Properties

Label 688.2.i
Level 688
Weight 2
Character orbit i
Rep. character \(\chi_{688}(49,\cdot)\)
Character field \(\Q(\zeta_{3})\)
Dimension 42
Newform subspaces 9
Sturm bound 176
Trace bound 5

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Defining parameters

Level: \( N \) \(=\) \( 688 = 2^{4} \cdot 43 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 688.i (of order \(3\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 43 \)
Character field: \(\Q(\zeta_{3})\)
Newform subspaces: \( 9 \)
Sturm bound: \(176\)
Trace bound: \(5\)
Distinguishing \(T_p\): \(3\), \(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(688, [\chi])\).

Total New Old
Modular forms 188 46 142
Cusp forms 164 42 122
Eisenstein series 24 4 20

Trace form

\( 42q + 3q^{3} - q^{5} - 3q^{7} - 24q^{9} + O(q^{10}) \) \( 42q + 3q^{3} - q^{5} - 3q^{7} - 24q^{9} + 12q^{11} - q^{13} - 5q^{15} + q^{17} + 11q^{19} - 6q^{21} - 3q^{23} - 18q^{25} - 18q^{27} - q^{29} - 7q^{31} - 4q^{33} + 6q^{35} + 7q^{37} + 18q^{39} + 4q^{45} - 8q^{47} - 20q^{49} - 26q^{51} + 7q^{53} + 24q^{55} + 13q^{57} - 24q^{59} - q^{61} - 28q^{63} + 6q^{65} + 33q^{67} + 9q^{69} - 3q^{71} + q^{73} - 36q^{75} + 45q^{79} - 29q^{81} - 5q^{83} + 6q^{85} + 78q^{87} - 11q^{89} + 27q^{91} + 5q^{93} - 21q^{95} - 40q^{97} - 16q^{99} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(688, [\chi])\) into newform subspaces

Label Dim. \(A\) Field CM Traces $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\)
688.2.i.a \(2\) \(5.494\) \(\Q(\sqrt{-3}) \) None \(0\) \(-3\) \(-1\) \(1\) \(q+(-3+3\zeta_{6})q^{3}+(-1+\zeta_{6})q^{5}+\zeta_{6}q^{7}+\cdots\)
688.2.i.b \(2\) \(5.494\) \(\Q(\sqrt{-3}) \) None \(0\) \(-1\) \(1\) \(5\) \(q+(-1+\zeta_{6})q^{3}+(1-\zeta_{6})q^{5}+5\zeta_{6}q^{7}+\cdots\)
688.2.i.c \(2\) \(5.494\) \(\Q(\sqrt{-3}) \) None \(0\) \(1\) \(-3\) \(-1\) \(q+(1-\zeta_{6})q^{3}+(-3+3\zeta_{6})q^{5}-\zeta_{6}q^{7}+\cdots\)
688.2.i.d \(2\) \(5.494\) \(\Q(\sqrt{-3}) \) None \(0\) \(1\) \(1\) \(3\) \(q+(1-\zeta_{6})q^{3}+(1-\zeta_{6})q^{5}+3\zeta_{6}q^{7}+\cdots\)
688.2.i.e \(4\) \(5.494\) \(\Q(\sqrt{-3}, \sqrt{5})\) None \(0\) \(-3\) \(2\) \(-4\) \(q+(-1-\beta _{1}-\beta _{3})q^{3}+(2-2\beta _{1}+2\beta _{3})q^{5}+\cdots\)
688.2.i.f \(4\) \(5.494\) \(\Q(\sqrt{-3}, \sqrt{17})\) None \(0\) \(1\) \(-3\) \(3\) \(q+\beta _{1}q^{3}+(\beta _{1}-2\beta _{2})q^{5}+(2+\beta _{1}-2\beta _{2}+\cdots)q^{7}+\cdots\)
688.2.i.g \(8\) \(5.494\) \(\mathbb{Q}[x]/(x^{8} - \cdots)\) None \(0\) \(-1\) \(0\) \(-4\) \(q+(\beta _{4}-\beta _{6})q^{3}+\beta _{5}q^{5}+(-1+\beta _{2}+\cdots)q^{7}+\cdots\)
688.2.i.h \(8\) \(5.494\) \(\mathbb{Q}[x]/(x^{8} - \cdots)\) None \(0\) \(3\) \(0\) \(-2\) \(q+(1-\beta _{1}+\beta _{2})q^{3}-\beta _{5}q^{5}+(\beta _{4}+\beta _{7})q^{7}+\cdots\)
688.2.i.i \(10\) \(5.494\) \(\mathbb{Q}[x]/(x^{10} + \cdots)\) None \(0\) \(5\) \(2\) \(-4\) \(q+(1-\beta _{1}+\beta _{7})q^{3}+(\beta _{1}-\beta _{6})q^{5}+(-\beta _{1}+\cdots)q^{7}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(688, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(688, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(43, [\chi])\)\(^{\oplus 5}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(86, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(172, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(344, [\chi])\)\(^{\oplus 2}\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ (\( ( 1 + 3 T^{2} )( 1 + 3 T + 3 T^{2} ) \))(\( 1 + T - 2 T^{2} + 3 T^{3} + 9 T^{4} \))(\( 1 - T - 2 T^{2} - 3 T^{3} + 9 T^{4} \))(\( 1 - T - 2 T^{2} - 3 T^{3} + 9 T^{4} \))(\( 1 + 3 T + 2 T^{2} + 3 T^{3} + 13 T^{4} + 9 T^{5} + 18 T^{6} + 81 T^{7} + 81 T^{8} \))(\( 1 - T - T^{2} + 4 T^{3} - 8 T^{4} + 12 T^{5} - 9 T^{6} - 27 T^{7} + 81 T^{8} \))(\( 1 + T - T^{2} - 10 T^{3} - 15 T^{4} + 4 T^{5} + 28 T^{6} + 51 T^{7} - 2 T^{8} + 153 T^{9} + 252 T^{10} + 108 T^{11} - 1215 T^{12} - 2430 T^{13} - 729 T^{14} + 2187 T^{15} + 6561 T^{16} \))(\( 1 - 3 T + T^{2} + 4 T^{3} - 7 T^{4} + 20 T^{5} - 18 T^{6} - 31 T^{7} + 82 T^{8} - 93 T^{9} - 162 T^{10} + 540 T^{11} - 567 T^{12} + 972 T^{13} + 729 T^{14} - 6561 T^{15} + 6561 T^{16} \))(\( 1 - 5 T + 8 T^{2} + 9 T^{3} - 51 T^{4} + 55 T^{5} + 74 T^{6} - 269 T^{7} + 229 T^{8} + 354 T^{9} - 1202 T^{10} + 1062 T^{11} + 2061 T^{12} - 7263 T^{13} + 5994 T^{14} + 13365 T^{15} - 37179 T^{16} + 19683 T^{17} + 52488 T^{18} - 98415 T^{19} + 59049 T^{20} \))
$5$ (\( 1 + T - 4 T^{2} + 5 T^{3} + 25 T^{4} \))(\( 1 - T - 4 T^{2} - 5 T^{3} + 25 T^{4} \))(\( 1 + 3 T + 4 T^{2} + 15 T^{3} + 25 T^{4} \))(\( 1 - T - 4 T^{2} - 5 T^{3} + 25 T^{4} \))(\( 1 - 2 T - 2 T^{2} + 8 T^{3} - 9 T^{4} + 40 T^{5} - 50 T^{6} - 250 T^{7} + 625 T^{8} \))(\( ( 1 - 7 T^{2} + 25 T^{4} )( 1 + 3 T + 8 T^{2} + 15 T^{3} + 25 T^{4} ) \))(\( 1 - 5 T^{2} - 20 T^{3} - 3 T^{4} + 100 T^{5} + 210 T^{6} - 220 T^{7} - 966 T^{8} - 1100 T^{9} + 5250 T^{10} + 12500 T^{11} - 1875 T^{12} - 62500 T^{13} - 78125 T^{14} + 390625 T^{16} \))(\( 1 - 9 T^{2} + 12 T^{3} + 29 T^{4} - 84 T^{5} + 18 T^{6} + 228 T^{7} - 126 T^{8} + 1140 T^{9} + 450 T^{10} - 10500 T^{11} + 18125 T^{12} + 37500 T^{13} - 140625 T^{14} + 390625 T^{16} \))(\( 1 - 2 T - 6 T^{2} + 28 T^{3} - 29 T^{4} - 48 T^{5} + 312 T^{6} - 714 T^{7} + 493 T^{8} + 2748 T^{9} - 10406 T^{10} + 13740 T^{11} + 12325 T^{12} - 89250 T^{13} + 195000 T^{14} - 150000 T^{15} - 453125 T^{16} + 2187500 T^{17} - 2343750 T^{18} - 3906250 T^{19} + 9765625 T^{20} \))
$7$ (\( ( 1 - 5 T + 7 T^{2} )( 1 + 4 T + 7 T^{2} ) \))(\( ( 1 - 4 T + 7 T^{2} )( 1 - T + 7 T^{2} ) \))(\( ( 1 - 4 T + 7 T^{2} )( 1 + 5 T + 7 T^{2} ) \))(\( 1 - 3 T + 2 T^{2} - 21 T^{3} + 49 T^{4} \))(\( 1 + 4 T + 3 T^{2} - 4 T^{3} + 8 T^{4} - 28 T^{5} + 147 T^{6} + 1372 T^{7} + 2401 T^{8} \))(\( 1 - 3 T - 3 T^{2} + 6 T^{3} + 32 T^{4} + 42 T^{5} - 147 T^{6} - 1029 T^{7} + 2401 T^{8} \))(\( ( 1 - 4 T + 7 T^{2} )^{4}( 1 + 5 T + 7 T^{2} )^{4} \))(\( 1 + 2 T - 2 T^{2} - 28 T^{3} - 63 T^{4} + 24 T^{5} + 354 T^{6} + 774 T^{7} - 740 T^{8} + 5418 T^{9} + 17346 T^{10} + 8232 T^{11} - 151263 T^{12} - 470596 T^{13} - 235298 T^{14} + 1647086 T^{15} + 5764801 T^{16} \))(\( 1 + 4 T - 3 T^{2} - 16 T^{3} - 156 T^{5} - 313 T^{6} + 684 T^{7} + 1547 T^{8} + 4020 T^{9} + 18432 T^{10} + 28140 T^{11} + 75803 T^{12} + 234612 T^{13} - 751513 T^{14} - 2621892 T^{15} - 13176688 T^{17} - 17294403 T^{18} + 161414428 T^{19} + 282475249 T^{20} \))
$11$ (\( ( 1 + 4 T + 11 T^{2} )^{2} \))(\( ( 1 + 11 T^{2} )^{2} \))(\( ( 1 + 11 T^{2} )^{2} \))(\( ( 1 + 11 T^{2} )^{2} \))(\( ( 1 - 5 T + 27 T^{2} - 55 T^{3} + 121 T^{4} )^{2} \))(\( ( 1 - T + 18 T^{2} - 11 T^{3} + 121 T^{4} )^{2} \))(\( ( 1 + T + 3 T^{2} + 17 T^{3} + 112 T^{4} + 187 T^{5} + 363 T^{6} + 1331 T^{7} + 14641 T^{8} )^{2} \))(\( ( 1 - T + 19 T^{2} + 27 T^{3} + 140 T^{4} + 297 T^{5} + 2299 T^{6} - 1331 T^{7} + 14641 T^{8} )^{2} \))(\( ( 1 - 4 T + 33 T^{2} - 119 T^{3} + 628 T^{4} - 1666 T^{5} + 6908 T^{6} - 14399 T^{7} + 43923 T^{8} - 58564 T^{9} + 161051 T^{10} )^{2} \))
$13$ (\( ( 1 - 2 T + 13 T^{2} )( 1 + 7 T + 13 T^{2} ) \))(\( 1 + 3 T - 4 T^{2} + 39 T^{3} + 169 T^{4} \))(\( 1 - T - 12 T^{2} - 13 T^{3} + 169 T^{4} \))(\( ( 1 - 7 T + 13 T^{2} )( 1 + 2 T + 13 T^{2} ) \))(\( 1 + 5 T - 6 T^{2} + 25 T^{3} + 467 T^{4} + 325 T^{5} - 1014 T^{6} + 10985 T^{7} + 28561 T^{8} \))(\( 1 + 9 T + 39 T^{2} + 144 T^{3} + 542 T^{4} + 1872 T^{5} + 6591 T^{6} + 19773 T^{7} + 28561 T^{8} \))(\( 1 - 9 T + 9 T^{2} + 56 T^{3} + 485 T^{4} - 2362 T^{5} - 3176 T^{6} - 9253 T^{7} + 184466 T^{8} - 120289 T^{9} - 536744 T^{10} - 5189314 T^{11} + 13852085 T^{12} + 20792408 T^{13} + 43441281 T^{14} - 564736653 T^{15} + 815730721 T^{16} \))(\( 1 - 5 T - 11 T^{2} + 4 T^{3} + 357 T^{4} + 282 T^{5} - 2592 T^{6} + 639 T^{7} - 18326 T^{8} + 8307 T^{9} - 438048 T^{10} + 619554 T^{11} + 10196277 T^{12} + 1485172 T^{13} - 53094899 T^{14} - 313742585 T^{15} + 815730721 T^{16} \))(\( 1 - T - 36 T^{2} + 21 T^{3} + 561 T^{4} - 91 T^{5} - 8304 T^{6} + 5005 T^{7} + 143951 T^{8} - 66842 T^{9} - 2083754 T^{10} - 868946 T^{11} + 24327719 T^{12} + 10995985 T^{13} - 237170544 T^{14} - 33787663 T^{15} + 2707839849 T^{16} + 1317718857 T^{17} - 29366305956 T^{18} - 10604499373 T^{19} + 137858491849 T^{20} \))
$17$ (\( 1 - T - 16 T^{2} - 17 T^{3} + 289 T^{4} \))(\( 1 + 3 T - 8 T^{2} + 51 T^{3} + 289 T^{4} \))(\( 1 + 3 T - 8 T^{2} + 51 T^{3} + 289 T^{4} \))(\( 1 + 3 T - 8 T^{2} + 51 T^{3} + 289 T^{4} \))(\( 1 + T - 2 T^{2} - 31 T^{3} - 297 T^{4} - 527 T^{5} - 578 T^{6} + 4913 T^{7} + 83521 T^{8} \))(\( 1 - 4 T - 5 T^{2} + 52 T^{3} - 120 T^{4} + 884 T^{5} - 1445 T^{6} - 19652 T^{7} + 83521 T^{8} \))(\( 1 - 5 T - 31 T^{2} + 208 T^{3} + 569 T^{4} - 4342 T^{5} - 4268 T^{6} + 30423 T^{7} + 58122 T^{8} + 517191 T^{9} - 1233452 T^{10} - 21332246 T^{11} + 47523449 T^{12} + 295330256 T^{13} - 748264639 T^{14} - 2051693365 T^{15} + 6975757441 T^{16} \))(\( 1 - T - 51 T^{2} + 20 T^{3} + 1481 T^{4} - 110 T^{5} - 30984 T^{6} + 975 T^{7} + 531786 T^{8} + 16575 T^{9} - 8954376 T^{10} - 540430 T^{11} + 123694601 T^{12} + 28397140 T^{13} - 1231016019 T^{14} - 410338673 T^{15} + 6975757441 T^{16} \))(\( 1 - 8 T^{2} - 94 T^{3} + 153 T^{4} + 415 T^{5} - 105 T^{6} + 8776 T^{7} - 21112 T^{8} + 247415 T^{9} - 1562530 T^{10} + 4206055 T^{11} - 6101368 T^{12} + 43116488 T^{13} - 8769705 T^{14} + 589240655 T^{15} + 3693048057 T^{16} - 38571835262 T^{17} - 55806059528 T^{18} + 2015993900449 T^{20} \))
$19$ (\( 1 + 3 T - 10 T^{2} + 57 T^{3} + 361 T^{4} \))(\( ( 1 - 7 T + 19 T^{2} )( 1 + 8 T + 19 T^{2} ) \))(\( ( 1 - T + 19 T^{2} )( 1 + 8 T + 19 T^{2} ) \))(\( ( 1 - 8 T + 19 T^{2} )( 1 + 7 T + 19 T^{2} ) \))(\( 1 - 2 T - 30 T^{2} + 8 T^{3} + 719 T^{4} + 152 T^{5} - 10830 T^{6} - 13718 T^{7} + 130321 T^{8} \))(\( 1 + 4 T - 9 T^{2} - 52 T^{3} - 40 T^{4} - 988 T^{5} - 3249 T^{6} + 27436 T^{7} + 130321 T^{8} \))(\( 1 - 16 T + 99 T^{2} - 436 T^{3} + 2793 T^{4} - 14752 T^{5} + 51310 T^{6} - 271292 T^{7} + 1553266 T^{8} - 5154548 T^{9} + 18522910 T^{10} - 101183968 T^{11} + 363986553 T^{12} - 1079579164 T^{13} + 4657542219 T^{14} - 14301947824 T^{15} + 16983563041 T^{16} \))(\( 1 - 6 T + 3 T^{2} + 218 T^{3} - 1239 T^{4} + 2144 T^{5} + 7738 T^{6} - 76440 T^{7} + 314782 T^{8} - 1452360 T^{9} + 2793418 T^{10} + 14705696 T^{11} - 161467719 T^{12} + 539789582 T^{13} + 141137643 T^{14} - 5363230434 T^{15} + 16983563041 T^{16} \))(\( 1 - T - 79 T^{2} + 86 T^{3} + 3544 T^{4} - 3530 T^{5} - 112013 T^{6} + 76951 T^{7} + 2783587 T^{8} - 694104 T^{9} - 57465104 T^{10} - 13187976 T^{11} + 1004874907 T^{12} + 527806909 T^{13} - 14597646173 T^{14} - 8740629470 T^{15} + 166730602264 T^{16} + 76872969554 T^{17} - 1341701480239 T^{18} - 322687697779 T^{19} + 6131066257801 T^{20} \))
$23$ (\( 1 - 3 T - 14 T^{2} - 69 T^{3} + 529 T^{4} \))(\( 1 + T - 22 T^{2} + 23 T^{3} + 529 T^{4} \))(\( 1 + 3 T - 14 T^{2} + 69 T^{3} + 529 T^{4} \))(\( 1 + 7 T + 26 T^{2} + 161 T^{3} + 529 T^{4} \))(\( 1 - 11 T + 46 T^{2} - 319 T^{3} + 2313 T^{4} - 7337 T^{5} + 24334 T^{6} - 133837 T^{7} + 279841 T^{8} \))(\( 1 - 3 T - T^{2} + 108 T^{3} - 636 T^{4} + 2484 T^{5} - 529 T^{6} - 36501 T^{7} + 279841 T^{8} \))(\( 1 - 3 T - 65 T^{2} + 62 T^{3} + 2733 T^{4} + 128 T^{5} - 82800 T^{6} + 415 T^{7} + 1947078 T^{8} + 9545 T^{9} - 43801200 T^{10} + 1557376 T^{11} + 764805453 T^{12} + 399053266 T^{13} - 9622332785 T^{14} - 10214476341 T^{15} + 78310985281 T^{16} \))(\( 1 - 3 T - 23 T^{2} + 300 T^{3} - 443 T^{4} - 5652 T^{5} + 31462 T^{6} + 31317 T^{7} - 742094 T^{8} + 720291 T^{9} + 16643398 T^{10} - 68767884 T^{11} - 123969563 T^{12} + 1930902900 T^{13} - 3404825447 T^{14} - 10214476341 T^{15} + 78310985281 T^{16} \))(\( 1 + 15 T + 68 T^{2} + 105 T^{3} + 997 T^{4} + 7511 T^{5} + 438 T^{6} - 180229 T^{7} - 1070987 T^{8} - 5270778 T^{9} - 26078778 T^{10} - 121227894 T^{11} - 566552123 T^{12} - 2192846243 T^{13} + 122570358 T^{14} + 48343372273 T^{15} + 147591781333 T^{16} + 357506671935 T^{17} + 5325146999108 T^{18} + 27017289921945 T^{19} + 41426511213649 T^{20} \))
$29$ (\( 1 + T - 28 T^{2} + 29 T^{3} + 841 T^{4} \))(\( 1 - 5 T - 4 T^{2} - 145 T^{3} + 841 T^{4} \))(\( 1 - 9 T + 52 T^{2} - 261 T^{3} + 841 T^{4} \))(\( 1 + 3 T - 20 T^{2} + 87 T^{3} + 841 T^{4} \))(\( ( 1 + 3 T - 20 T^{2} + 87 T^{3} + 841 T^{4} )^{2} \))(\( 1 + 5 T - T^{2} - 160 T^{3} - 890 T^{4} - 4640 T^{5} - 841 T^{6} + 121945 T^{7} + 707281 T^{8} \))(\( 1 - 54 T^{2} + 208 T^{3} + 1173 T^{4} - 8632 T^{5} + 7522 T^{6} + 144456 T^{7} - 435252 T^{8} + 4189224 T^{9} + 6326002 T^{10} - 210525848 T^{11} + 829640613 T^{12} + 4266318992 T^{13} - 32120459334 T^{14} + 500246412961 T^{16} \))(\( 1 + 8 T - 26 T^{2} - 432 T^{3} - 247 T^{4} + 8720 T^{5} + 13014 T^{6} - 52424 T^{7} - 38972 T^{8} - 1520296 T^{9} + 10944774 T^{10} + 212672080 T^{11} - 174698407 T^{12} - 8860816368 T^{13} - 15465406346 T^{14} + 137999010472 T^{15} + 500246412961 T^{16} \))(\( 1 - 8 T - 51 T^{2} + 616 T^{3} + 1164 T^{4} - 24720 T^{5} + 3979 T^{6} + 583672 T^{7} - 1056129 T^{8} - 5684680 T^{9} + 33423336 T^{10} - 164855720 T^{11} - 888204489 T^{12} + 14235176408 T^{13} + 2814271099 T^{14} - 507035603280 T^{15} + 692374345644 T^{16} + 10625923806344 T^{17} - 25512567061011 T^{18} - 116057167806952 T^{19} + 420707233300201 T^{20} \))
$31$ (\( ( 1 - 11 T + 31 T^{2} )( 1 + 4 T + 31 T^{2} ) \))(\( 1 + 5 T - 6 T^{2} + 155 T^{3} + 961 T^{4} \))(\( ( 1 - 4 T + 31 T^{2} )( 1 + 11 T + 31 T^{2} ) \))(\( 1 - 5 T - 6 T^{2} - 155 T^{3} + 961 T^{4} \))(\( ( 1 - 31 T^{2} + 961 T^{4} )^{2} \))(\( 1 - 13 T + 69 T^{2} - 494 T^{3} + 3800 T^{4} - 15314 T^{5} + 66309 T^{6} - 387283 T^{7} + 923521 T^{8} \))(\( 1 + 14 T + 55 T^{2} + 74 T^{3} + 785 T^{4} + 1112 T^{5} - 56034 T^{6} - 484612 T^{7} - 2734974 T^{8} - 15022972 T^{9} - 53848674 T^{10} + 33127592 T^{11} + 724963985 T^{12} + 2118557174 T^{13} + 48812702455 T^{14} + 385176597554 T^{15} + 852891037441 T^{16} \))(\( 1 + 16 T + 71 T^{2} - 16 T^{3} + 933 T^{4} + 17920 T^{5} + 77126 T^{6} + 366048 T^{7} + 2747038 T^{8} + 11347488 T^{9} + 74118086 T^{10} + 533854720 T^{11} + 861645093 T^{12} - 458066416 T^{13} + 63012761351 T^{14} + 440201825776 T^{15} + 852891037441 T^{16} \))(\( 1 - 10 T - 10 T^{2} + 336 T^{3} - 281 T^{4} - 768 T^{5} + 7796 T^{6} - 358866 T^{7} + 1307797 T^{8} + 7100848 T^{9} - 67967182 T^{10} + 220126288 T^{11} + 1256792917 T^{12} - 10690977006 T^{13} + 7199769716 T^{14} - 21987187968 T^{15} - 249388534361 T^{16} + 9244238341296 T^{17} - 8528910374410 T^{18} - 264396221606710 T^{19} + 819628286980801 T^{20} \))
$37$ (\( 1 + 5 T - 12 T^{2} + 185 T^{3} + 1369 T^{4} \))(\( ( 1 - 11 T + 37 T^{2} )( 1 + 10 T + 37 T^{2} ) \))(\( ( 1 + T + 37 T^{2} )( 1 + 10 T + 37 T^{2} ) \))(\( 1 - 9 T + 44 T^{2} - 333 T^{3} + 1369 T^{4} \))(\( 1 - 3 T - 56 T^{2} + 27 T^{3} + 2523 T^{4} + 999 T^{5} - 76664 T^{6} - 151959 T^{7} + 1874161 T^{8} \))(\( 1 - 5 T - 51 T^{2} - 10 T^{3} + 3482 T^{4} - 370 T^{5} - 69819 T^{6} - 253265 T^{7} + 1874161 T^{8} \))(\( 1 - 9 T - 45 T^{2} + 554 T^{3} + 1363 T^{4} - 13066 T^{5} - 95270 T^{6} + 23185 T^{7} + 6085838 T^{8} + 857845 T^{9} - 130424630 T^{10} - 661832098 T^{11} + 2554481443 T^{12} + 38416552178 T^{13} - 115457688405 T^{14} - 854386894197 T^{15} + 3512479453921 T^{16} \))(\( 1 + 11 T - 49 T^{2} - 578 T^{3} + 5595 T^{4} + 36866 T^{5} - 207502 T^{6} - 235323 T^{7} + 12232534 T^{8} - 8706951 T^{9} - 284070238 T^{10} + 1867373498 T^{11} + 10485930795 T^{12} - 40080807146 T^{13} - 125720594041 T^{14} + 1044250648463 T^{15} + 3512479453921 T^{16} \))(\( 1 - 7 T - 88 T^{2} + 347 T^{3} + 6219 T^{4} - 6921 T^{5} - 280430 T^{6} - 179321 T^{7} + 9749429 T^{8} + 8035642 T^{9} - 348448214 T^{10} + 297318754 T^{11} + 13346968301 T^{12} - 9083146613 T^{13} - 525570969230 T^{14} - 479929526397 T^{15} + 15956252537571 T^{16} + 32941361365151 T^{17} - 309098191945048 T^{18} - 909732178565539 T^{19} + 4808584372417849 T^{20} \))
$41$ (\( ( 1 + 2 T + 41 T^{2} )^{2} \))(\( ( 1 - 6 T + 41 T^{2} )^{2} \))(\( ( 1 - 6 T + 41 T^{2} )^{2} \))(\( ( 1 + 10 T + 41 T^{2} )^{2} \))(\( ( 1 - 10 T + 87 T^{2} - 410 T^{3} + 1681 T^{4} )^{2} \))(\( ( 1 - 7 T + 56 T^{2} - 287 T^{3} + 1681 T^{4} )^{2} \))(\( ( 1 + 2 T + 109 T^{2} + 290 T^{3} + 5660 T^{4} + 11890 T^{5} + 183229 T^{6} + 137842 T^{7} + 2825761 T^{8} )^{2} \))(\( ( 1 + 14 T + 201 T^{2} + 1602 T^{3} + 12832 T^{4} + 65682 T^{5} + 337881 T^{6} + 964894 T^{7} + 2825761 T^{8} )^{2} \))(\( ( 1 + T + 100 T^{2} + 215 T^{3} + 6223 T^{4} + 10136 T^{5} + 255143 T^{6} + 361415 T^{7} + 6892100 T^{8} + 2825761 T^{9} + 115856201 T^{10} )^{2} \))
$43$ (\( 1 + 8 T + 43 T^{2} \))(\( 1 - 8 T + 43 T^{2} \))(\( 1 + 8 T + 43 T^{2} \))(\( 1 - 8 T + 43 T^{2} \))(\( ( 1 - 13 T + 43 T^{2} )^{2} \))(\( 1 + 11 T + 78 T^{2} + 473 T^{3} + 1849 T^{4} \))(\( 1 + 4 T + 67 T^{2} + 692 T^{3} + 2096 T^{4} + 29756 T^{5} + 123883 T^{6} + 318028 T^{7} + 3418801 T^{8} \))(\( 1 + 35 T^{2} + 144 T^{3} - 372 T^{4} + 6192 T^{5} + 64715 T^{6} + 3418801 T^{8} \))(\( 1 + 11 T + 114 T^{2} + 1101 T^{3} + 8757 T^{4} + 56040 T^{5} + 376551 T^{6} + 2035749 T^{7} + 9063798 T^{8} + 37606811 T^{9} + 147008443 T^{10} \))
$47$ (\( ( 1 + 12 T + 47 T^{2} )^{2} \))(\( ( 1 + 47 T^{2} )^{2} \))(\( ( 1 + 47 T^{2} )^{2} \))(\( ( 1 - 8 T + 47 T^{2} )^{2} \))(\( ( 1 + 9 T + 103 T^{2} + 423 T^{3} + 2209 T^{4} )^{2} \))(\( ( 1 - 6 T + 86 T^{2} - 282 T^{3} + 2209 T^{4} )^{2} \))(\( ( 1 - T + 147 T^{2} - 125 T^{3} + 9688 T^{4} - 5875 T^{5} + 324723 T^{6} - 103823 T^{7} + 4879681 T^{8} )^{2} \))(\( ( 1 - 3 T + 63 T^{2} - 231 T^{3} + 5536 T^{4} - 10857 T^{5} + 139167 T^{6} - 311469 T^{7} + 4879681 T^{8} )^{2} \))(\( ( 1 + T - 12 T^{2} + 22 T^{3} + 1039 T^{4} + 11346 T^{5} + 48833 T^{6} + 48598 T^{7} - 1245876 T^{8} + 4879681 T^{9} + 229345007 T^{10} )^{2} \))
$53$ (\( 1 - 7 T - 4 T^{2} - 371 T^{3} + 2809 T^{4} \))(\( 1 - 5 T - 28 T^{2} - 265 T^{3} + 2809 T^{4} \))(\( 1 - 9 T + 28 T^{2} - 477 T^{3} + 2809 T^{4} \))(\( 1 - 5 T - 28 T^{2} - 265 T^{3} + 2809 T^{4} \))(\( 1 - 5 T - 86 T^{2} - 25 T^{3} + 8187 T^{4} - 1325 T^{5} - 241574 T^{6} - 744385 T^{7} + 7890481 T^{8} \))(\( 1 + 17 T + 115 T^{2} + 1156 T^{3} + 12150 T^{4} + 61268 T^{5} + 323035 T^{6} + 2530909 T^{7} + 7890481 T^{8} \))(\( 1 + 9 T - 83 T^{2} - 940 T^{3} + 4389 T^{4} + 52706 T^{5} - 103740 T^{6} - 1458815 T^{7} - 1291038 T^{8} - 77317195 T^{9} - 291405660 T^{10} + 7846711162 T^{11} + 34631321109 T^{12} - 393103763420 T^{13} - 1839641973707 T^{14} + 10572400258533 T^{15} + 62259690411361 T^{16} \))(\( 1 - 7 T - 75 T^{2} - 268 T^{3} + 7373 T^{4} + 29542 T^{5} - 56640 T^{6} - 1308051 T^{7} - 6416310 T^{8} - 69326703 T^{9} - 159101760 T^{10} + 4398124334 T^{11} + 58176516413 T^{12} - 112076392124 T^{13} - 1662327084675 T^{14} - 8222977978859 T^{15} + 62259690411361 T^{16} \))(\( 1 + 5 T - 180 T^{2} - 649 T^{3} + 18809 T^{4} + 43175 T^{5} - 1493000 T^{6} - 2055705 T^{7} + 96588391 T^{8} + 45226898 T^{9} - 5434118922 T^{10} + 2397025594 T^{11} + 271316790319 T^{12} - 306047193285 T^{13} - 11780488133000 T^{14} + 18055590410275 T^{15} + 416889468475361 T^{16} - 762387529754213 T^{17} - 11206744274044980 T^{18} + 16498817959010665 T^{19} + 174887470365513049 T^{20} \))
$59$ (\( ( 1 + 12 T + 59 T^{2} )^{2} \))(\( ( 1 + 4 T + 59 T^{2} )^{2} \))(\( ( 1 - 12 T + 59 T^{2} )^{2} \))(\( ( 1 + 12 T + 59 T^{2} )^{2} \))(\( ( 1 - T + 87 T^{2} - 59 T^{3} + 3481 T^{4} )^{2} \))(\( ( 1 - 5 T + 18 T^{2} - 295 T^{3} + 3481 T^{4} )^{2} \))(\( ( 1 - T + 215 T^{2} - 201 T^{3} + 18424 T^{4} - 11859 T^{5} + 748415 T^{6} - 205379 T^{7} + 12117361 T^{8} )^{2} \))(\( ( 1 + 13 T + 235 T^{2} + 2181 T^{3} + 20624 T^{4} + 128679 T^{5} + 818035 T^{6} + 2669927 T^{7} + 12117361 T^{8} )^{2} \))(\( ( 1 - 10 T + 195 T^{2} - 1159 T^{3} + 13318 T^{4} - 63318 T^{5} + 785762 T^{6} - 4034479 T^{7} + 40048905 T^{8} - 121173610 T^{9} + 714924299 T^{10} )^{2} \))
$61$ (\( 1 - 15 T + 164 T^{2} - 915 T^{3} + 3721 T^{4} \))(\( 1 - 5 T - 36 T^{2} - 305 T^{3} + 3721 T^{4} \))(\( ( 1 - 14 T + 61 T^{2} )( 1 + 13 T + 61 T^{2} ) \))(\( ( 1 - 14 T + 61 T^{2} )( 1 + T + 61 T^{2} ) \))(\( 1 - T - 110 T^{2} + 11 T^{3} + 8539 T^{4} + 671 T^{5} - 409310 T^{6} - 226981 T^{7} + 13845841 T^{8} \))(\( 1 + T - 117 T^{2} - 4 T^{3} + 10142 T^{4} - 244 T^{5} - 435357 T^{6} + 226981 T^{7} + 13845841 T^{8} \))(\( 1 - 15 T - 51 T^{2} + 916 T^{3} + 12205 T^{4} - 70406 T^{5} - 1058420 T^{6} + 456233 T^{7} + 93661490 T^{8} + 27830213 T^{9} - 3938380820 T^{10} - 15980824286 T^{11} + 168988489405 T^{12} + 773650211716 T^{13} - 2627539092411 T^{14} - 47141142540315 T^{15} + 191707312997281 T^{16} \))(\( 1 + 29 T + 305 T^{2} + 2656 T^{3} + 41301 T^{4} + 418166 T^{5} + 2426828 T^{6} + 23998461 T^{7} + 260930650 T^{8} + 1463906121 T^{9} + 9030226988 T^{10} + 94915736846 T^{11} + 571847079141 T^{12} + 2243247775456 T^{13} + 15713714180105 T^{14} + 91139542244609 T^{15} + 191707312997281 T^{16} \))(\( 1 + 21 T + 92 T^{2} - 1337 T^{3} - 14127 T^{4} - 2729 T^{5} + 755840 T^{6} + 8506031 T^{7} + 47206655 T^{8} - 314452486 T^{9} - 6062195338 T^{10} - 19181601646 T^{11} + 175655963255 T^{12} + 1930707422411 T^{13} + 10465240461440 T^{14} - 2304903305429 T^{15} - 727828328597847 T^{16} - 4201847171760077 T^{17} + 17637072795749852 T^{18} + 245577067949516961 T^{19} + 713342911662882601 T^{20} \))
$67$ (\( 1 - 13 T + 102 T^{2} - 871 T^{3} + 4489 T^{4} \))(\( ( 1 - 16 T + 67 T^{2} )( 1 + 5 T + 67 T^{2} ) \))(\( ( 1 - 16 T + 67 T^{2} )( 1 + 11 T + 67 T^{2} ) \))(\( 1 + 3 T - 58 T^{2} + 201 T^{3} + 4489 T^{4} \))(\( 1 + T - 122 T^{2} - 11 T^{3} + 10573 T^{4} - 737 T^{5} - 547658 T^{6} + 300763 T^{7} + 20151121 T^{8} \))(\( ( 1 - 16 T + 67 T^{2} )^{2}( 1 + 11 T + 67 T^{2} )^{2} \))(\( 1 + 15 T - 65 T^{2} - 1094 T^{3} + 13605 T^{4} + 81804 T^{5} - 1325556 T^{6} - 990195 T^{7} + 121257526 T^{8} - 66343065 T^{9} - 5950420884 T^{10} + 24603616452 T^{11} + 274156001205 T^{12} - 1477036867058 T^{13} - 5879794840985 T^{14} + 90910674079845 T^{15} + 406067677556641 T^{16} \))(\( 1 + 7 T - 171 T^{2} - 1048 T^{3} + 18413 T^{4} + 78372 T^{5} - 1561670 T^{6} - 2046593 T^{7} + 117891882 T^{8} - 137121731 T^{9} - 7010336630 T^{10} + 23571397836 T^{11} + 371042590973 T^{12} - 1414931112136 T^{13} - 15468383350899 T^{14} + 42424981237261 T^{15} + 406067677556641 T^{16} \))(\( 1 - 20 T + 20 T^{2} + 454 T^{3} + 19359 T^{4} - 94011 T^{5} - 1606337 T^{6} + 4367090 T^{7} + 71695364 T^{8} + 365406227 T^{9} - 9837576974 T^{10} + 24482217209 T^{11} + 321840488996 T^{12} + 1313459089670 T^{13} - 32369491253777 T^{14} - 126926611434177 T^{15} + 1751183820409671 T^{16} + 2751563068816642 T^{17} + 8121353551132820 T^{18} - 544130687925898940 T^{19} + 1822837804551761449 T^{20} \))
$71$ (\( 1 - 5 T - 46 T^{2} - 355 T^{3} + 5041 T^{4} \))(\( 1 + 15 T + 154 T^{2} + 1065 T^{3} + 5041 T^{4} \))(\( 1 - 3 T - 62 T^{2} - 213 T^{3} + 5041 T^{4} \))(\( 1 + T - 70 T^{2} + 71 T^{3} + 5041 T^{4} \))(\( 1 - 3 T + 16 T^{2} + 447 T^{3} - 5631 T^{4} + 31737 T^{5} + 80656 T^{6} - 1073733 T^{7} + 25411681 T^{8} \))(\( 1 + 7 T - 101 T^{2} + 56 T^{3} + 13980 T^{4} + 3976 T^{5} - 509141 T^{6} + 2505377 T^{7} + 25411681 T^{8} \))(\( 1 - 5 T - 159 T^{2} + 1268 T^{3} + 11491 T^{4} - 110020 T^{5} - 445126 T^{6} + 3673627 T^{7} + 22281442 T^{8} + 260827517 T^{9} - 2243880166 T^{10} - 39377368220 T^{11} + 292005626371 T^{12} + 2287762817068 T^{13} - 20367945143439 T^{14} - 45475600791955 T^{15} + 645753531245761 T^{16} \))(\( 1 + 7 T - 69 T^{2} - 2 T^{3} + 2231 T^{4} - 44884 T^{5} + 117504 T^{6} + 3373389 T^{7} + 4655622 T^{8} + 239510619 T^{9} + 592337664 T^{10} - 16064477324 T^{11} + 56693460311 T^{12} - 3608458702 T^{13} - 8838919590549 T^{14} + 63665841108737 T^{15} + 645753531245761 T^{16} \))(\( 1 - 11 T - 86 T^{2} + 2261 T^{3} - 3263 T^{4} - 214333 T^{5} + 1463240 T^{6} + 10991945 T^{7} - 162413475 T^{8} - 250592522 T^{9} + 12178138542 T^{10} - 17792069062 T^{11} - 818726327475 T^{12} + 3934138026895 T^{13} + 37183388106440 T^{14} - 386705889487883 T^{15} - 417991226434223 T^{16} + 20564066678122051 T^{17} - 55534803687135446 T^{18} - 504333507902939341 T^{19} + 3255243551009881201 T^{20} \))
$73$ (\( 1 + 11 T + 48 T^{2} + 803 T^{3} + 5329 T^{4} \))(\( 1 - 13 T + 96 T^{2} - 949 T^{3} + 5329 T^{4} \))(\( 1 + 11 T + 48 T^{2} + 803 T^{3} + 5329 T^{4} \))(\( 1 + 11 T + 48 T^{2} + 803 T^{3} + 5329 T^{4} \))(\( 1 - 3 T - 128 T^{2} + 27 T^{3} + 12783 T^{4} + 1971 T^{5} - 682112 T^{6} - 1167051 T^{7} + 28398241 T^{8} \))(\( 1 + 10 T - 3 T^{2} - 430 T^{3} - 2020 T^{4} - 31390 T^{5} - 15987 T^{6} + 3890170 T^{7} + 28398241 T^{8} \))(\( 1 - 5 T - 89 T^{2} - 1914 T^{3} + 14787 T^{4} + 141354 T^{5} + 1733010 T^{6} - 15113015 T^{7} - 108706946 T^{8} - 1103250095 T^{9} + 9235210290 T^{10} + 54989109018 T^{11} + 419924789667 T^{12} - 3967859029002 T^{13} - 13468746139721 T^{14} - 55236992595485 T^{15} + 806460091894081 T^{16} \))(\( 1 - 25 T + 227 T^{2} - 254 T^{3} - 10869 T^{4} + 120242 T^{5} - 235666 T^{6} - 14099295 T^{7} + 209024830 T^{8} - 1029248535 T^{9} - 1255864114 T^{10} + 46776182114 T^{11} - 308660481429 T^{12} - 526560184622 T^{13} + 34352869367603 T^{14} - 276184962977425 T^{15} + 806460091894081 T^{16} \))(\( 1 + 2 T - 292 T^{2} + 110 T^{3} + 49419 T^{4} - 65805 T^{5} - 5626607 T^{6} + 7758346 T^{7} + 494157350 T^{8} - 312773261 T^{9} - 37976489726 T^{10} - 22832448053 T^{11} + 2633364518150 T^{12} + 3018128485882 T^{13} - 159785741598287 T^{14} - 136418476177365 T^{15} + 7478786128976091 T^{16} + 1215213837100670 T^{17} - 235486346833071652 T^{18} + 117743173416535826 T^{19} + 4297625829703557649 T^{20} \))
$79$ (\( 1 - T - 78 T^{2} - 79 T^{3} + 6241 T^{4} \))(\( 1 + 11 T + 42 T^{2} + 869 T^{3} + 6241 T^{4} \))(\( 1 + T - 78 T^{2} + 79 T^{3} + 6241 T^{4} \))(\( 1 + 5 T - 54 T^{2} + 395 T^{3} + 6241 T^{4} \))(\( 1 + 5 T - 138 T^{2} + 25 T^{3} + 18353 T^{4} + 1975 T^{5} - 861258 T^{6} + 2465195 T^{7} + 38950081 T^{8} \))(\( 1 - 17 T + 63 T^{2} - 1156 T^{3} + 22004 T^{4} - 91324 T^{5} + 393183 T^{6} - 8381663 T^{7} + 38950081 T^{8} \))(\( 1 - 17 T + 31 T^{2} + 882 T^{3} - 6939 T^{4} + 88992 T^{5} - 790200 T^{6} - 4043963 T^{7} + 107627638 T^{8} - 319473077 T^{9} - 4931638200 T^{10} + 43876526688 T^{11} - 270274612059 T^{12} + 2713963743918 T^{13} + 7535711121151 T^{14} - 326466452764703 T^{15} + 1517108809906561 T^{16} \))(\( 1 + 3 T - 159 T^{2} + 1532 T^{3} + 18381 T^{4} - 195268 T^{5} + 570310 T^{6} + 12416307 T^{7} - 97474622 T^{8} + 980888253 T^{9} + 3559304710 T^{10} - 96274739452 T^{11} + 715941438861 T^{12} + 4714050403268 T^{13} - 38650905427839 T^{14} + 57611726958477 T^{15} + 1517108809906561 T^{16} \))(\( 1 - 35 T + 348 T^{2} - 2053 T^{3} + 80421 T^{4} - 1276371 T^{5} + 5899214 T^{6} - 65244183 T^{7} + 1631329493 T^{8} - 11536311390 T^{9} + 36580682262 T^{10} - 911368599810 T^{11} + 10181127365813 T^{12} - 32167926742137 T^{13} + 229774863136334 T^{14} - 3927465553048029 T^{15} + 19549336260454341 T^{16} - 39425625148584427 T^{17} + 527953865847483228 T^{18} - 4194805859391641165 T^{19} + 9468276082626847201 T^{20} \))
$83$ (\( 1 + 3 T - 74 T^{2} + 249 T^{3} + 6889 T^{4} \))(\( 1 + 9 T - 2 T^{2} + 747 T^{3} + 6889 T^{4} \))(\( 1 + 15 T + 142 T^{2} + 1245 T^{3} + 6889 T^{4} \))(\( 1 - 9 T - 2 T^{2} - 747 T^{3} + 6889 T^{4} \))(\( 1 - 3 T + 52 T^{2} + 627 T^{3} - 5787 T^{4} + 52041 T^{5} + 358228 T^{6} - 1715361 T^{7} + 47458321 T^{8} \))(\( 1 - 5 T - 109 T^{2} + 160 T^{3} + 8992 T^{4} + 13280 T^{5} - 750901 T^{6} - 2858935 T^{7} + 47458321 T^{8} \))(\( 1 + 7 T - 67 T^{2} - 1664 T^{3} - 9889 T^{4} + 40856 T^{5} + 708850 T^{6} + 3930051 T^{7} + 4142010 T^{8} + 326194233 T^{9} + 4883267650 T^{10} + 23360929672 T^{11} - 469315336369 T^{12} - 6554563629952 T^{13} - 21905005015723 T^{14} + 189952356927389 T^{15} + 2252292232139041 T^{16} \))(\( 1 + 7 T - 197 T^{2} - 1410 T^{3} + 20963 T^{4} + 114940 T^{5} - 2101488 T^{6} - 3310303 T^{7} + 205581814 T^{8} - 274755149 T^{9} - 14477150832 T^{10} + 65721197780 T^{11} + 994868783123 T^{12} - 5554047306630 T^{13} - 64407253553693 T^{14} + 189952356927389 T^{15} + 2252292232139041 T^{16} \))(\( 1 - 19 T - 66 T^{2} + 1433 T^{3} + 28097 T^{4} - 146545 T^{5} - 3744212 T^{6} + 10407189 T^{7} + 294043885 T^{8} + 226357178 T^{9} - 30936509178 T^{10} + 18787645774 T^{11} + 2025668323765 T^{12} + 5950695376743 T^{13} - 177694014988052 T^{14} - 577246711028435 T^{15} + 9186043670548793 T^{16} + 38885961068135491 T^{17} - 148651287321176706 T^{18} - 3551864850083267657 T^{19} + 15516041187205853449 T^{20} \))
$89$ (\( 1 + 15 T + 136 T^{2} + 1335 T^{3} + 7921 T^{4} \))(\( 1 - T - 88 T^{2} - 89 T^{3} + 7921 T^{4} \))(\( 1 - 9 T - 8 T^{2} - 801 T^{3} + 7921 T^{4} \))(\( 1 - T - 88 T^{2} - 89 T^{3} + 7921 T^{4} \))(\( 1 + 3 T - 160 T^{2} - 27 T^{3} + 19839 T^{4} - 2403 T^{5} - 1267360 T^{6} + 2114907 T^{7} + 62742241 T^{8} \))(\( ( 1 + 7 T - 40 T^{2} + 623 T^{3} + 7921 T^{4} )^{2} \))(\( 1 - 3 T - 309 T^{2} + 518 T^{3} + 57891 T^{4} - 53402 T^{5} - 7663010 T^{6} + 1794603 T^{7} + 783783678 T^{8} + 159719667 T^{9} - 60698702210 T^{10} - 37646754538 T^{11} + 3632211073731 T^{12} + 2892542794582 T^{13} - 153567218906949 T^{14} - 132694004686587 T^{15} + 3936588805702081 T^{16} \))(\( 1 + 5 T - 125 T^{2} + 798 T^{3} + 11579 T^{4} - 116146 T^{5} + 731406 T^{6} + 9841411 T^{7} - 83863874 T^{8} + 875885579 T^{9} + 5793466926 T^{10} - 81879329474 T^{11} + 726492408539 T^{12} + 4456079440302 T^{13} - 62122661370125 T^{14} + 221156674477645 T^{15} + 3936588805702081 T^{16} \))(\( 1 - 12 T - 226 T^{2} + 2570 T^{3} + 35897 T^{4} - 320539 T^{5} - 4291883 T^{6} + 27048096 T^{7} + 402445144 T^{8} - 958603859 T^{9} - 35993489098 T^{10} - 85315743451 T^{11} + 3187767985624 T^{12} + 19068069189024 T^{13} - 269282357529803 T^{14} - 1789908831723011 T^{15} + 17840137401627017 T^{16} + 113674530681509530 T^{17} - 889669070088670306 T^{18} - 4204276844489822508 T^{19} + 31181719929966183601 T^{20} \))
$97$ (\( ( 1 + 18 T + 97 T^{2} )^{2} \))(\( ( 1 + 10 T + 97 T^{2} )^{2} \))(\( ( 1 + 10 T + 97 T^{2} )^{2} \))(\( ( 1 + 2 T + 97 T^{2} )^{2} \))(\( ( 1 + 14 T + 238 T^{2} + 1358 T^{3} + 9409 T^{4} )^{2} \))(\( ( 1 - 19 T + 280 T^{2} - 1843 T^{3} + 9409 T^{4} )^{2} \))(\( ( 1 + 30 T + 476 T^{2} + 5170 T^{3} + 49894 T^{4} + 501490 T^{5} + 4478684 T^{6} + 27380190 T^{7} + 88529281 T^{8} )^{2} \))(\( ( 1 - 14 T + 368 T^{2} - 3978 T^{3} + 52702 T^{4} - 385866 T^{5} + 3462512 T^{6} - 12777422 T^{7} + 88529281 T^{8} )^{2} \))(\( ( 1 - 31 T + 679 T^{2} - 10828 T^{3} + 143944 T^{4} - 1530442 T^{5} + 13962568 T^{6} - 101880652 T^{7} + 619704967 T^{8} - 2744407711 T^{9} + 8587340257 T^{10} )^{2} \))
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