Properties

Label 4020.2.q
Level 4020
Weight 2
Character orbit q
Rep. character \(\chi_{4020}(841,\cdot)\)
Character field \(\Q(\zeta_{3})\)
Dimension 92
Newform subspaces 13
Sturm bound 1632
Trace bound 11

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Defining parameters

Level: \( N \) = \( 4020 = 2^{2} \cdot 3 \cdot 5 \cdot 67 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 4020.q (of order \(3\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) = \( 67 \)
Character field: \(\Q(\zeta_{3})\)
Newform subspaces: \( 13 \)
Sturm bound: \(1632\)
Trace bound: \(11\)
Distinguishing \(T_p\): \(7\), \(11\), \(17\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(4020, [\chi])\).

Total New Old
Modular forms 1656 92 1564
Cusp forms 1608 92 1516
Eisenstein series 48 0 48

Trace form

\( 92q + 4q^{3} - 4q^{7} + 92q^{9} + O(q^{10}) \) \( 92q + 4q^{3} - 4q^{7} + 92q^{9} - 4q^{11} - 10q^{13} + 8q^{17} - 4q^{21} + 12q^{23} + 92q^{25} + 4q^{27} + 20q^{29} - 2q^{31} - 4q^{35} - 4q^{37} + 2q^{39} + 4q^{41} - 20q^{43} - 20q^{47} - 50q^{49} + 4q^{51} - 16q^{53} + 12q^{55} - 16q^{57} - 56q^{59} + 10q^{61} - 4q^{63} + 8q^{65} - 46q^{67} - 8q^{69} - 8q^{71} + 10q^{73} + 4q^{75} + 8q^{77} - 10q^{79} + 92q^{81} - 4q^{83} - 12q^{85} + 4q^{87} + 64q^{89} - 8q^{91} - 14q^{93} - 30q^{97} - 4q^{99} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(4020, [\chi])\) into newform subspaces

Label Dim. \(A\) Field CM Traces $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\)
4020.2.q.a \(2\) \(32.100\) \(\Q(\sqrt{-3}) \) None \(0\) \(-2\) \(2\) \(-3\) \(q-q^{3}+q^{5}-3\zeta_{6}q^{7}+q^{9}-5\zeta_{6}q^{11}+\cdots\)
4020.2.q.b \(2\) \(32.100\) \(\Q(\sqrt{-3}) \) None \(0\) \(-2\) \(2\) \(-3\) \(q-q^{3}+q^{5}-3\zeta_{6}q^{7}+q^{9}+5\zeta_{6}q^{11}+\cdots\)
4020.2.q.c \(2\) \(32.100\) \(\Q(\sqrt{-3}) \) None \(0\) \(-2\) \(2\) \(1\) \(q-q^{3}+q^{5}+\zeta_{6}q^{7}+q^{9}-3\zeta_{6}q^{11}+\cdots\)
4020.2.q.d \(2\) \(32.100\) \(\Q(\sqrt{-3}) \) None \(0\) \(-2\) \(2\) \(1\) \(q-q^{3}+q^{5}+\zeta_{6}q^{7}+q^{9}+\zeta_{6}q^{11}+\cdots\)
4020.2.q.e \(2\) \(32.100\) \(\Q(\sqrt{-3}) \) None \(0\) \(2\) \(-2\) \(-1\) \(q+q^{3}-q^{5}-\zeta_{6}q^{7}+q^{9}-3\zeta_{6}q^{11}+\cdots\)
4020.2.q.f \(2\) \(32.100\) \(\Q(\sqrt{-3}) \) None \(0\) \(2\) \(-2\) \(-1\) \(q+q^{3}-q^{5}-\zeta_{6}q^{7}+q^{9}+\zeta_{6}q^{11}+\cdots\)
4020.2.q.g \(2\) \(32.100\) \(\Q(\sqrt{-3}) \) None \(0\) \(2\) \(-2\) \(-1\) \(q+q^{3}-q^{5}-\zeta_{6}q^{7}+q^{9}+5\zeta_{6}q^{11}+\cdots\)
4020.2.q.h \(2\) \(32.100\) \(\Q(\sqrt{-3}) \) None \(0\) \(2\) \(-2\) \(1\) \(q+q^{3}-q^{5}+\zeta_{6}q^{7}+q^{9}-3\zeta_{6}q^{11}+\cdots\)
4020.2.q.i \(4\) \(32.100\) \(\Q(\sqrt{-3}, \sqrt{-11})\) None \(0\) \(4\) \(-4\) \(-1\) \(q+q^{3}-q^{5}+(-\beta _{1}+\beta _{3})q^{7}+q^{9}+\cdots\)
4020.2.q.j \(12\) \(32.100\) \(\mathbb{Q}[x]/(x^{12} - \cdots)\) None \(0\) \(12\) \(-12\) \(2\) \(q+q^{3}-q^{5}+(\beta _{1}-\beta _{6}-\beta _{9})q^{7}+q^{9}+\cdots\)
4020.2.q.k \(14\) \(32.100\) \(\mathbb{Q}[x]/(x^{14} - \cdots)\) None \(0\) \(-14\) \(14\) \(3\) \(q-q^{3}+q^{5}-\beta _{6}q^{7}+q^{9}+(1-\beta _{5}+\cdots)q^{11}+\cdots\)
4020.2.q.l \(22\) \(32.100\) None \(0\) \(-22\) \(-22\) \(1\)
4020.2.q.m \(24\) \(32.100\) None \(0\) \(24\) \(24\) \(-3\)

Decomposition of \(S_{2}^{\mathrm{old}}(4020, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(4020, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(67, [\chi])\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(134, [\chi])\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(201, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(268, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(335, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(402, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(670, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(804, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(1005, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(1340, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(2010, [\chi])\)\(^{\oplus 2}\)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ 1
$3$ (\( ( 1 + T )^{2} \))(\( ( 1 + T )^{2} \))(\( ( 1 + T )^{2} \))(\( ( 1 + T )^{2} \))(\( ( 1 - T )^{2} \))(\( ( 1 - T )^{2} \))(\( ( 1 - T )^{2} \))(\( ( 1 - T )^{2} \))(\( ( 1 - T )^{4} \))(\( ( 1 - T )^{12} \))(\( ( 1 + T )^{14} \))
$5$ (\( ( 1 - T )^{2} \))(\( ( 1 - T )^{2} \))(\( ( 1 - T )^{2} \))(\( ( 1 - T )^{2} \))(\( ( 1 + T )^{2} \))(\( ( 1 + T )^{2} \))(\( ( 1 + T )^{2} \))(\( ( 1 + T )^{2} \))(\( ( 1 + T )^{4} \))(\( ( 1 + T )^{12} \))(\( ( 1 - T )^{14} \))
$7$ (\( 1 + 3 T + 2 T^{2} + 21 T^{3} + 49 T^{4} \))(\( 1 + 3 T + 2 T^{2} + 21 T^{3} + 49 T^{4} \))(\( ( 1 - 5 T + 7 T^{2} )( 1 + 4 T + 7 T^{2} ) \))(\( ( 1 - 5 T + 7 T^{2} )( 1 + 4 T + 7 T^{2} ) \))(\( ( 1 - 4 T + 7 T^{2} )( 1 + 5 T + 7 T^{2} ) \))(\( ( 1 - 4 T + 7 T^{2} )( 1 + 5 T + 7 T^{2} ) \))(\( ( 1 - 4 T + 7 T^{2} )( 1 + 5 T + 7 T^{2} ) \))(\( ( 1 - 5 T + 7 T^{2} )( 1 + 4 T + 7 T^{2} ) \))(\( 1 + T - 5 T^{2} - 8 T^{3} - 20 T^{4} - 56 T^{5} - 245 T^{6} + 343 T^{7} + 2401 T^{8} \))(\( 1 - 2 T + 4 T^{2} - 48 T^{4} + 107 T^{5} + 396 T^{6} - 1161 T^{7} + 2332 T^{8} - 8784 T^{9} - 6359 T^{10} - 549 T^{11} + 84688 T^{12} - 3843 T^{13} - 311591 T^{14} - 3012912 T^{15} + 5599132 T^{16} - 19512927 T^{17} + 46589004 T^{18} + 88119101 T^{19} - 276710448 T^{20} + 1129900996 T^{22} - 3954653486 T^{23} + 13841287201 T^{24} \))(\( 1 - 3 T - 20 T^{2} + 57 T^{3} + 177 T^{4} - 389 T^{5} - 1803 T^{6} + 3303 T^{7} + 18070 T^{8} - 36377 T^{9} - 110178 T^{10} + 123265 T^{11} + 781213 T^{12} + 114570 T^{13} - 6762664 T^{14} + 801990 T^{15} + 38279437 T^{16} + 42279895 T^{17} - 264537378 T^{18} - 611388239 T^{19} + 2125917430 T^{20} + 2720162529 T^{21} - 10393936203 T^{22} - 15697553123 T^{23} + 49998119073 T^{24} + 112707624351 T^{25} - 276825744020 T^{26} - 290667031221 T^{27} + 678223072849 T^{28} \))
$11$ (\( 1 + 5 T + 14 T^{2} + 55 T^{3} + 121 T^{4} \))(\( 1 - 5 T + 14 T^{2} - 55 T^{3} + 121 T^{4} \))(\( 1 + 3 T - 2 T^{2} + 33 T^{3} + 121 T^{4} \))(\( 1 - T - 10 T^{2} - 11 T^{3} + 121 T^{4} \))(\( 1 + 3 T - 2 T^{2} + 33 T^{3} + 121 T^{4} \))(\( 1 - T - 10 T^{2} - 11 T^{3} + 121 T^{4} \))(\( 1 - 5 T + 14 T^{2} - 55 T^{3} + 121 T^{4} \))(\( 1 + 3 T - 2 T^{2} + 33 T^{3} + 121 T^{4} \))(\( 1 - 3 T - 7 T^{2} + 18 T^{3} + 36 T^{4} + 198 T^{5} - 847 T^{6} - 3993 T^{7} + 14641 T^{8} \))(\( 1 + 5 T - 10 T^{2} - 39 T^{3} + 71 T^{4} - 535 T^{5} - 1716 T^{6} - 1332 T^{7} - 1883 T^{8} + 75951 T^{9} - 168035 T^{10} - 92937 T^{11} + 7017904 T^{12} - 1022307 T^{13} - 20332235 T^{14} + 101090781 T^{15} - 27569003 T^{16} - 214519932 T^{17} - 3039998676 T^{18} - 10425636485 T^{19} + 15219480551 T^{20} - 91959959949 T^{21} - 259374246010 T^{22} + 1426558353055 T^{23} + 3138428376721 T^{24} \))(\( 1 - 6 T - 11 T^{2} + 16 T^{3} + 550 T^{4} + 451 T^{5} - 6158 T^{6} - 23818 T^{7} + 26095 T^{8} + 323347 T^{9} + 407400 T^{10} - 2104801 T^{11} - 9984167 T^{12} + 14819849 T^{13} + 70409980 T^{14} + 163018339 T^{15} - 1208084207 T^{16} - 2801490131 T^{17} + 5964743400 T^{18} + 52075357697 T^{19} + 46228884295 T^{20} - 464145438878 T^{21} - 1320021989198 T^{22} + 1063434408641 T^{23} + 14265583530550 T^{24} + 4564986729776 T^{25} - 34522712143931 T^{26} - 207136272863586 T^{27} + 379749833583241 T^{28} \))
$13$ (\( 1 + T - 12 T^{2} + 13 T^{3} + 169 T^{4} \))(\( ( 1 - 2 T + 13 T^{2} )( 1 + 7 T + 13 T^{2} ) \))(\( ( 1 - 2 T + 13 T^{2} )( 1 + 7 T + 13 T^{2} ) \))(\( 1 - 3 T - 4 T^{2} - 39 T^{3} + 169 T^{4} \))(\( 1 - T - 12 T^{2} - 13 T^{3} + 169 T^{4} \))(\( ( 1 + 2 T + 13 T^{2} )( 1 + 5 T + 13 T^{2} ) \))(\( 1 - T - 12 T^{2} - 13 T^{3} + 169 T^{4} \))(\( ( 1 - 2 T + 13 T^{2} )( 1 + 7 T + 13 T^{2} ) \))(\( ( 1 - T - 12 T^{2} - 13 T^{3} + 169 T^{4} )^{2} \))(\( ( 1 - T - 12 T^{2} - 13 T^{3} + 169 T^{4} )^{6} \))(\( 1 - 9 T - 3 T^{2} + 234 T^{3} - 216 T^{4} - 3440 T^{5} + 5289 T^{6} + 22909 T^{7} - 40116 T^{8} + 156307 T^{9} - 627413 T^{10} - 5854224 T^{11} + 29555027 T^{12} + 38130675 T^{13} - 498920098 T^{14} + 495698775 T^{15} + 4994799563 T^{16} - 12861730128 T^{17} - 17919542693 T^{18} + 58035694951 T^{19} - 193632269844 T^{20} + 1437505775953 T^{21} + 4314399783369 T^{22} - 36479477843120 T^{23} - 29777434239384 T^{24} + 419365532204658 T^{25} - 69894255367443 T^{26} - 2725875959330277 T^{27} + 3937376385699289 T^{28} \))
$17$ (\( 1 - 7 T + 32 T^{2} - 119 T^{3} + 289 T^{4} \))(\( 1 + 3 T - 8 T^{2} + 51 T^{3} + 289 T^{4} \))(\( 1 - T - 16 T^{2} - 17 T^{3} + 289 T^{4} \))(\( 1 - 5 T + 8 T^{2} - 85 T^{3} + 289 T^{4} \))(\( 1 + T - 16 T^{2} + 17 T^{3} + 289 T^{4} \))(\( 1 - 3 T - 8 T^{2} - 51 T^{3} + 289 T^{4} \))(\( 1 - 7 T + 32 T^{2} - 119 T^{3} + 289 T^{4} \))(\( 1 + 3 T - 8 T^{2} + 51 T^{3} + 289 T^{4} \))(\( 1 + 9 T + 35 T^{2} + 108 T^{3} + 450 T^{4} + 1836 T^{5} + 10115 T^{6} + 44217 T^{7} + 83521 T^{8} \))(\( 1 - 9 T + 4 T^{2} + 355 T^{3} - 1746 T^{4} - 1694 T^{5} + 47899 T^{6} - 146091 T^{7} - 390577 T^{8} + 4607833 T^{9} - 10879892 T^{10} - 42482959 T^{11} + 376180152 T^{12} - 722210303 T^{13} - 3144288788 T^{14} + 22638283529 T^{15} - 32621381617 T^{16} - 207428328987 T^{17} + 1156165417531 T^{18} - 695113712062 T^{19} - 12179672491986 T^{20} + 42098696156435 T^{21} + 8063975601796 T^{22} - 308447066768697 T^{23} + 582622237229761 T^{24} \))(\( 1 + 12 T + 13 T^{2} - 120 T^{3} + 1763 T^{4} + 8439 T^{5} - 29854 T^{6} + 48731 T^{7} + 1353310 T^{8} - 587405 T^{9} - 5786839 T^{10} + 84344359 T^{11} + 106817487 T^{12} - 21465324 T^{13} + 3964551862 T^{14} - 364910508 T^{15} + 30870253743 T^{16} + 414383835767 T^{17} - 483322580119 T^{18} - 834031101085 T^{19} + 32665613503390 T^{20} + 19996213873963 T^{21} - 208254262643614 T^{22} + 1000763089758183 T^{23} + 3554197246491587 T^{24} - 4112627556915960 T^{25} + 7574089083986893 T^{26} + 118854936394871244 T^{27} + 168377826559400929 T^{28} \))
$19$ (\( 1 - 5 T + 6 T^{2} - 95 T^{3} + 361 T^{4} \))(\( ( 1 - 8 T + 19 T^{2} )( 1 + 7 T + 19 T^{2} ) \))(\( ( 1 - T + 19 T^{2} )( 1 + 8 T + 19 T^{2} ) \))(\( ( 1 - T + 19 T^{2} )( 1 + 8 T + 19 T^{2} ) \))(\( 1 - 5 T + 6 T^{2} - 95 T^{3} + 361 T^{4} \))(\( 1 - 5 T + 6 T^{2} - 95 T^{3} + 361 T^{4} \))(\( 1 - 5 T + 6 T^{2} - 95 T^{3} + 361 T^{4} \))(\( ( 1 - 8 T + 19 T^{2} )( 1 + 7 T + 19 T^{2} ) \))(\( 1 + 7 T + 7 T^{2} + 28 T^{3} + 472 T^{4} + 532 T^{5} + 2527 T^{6} + 48013 T^{7} + 130321 T^{8} \))(\( 1 + 11 T + 33 T^{2} - 388 T^{4} - 4838 T^{5} - 28432 T^{6} - 111029 T^{7} - 333428 T^{8} - 54312 T^{9} + 4625736 T^{10} + 47081975 T^{11} + 298793580 T^{12} + 894557525 T^{13} + 1669890696 T^{14} - 372526008 T^{15} - 43452670388 T^{16} - 274918795871 T^{17} - 1337608488592 T^{18} - 4324551473282 T^{19} - 6589622459908 T^{20} + 202325186507433 T^{22} + 1281392847880409 T^{23} + 2213314919066161 T^{24} \))(\( 1 - 14 T + 8 T^{2} + 582 T^{3} + 222 T^{4} - 25358 T^{5} - 8233 T^{6} + 614419 T^{7} + 1938247 T^{8} - 16399564 T^{9} - 65078281 T^{10} + 228533498 T^{11} + 2092709725 T^{12} - 2476014457 T^{13} - 40027234346 T^{14} - 47044274683 T^{15} + 755468210725 T^{16} + 1567511262782 T^{17} - 8481066658201 T^{18} - 40606944020836 T^{19} + 91186537710607 T^{20} + 549211780004641 T^{21} - 139825674516553 T^{22} - 8182714640279882 T^{23} + 1361096709231822 T^{24} + 67797330678763458 T^{25} + 17706519352529288 T^{26} - 588741768471598826 T^{27} + 799006685782884121 T^{28} \))
$23$ (\( 1 + T - 22 T^{2} + 23 T^{3} + 529 T^{4} \))(\( 1 + 7 T + 26 T^{2} + 161 T^{3} + 529 T^{4} \))(\( 1 - 9 T + 58 T^{2} - 207 T^{3} + 529 T^{4} \))(\( 1 + 3 T - 14 T^{2} + 69 T^{3} + 529 T^{4} \))(\( 1 + T - 22 T^{2} + 23 T^{3} + 529 T^{4} \))(\( 1 - 3 T - 14 T^{2} - 69 T^{3} + 529 T^{4} \))(\( 1 + 9 T + 58 T^{2} + 207 T^{3} + 529 T^{4} \))(\( 1 + 3 T - 14 T^{2} + 69 T^{3} + 529 T^{4} \))(\( ( 1 + 9 T + 23 T^{2} )^{2}( 1 - 9 T + 58 T^{2} - 207 T^{3} + 529 T^{4} ) \))(\( 1 - 19 T + 120 T^{2} - 95 T^{3} - 1424 T^{4} - 2326 T^{5} + 61689 T^{6} - 352863 T^{7} + 2083525 T^{8} - 6539565 T^{9} - 1493768 T^{10} - 115618269 T^{11} + 1361354230 T^{12} - 2659220187 T^{13} - 790203272 T^{14} - 79566887355 T^{15} + 583055719525 T^{16} - 2271147300009 T^{17} + 9132185956521 T^{18} - 7919623989722 T^{19} - 111514843040144 T^{20} - 171109502838985 T^{21} + 4971181345637880 T^{22} - 18103385400364613 T^{23} + 21914624432020321 T^{24} \))(\( 1 - 6 T - 55 T^{2} + 462 T^{3} + 753 T^{4} - 13307 T^{5} + 10046 T^{6} + 152185 T^{7} - 95636 T^{8} - 824917 T^{9} - 12768145 T^{10} + 51517171 T^{11} + 264832895 T^{12} - 956116592 T^{13} - 2075704102 T^{14} - 21990681616 T^{15} + 140096601455 T^{16} + 626809419557 T^{17} - 3573050464945 T^{18} - 5309448758531 T^{19} - 14157560280404 T^{20} + 518163360651695 T^{21} + 786712158132926 T^{22} - 23967938466088141 T^{23} + 31194162943877697 T^{24} + 440198108156234274 T^{25} - 1205304343761117655 T^{26} - 3024218171618804298 T^{27} + 11592836324538749809 T^{28} \))
$29$ (\( 1 + 3 T - 20 T^{2} + 87 T^{3} + 841 T^{4} \))(\( 1 + 5 T - 4 T^{2} + 145 T^{3} + 841 T^{4} \))(\( 1 + 5 T - 4 T^{2} + 145 T^{3} + 841 T^{4} \))(\( 1 - 7 T + 20 T^{2} - 203 T^{3} + 841 T^{4} \))(\( 1 + 5 T - 4 T^{2} + 145 T^{3} + 841 T^{4} \))(\( 1 + T - 28 T^{2} + 29 T^{3} + 841 T^{4} \))(\( 1 - 3 T - 20 T^{2} - 87 T^{3} + 841 T^{4} \))(\( 1 - 3 T - 20 T^{2} - 87 T^{3} + 841 T^{4} \))(\( 1 + 3 T - 43 T^{2} - 18 T^{3} + 1602 T^{4} - 522 T^{5} - 36163 T^{6} + 73167 T^{7} + 707281 T^{8} \))(\( 1 - 8 T + 32 T^{2} - 368 T^{3} + 1206 T^{4} - 1655 T^{5} + 89070 T^{6} - 339779 T^{7} + 1355490 T^{8} - 22458518 T^{9} + 53699035 T^{10} - 227990417 T^{11} + 3614850538 T^{12} - 6611722093 T^{13} + 45160888435 T^{14} - 547740795502 T^{15} + 958712322690 T^{16} - 6969257696071 T^{17} + 52980913201470 T^{18} - 28548545291395 T^{19} + 603297174030966 T^{20} - 5338629719119792 T^{21} + 13462631465606432 T^{22} - 97604078125646632 T^{23} + 353814783205469041 T^{24} \))(\( 1 + T - 94 T^{2} + 111 T^{3} + 4141 T^{4} - 11695 T^{5} - 81857 T^{6} + 347769 T^{7} + 213802 T^{8} + 2744015 T^{9} + 736264 T^{10} - 378858495 T^{11} + 1902063729 T^{12} + 6351771646 T^{13} - 90593298388 T^{14} + 184201377734 T^{15} + 1599635596089 T^{16} - 9239979834555 T^{17} + 520745538184 T^{18} + 56282900523235 T^{19} + 127174415676442 T^{20} + 5998972234104621 T^{21} - 40948670625748577 T^{22} - 169661072187787955 T^{23} + 1742148653096132341 T^{24} + 1354256583993347019 T^{25} - 33258589621314089854 T^{26} + 10260628712958602189 T^{27} + \)\(29\!\cdots\!81\)\( T^{28} \))
$31$ (\( 1 - 3 T - 22 T^{2} - 93 T^{3} + 961 T^{4} \))(\( 1 + 5 T - 6 T^{2} + 155 T^{3} + 961 T^{4} \))(\( ( 1 - 11 T + 31 T^{2} )( 1 + 4 T + 31 T^{2} ) \))(\( 1 + T - 30 T^{2} + 31 T^{3} + 961 T^{4} \))(\( ( 1 - 11 T + 31 T^{2} )( 1 + 4 T + 31 T^{2} ) \))(\( 1 + T - 30 T^{2} + 31 T^{3} + 961 T^{4} \))(\( ( 1 - 11 T + 31 T^{2} )( 1 + 4 T + 31 T^{2} ) \))(\( 1 + 5 T - 6 T^{2} + 155 T^{3} + 961 T^{4} \))(\( 1 + 4 T - 17 T^{2} - 116 T^{3} - 368 T^{4} - 3596 T^{5} - 16337 T^{6} + 119164 T^{7} + 923521 T^{8} \))(\( 1 + 4 T - 109 T^{2} - 606 T^{3} + 5492 T^{4} + 36266 T^{5} - 184734 T^{6} - 1057922 T^{7} + 6973790 T^{8} + 13628888 T^{9} - 332452362 T^{10} - 40779865 T^{11} + 12559806528 T^{12} - 1264175815 T^{13} - 319486719882 T^{14} + 406018202408 T^{15} + 6440441514590 T^{16} - 30287408684222 T^{17} - 163952105005854 T^{18} + 997772463349526 T^{19} + 4684077577625972 T^{20} - 16022411029366626 T^{21} - 89339483280907309 T^{22} + 101633907585619324 T^{23} + 787662783788549761 T^{24} \))(\( 1 - 7 T - 39 T^{2} + 660 T^{3} - 2351 T^{4} - 13913 T^{5} + 146821 T^{6} - 144326 T^{7} - 3076282 T^{8} + 12618238 T^{9} + 70496210 T^{10} - 862571403 T^{11} + 1968366689 T^{12} + 19721322013 T^{13} - 189458995778 T^{14} + 611360982403 T^{15} + 1891600388129 T^{16} - 25696864666773 T^{17} + 65104730355410 T^{18} + 361249441055938 T^{19} - 2730211598794042 T^{20} - 3970785544184186 T^{21} + 125222315008125061 T^{22} - 367854463121415623 T^{23} - 1926946102691863151 T^{24} + 16769594751627188460 T^{25} - 30718848567753440679 T^{26} - \)\(17\!\cdots\!37\)\( T^{27} + \)\(75\!\cdots\!21\)\( T^{28} \))
$37$ (\( ( 1 - 10 T + 37 T^{2} )( 1 + 11 T + 37 T^{2} ) \))(\( ( 1 - 10 T + 37 T^{2} )( 1 - T + 37 T^{2} ) \))(\( 1 + 5 T - 12 T^{2} + 185 T^{3} + 1369 T^{4} \))(\( 1 - 3 T - 28 T^{2} - 111 T^{3} + 1369 T^{4} \))(\( 1 + 7 T + 12 T^{2} + 259 T^{3} + 1369 T^{4} \))(\( ( 1 - 11 T + 37 T^{2} )( 1 + 10 T + 37 T^{2} ) \))(\( 1 + 7 T + 12 T^{2} + 259 T^{3} + 1369 T^{4} \))(\( 1 - 7 T + 12 T^{2} - 259 T^{3} + 1369 T^{4} \))(\( 1 + T + T^{2} - 74 T^{3} - 1406 T^{4} - 2738 T^{5} + 1369 T^{6} + 50653 T^{7} + 1874161 T^{8} \))(\( 1 - 5 T - 35 T^{2} + 148 T^{3} + 910 T^{4} + 4694 T^{5} + 48184 T^{6} - 753797 T^{7} - 2440934 T^{8} + 39099266 T^{9} + 52006098 T^{10} - 569825299 T^{11} - 258124762 T^{12} - 21083536063 T^{13} + 71196348162 T^{14} + 1980495120698 T^{15} - 4574703306374 T^{16} - 52271266754729 T^{17} + 123626961291256 T^{18} + 445610231262302 T^{19} + 3196356303068110 T^{20} + 19234337489671396 T^{21} - 168300453034624715 T^{22} - 889588108897302065 T^{23} + 6582952005840035281 T^{24} \))(\( 1 + 2 T - 144 T^{2} - 58 T^{3} + 11562 T^{4} - 8564 T^{5} - 567933 T^{6} + 1219349 T^{7} + 16972491 T^{8} - 72220508 T^{9} - 82031967 T^{10} + 2604100054 T^{11} - 19609237189 T^{12} - 38834898747 T^{13} + 1110997548598 T^{14} - 1436891253639 T^{15} - 26845045711741 T^{16} + 131905480035262 T^{17} - 153741113304687 T^{18} - 5008055801270156 T^{19} + 43546768385214819 T^{20} + 115755089450246417 T^{21} - 1994852993703715293 T^{22} - 1112992339605039428 T^{23} + 55596852513895170138 T^{24} - 10319222063208703954 T^{25} - \)\(94\!\cdots\!64\)\( T^{26} + \)\(48\!\cdots\!94\)\( T^{27} + \)\(90\!\cdots\!89\)\( T^{28} \))
$41$ (\( 1 - 5 T - 16 T^{2} - 205 T^{3} + 1681 T^{4} \))(\( 1 - 11 T + 80 T^{2} - 451 T^{3} + 1681 T^{4} \))(\( 1 + T - 40 T^{2} + 41 T^{3} + 1681 T^{4} \))(\( 1 - 3 T - 32 T^{2} - 123 T^{3} + 1681 T^{4} \))(\( 1 + 5 T - 16 T^{2} + 205 T^{3} + 1681 T^{4} \))(\( 1 - 7 T + 8 T^{2} - 287 T^{3} + 1681 T^{4} \))(\( 1 - 3 T - 32 T^{2} - 123 T^{3} + 1681 T^{4} \))(\( 1 + 9 T + 40 T^{2} + 369 T^{3} + 1681 T^{4} \))(\( 1 - 15 T + 95 T^{2} - 720 T^{3} + 5994 T^{4} - 29520 T^{5} + 159695 T^{6} - 1033815 T^{7} + 2825761 T^{8} \))(\( 1 + 9 T - 112 T^{2} - 853 T^{3} + 9803 T^{4} + 50521 T^{5} - 565952 T^{6} - 1758476 T^{7} + 24242199 T^{8} + 40372655 T^{9} - 791971767 T^{10} - 501901089 T^{11} + 26593085586 T^{12} - 20577944649 T^{13} - 1331304540327 T^{14} + 2782523755255 T^{15} + 68502660488439 T^{16} - 203730348909676 T^{17} - 2688330995402432 T^{18} + 9839180670742001 T^{19} + 78276222021073163 T^{20} - 279256790038048733 T^{21} - 1503337842737068912 T^{22} + 4952961285446235969 T^{23} + 22563490300366186081 T^{24} \))(\( 1 + 18 T - 29 T^{2} - 2200 T^{3} - 1130 T^{4} + 183989 T^{5} + 379300 T^{6} - 8058104 T^{7} - 4062155 T^{8} + 263971859 T^{9} - 1279880946 T^{10} - 2948011895 T^{11} + 160106068045 T^{12} + 57760123657 T^{13} - 7992127618244 T^{14} + 2368165069937 T^{15} + 269138300383645 T^{16} - 203179927815295 T^{17} - 3616637661849906 T^{18} + 30582776754647659 T^{19} - 19295659693099355 T^{20} - 1569350193377581624 T^{21} + 3028682139405595300 T^{22} + 60234674727210490429 T^{23} - 15167605020472213130 T^{24} - \)\(12\!\cdots\!00\)\( T^{25} - \)\(65\!\cdots\!49\)\( T^{26} + \)\(16\!\cdots\!78\)\( T^{27} + \)\(37\!\cdots\!61\)\( T^{28} \))
$43$ (\( ( 1 + 4 T + 43 T^{2} )^{2} \))(\( ( 1 - 4 T + 43 T^{2} )^{2} \))(\( ( 1 + 4 T + 43 T^{2} )^{2} \))(\( ( 1 - 4 T + 43 T^{2} )^{2} \))(\( ( 1 + 8 T + 43 T^{2} )^{2} \))(\( ( 1 - 8 T + 43 T^{2} )^{2} \))(\( ( 1 + 8 T + 43 T^{2} )^{2} \))(\( ( 1 + 4 T + 43 T^{2} )^{2} \))(\( ( 1 - T + 78 T^{2} - 43 T^{3} + 1849 T^{4} )^{2} \))(\( ( 1 + 7 T + 202 T^{2} + 1304 T^{3} + 19362 T^{4} + 102444 T^{5} + 1076954 T^{6} + 4405092 T^{7} + 35800338 T^{8} + 103677128 T^{9} + 690597802 T^{10} + 1029059101 T^{11} + 6321363049 T^{12} )^{2} \))(\( ( 1 + 3 T + 165 T^{2} + 625 T^{3} + 14336 T^{4} + 57190 T^{5} + 833372 T^{6} + 3148724 T^{7} + 35834996 T^{8} + 105744310 T^{9} + 1139812352 T^{10} + 2136750625 T^{11} + 24256393095 T^{12} + 18964089147 T^{13} + 271818611107 T^{14} )^{2} \))
$47$ (\( 1 + 3 T - 38 T^{2} + 141 T^{3} + 2209 T^{4} \))(\( 1 + 13 T + 122 T^{2} + 611 T^{3} + 2209 T^{4} \))(\( 1 + T - 46 T^{2} + 47 T^{3} + 2209 T^{4} \))(\( 1 - 3 T - 38 T^{2} - 141 T^{3} + 2209 T^{4} \))(\( 1 - T - 46 T^{2} - 47 T^{3} + 2209 T^{4} \))(\( 1 + 3 T - 38 T^{2} + 141 T^{3} + 2209 T^{4} \))(\( 1 + 7 T + 2 T^{2} + 329 T^{3} + 2209 T^{4} \))(\( 1 - 3 T - 38 T^{2} - 141 T^{3} + 2209 T^{4} \))(\( 1 - 9 T - 25 T^{2} - 108 T^{3} + 5220 T^{4} - 5076 T^{5} - 55225 T^{6} - 934407 T^{7} + 4879681 T^{8} \))(\( 1 + 6 T - 23 T^{2} - 1160 T^{3} - 6572 T^{4} + 16868 T^{5} + 616228 T^{6} + 3244950 T^{7} - 6853044 T^{8} - 219512470 T^{9} - 865976942 T^{10} + 3383470903 T^{11} + 68645946092 T^{12} + 159023132441 T^{13} - 1912943064878 T^{14} - 22790443172810 T^{15} - 33440668598964 T^{16} + 744213080464650 T^{17} + 6642454303759012 T^{18} + 8545718795969884 T^{19} - 156487775941093292 T^{20} - 1298191348799209720 T^{21} - 1209780041424091127 T^{22} + 14832955290504073818 T^{23} + \)\(11\!\cdots\!41\)\( T^{24} \))(\( 1 - 7 T - 153 T^{2} + 1362 T^{3} + 9935 T^{4} - 114473 T^{5} - 354685 T^{6} + 5522720 T^{7} + 11215060 T^{8} - 156793834 T^{9} - 900317666 T^{10} + 2422525493 T^{11} + 78249322737 T^{12} - 17047291839 T^{13} - 4482369027754 T^{14} - 801222716433 T^{15} + 172852753926033 T^{16} + 251513864259739 T^{17} - 4393263008744546 T^{18} - 35959882956286838 T^{19} + 120889546667654740 T^{20} + 2797937639843419360 T^{21} - 8445506209626700285 T^{22} - \)\(12\!\cdots\!91\)\( T^{23} + \)\(52\!\cdots\!15\)\( T^{24} + \)\(33\!\cdots\!86\)\( T^{25} - \)\(17\!\cdots\!73\)\( T^{26} - \)\(38\!\cdots\!89\)\( T^{27} + \)\(25\!\cdots\!69\)\( T^{28} \))
$53$ (\( ( 1 - 6 T + 53 T^{2} )^{2} \))(\( ( 1 + 14 T + 53 T^{2} )^{2} \))(\( ( 1 + 10 T + 53 T^{2} )^{2} \))(\( ( 1 - 6 T + 53 T^{2} )^{2} \))(\( ( 1 - 6 T + 53 T^{2} )^{2} \))(\( ( 1 + 2 T + 53 T^{2} )^{2} \))(\( ( 1 - 6 T + 53 T^{2} )^{2} \))(\( ( 1 + 6 T + 53 T^{2} )^{2} \))(\( ( 1 + 6 T + 82 T^{2} + 318 T^{3} + 2809 T^{4} )^{2} \))(\( ( 1 + 10 T + 292 T^{2} + 2399 T^{3} + 37181 T^{4} + 241012 T^{5} + 2598106 T^{6} + 12773636 T^{7} + 104441429 T^{8} + 357155923 T^{9} + 2304020452 T^{10} + 4181954930 T^{11} + 22164361129 T^{12} )^{2} \))(\( ( 1 + 6 T + 247 T^{2} + 1145 T^{3} + 30041 T^{4} + 116883 T^{5} + 2344387 T^{6} + 7652092 T^{7} + 124252511 T^{8} + 328324347 T^{9} + 4472413957 T^{10} + 9034600745 T^{11} + 103294286771 T^{12} + 132986166774 T^{13} + 1174711139837 T^{14} )^{2} \))
$59$ (\( ( 1 - 4 T + 59 T^{2} )^{2} \))(\( ( 1 + 59 T^{2} )^{2} \))(\( ( 1 + 12 T + 59 T^{2} )^{2} \))(\( ( 1 + 4 T + 59 T^{2} )^{2} \))(\( ( 1 + 12 T + 59 T^{2} )^{2} \))(\( ( 1 - 4 T + 59 T^{2} )^{2} \))(\( ( 1 - 4 T + 59 T^{2} )^{2} \))(\( ( 1 + 12 T + 59 T^{2} )^{2} \))(\( ( 1 - 6 T + 94 T^{2} - 354 T^{3} + 3481 T^{4} )^{2} \))(\( ( 1 - 3 T + 70 T^{2} - 98 T^{3} + 1610 T^{4} - 9168 T^{5} + 82294 T^{6} - 540912 T^{7} + 5604410 T^{8} - 20127142 T^{9} + 848215270 T^{10} - 2144772897 T^{11} + 42180533641 T^{12} )^{2} \))(\( ( 1 + T + 289 T^{2} + 403 T^{3} + 39692 T^{4} + 63484 T^{5} + 3413740 T^{6} + 5088680 T^{7} + 201410660 T^{8} + 220987804 T^{9} + 8151903268 T^{10} + 4883296483 T^{11} + 206613122411 T^{12} + 42180533641 T^{13} + 2488651484819 T^{14} )^{2} \))
$61$ (\( 1 + 7 T - 12 T^{2} + 427 T^{3} + 3721 T^{4} \))(\( 1 - 9 T + 20 T^{2} - 549 T^{3} + 3721 T^{4} \))(\( 1 + 7 T - 12 T^{2} + 427 T^{3} + 3721 T^{4} \))(\( 1 + 7 T - 12 T^{2} + 427 T^{3} + 3721 T^{4} \))(\( 1 + 3 T - 52 T^{2} + 183 T^{3} + 3721 T^{4} \))(\( 1 + 11 T + 60 T^{2} + 671 T^{3} + 3721 T^{4} \))(\( ( 1 - 14 T + 61 T^{2} )( 1 + T + 61 T^{2} ) \))(\( ( 1 - 14 T + 61 T^{2} )( 1 + 13 T + 61 T^{2} ) \))(\( ( 1 + 11 T + 60 T^{2} + 671 T^{3} + 3721 T^{4} )^{2} \))(\( 1 - 27 T + 151 T^{2} + 1702 T^{3} - 6246 T^{4} - 320230 T^{5} + 2445536 T^{6} + 10370959 T^{7} - 96143610 T^{8} - 1233847964 T^{9} + 14003131220 T^{10} - 9804089745 T^{11} - 438692150218 T^{12} - 598049474445 T^{13} + 52105651269620 T^{14} - 280060044716684 T^{15} - 1331189137226010 T^{16} + 8759273609222659 T^{17} + 125994930233302496 T^{18} - 1006400538379004830 T^{19} - 1197403876981017126 T^{20} + 19903436650003707982 T^{21} + \)\(10\!\cdots\!51\)\( T^{22} - \)\(11\!\cdots\!47\)\( T^{23} + \)\(26\!\cdots\!21\)\( T^{24} \))(\( 1 - 250 T^{2} - 1082 T^{3} + 35548 T^{4} + 260866 T^{5} - 2717433 T^{6} - 36918565 T^{7} + 78765775 T^{8} + 3261082584 T^{9} + 10157211285 T^{10} - 189570224500 T^{11} - 1612043826243 T^{12} + 4675152702057 T^{13} + 127013861698650 T^{14} + 285184314825477 T^{15} - 5998415077450203 T^{16} - 43028839127234500 T^{17} + 140635132455515685 T^{18} + 2754298287701921784 T^{19} + 4058042214834294775 T^{20} - \)\(11\!\cdots\!65\)\( T^{21} - \)\(52\!\cdots\!73\)\( T^{22} + \)\(30\!\cdots\!06\)\( T^{23} + \)\(25\!\cdots\!48\)\( T^{24} - \)\(47\!\cdots\!02\)\( T^{25} - \)\(66\!\cdots\!50\)\( T^{26} + \)\(98\!\cdots\!41\)\( T^{28} \))
$67$ (\( 1 + 16 T + 67 T^{2} \))(\( 1 + 16 T + 67 T^{2} \))(\( 1 + 16 T + 67 T^{2} \))(\( 1 - 16 T + 67 T^{2} \))(\( 1 + 16 T + 67 T^{2} \))(\( 1 - 16 T + 67 T^{2} \))(\( 1 - 16 T + 67 T^{2} \))(\( 1 + 16 T + 67 T^{2} \))(\( 1 - 13 T + 102 T^{2} - 871 T^{3} + 4489 T^{4} \))(\( 1 - 3 T - 9 T^{2} + 945 T^{3} - 6138 T^{4} - 52035 T^{5} + 480724 T^{6} - 3486345 T^{7} - 27553482 T^{8} + 284221035 T^{9} - 181360089 T^{10} - 4050375321 T^{11} + 90458382169 T^{12} \))(\( 1 - 25 T + 298 T^{2} - 2382 T^{3} + 12789 T^{4} - 5772 T^{5} - 884312 T^{6} + 10995854 T^{7} - 59248904 T^{8} - 25910508 T^{9} + 3846458007 T^{10} - 47999970222 T^{11} + 402337281886 T^{12} - 2261459554225 T^{13} + 6060711605323 T^{14} \))
$71$ (\( 1 + T - 70 T^{2} + 71 T^{3} + 5041 T^{4} \))(\( 1 + 15 T + 154 T^{2} + 1065 T^{3} + 5041 T^{4} \))(\( 1 + 3 T - 62 T^{2} + 213 T^{3} + 5041 T^{4} \))(\( 1 + 15 T + 154 T^{2} + 1065 T^{3} + 5041 T^{4} \))(\( 1 + 15 T + 154 T^{2} + 1065 T^{3} + 5041 T^{4} \))(\( 1 - 13 T + 98 T^{2} - 923 T^{3} + 5041 T^{4} \))(\( 1 + 7 T - 22 T^{2} + 497 T^{3} + 5041 T^{4} \))(\( 1 + 3 T - 62 T^{2} + 213 T^{3} + 5041 T^{4} \))(\( 1 + 9 T - 7 T^{2} - 486 T^{3} - 3048 T^{4} - 34506 T^{5} - 35287 T^{6} + 3221199 T^{7} + 25411681 T^{8} \))(\( 1 - 25 T + 66 T^{2} + 2427 T^{3} - 921 T^{4} - 287069 T^{5} - 349824 T^{6} + 22779568 T^{7} + 109032831 T^{8} - 1504395045 T^{9} - 10745043831 T^{10} + 19122129269 T^{11} + 1224386041720 T^{12} + 1357671178099 T^{13} - 54165765952071 T^{14} - 538439534950995 T^{15} + 2770707519898911 T^{16} + 41099565188700368 T^{17} - 44812553722379904 T^{18} - 2610927048749145979 T^{19} - 594739002277345881 T^{20} + \)\(11\!\cdots\!37\)\( T^{21} + \)\(21\!\cdots\!66\)\( T^{22} - \)\(57\!\cdots\!75\)\( T^{23} + \)\(16\!\cdots\!41\)\( T^{24} \))(\( 1 - 22 T - 17 T^{2} + 820 T^{3} + 49824 T^{4} - 364769 T^{5} - 2625066 T^{6} - 31439174 T^{7} + 604960277 T^{8} + 1291494445 T^{9} + 13047003346 T^{10} - 519002271697 T^{11} - 392108245783 T^{12} - 2884405046697 T^{13} + 359176316454548 T^{14} - 204792758315487 T^{15} - 1976617666992103 T^{16} - 185756622065344967 T^{17} + 331546287034484626 T^{18} + 2330152184322455195 T^{19} + 77495583244626806117 T^{20} - \)\(28\!\cdots\!34\)\( T^{21} - \)\(16\!\cdots\!26\)\( T^{22} - \)\(16\!\cdots\!39\)\( T^{23} + \)\(16\!\cdots\!24\)\( T^{24} + \)\(18\!\cdots\!20\)\( T^{25} - \)\(27\!\cdots\!97\)\( T^{26} - \)\(25\!\cdots\!42\)\( T^{27} + \)\(82\!\cdots\!81\)\( T^{28} \))
$73$ (\( 1 + 5 T - 48 T^{2} + 365 T^{3} + 5329 T^{4} \))(\( 1 - 3 T - 64 T^{2} - 219 T^{3} + 5329 T^{4} \))(\( 1 - 11 T + 48 T^{2} - 803 T^{3} + 5329 T^{4} \))(\( 1 - 3 T - 64 T^{2} - 219 T^{3} + 5329 T^{4} \))(\( 1 + 11 T + 48 T^{2} + 803 T^{3} + 5329 T^{4} \))(\( 1 + 11 T + 48 T^{2} + 803 T^{3} + 5329 T^{4} \))(\( 1 - 5 T - 48 T^{2} - 365 T^{3} + 5329 T^{4} \))(\( ( 1 - 17 T + 73 T^{2} )( 1 + 10 T + 73 T^{2} ) \))(\( ( 1 - 13 T + 96 T^{2} - 949 T^{3} + 5329 T^{4} )^{2} \))(\( 1 + 8 T - 13 T^{2} - 410 T^{3} - 9332 T^{4} - 91230 T^{5} + 69722 T^{6} + 6044840 T^{7} + 55975786 T^{8} + 340615830 T^{9} + 20999164 T^{10} - 25331655699 T^{11} - 349115977166 T^{12} - 1849210866027 T^{13} + 111904544956 T^{14} + 132505348339110 T^{15} + 1589613860992426 T^{16} + 12531386088230120 T^{17} + 10551324925321658 T^{18} - 1007854166897219310 T^{19} - 7525885577555563892 T^{20} - 24137350550389844330 T^{21} - 55869135786146249437 T^{22} + \)\(25\!\cdots\!16\)\( T^{23} + \)\(22\!\cdots\!21\)\( T^{24} \))(\( 1 + 15 T - 305 T^{2} - 3966 T^{3} + 75355 T^{4} + 692695 T^{5} - 13270379 T^{6} - 81309244 T^{7} + 1857676842 T^{8} + 7051146754 T^{9} - 209843950998 T^{10} - 423003183473 T^{11} + 19720732477631 T^{12} + 12001986433031 T^{13} - 1563096084841678 T^{14} + 876145009611263 T^{15} + 105091783373295599 T^{16} - 164555429425116041 T^{17} - 5959199092833394518 T^{18} + 14617532033791559122 T^{19} + \)\(28\!\cdots\!38\)\( T^{20} - \)\(89\!\cdots\!68\)\( T^{21} - \)\(10\!\cdots\!99\)\( T^{22} + \)\(40\!\cdots\!35\)\( T^{23} + \)\(32\!\cdots\!95\)\( T^{24} - \)\(12\!\cdots\!82\)\( T^{25} - \)\(69\!\cdots\!05\)\( T^{26} + \)\(25\!\cdots\!95\)\( T^{27} + \)\(12\!\cdots\!09\)\( T^{28} \))
$79$ (\( 1 + 9 T + 2 T^{2} + 711 T^{3} + 6241 T^{4} \))(\( 1 - 3 T - 70 T^{2} - 237 T^{3} + 6241 T^{4} \))(\( 1 + 5 T - 54 T^{2} + 395 T^{3} + 6241 T^{4} \))(\( 1 - 3 T - 70 T^{2} - 237 T^{3} + 6241 T^{4} \))(\( 1 - 3 T - 70 T^{2} - 237 T^{3} + 6241 T^{4} \))(\( 1 - 11 T + 42 T^{2} - 869 T^{3} + 6241 T^{4} \))(\( ( 1 - 4 T + 79 T^{2} )( 1 + 17 T + 79 T^{2} ) \))(\( ( 1 + 4 T + 79 T^{2} )( 1 + 13 T + 79 T^{2} ) \))(\( 1 - 20 T + 175 T^{2} - 1340 T^{3} + 12784 T^{4} - 105860 T^{5} + 1092175 T^{6} - 9860780 T^{7} + 38950081 T^{8} \))(\( 1 + 2 T - 254 T^{2} + 354 T^{3} + 30644 T^{4} - 104573 T^{5} - 2255531 T^{6} + 2220835 T^{7} + 156777095 T^{8} + 785447464 T^{9} - 19907349612 T^{10} - 43426772841 T^{11} + 2128421622846 T^{12} - 3430715054439 T^{13} - 124241768928492 T^{14} + 387256232203096 T^{15} + 6106480549194695 T^{16} + 6833634547873165 T^{17} - 548291291638736651 T^{18} - 2008210374409605107 T^{19} + 46490282370776655284 T^{20} + 42427464977846884926 T^{21} - \)\(24\!\cdots\!54\)\( T^{22} + \)\(14\!\cdots\!58\)\( T^{23} + \)\(59\!\cdots\!41\)\( T^{24} \))(\( 1 - 9 T - 290 T^{2} + 971 T^{3} + 58957 T^{4} + 39935 T^{5} - 7304652 T^{6} - 31245424 T^{7} + 627140605 T^{8} + 4416045423 T^{9} - 33081571720 T^{10} - 353555917085 T^{11} + 956394409296 T^{12} + 11107915331417 T^{13} - 6705152277818 T^{14} + 877525311181943 T^{15} + 5968857508416336 T^{16} - 174316855803671315 T^{17} - 1288529898101309320 T^{18} + 13588420827116811777 T^{19} + \)\(15\!\cdots\!05\)\( T^{20} - \)\(60\!\cdots\!16\)\( T^{21} - \)\(11\!\cdots\!72\)\( T^{22} + \)\(47\!\cdots\!65\)\( T^{23} + \)\(55\!\cdots\!57\)\( T^{24} + \)\(72\!\cdots\!09\)\( T^{25} - \)\(17\!\cdots\!90\)\( T^{26} - \)\(42\!\cdots\!51\)\( T^{27} + \)\(36\!\cdots\!81\)\( T^{28} \))
$83$ (\( 1 + 9 T - 2 T^{2} + 747 T^{3} + 6889 T^{4} \))(\( 1 + 11 T + 38 T^{2} + 913 T^{3} + 6889 T^{4} \))(\( 1 - 9 T - 2 T^{2} - 747 T^{3} + 6889 T^{4} \))(\( 1 + 11 T + 38 T^{2} + 913 T^{3} + 6889 T^{4} \))(\( 1 - 7 T - 34 T^{2} - 581 T^{3} + 6889 T^{4} \))(\( 1 - 11 T + 38 T^{2} - 913 T^{3} + 6889 T^{4} \))(\( 1 + 17 T + 206 T^{2} + 1411 T^{3} + 6889 T^{4} \))(\( 1 + 15 T + 142 T^{2} + 1245 T^{3} + 6889 T^{4} \))(\( 1 - 3 T - 85 T^{2} + 216 T^{3} + 1200 T^{4} + 17928 T^{5} - 585565 T^{6} - 1715361 T^{7} + 47458321 T^{8} \))(\( 1 - 20 T + 131 T^{2} - 1714 T^{3} + 14638 T^{4} + 93416 T^{5} - 900524 T^{6} + 12623066 T^{7} - 198497842 T^{8} + 47393114 T^{9} + 2206202642 T^{10} - 56276030963 T^{11} + 1636885001364 T^{12} - 4670910569929 T^{13} + 15198530000738 T^{14} + 27098766474718 T^{15} - 9420374303443282 T^{16} + 49722770013271438 T^{17} - 294417652787745356 T^{18} + 2534941339246995832 T^{19} + 32969053694051282158 T^{20} - \)\(32\!\cdots\!42\)\( T^{21} + \)\(20\!\cdots\!19\)\( T^{22} - \)\(25\!\cdots\!40\)\( T^{23} + \)\(10\!\cdots\!61\)\( T^{24} \))(\( 1 + T - 299 T^{2} + 1040 T^{3} + 46121 T^{4} - 338171 T^{5} - 3970157 T^{6} + 54632160 T^{7} + 144189246 T^{8} - 5524282328 T^{9} + 13760244348 T^{10} + 359721042959 T^{11} - 3019786479455 T^{12} - 11121758694339 T^{13} + 306888352958254 T^{14} - 923105971630137 T^{15} - 20803309056965495 T^{16} + 205683815990397733 T^{17} + 653038093305819708 T^{18} - 21760372613398656904 T^{19} + 47141285923034589774 T^{20} + \)\(14\!\cdots\!20\)\( T^{21} - \)\(89\!\cdots\!37\)\( T^{22} - \)\(63\!\cdots\!13\)\( T^{23} + \)\(71\!\cdots\!29\)\( T^{24} + \)\(13\!\cdots\!80\)\( T^{25} - \)\(31\!\cdots\!39\)\( T^{26} + \)\(88\!\cdots\!63\)\( T^{27} + \)\(73\!\cdots\!29\)\( T^{28} \))
$89$ (\( ( 1 + 10 T + 89 T^{2} )^{2} \))(\( ( 1 - 2 T + 89 T^{2} )^{2} \))(\( ( 1 - 10 T + 89 T^{2} )^{2} \))(\( ( 1 + 6 T + 89 T^{2} )^{2} \))(\( ( 1 + 14 T + 89 T^{2} )^{2} \))(\( ( 1 - 2 T + 89 T^{2} )^{2} \))(\( ( 1 - 18 T + 89 T^{2} )^{2} \))(\( ( 1 + 6 T + 89 T^{2} )^{2} \))(\( ( 1 + 46 T^{2} + 7921 T^{4} )^{2} \))(\( ( 1 - 6 T + 278 T^{2} - 1201 T^{3} + 41023 T^{4} - 169406 T^{5} + 4354474 T^{6} - 15077134 T^{7} + 324943183 T^{8} - 846667769 T^{9} + 17442342998 T^{10} - 33504356694 T^{11} + 496981290961 T^{12} )^{2} \))(\( ( 1 + 6 T + 509 T^{2} + 2393 T^{3} + 116955 T^{4} + 439711 T^{5} + 16028883 T^{6} + 48830780 T^{7} + 1426570587 T^{8} + 3482950831 T^{9} + 82449649395 T^{10} + 150142182713 T^{11} + 2842286259541 T^{12} + 2981887745766 T^{13} + 44231334895529 T^{14} )^{2} \))
$97$ (\( 1 - 7 T - 48 T^{2} - 679 T^{3} + 9409 T^{4} \))(\( 1 + 9 T - 16 T^{2} + 873 T^{3} + 9409 T^{4} \))(\( 1 + 13 T + 72 T^{2} + 1261 T^{3} + 9409 T^{4} \))(\( 1 + 13 T + 72 T^{2} + 1261 T^{3} + 9409 T^{4} \))(\( 1 - 17 T + 192 T^{2} - 1649 T^{3} + 9409 T^{4} \))(\( 1 + 7 T - 48 T^{2} + 679 T^{3} + 9409 T^{4} \))(\( 1 - T - 96 T^{2} - 97 T^{3} + 9409 T^{4} \))(\( 1 + 17 T + 192 T^{2} + 1649 T^{3} + 9409 T^{4} \))(\( ( 1 - T - 96 T^{2} - 97 T^{3} + 9409 T^{4} )^{2} \))(\( 1 + T - 74 T^{2} - 3085 T^{3} + 512 T^{4} + 161294 T^{5} + 5154103 T^{6} + 4425829 T^{7} - 201336043 T^{8} - 7008527843 T^{9} - 13501932242 T^{10} + 195913282487 T^{11} + 7566580882136 T^{12} + 19003588401239 T^{13} - 127039680464978 T^{14} - 6396494132054339 T^{15} - 17824135126175083 T^{16} + 38006099542298053 T^{17} + 4293223509520573687 T^{18} + 13032278496612758222 T^{19} + 4012766000321004032 T^{20} - \)\(23\!\cdots\!45\)\( T^{21} - \)\(54\!\cdots\!26\)\( T^{22} + \)\(71\!\cdots\!53\)\( T^{23} + \)\(69\!\cdots\!41\)\( T^{24} \))(\( 1 - 16 T - 305 T^{2} + 4468 T^{3} + 66659 T^{4} - 627781 T^{5} - 13361122 T^{6} + 73927187 T^{7} + 2109835734 T^{8} - 7502842347 T^{9} - 269438721123 T^{10} + 493586727995 T^{11} + 30832175504259 T^{12} - 14324354895126 T^{13} - 3192872878082918 T^{14} - 1389462424827222 T^{15} + 290099939319572931 T^{16} + 450483279799380635 T^{17} - 23853216254578702563 T^{18} - 64429460128317463179 T^{19} + \)\(17\!\cdots\!86\)\( T^{20} + \)\(59\!\cdots\!31\)\( T^{21} - \)\(10\!\cdots\!42\)\( T^{22} - \)\(47\!\cdots\!77\)\( T^{23} + \)\(49\!\cdots\!91\)\( T^{24} + \)\(31\!\cdots\!04\)\( T^{25} - \)\(21\!\cdots\!05\)\( T^{26} - \)\(10\!\cdots\!32\)\( T^{27} + \)\(65\!\cdots\!69\)\( T^{28} \))
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