076 T^{7} + 16207687 T^{8} - 16731116 T^{9} - 585561221 T^{10} + 1773750856 T^{11} + 17847506476 T^{12} - 52367002852 T^{13} - 185254065399 T^{14} + 2337051286572 T^{15} + 7984925229121 T^{16} \))
$43$ (\( 1 - 10 T + 57 T^{2} - 430 T^{3} + 1849 T^{4} \))(\( 1 - 4 T - 27 T^{2} - 172 T^{3} + 1849 T^{4} \))(\( ( 1 - 5 T + 43 T^{2} )( 1 + 13 T + 43 T^{2} ) \))(\( 1 + 4 T - 27 T^{2} + 172 T^{3} + 1849 T^{4} \))(\( ( 1 - 5 T + 43 T^{2} )( 1 + 13 T + 43 T^{2} ) \))(\( 1 + 4 T - 27 T^{2} + 172 T^{3} + 1849 T^{4} \))(\( 1 - 10 T + 57 T^{2} - 430 T^{3} + 1849 T^{4} \))(\( 1 - 14 T + 66 T^{2} - 616 T^{3} + 6623 T^{4} - 26488 T^{5} + 122034 T^{6} - 1113098 T^{7} + 3418801 T^{8} \))(\( 1 - 14 T + 66 T^{2} - 616 T^{3} + 6623 T^{4} - 26488 T^{5} + 122034 T^{6} - 1113098 T^{7} + 3418801 T^{8} \))(\( 1 + 12 T + 50 T^{2} + 96 T^{3} + 795 T^{4} + 4128 T^{5} + 92450 T^{6} + 954084 T^{7} + 3418801 T^{8} \))(\( 1 + 9 T - 63 T^{2} - 218 T^{3} + 7731 T^{4} + 14193 T^{5} - 309354 T^{6} + 610299 T^{7} + 14294619 T^{8} - 17332526 T^{9} - 215384463 T^{10} + 1323075987 T^{11} + 6321363049 T^{12} \))(\( 1 + 9 T - 63 T^{2} - 218 T^{3} + 7731 T^{4} + 14193 T^{5} - 309354 T^{6} + 610299 T^{7} + 14294619 T^{8} - 17332526 T^{9} - 215384463 T^{10} + 1323075987 T^{11} + 6321363049 T^{12} \))(\( 1 - 10 T - 82 T^{2} + 560 T^{3} + 9370 T^{4} - 30670 T^{5} - 579992 T^{6} + 359370 T^{7} + 30861339 T^{8} + 15452910 T^{9} - 1072405208 T^{10} - 2438479690 T^{11} + 32034165370 T^{12} + 82324728080 T^{13} - 518351770018 T^{14} - 2718186111070 T^{15} + 11688200277601 T^{16} \))(\( 1 - 10 T - 82 T^{2} + 560 T^{3} + 9370 T^{4} - 30670 T^{5} - 579992 T^{6} + 359370 T^{7} + 30861339 T^{8} + 15452910 T^{9} - 1072405208 T^{10} - 2438479690 T^{11} + 32034165370 T^{12} + 82324728080 T^{13} - 518351770018 T^{14} - 2718186111070 T^{15} + 11688200277601 T^{16} \))
$47$ (\( 1 - 8 T + 17 T^{2} - 376 T^{3} + 2209 T^{4} \))(\( 1 - 47 T^{2} + 2209 T^{4} \))(\( 1 - 47 T^{2} + 2209 T^{4} \))(\( 1 + 8 T + 17 T^{2} + 376 T^{3} + 2209 T^{4} \))(\( 1 - 47 T^{2} + 2209 T^{4} \))(\( 1 + 8 T + 17 T^{2} + 376 T^{3} + 2209 T^{4} \))(\( 1 - 8 T + 17 T^{2} - 376 T^{3} + 2209 T^{4} \))(\( 1 - 74 T^{2} + 3267 T^{4} - 163466 T^{6} + 4879681 T^{8} \))(\( 1 - 74 T^{2} + 3267 T^{4} - 163466 T^{6} + 4879681 T^{8} \))(\( 1 + 14 T + 60 T^{2} + 588 T^{3} + 7075 T^{4} + 27636 T^{5} + 132540 T^{6} + 1453522 T^{7} + 4879681 T^{8} \))(\( 1 - 6 T - 33 T^{2} + 122 T^{3} - 90 T^{4} + 7170 T^{5} - 23821 T^{6} + 336990 T^{7} - 198810 T^{8} + 12666406 T^{9} - 161029473 T^{10} - 1376070042 T^{11} + 10779215329 T^{12} \))(\( 1 - 6 T - 33 T^{2} + 122 T^{3} - 90 T^{4} + 7170 T^{5} - 23821 T^{6} + 336990 T^{7} - 198810 T^{8} + 12666406 T^{9} - 161029473 T^{10} - 1376070042 T^{11} + 10779215329 T^{12} \))(\( 1 - 12 T - 48 T^{2} + 856 T^{3} + 4198 T^{4} - 48676 T^{5} - 178016 T^{6} + 524948 T^{7} + 14112403 T^{8} + 24672556 T^{9} - 393237344 T^{10} - 5053688348 T^{11} + 20484900838 T^{12} + 196319325992 T^{13} - 517402335792 T^{14} - 6079477445556 T^{15} + 23811286661761 T^{16} \))(\( 1 - 12 T - 48 T^{2} + 856 T^{3} + 4198 T^{4} - 48676 T^{5} - 178016 T^{6} + 524948 T^{7} + 14112403 T^{8} + 24672556 T^{9} - 393237344 T^{10} - 5053688348 T^{11} + 20484900838 T^{12} + 196319325992 T^{13} - 517402335792 T^{14} - 6079477445556 T^{15} + 23811286661761 T^{16} \))
$53$ (\( 1 - 3 T - 44 T^{2} - 159 T^{3} + 2809 T^{4} \))(\( 1 - 6 T - 17 T^{2} - 318 T^{3} + 2809 T^{4} \))(\( 1 + 3 T - 44 T^{2} + 159 T^{3} + 2809 T^{4} \))(\( 1 - T - 52 T^{2} - 53 T^{3} + 2809 T^{4} \))(\( 1 - 3 T - 44 T^{2} - 159 T^{3} + 2809 T^{4} \))(\( 1 + T - 52 T^{2} + 53 T^{3} + 2809 T^{4} \))(\( 1 + 3 T - 44 T^{2} + 159 T^{3} + 2809 T^{4} \))(\( 1 + T - 74 T^{2} - 31 T^{3} + 2763 T^{4} - 1643 T^{5} - 207866 T^{6} + 148877 T^{7} + 7890481 T^{8} \))(\( 1 - T - 74 T^{2} + 31 T^{3} + 2763 T^{4} + 1643 T^{5} - 207866 T^{6} - 148877 T^{7} + 7890481 T^{8} \))(\( 1 + 4 T + 18 T^{2} - 432 T^{3} - 3653 T^{4} - 22896 T^{5} + 50562 T^{6} + 595508 T^{7} + 7890481 T^{8} \))(\( 1 - 75 T^{2} + 272 T^{3} + 1650 T^{4} - 10200 T^{5} + 18125 T^{6} - 540600 T^{7} + 4634850 T^{8} + 40494544 T^{9} - 591786075 T^{10} + 22164361129 T^{12} \))(\( 1 - 75 T^{2} - 272 T^{3} + 1650 T^{4} + 10200 T^{5} + 18125 T^{6} + 540600 T^{7} + 4634850 T^{8} - 40494544 T^{9} - 591786075 T^{10} + 22164361129 T^{12} \))(\( 1 + 8 T - 117 T^{2} - 900 T^{3} + 9620 T^{4} + 54732 T^{5} - 591211 T^{6} - 1124590 T^{7} + 35126499 T^{8} - 59603270 T^{9} - 1660711699 T^{10} + 8148335964 T^{11} + 75906427220 T^{12} - 376375943700 T^{13} - 2593230252093 T^{14} + 9397689118696 T^{15} + 62259690411361 T^{16} \))(\( 1 - 8 T - 117 T^{2} + 900 T^{3} + 9620 T^{4} - 54732 T^{5} - 591211 T^{6} + 1124590 T^{7} + 35126499 T^{8} + 59603270 T^{9} - 1660711699 T^{10} - 8148335964 T^{11} + 75906427220 T^{12} + 376375943700 T^{13} - 2593230252093 T^{14} - 9397689118696 T^{15} + 62259690411361 T^{16} \))
$59$ (\( 1 + 3 T - 50 T^{2} + 177 T^{3} + 3481 T^{4} \))(\( 1 + 9 T + 22 T^{2} + 531 T^{3} + 3481 T^{4} \))(\( 1 - 9 T + 22 T^{2} - 531 T^{3} + 3481 T^{4} \))(\( 1 + 15 T + 166 T^{2} + 885 T^{3} + 3481 T^{4} \))(\( 1 + 9 T + 22 T^{2} + 531 T^{3} + 3481 T^{4} \))(\( 1 - 15 T + 166 T^{2} - 885 T^{3} + 3481 T^{4} \))(\( 1 - 3 T - 50 T^{2} - 177 T^{3} + 3481 T^{4} \))(\( 1 - 8 T - 50 T^{2} + 32 T^{3} + 5739 T^{4} + 1888 T^{5} - 174050 T^{6} - 1643032 T^{7} + 12117361 T^{8} \))(\( 1 + 8 T - 50 T^{2} - 32 T^{3} + 5739 T^{4} - 1888 T^{5} - 174050 T^{6} + 1643032 T^{7} + 12117361 T^{8} \))(\( 1 - 55 T^{2} - 456 T^{4} - 191455 T^{6} + 12117361 T^{8} \))(\( 1 - 6 T - 132 T^{2} + 414 T^{3} + 14046 T^{4} - 21390 T^{5} - 874253 T^{6} - 1262010 T^{7} + 48894126 T^{8} + 85026906 T^{9} - 1599491652 T^{10} - 4289545794 T^{11} + 42180533641 T^{12} \))(\( 1 + 6 T - 132 T^{2} - 414 T^{3} + 14046 T^{4} + 21390 T^{5} - 874253 T^{6} + 1262010 T^{7} + 48894126 T^{8} - 85026906 T^{9} - 1599491652 T^{10} + 4289545794 T^{11} + 42180533641 T^{12} \))(\( 1 - 8 T - 176 T^{2} + 736 T^{3} + 25782 T^{4} - 57968 T^{5} - 2278912 T^{6} + 1136776 T^{7} + 158799731 T^{8} + 67069784 T^{9} - 7932892672 T^{10} - 11905409872 T^{11} + 312409801302 T^{12} + 526184284064 T^{13} - 7423773920816 T^{14} - 19909211878552 T^{15} + 146830437604321 T^{16} \))(\( 1 + 8 T - 176 T^{2} - 736 T^{3} + 25782 T^{4} + 57968 T^{5} - 2278912 T^{6} - 1136776 T^{7} + 158799731 T^{8} - 67069784 T^{9} - 7932892672 T^{10} + 11905409872 T^{11} + 312409801302 T^{12} - 526184284064 T^{13} - 7423773920816 T^{14} + 19909211878552 T^{15} + 146830437604321 T^{16} \))
$61$ (\( 1 - 61 T^{2} + 3721 T^{4} \))(\( 1 - 4 T - 45 T^{2} - 244 T^{3} + 3721 T^{4} \))(\( 1 - 10 T + 39 T^{2} - 610 T^{3} + 3721 T^{4} \))(\( 1 + 2 T - 57 T^{2} + 122 T^{3} + 3721 T^{4} \))(\( 1 - 10 T + 39 T^{2} - 610 T^{3} + 3721 T^{4} \))(\( 1 + 2 T - 57 T^{2} + 122 T^{3} + 3721 T^{4} \))(\( 1 - 61 T^{2} + 3721 T^{4} \))(\( 1 + 5 T - 102 T^{2} + 25 T^{3} + 10883 T^{4} + 1525 T^{5} - 379542 T^{6} + 1134905 T^{7} + 13845841 T^{8} \))(\( 1 + 5 T - 102 T^{2} + 25 T^{3} + 10883 T^{4} + 1525 T^{5} - 379542 T^{6} + 1134905 T^{7} + 13845841 T^{8} \))(\( 1 - 14 T + 88 T^{2} + 196 T^{3} - 4013 T^{4} + 11956 T^{5} + 327448 T^{6} - 3177734 T^{7} + 13845841 T^{8} \))(\( 1 + 18 T + 45 T^{2} + 326 T^{3} + 18558 T^{4} + 109386 T^{5} + 13161 T^{6} + 6672546 T^{7} + 69054318 T^{8} + 73995806 T^{9} + 623062845 T^{10} + 15202733418 T^{11} + 51520374361 T^{12} \))(\( 1 + 18 T + 45 T^{2} + 326 T^{3} + 18558 T^{4} + 109386 T^{5} + 13161 T^{6} + 6672546 T^{7} + 69054318 T^{8} + 73995806 T^{9} + 623062845 T^{10} + 15202733418 T^{11} + 51520374361 T^{12} \))(\( 1 - 129 T^{2} + 300 T^{3} + 7320 T^{4} - 28500 T^{5} - 219891 T^{6} + 962550 T^{7} + 9741539 T^{8} + 58715550 T^{9} - 818214411 T^{10} - 6468958500 T^{11} + 101351556120 T^{12} + 253378890300 T^{13} - 6646128292569 T^{14} + 191707312997281 T^{16} \))(\( 1 - 129 T^{2} + 300 T^{3} + 7320 T^{4} - 28500 T^{5} - 219891 T^{6} + 962550 T^{7} + 9741539 T^{8} + 58715550 T^{9} - 818214411 T^{10} - 6468958500 T^{11} + 101351556120 T^{12} + 253378890300 T^{13} - 6646128292569 T^{14} + 191707312997281 T^{16} \))
$67$ (\( 1 + 15 T + 158 T^{2} + 1005 T^{3} + 4489 T^{4} \))(\( 1 + 7 T - 18 T^{2} + 469 T^{3} + 4489 T^{4} \))(\( ( 1 - 16 T + 67 T^{2} )( 1 + 11 T + 67 T^{2} ) \))(\( 1 + 3 T - 58 T^{2} + 201 T^{3} + 4489 T^{4} \))(\( ( 1 - 11 T + 67 T^{2} )( 1 + 16 T + 67 T^{2} ) \))(\( 1 - 3 T - 58 T^{2} - 201 T^{3} + 4489 T^{4} \))(\( 1 - 15 T + 158 T^{2} - 1005 T^{3} + 4489 T^{4} \))(\( 1 - 10 T - 14 T^{2} + 200 T^{3} + 1807 T^{4} + 13400 T^{5} - 62846 T^{6} - 3007630 T^{7} + 20151121 T^{8} \))(\( 1 + 10 T - 14 T^{2} - 200 T^{3} + 1807 T^{4} - 13400 T^{5} - 62846 T^{6} + 3007630 T^{7} + 20151121 T^{8} \))(\( 1 + 4 T - 115 T^{2} - 12 T^{3} + 11600 T^{4} - 804 T^{5} - 516235 T^{6} + 1203052 T^{7} + 20151121 T^{8} \))(\( 1 - 18 T + 42 T^{2} - 90 T^{3} + 15756 T^{4} - 103644 T^{5} + 34031 T^{6} - 6944148 T^{7} + 70728684 T^{8} - 27068670 T^{9} + 846347082 T^{10} - 24302251926 T^{11} + 90458382169 T^{12} \))(\( 1 + 18 T + 42 T^{2} + 90 T^{3} + 15756 T^{4} + 103644 T^{5} + 34031 T^{6} + 6944148 T^{7} + 70728684 T^{8} + 27068670 T^{9} + 846347082 T^{10} + 24302251926 T^{11} + 90458382169 T^{12} \))(\( 1 + 10 T - 138 T^{2} - 1360 T^{3} + 15050 T^{4} + 112470 T^{5} - 1068008 T^{6} - 3134810 T^{7} + 75569499 T^{8} - 210032270 T^{9} - 4794287912 T^{10} + 33826814610 T^{11} + 303274371050 T^{12} - 1836170145520 T^{13} - 12483256739322 T^{14} + 60607116053230 T^{15} + 406067677556641 T^{16} \))(\( 1 - 10 T - 138 T^{2} + 1360 T^{3} + 15050 T^{4} - 112470 T^{5} - 1068008 T^{6} + 3134810 T^{7} + 75569499 T^{8} + 210032270 T^{9} - 4794287912 T^{10} - 33826814610 T^{11} + 303274371050 T^{12} + 1836170145520 T^{13} - 12483256739322 T^{14} - 60607116053230 T^{15} + 406067677556641 T^{16} \))
$71$ (\( 1 - 71 T^{2} + 5041 T^{4} \))(\( 1 + 6 T - 35 T^{2} + 426 T^{3} + 5041 T^{4} \))(\( 1 + 6 T - 35 T^{2} + 426 T^{3} + 5041 T^{4} \))(\( 1 + 2 T - 67 T^{2} + 142 T^{3} + 5041 T^{4} \))(\( 1 - 6 T - 35 T^{2} - 426 T^{3} + 5041 T^{4} \))(\( 1 - 2 T - 67 T^{2} - 142 T^{3} + 5041 T^{4} \))(\( 1 - 71 T^{2} + 5041 T^{4} \))(\( 1 + 12 T + 46 T^{2} - 528 T^{3} - 5661 T^{4} - 37488 T^{5} + 231886 T^{6} + 4294932 T^{7} + 25411681 T^{8} \))(\( 1 - 12 T + 46 T^{2} + 528 T^{3} - 5661 T^{4} + 37488 T^{5} + 231886 T^{6} - 4294932 T^{7} + 25411681 T^{8} \))(\( 1 + 16 T + 78 T^{2} + 576 T^{3} + 8467 T^{4} + 40896 T^{5} + 393198 T^{6} + 5726576 T^{7} + 25411681 T^{8} \))(\( 1 - 12 T - 81 T^{2} + 404 T^{3} + 16026 T^{4} - 26292 T^{5} - 1195753 T^{6} - 1866732 T^{7} + 80787066 T^{8} + 144596044 T^{9} - 2058346161 T^{10} - 21650752212 T^{11} + 128100283921 T^{12} \))(\( 1 + 12 T - 81 T^{2} - 404 T^{3} + 16026 T^{4} + 26292 T^{5} - 1195753 T^{6} + 1866732 T^{7} + 80787066 T^{8} - 144596044 T^{9} - 2058346161 T^{10} + 21650752212 T^{11} + 128100283921 T^{12} \))(\( 1 - 184 T^{2} - 720 T^{3} + 17830 T^{4} + 91800 T^{5} - 964096 T^{6} - 3636720 T^{7} + 51551299 T^{8} - 258207120 T^{9} - 4860007936 T^{10} + 32856229800 T^{11} + 453090272230 T^{12} - 1299045132720 T^{13} - 23570452241464 T^{14} + 645753531245761 T^{16} \))(\( 1 - 184 T^{2} + 720 T^{3} + 17830 T^{4} - 91800 T^{5} - 964096 T^{6} + 3636720 T^{7} + 51551299 T^{8} + 258207120 T^{9} - 4860007936 T^{10} - 32856229800 T^{11} + 453090272230 T^{12} + 1299045132720 T^{13} - 23570452241464 T^{14} + 645753531245761 T^{16} \))
$73$ (\( 1 - 11 T + 48 T^{2} - 803 T^{3} + 5329 T^{4} \))(\( 1 - T - 72 T^{2} - 73 T^{3} + 5329 T^{4} \))(\( ( 1 - 17 T + 73 T^{2} )( 1 + 10 T + 73 T^{2} ) \))(\( 1 + 9 T + 8 T^{2} + 657 T^{3} + 5329 T^{4} \))(\( ( 1 - 17 T + 73 T^{2} )( 1 + 10 T + 73 T^{2} ) \))(\( 1 + 9 T + 8 T^{2} + 657 T^{3} + 5329 T^{4} \))(\( 1 - 11 T + 48 T^{2} - 803 T^{3} + 5329 T^{4} \))(\( 1 - 9 T - 84 T^{2} - 171 T^{3} + 15983 T^{4} - 12483 T^{5} - 447636 T^{6} - 3501153 T^{7} + 28398241 T^{8} \))(\( 1 - 9 T - 84 T^{2} - 171 T^{3} + 15983 T^{4} - 12483 T^{5} - 447636 T^{6} - 3501153 T^{7} + 28398241 T^{8} \))(\( 1 + 14 T + 29 T^{2} + 294 T^{3} + 8252 T^{4} + 21462 T^{5} + 154541 T^{6} + 5446238 T^{7} + 28398241 T^{8} \))(\( 1 + 12 T - 12 T^{2} - 1518 T^{3} - 7392 T^{4} + 55056 T^{5} + 1107155 T^{6} + 4019088 T^{7} - 39391968 T^{8} - 590527806 T^{9} - 340778892 T^{10} + 24876859116 T^{11} + 151334226289 T^{12} \))(\( 1 + 12 T - 12 T^{2} - 1518 T^{3} - 7392 T^{4} + 55056 T^{5} + 1107155 T^{6} + 4019088 T^{7} - 39391968 T^{8} - 590527806 T^{9} - 340778892 T^{10} + 24876859116 T^{11} + 151334226289 T^{12} \))(\( 1 - 14 T - 147 T^{2} + 1318 T^{3} + 35528 T^{4} - 173092 T^{5} - 3581409 T^{6} + 2002724 T^{7} + 356599063 T^{8} + 146198852 T^{9} - 19085328561 T^{10} - 67335730564 T^{11} + 1008932706248 T^{12} + 2732308359574 T^{13} - 22246131264483 T^{14} - 154663579267358 T^{15} + 806460091894081 T^{16} \))(\( 1 - 14 T - 147 T^{2} + 1318 T^{3} + 35528 T^{4} - 173092 T^{5} - 3581409 T^{6} + 2002724 T^{7} + 356599063 T^{8} + 146198852 T^{9} - 19085328561 T^{10} - 67335730564 T^{11} + 1008932706248 T^{12} + 2732308359574 T^{13} - 22246131264483 T^{14} - 154663579267358 T^{15} + 806460091894081 T^{16} \))
$79$ (\( 1 - 12 T + 65 T^{2} - 948 T^{3} + 6241 T^{4} \))(\( ( 1 - 13 T + 79 T^{2} )( 1 + 17 T + 79 T^{2} ) \))(\( 1 + 10 T + 21 T^{2} + 790 T^{3} + 6241 T^{4} \))(\( 1 - 10 T + 21 T^{2} - 790 T^{3} + 6241 T^{4} \))(\( 1 - 10 T + 21 T^{2} - 790 T^{3} + 6241 T^{4} \))(\( 1 + 10 T + 21 T^{2} + 790 T^{3} + 6241 T^{4} \))(\( 1 + 12 T + 65 T^{2} + 948 T^{3} + 6241 T^{4} \))(\( 1 + 10 T - 78 T^{2} + 200 T^{3} + 17543 T^{4} + 15800 T^{5} - 486798 T^{6} + 4930390 T^{7} + 38950081 T^{8} \))(\( 1 - 10 T - 78 T^{2} - 200 T^{3} + 17543 T^{4} - 15800 T^{5} - 486798 T^{6} - 4930390 T^{7} + 38950081 T^{8} \))(\( ( 1 - 13 T + 79 T^{2} )^{2}( 1 + 17 T + 79 T^{2} )^{2} \))(\( 1 - 18 T - 9 T^{2} - 34 T^{3} + 30942 T^{4} - 157626 T^{5} - 915837 T^{6} - 12452454 T^{7} + 193109022 T^{8} - 16763326 T^{9} - 350550729 T^{10} - 55387015182 T^{11} + 243087455521 T^{12} \))(\( 1 + 18 T - 9 T^{2} + 34 T^{3} + 30942 T^{4} + 157626 T^{5} - 915837 T^{6} + 12452454 T^{7} + 193109022 T^{8} + 16763326 T^{9} - 350550729 T^{10} + 55387015182 T^{11} + 243087455521 T^{12} \))(\( 1 + 22 T + 74 T^{2} + 56 T^{3} + 21922 T^{4} + 98442 T^{5} - 2089112 T^{6} - 9976614 T^{7} + 65507451 T^{8} - 788152506 T^{9} - 13038147992 T^{10} + 48535745238 T^{11} + 853863675682 T^{12} + 172315158344 T^{13} + 17988471708554 T^{14} + 422485997695498 T^{15} + 1517108809906561 T^{16} \))(\( 1 - 22 T + 74 T^{2} - 56 T^{3} + 21922 T^{4} - 98442 T^{5} - 2089112 T^{6} + 9976614 T^{7} + 65507451 T^{8} + 788152506 T^{9} - 13038147992 T^{10} - 48535745238 T^{11} + 853863675682 T^{12} - 172315158344 T^{13} + 17988471708554 T^{14} - 422485997695498 T^{15} + 1517108809906561 T^{16} \))
$83$ (\( ( 1 - 2 T + 83 T^{2} )^{2} \))(\( ( 1 - 3 T + 83 T^{2} )^{2} \))(\( ( 1 + 6 T + 83 T^{2} )^{2} \))(\( ( 1 + 6 T + 83 T^{2} )^{2} \))(\( ( 1 + 6 T + 83 T^{2} )^{2} \))(\( ( 1 + 6 T + 83 T^{2} )^{2} \))(\( ( 1 - 2 T + 83 T^{2} )^{2} \))(\( ( 1 - 8 T + 162 T^{2} - 664 T^{3} + 6889 T^{4} )^{2} \))(\( ( 1 - 8 T + 162 T^{2} - 664 T^{3} + 6889 T^{4} )^{2} \))(\( ( 1 + 103 T^{2} + 6889 T^{4} )^{2} \))(\( ( 1 - 6 T + 222 T^{2} - 945 T^{3} + 18426 T^{4} - 41334 T^{5} + 571787 T^{6} )^{2} \))(\( ( 1 - 6 T + 222 T^{2} - 945 T^{3} + 18426 T^{4} - 41334 T^{5} + 571787 T^{6} )^{2} \))(\( ( 1 + 12 T + 136 T^{2} + 156 T^{3} + 1914 T^{4} + 12948 T^{5} + 936904 T^{6} + 6861444 T^{7} + 47458321 T^{8} )^{2} \))(\( ( 1 + 12 T + 136 T^{2} + 156 T^{3} + 1914 T^{4} + 12948 T^{5} + 936904 T^{6} + 6861444 T^{7} + 47458321 T^{8} )^{2} \))
$89$ (\( 1 + 6 T - 53 T^{2} + 534 T^{3} + 7921 T^{4} \))(\( 1 - 6 T - 53 T^{2} - 534 T^{3} + 7921 T^{4} \))(\( 1 + 12 T + 55 T^{2} + 1068 T^{3} + 7921 T^{4} \))(\( 1 - 89 T^{2} + 7921 T^{4} \))(\( 1 - 12 T + 55 T^{2} - 1068 T^{3} + 7921 T^{4} \))(\( 1 - 89 T^{2} + 7921 T^{4} \))(\( 1 - 6 T - 53 T^{2} - 534 T^{3} + 7921 T^{4} \))(\( 1 - 11 T - 76 T^{2} - 209 T^{3} + 20119 T^{4} - 18601 T^{5} - 601996 T^{6} - 7754659 T^{7} + 62742241 T^{8} \))(\( 1 + 11 T - 76 T^{2} + 209 T^{3} + 20119 T^{4} + 18601 T^{5} - 601996 T^{6} + 7754659 T^{7} + 62742241 T^{8} \))(\( ( 1 - 89 T^{2} + 7921 T^{4} )^{2} \))(\( 1 + 9 T - 69 T^{2} + 108 T^{3} + 3561 T^{4} - 75861 T^{5} - 848522 T^{6} - 6751629 T^{7} + 28206681 T^{8} + 76136652 T^{9} - 4329214629 T^{10} + 50256535041 T^{11} + 496981290961 T^{12} \))(\( 1 - 9 T - 69 T^{2} - 108 T^{3} + 3561 T^{4} + 75861 T^{5} - 848522 T^{6} + 6751629 T^{7} + 28206681 T^{8} - 76136652 T^{9} - 4329214629 T^{10} - 50256535041 T^{11} + 496981290961 T^{12} \))(\( 1 - 16 T - 71 T^{2} + 1864 T^{3} + 8956 T^{4} - 157408 T^{5} - 757749 T^{6} + 7597372 T^{7} + 28710247 T^{8} + 676166108 T^{9} - 6002129829 T^{10} - 110967760352 T^{11} + 561919510396 T^{12} + 10408686812936 T^{13} - 35285671658231 T^{14} - 707701358328464 T^{15} + 3936588805702081 T^{16} \))(\( 1 + 16 T - 71 T^{2} - 1864 T^{3} + 8956 T^{4} + 157408 T^{5} - 757749 T^{6} - 7597372 T^{7} + 28710247 T^{8} - 676166108 T^{9} - 6002129829 T^{10} + 110967760352 T^{11} + 561919510396 T^{12} - 10408686812936 T^{13} - 35285671658231 T^{14} + 707701358328464 T^{15} + 3936588805702081 T^{16} \))
$97$ (\( 1 + 12 T + 47 T^{2} + 1164 T^{3} + 9409 T^{4} \))(\( 1 - 17 T + 192 T^{2} - 1649 T^{3} + 9409 T^{4} \))(\( 1 + 10 T + 3 T^{2} + 970 T^{3} + 9409 T^{4} \))(\( 1 - 2 T - 93 T^{2} - 194 T^{3} + 9409 T^{4} \))(\( 1 - 10 T + 3 T^{2} - 970 T^{3} + 9409 T^{4} \))(\( 1 + 2 T - 93 T^{2} + 194 T^{3} + 9409 T^{4} \))(\( 1 - 12 T + 47 T^{2} - 1164 T^{3} + 9409 T^{4} \))(\( 1 - 15 T - 24 T^{2} - 825 T^{3} + 30767 T^{4} - 80025 T^{5} - 225816 T^{6} - 13690095 T^{7} + 88529281 T^{8} \))(\( 1 + 15 T - 24 T^{2} + 825 T^{3} + 30767 T^{4} + 80025 T^{5} - 225816 T^{6} + 13690095 T^{7} + 88529281 T^{8} \))(\( 1 + 18 T + 77 T^{2} + 954 T^{3} + 20172 T^{4} + 92538 T^{5} + 724493 T^{6} + 16428114 T^{7} + 88529281 T^{8} \))(\( 1 - 288 T^{2} - 2 T^{3} + 55008 T^{4} + 288 T^{5} - 6220221 T^{6} + 27936 T^{7} + 517570272 T^{8} - 1825346 T^{9} - 25496432928 T^{10} + 832972004929 T^{12} \))(\( 1 - 288 T^{2} + 2 T^{3} + 55008 T^{4} - 288 T^{5} - 6220221 T^{6} - 27936 T^{7} + 517570272 T^{8} + 1825346 T^{9} - 25496432928 T^{10} + 832972004929 T^{12} \))(\( 1 + 28 T + 297 T^{2} + 1236 T^{3} - 4112 T^{4} - 169336 T^{5} - 2195481 T^{6} - 12285432 T^{7} - 36943377 T^{8} - 1191686904 T^{9} - 20657280729 T^{10} - 154548395128 T^{11} - 364032403472 T^{12} + 10613952557652 T^{13} + 247392685463913 T^{14} + 2262351965387164 T^{15} + 7837433594376961 T^{16} \))(\( 1 - 28 T + 297 T^{2} - 1236 T^{3} - 4112 T^{4} + 169336 T^{5} - 2195481 T^{6} + 12285432 T^{7} - 36943377 T^{8} + 1191686904 T^{9} - 20657280729 T^{10} + 154548395128 T^{11} - 364032403472 T^{12} - 10613952557652 T^{13} + 247392685463913 T^{14} - 2262351965387164 T^{15} + 7837433594376961 T^{16} \))
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