Properties

Label 2100.2.bi
Level 2100
Weight 2
Character orbit bi
Rep. character \(\chi_{2100}(101,\cdot)\)
Character field \(\Q(\zeta_{6})\)
Dimension 102
Newform subspaces 14
Sturm bound 960
Trace bound 11

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Defining parameters

Level: \( N \) \(=\) \( 2100 = 2^{2} \cdot 3 \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2100.bi (of order \(6\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 21 \)
Character field: \(\Q(\zeta_{6})\)
Newform subspaces: \( 14 \)
Sturm bound: \(960\)
Trace bound: \(11\)
Distinguishing \(T_p\): \(11\), \(13\), \(19\), \(37\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(2100, [\chi])\).

Total New Old
Modular forms 1032 102 930
Cusp forms 888 102 786
Eisenstein series 144 0 144

Trace form

\( 102q + 7q^{7} + O(q^{10}) \) \( 102q + 7q^{7} - 15q^{19} - 11q^{21} - 21q^{31} - 3q^{33} + 5q^{37} + 3q^{39} + 10q^{43} - 3q^{49} - 7q^{51} + 24q^{61} + 50q^{63} + q^{67} + 51q^{73} + 33q^{79} - 32q^{81} + 18q^{87} + 73q^{91} + 22q^{93} - 14q^{99} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(2100, [\chi])\) into newform subspaces

Label Dim. \(A\) Field CM Traces $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\)
2100.2.bi.a \(2\) \(16.769\) \(\Q(\sqrt{-3}) \) \(\Q(\sqrt{-3}) \) \(0\) \(-3\) \(0\) \(-5\) \(q+(-1-\zeta_{6})q^{3}+(-3+\zeta_{6})q^{7}+3\zeta_{6}q^{9}+\cdots\)
2100.2.bi.b \(2\) \(16.769\) \(\Q(\sqrt{-3}) \) \(\Q(\sqrt{-3}) \) \(0\) \(-3\) \(0\) \(-1\) \(q+(-1-\zeta_{6})q^{3}+(1-3\zeta_{6})q^{7}+3\zeta_{6}q^{9}+\cdots\)
2100.2.bi.c \(2\) \(16.769\) \(\Q(\sqrt{-3}) \) \(\Q(\sqrt{-3}) \) \(0\) \(-3\) \(0\) \(4\) \(q+(-1-\zeta_{6})q^{3}+(1+2\zeta_{6})q^{7}+3\zeta_{6}q^{9}+\cdots\)
2100.2.bi.d \(2\) \(16.769\) \(\Q(\sqrt{-3}) \) \(\Q(\sqrt{-3}) \) \(0\) \(-3\) \(0\) \(5\) \(q+(-1-\zeta_{6})q^{3}+(2+\zeta_{6})q^{7}+3\zeta_{6}q^{9}+\cdots\)
2100.2.bi.e \(2\) \(16.769\) \(\Q(\sqrt{-3}) \) None \(0\) \(0\) \(0\) \(-4\) \(q+(-1+2\zeta_{6})q^{3}+(-1-2\zeta_{6})q^{7}+\cdots\)
2100.2.bi.f \(2\) \(16.769\) \(\Q(\sqrt{-3}) \) None \(0\) \(3\) \(0\) \(-4\) \(q+(2-\zeta_{6})q^{3}+(-1-2\zeta_{6})q^{7}+(3-3\zeta_{6})q^{9}+\cdots\)
2100.2.bi.g \(2\) \(16.769\) \(\Q(\sqrt{-3}) \) \(\Q(\sqrt{-3}) \) \(0\) \(3\) \(0\) \(-4\) \(q+(1+\zeta_{6})q^{3}+(-1-2\zeta_{6})q^{7}+3\zeta_{6}q^{9}+\cdots\)
2100.2.bi.h \(2\) \(16.769\) \(\Q(\sqrt{-3}) \) \(\Q(\sqrt{-3}) \) \(0\) \(3\) \(0\) \(1\) \(q+(1+\zeta_{6})q^{3}+(-1+3\zeta_{6})q^{7}+3\zeta_{6}q^{9}+\cdots\)
2100.2.bi.i \(2\) \(16.769\) \(\Q(\sqrt{-3}) \) \(\Q(\sqrt{-3}) \) \(0\) \(3\) \(0\) \(5\) \(q+(1+\zeta_{6})q^{3}+(3-\zeta_{6})q^{7}+3\zeta_{6}q^{9}+\cdots\)
2100.2.bi.j \(10\) \(16.769\) 10.0.\(\cdots\).1 None \(0\) \(-1\) \(0\) \(5\) \(q+(-\beta _{5}+\beta _{8})q^{3}+(\beta _{1}+\beta _{3}-\beta _{4}-\beta _{6}+\cdots)q^{7}+\cdots\)
2100.2.bi.k \(10\) \(16.769\) 10.0.\(\cdots\).1 None \(0\) \(1\) \(0\) \(5\) \(q-\beta _{5}q^{3}+(1-\beta _{2}-\beta _{3}+\beta _{4}-\beta _{5}+\cdots)q^{7}+\cdots\)
2100.2.bi.l \(16\) \(16.769\) \(\mathbb{Q}[x]/(x^{16} - \cdots)\) None \(0\) \(-3\) \(0\) \(-6\) \(q-\beta _{2}q^{3}+(-1+\beta _{7}+\beta _{10})q^{7}+(-\beta _{10}+\cdots)q^{9}+\cdots\)
2100.2.bi.m \(16\) \(16.769\) \(\mathbb{Q}[x]/(x^{16} - \cdots)\) None \(0\) \(3\) \(0\) \(6\) \(q+\beta _{2}q^{3}+(1-\beta _{7}-\beta _{10})q^{7}+(-\beta _{10}+\cdots)q^{9}+\cdots\)
2100.2.bi.n \(32\) \(16.769\) None \(0\) \(0\) \(0\) \(0\)

Decomposition of \(S_{2}^{\mathrm{old}}(2100, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(2100, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(21, [\chi])\)\(^{\oplus 9}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(42, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(84, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(105, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(210, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(420, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(525, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(1050, [\chi])\)\(^{\oplus 2}\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ (\( 1 + 3 T + 3 T^{2} \))(\( 1 + 3 T + 3 T^{2} \))(\( 1 + 3 T + 3 T^{2} \))(\( 1 + 3 T + 3 T^{2} \))(\( 1 + 3 T^{2} \))(\( 1 - 3 T + 3 T^{2} \))(\( 1 - 3 T + 3 T^{2} \))(\( 1 - 3 T + 3 T^{2} \))(\( 1 - 3 T + 3 T^{2} \))(\( 1 + T - T^{2} - 4 T^{3} + T^{4} + 21 T^{5} + 3 T^{6} - 36 T^{7} - 27 T^{8} + 81 T^{9} + 243 T^{10} \))(\( 1 - T + 2 T^{2} + T^{3} - 11 T^{4} + 12 T^{5} - 33 T^{6} + 9 T^{7} + 54 T^{8} - 81 T^{9} + 243 T^{10} \))(\( 1 + 3 T + 9 T^{2} + 18 T^{3} + 32 T^{4} + 36 T^{5} - 51 T^{7} - 167 T^{8} - 153 T^{9} + 972 T^{11} + 2592 T^{12} + 4374 T^{13} + 6561 T^{14} + 6561 T^{15} + 6561 T^{16} \))(\( 1 - 3 T + 9 T^{2} - 18 T^{3} + 32 T^{4} - 36 T^{5} + 51 T^{7} - 167 T^{8} + 153 T^{9} - 972 T^{11} + 2592 T^{12} - 4374 T^{13} + 6561 T^{14} - 6561 T^{15} + 6561 T^{16} \))
$5$ 1
$7$ (\( 1 + 5 T + 7 T^{2} \))(\( 1 + T + 7 T^{2} \))(\( 1 - 4 T + 7 T^{2} \))(\( 1 - 5 T + 7 T^{2} \))(\( 1 + 4 T + 7 T^{2} \))(\( 1 + 4 T + 7 T^{2} \))(\( 1 + 4 T + 7 T^{2} \))(\( 1 - T + 7 T^{2} \))(\( 1 - 5 T + 7 T^{2} \))(\( 1 - 5 T + 19 T^{2} - 6 T^{3} - 91 T^{4} + 533 T^{5} - 637 T^{6} - 294 T^{7} + 6517 T^{8} - 12005 T^{9} + 16807 T^{10} \))(\( 1 - 5 T + 19 T^{2} - 6 T^{3} - 91 T^{4} + 533 T^{5} - 637 T^{6} - 294 T^{7} + 6517 T^{8} - 12005 T^{9} + 16807 T^{10} \))(\( ( 1 + 3 T + 9 T^{2} + 39 T^{3} + 95 T^{4} + 273 T^{5} + 441 T^{6} + 1029 T^{7} + 2401 T^{8} )^{2} \))(\( ( 1 - 3 T + 9 T^{2} - 39 T^{3} + 95 T^{4} - 273 T^{5} + 441 T^{6} - 1029 T^{7} + 2401 T^{8} )^{2} \))
$11$ (\( 1 + 11 T^{2} + 121 T^{4} \))(\( 1 + 11 T^{2} + 121 T^{4} \))(\( 1 + 11 T^{2} + 121 T^{4} \))(\( 1 + 11 T^{2} + 121 T^{4} \))(\( 1 - 9 T + 38 T^{2} - 99 T^{3} + 121 T^{4} \))(\( 1 + 9 T + 38 T^{2} + 99 T^{3} + 121 T^{4} \))(\( 1 + 11 T^{2} + 121 T^{4} \))(\( 1 + 11 T^{2} + 121 T^{4} \))(\( 1 + 11 T^{2} + 121 T^{4} \))(\( 1 - 6 T + 51 T^{2} - 234 T^{3} + 1195 T^{4} - 3666 T^{5} + 12622 T^{6} - 19434 T^{7} + 25645 T^{8} + 184704 T^{9} - 509783 T^{10} + 2031744 T^{11} + 3103045 T^{12} - 25866654 T^{13} + 184798702 T^{14} - 590412966 T^{15} + 2117015395 T^{16} - 4559998014 T^{17} + 10932302931 T^{18} - 14147686146 T^{19} + 25937424601 T^{20} \))(\( 1 + 6 T + 51 T^{2} + 234 T^{3} + 1195 T^{4} + 3666 T^{5} + 12622 T^{6} + 19434 T^{7} + 25645 T^{8} - 184704 T^{9} - 509783 T^{10} - 2031744 T^{11} + 3103045 T^{12} + 25866654 T^{13} + 184798702 T^{14} + 590412966 T^{15} + 2117015395 T^{16} + 4559998014 T^{17} + 10932302931 T^{18} + 14147686146 T^{19} + 25937424601 T^{20} \))(\( 1 + 14 T^{2} - 69 T^{4} - 2576 T^{6} - 25195 T^{8} - 226506 T^{10} - 1311833 T^{12} + 39166148 T^{14} + 900447201 T^{16} + 4739103908 T^{18} - 19206546953 T^{20} - 401269195866 T^{22} - 5400772006795 T^{24} - 66814805772176 T^{26} - 216551557993749 T^{28} + 5316497670165374 T^{30} + 45949729863572161 T^{32} \))(\( 1 + 14 T^{2} - 69 T^{4} - 2576 T^{6} - 25195 T^{8} - 226506 T^{10} - 1311833 T^{12} + 39166148 T^{14} + 900447201 T^{16} + 4739103908 T^{18} - 19206546953 T^{20} - 401269195866 T^{22} - 5400772006795 T^{24} - 66814805772176 T^{26} - 216551557993749 T^{28} + 5316497670165374 T^{30} + 45949729863572161 T^{32} \))
$13$ (\( ( 1 - 5 T + 13 T^{2} )( 1 + 5 T + 13 T^{2} ) \))(\( ( 1 - 7 T + 13 T^{2} )( 1 + 7 T + 13 T^{2} ) \))(\( ( 1 - 2 T + 13 T^{2} )( 1 + 2 T + 13 T^{2} ) \))(\( ( 1 - 5 T + 13 T^{2} )( 1 + 5 T + 13 T^{2} ) \))(\( ( 1 - 13 T^{2} )^{2} \))(\( ( 1 - 13 T^{2} )^{2} \))(\( ( 1 - 2 T + 13 T^{2} )( 1 + 2 T + 13 T^{2} ) \))(\( ( 1 - 7 T + 13 T^{2} )( 1 + 7 T + 13 T^{2} ) \))(\( ( 1 - 5 T + 13 T^{2} )( 1 + 5 T + 13 T^{2} ) \))(\( 1 - 49 T^{2} + 1364 T^{4} - 28371 T^{6} + 474775 T^{8} - 6687888 T^{10} + 80236975 T^{12} - 810304131 T^{14} + 6583767476 T^{16} - 39970805329 T^{18} + 137858491849 T^{20} \))(\( 1 - 49 T^{2} + 1364 T^{4} - 28371 T^{6} + 474775 T^{8} - 6687888 T^{10} + 80236975 T^{12} - 810304131 T^{14} + 6583767476 T^{16} - 39970805329 T^{18} + 137858491849 T^{20} \))(\( ( 1 - 83 T^{2} + 3165 T^{4} - 73411 T^{6} + 1146521 T^{8} - 12406459 T^{10} + 90395565 T^{12} - 400625147 T^{14} + 815730721 T^{16} )^{2} \))(\( ( 1 - 83 T^{2} + 3165 T^{4} - 73411 T^{6} + 1146521 T^{8} - 12406459 T^{10} + 90395565 T^{12} - 400625147 T^{14} + 815730721 T^{16} )^{2} \))
$17$ (\( 1 - 17 T^{2} + 289 T^{4} \))(\( 1 - 17 T^{2} + 289 T^{4} \))(\( 1 - 17 T^{2} + 289 T^{4} \))(\( 1 - 17 T^{2} + 289 T^{4} \))(\( 1 - 3 T - 8 T^{2} - 51 T^{3} + 289 T^{4} \))(\( 1 + 3 T - 8 T^{2} + 51 T^{3} + 289 T^{4} \))(\( 1 - 17 T^{2} + 289 T^{4} \))(\( 1 - 17 T^{2} + 289 T^{4} \))(\( 1 - 17 T^{2} + 289 T^{4} \))(\( 1 - 6 T - 15 T^{2} + 138 T^{3} + 193 T^{4} - 2754 T^{5} + 4928 T^{6} + 30306 T^{7} - 220151 T^{8} - 325680 T^{9} + 6075881 T^{10} - 5536560 T^{11} - 63623639 T^{12} + 148893378 T^{13} + 411591488 T^{14} - 3910286178 T^{15} + 4658550817 T^{16} + 56626736874 T^{17} - 104636361615 T^{18} - 711527258982 T^{19} + 2015993900449 T^{20} \))(\( 1 + 6 T - 15 T^{2} - 138 T^{3} + 193 T^{4} + 2754 T^{5} + 4928 T^{6} - 30306 T^{7} - 220151 T^{8} + 325680 T^{9} + 6075881 T^{10} + 5536560 T^{11} - 63623639 T^{12} - 148893378 T^{13} + 411591488 T^{14} + 3910286178 T^{15} + 4658550817 T^{16} - 56626736874 T^{17} - 104636361615 T^{18} + 711527258982 T^{19} + 2015993900449 T^{20} \))(\( 1 - 14 T^{2} + 267 T^{4} + 10052 T^{6} - 224695 T^{8} + 3669918 T^{10} + 2136955 T^{12} - 1194075764 T^{14} + 23802887097 T^{16} - 345087895796 T^{18} + 178480618555 T^{20} + 88582898949342 T^{22} - 1567417818205495 T^{24} + 20264770687313348 T^{26} + 155560137340346187 T^{28} - 2357289571831613006 T^{30} + 48661191875666868481 T^{32} \))(\( 1 - 14 T^{2} + 267 T^{4} + 10052 T^{6} - 224695 T^{8} + 3669918 T^{10} + 2136955 T^{12} - 1194075764 T^{14} + 23802887097 T^{16} - 345087895796 T^{18} + 178480618555 T^{20} + 88582898949342 T^{22} - 1567417818205495 T^{24} + 20264770687313348 T^{26} + 155560137340346187 T^{28} - 2357289571831613006 T^{30} + 48661191875666868481 T^{32} \))
$19$ (\( ( 1 - 8 T + 19 T^{2} )( 1 - 7 T + 19 T^{2} ) \))(\( ( 1 - 8 T + 19 T^{2} )( 1 - T + 19 T^{2} ) \))(\( ( 1 - T + 19 T^{2} )( 1 + 7 T + 19 T^{2} ) \))(\( ( 1 + 7 T + 19 T^{2} )( 1 + 8 T + 19 T^{2} ) \))(\( 1 - 3 T + 22 T^{2} - 57 T^{3} + 361 T^{4} \))(\( 1 - 3 T + 22 T^{2} - 57 T^{3} + 361 T^{4} \))(\( ( 1 - T + 19 T^{2} )( 1 + 7 T + 19 T^{2} ) \))(\( ( 1 - 8 T + 19 T^{2} )( 1 - T + 19 T^{2} ) \))(\( ( 1 - 8 T + 19 T^{2} )( 1 - 7 T + 19 T^{2} ) \))(\( 1 - 3 T + 83 T^{2} - 240 T^{3} + 3800 T^{4} - 10380 T^{5} + 124845 T^{6} - 315315 T^{7} + 3213415 T^{8} - 7312140 T^{9} + 67225536 T^{10} - 138930660 T^{11} + 1160042815 T^{12} - 2162745585 T^{13} + 16269925245 T^{14} - 25701907620 T^{15} + 178774347800 T^{16} - 214529217360 T^{17} + 1409635732403 T^{18} - 968063093337 T^{19} + 6131066257801 T^{20} \))(\( 1 - 3 T + 83 T^{2} - 240 T^{3} + 3800 T^{4} - 10380 T^{5} + 124845 T^{6} - 315315 T^{7} + 3213415 T^{8} - 7312140 T^{9} + 67225536 T^{10} - 138930660 T^{11} + 1160042815 T^{12} - 2162745585 T^{13} + 16269925245 T^{14} - 25701907620 T^{15} + 178774347800 T^{16} - 214529217360 T^{17} + 1409635732403 T^{18} - 968063093337 T^{19} + 6131066257801 T^{20} \))(\( ( 1 + 9 T + 64 T^{2} + 333 T^{3} + 1299 T^{4} + 3186 T^{5} - 4123 T^{6} - 75567 T^{7} - 435298 T^{8} - 1435773 T^{9} - 1488403 T^{10} + 21852774 T^{11} + 169286979 T^{12} + 824540967 T^{13} + 3010936384 T^{14} + 8044845651 T^{15} + 16983563041 T^{16} )^{2} \))(\( ( 1 + 9 T + 64 T^{2} + 333 T^{3} + 1299 T^{4} + 3186 T^{5} - 4123 T^{6} - 75567 T^{7} - 435298 T^{8} - 1435773 T^{9} - 1488403 T^{10} + 21852774 T^{11} + 169286979 T^{12} + 824540967 T^{13} + 3010936384 T^{14} + 8044845651 T^{15} + 16983563041 T^{16} )^{2} \))
$23$ (\( 1 + 23 T^{2} + 529 T^{4} \))(\( 1 + 23 T^{2} + 529 T^{4} \))(\( 1 + 23 T^{2} + 529 T^{4} \))(\( 1 + 23 T^{2} + 529 T^{4} \))(\( 1 - 9 T + 50 T^{2} - 207 T^{3} + 529 T^{4} \))(\( 1 + 9 T + 50 T^{2} + 207 T^{3} + 529 T^{4} \))(\( 1 + 23 T^{2} + 529 T^{4} \))(\( 1 + 23 T^{2} + 529 T^{4} \))(\( 1 + 23 T^{2} + 529 T^{4} \))(\( 1 - 24 T + 338 T^{2} - 3504 T^{3} + 29201 T^{4} - 203310 T^{5} + 1216412 T^{6} - 6430392 T^{7} + 31045441 T^{8} - 143820006 T^{9} + 675418266 T^{10} - 3307860138 T^{11} + 16423038289 T^{12} - 78238579464 T^{13} + 340401950492 T^{14} - 1308572895330 T^{15} + 4322795994689 T^{16} - 11930508366288 T^{17} + 26469113024978 T^{18} - 43227663875112 T^{19} + 41426511213649 T^{20} \))(\( 1 + 24 T + 338 T^{2} + 3504 T^{3} + 29201 T^{4} + 203310 T^{5} + 1216412 T^{6} + 6430392 T^{7} + 31045441 T^{8} + 143820006 T^{9} + 675418266 T^{10} + 3307860138 T^{11} + 16423038289 T^{12} + 78238579464 T^{13} + 340401950492 T^{14} + 1308572895330 T^{15} + 4322795994689 T^{16} + 11930508366288 T^{17} + 26469113024978 T^{18} + 43227663875112 T^{19} + 41426511213649 T^{20} \))(\( 1 + 105 T^{2} + 5189 T^{4} + 177324 T^{6} + 5348524 T^{8} + 157563000 T^{10} + 4443717224 T^{12} + 116142079557 T^{14} + 2787310422721 T^{16} + 61439160085653 T^{18} + 1243534271681384 T^{20} + 23324978778507000 T^{22} + 418848184239075244 T^{24} + 7345914674449095276 T^{26} + \)\(11\!\cdots\!69\)\( T^{28} + \)\(12\!\cdots\!45\)\( T^{30} + \)\(61\!\cdots\!61\)\( T^{32} \))(\( 1 + 105 T^{2} + 5189 T^{4} + 177324 T^{6} + 5348524 T^{8} + 157563000 T^{10} + 4443717224 T^{12} + 116142079557 T^{14} + 2787310422721 T^{16} + 61439160085653 T^{18} + 1243534271681384 T^{20} + 23324978778507000 T^{22} + 418848184239075244 T^{24} + 7345914674449095276 T^{26} + \)\(11\!\cdots\!69\)\( T^{28} + \)\(12\!\cdots\!45\)\( T^{30} + \)\(61\!\cdots\!61\)\( T^{32} \))
$29$ (\( ( 1 - 29 T^{2} )^{2} \))(\( ( 1 - 29 T^{2} )^{2} \))(\( ( 1 - 29 T^{2} )^{2} \))(\( ( 1 - 29 T^{2} )^{2} \))(\( ( 1 - 29 T^{2} )^{2} \))(\( ( 1 - 29 T^{2} )^{2} \))(\( ( 1 - 29 T^{2} )^{2} \))(\( ( 1 - 29 T^{2} )^{2} \))(\( ( 1 - 29 T^{2} )^{2} \))(\( 1 - 148 T^{2} + 11438 T^{4} - 619594 T^{6} + 25499257 T^{8} - 826380936 T^{10} + 21444875137 T^{12} - 438227063914 T^{14} + 6803589145598 T^{16} - 74036469118228 T^{18} + 420707233300201 T^{20} \))(\( 1 - 148 T^{2} + 11438 T^{4} - 619594 T^{6} + 25499257 T^{8} - 826380936 T^{10} + 21444875137 T^{12} - 438227063914 T^{14} + 6803589145598 T^{16} - 74036469118228 T^{18} + 420707233300201 T^{20} \))(\( ( 1 - 90 T^{2} + 2929 T^{4} - 16011 T^{6} - 944082 T^{8} - 13465251 T^{10} + 2071626049 T^{12} - 53534098890 T^{14} + 500246412961 T^{16} )^{2} \))(\( ( 1 - 90 T^{2} + 2929 T^{4} - 16011 T^{6} - 944082 T^{8} - 13465251 T^{10} + 2071626049 T^{12} - 53534098890 T^{14} + 500246412961 T^{16} )^{2} \))
$31$ (\( ( 1 + 7 T + 31 T^{2} )( 1 + 11 T + 31 T^{2} ) \))(\( ( 1 - 11 T + 31 T^{2} )( 1 - 7 T + 31 T^{2} ) \))(\( ( 1 - 7 T + 31 T^{2} )( 1 + 4 T + 31 T^{2} ) \))(\( ( 1 - 4 T + 31 T^{2} )( 1 + 7 T + 31 T^{2} ) \))(\( ( 1 - 4 T + 31 T^{2} )( 1 + 7 T + 31 T^{2} ) \))(\( ( 1 - 4 T + 31 T^{2} )( 1 + 7 T + 31 T^{2} ) \))(\( ( 1 - 7 T + 31 T^{2} )( 1 + 4 T + 31 T^{2} ) \))(\( ( 1 - 11 T + 31 T^{2} )( 1 - 7 T + 31 T^{2} ) \))(\( ( 1 + 7 T + 31 T^{2} )( 1 + 11 T + 31 T^{2} ) \))(\( 1 - 15 T + 167 T^{2} - 1380 T^{3} + 9668 T^{4} - 61404 T^{5} + 363093 T^{6} - 2129475 T^{7} + 12161347 T^{8} - 68552376 T^{9} + 387742776 T^{10} - 2125123656 T^{11} + 11687054467 T^{12} - 63439189725 T^{13} + 335324010453 T^{14} - 1757944388004 T^{15} + 8580385587908 T^{16} - 37967407473180 T^{17} + 142432803252647 T^{18} - 396594332410065 T^{19} + 819628286980801 T^{20} \))(\( 1 - 15 T + 167 T^{2} - 1380 T^{3} + 9668 T^{4} - 61404 T^{5} + 363093 T^{6} - 2129475 T^{7} + 12161347 T^{8} - 68552376 T^{9} + 387742776 T^{10} - 2125123656 T^{11} + 11687054467 T^{12} - 63439189725 T^{13} + 335324010453 T^{14} - 1757944388004 T^{15} + 8580385587908 T^{16} - 37967407473180 T^{17} + 142432803252647 T^{18} - 396594332410065 T^{19} + 819628286980801 T^{20} \))(\( ( 1 + 9 T + 97 T^{2} + 630 T^{3} + 3786 T^{4} + 11934 T^{5} + 10454 T^{6} - 251757 T^{7} - 2144539 T^{8} - 7804467 T^{9} + 10046294 T^{10} + 355525794 T^{11} + 3496450506 T^{12} + 18036365130 T^{13} + 86087857057 T^{14} + 247613526999 T^{15} + 852891037441 T^{16} )^{2} \))(\( ( 1 + 9 T + 97 T^{2} + 630 T^{3} + 3786 T^{4} + 11934 T^{5} + 10454 T^{6} - 251757 T^{7} - 2144539 T^{8} - 7804467 T^{9} + 10046294 T^{10} + 355525794 T^{11} + 3496450506 T^{12} + 18036365130 T^{13} + 86087857057 T^{14} + 247613526999 T^{15} + 852891037441 T^{16} )^{2} \))
$37$ (\( ( 1 - 10 T + 37 T^{2} )( 1 + 11 T + 37 T^{2} ) \))(\( ( 1 + T + 37 T^{2} )( 1 + 10 T + 37 T^{2} ) \))(\( ( 1 - 10 T + 37 T^{2} )( 1 + 11 T + 37 T^{2} ) \))(\( ( 1 + T + 37 T^{2} )( 1 + 10 T + 37 T^{2} ) \))(\( 1 - 7 T + 12 T^{2} - 259 T^{3} + 1369 T^{4} \))(\( 1 - 7 T + 12 T^{2} - 259 T^{3} + 1369 T^{4} \))(\( ( 1 - 11 T + 37 T^{2} )( 1 + 10 T + 37 T^{2} ) \))(\( ( 1 - 10 T + 37 T^{2} )( 1 - T + 37 T^{2} ) \))(\( ( 1 - 11 T + 37 T^{2} )( 1 + 10 T + 37 T^{2} ) \))(\( 1 - T - 117 T^{2} + 176 T^{3} + 6668 T^{4} - 11380 T^{5} - 299439 T^{6} + 438173 T^{7} + 12712375 T^{8} - 7307172 T^{9} - 492022188 T^{10} - 270365364 T^{11} + 17403241375 T^{12} + 22194776969 T^{13} - 561196895679 T^{14} - 789134230660 T^{15} + 17108263695212 T^{16} + 16708010375408 T^{17} - 410960096108757 T^{18} - 129961739795077 T^{19} + 4808584372417849 T^{20} \))(\( 1 - T - 117 T^{2} + 176 T^{3} + 6668 T^{4} - 11380 T^{5} - 299439 T^{6} + 438173 T^{7} + 12712375 T^{8} - 7307172 T^{9} - 492022188 T^{10} - 270365364 T^{11} + 17403241375 T^{12} + 22194776969 T^{13} - 561196895679 T^{14} - 789134230660 T^{15} + 17108263695212 T^{16} + 16708010375408 T^{17} - 410960096108757 T^{18} - 129961739795077 T^{19} + 4808584372417849 T^{20} \))(\( ( 1 - 3 T - 93 T^{3} - 1331 T^{4} + 10584 T^{5} + 15795 T^{6} - 161037 T^{7} + 1169532 T^{8} - 5958369 T^{9} + 21623355 T^{10} + 536111352 T^{11} - 2494508291 T^{12} - 6448988001 T^{13} - 284795631399 T^{15} + 3512479453921 T^{16} )^{2} \))(\( ( 1 + 3 T + 93 T^{3} - 1331 T^{4} - 10584 T^{5} + 15795 T^{6} + 161037 T^{7} + 1169532 T^{8} + 5958369 T^{9} + 21623355 T^{10} - 536111352 T^{11} - 2494508291 T^{12} + 6448988001 T^{13} + 284795631399 T^{15} + 3512479453921 T^{16} )^{2} \))
$41$ (\( ( 1 + 41 T^{2} )^{2} \))(\( ( 1 + 41 T^{2} )^{2} \))(\( ( 1 + 41 T^{2} )^{2} \))(\( ( 1 + 41 T^{2} )^{2} \))(\( ( 1 - 6 T + 41 T^{2} )^{2} \))(\( ( 1 + 6 T + 41 T^{2} )^{2} \))(\( ( 1 + 41 T^{2} )^{2} \))(\( ( 1 + 41 T^{2} )^{2} \))(\( ( 1 + 41 T^{2} )^{2} \))(\( ( 1 - 4 T + 90 T^{2} - 592 T^{3} + 3749 T^{4} - 36434 T^{5} + 153709 T^{6} - 995152 T^{7} + 6202890 T^{8} - 11303044 T^{9} + 115856201 T^{10} )^{2} \))(\( ( 1 + 4 T + 90 T^{2} + 592 T^{3} + 3749 T^{4} + 36434 T^{5} + 153709 T^{6} + 995152 T^{7} + 6202890 T^{8} + 11303044 T^{9} + 115856201 T^{10} )^{2} \))(\( ( 1 + 175 T^{2} + 13716 T^{4} + 685412 T^{6} + 28562204 T^{8} + 1152177572 T^{10} + 38758137876 T^{12} + 831268242175 T^{14} + 7984925229121 T^{16} )^{2} \))(\( ( 1 + 175 T^{2} + 13716 T^{4} + 685412 T^{6} + 28562204 T^{8} + 1152177572 T^{10} + 38758137876 T^{12} + 831268242175 T^{14} + 7984925229121 T^{16} )^{2} \))
$43$ (\( ( 1 + 8 T + 43 T^{2} )^{2} \))(\( ( 1 - 8 T + 43 T^{2} )^{2} \))(\( ( 1 - 13 T + 43 T^{2} )^{2} \))(\( ( 1 + 13 T + 43 T^{2} )^{2} \))(\( ( 1 + 4 T + 43 T^{2} )^{2} \))(\( ( 1 + 4 T + 43 T^{2} )^{2} \))(\( ( 1 + 13 T + 43 T^{2} )^{2} \))(\( ( 1 + 8 T + 43 T^{2} )^{2} \))(\( ( 1 - 8 T + 43 T^{2} )^{2} \))(\( ( 1 - 13 T + 173 T^{2} - 1668 T^{3} + 14545 T^{4} - 96933 T^{5} + 625435 T^{6} - 3084132 T^{7} + 13754711 T^{8} - 44444413 T^{9} + 147008443 T^{10} )^{2} \))(\( ( 1 - 13 T + 173 T^{2} - 1668 T^{3} + 14545 T^{4} - 96933 T^{5} + 625435 T^{6} - 3084132 T^{7} + 13754711 T^{8} - 44444413 T^{9} + 147008443 T^{10} )^{2} \))(\( ( 1 + T + 150 T^{2} + 110 T^{3} + 9290 T^{4} + 4730 T^{5} + 277350 T^{6} + 79507 T^{7} + 3418801 T^{8} )^{4} \))(\( ( 1 - T + 150 T^{2} - 110 T^{3} + 9290 T^{4} - 4730 T^{5} + 277350 T^{6} - 79507 T^{7} + 3418801 T^{8} )^{4} \))
$47$ (\( 1 - 47 T^{2} + 2209 T^{4} \))(\( 1 - 47 T^{2} + 2209 T^{4} \))(\( 1 - 47 T^{2} + 2209 T^{4} \))(\( 1 - 47 T^{2} + 2209 T^{4} \))(\( 1 - 3 T - 38 T^{2} - 141 T^{3} + 2209 T^{4} \))(\( 1 + 3 T - 38 T^{2} + 141 T^{3} + 2209 T^{4} \))(\( 1 - 47 T^{2} + 2209 T^{4} \))(\( 1 - 47 T^{2} + 2209 T^{4} \))(\( 1 - 47 T^{2} + 2209 T^{4} \))(\( 1 - 14 T + 49 T^{2} + 70 T^{3} - 1449 T^{4} + 16780 T^{5} - 75362 T^{6} - 133028 T^{7} + 1885261 T^{8} - 57961722 T^{9} + 732803503 T^{10} - 2724200934 T^{11} + 4164541549 T^{12} - 13811366044 T^{13} - 367742519522 T^{14} + 3848409217460 T^{15} - 15619083011721 T^{16} + 35463618432410 T^{17} + 1166753046426289 T^{18} - 15667826623438738 T^{19} + 52599132235830049 T^{20} \))(\( 1 + 14 T + 49 T^{2} - 70 T^{3} - 1449 T^{4} - 16780 T^{5} - 75362 T^{6} + 133028 T^{7} + 1885261 T^{8} + 57961722 T^{9} + 732803503 T^{10} + 2724200934 T^{11} + 4164541549 T^{12} + 13811366044 T^{13} - 367742519522 T^{14} - 3848409217460 T^{15} - 15619083011721 T^{16} - 35463618432410 T^{17} + 1166753046426289 T^{18} + 15667826623438738 T^{19} + 52599132235830049 T^{20} \))(\( 1 - 34 T^{2} - 1025 T^{4} + 161680 T^{6} - 9103783 T^{8} + 152237258 T^{10} + 4847483963 T^{12} - 753776826296 T^{14} + 44233333645801 T^{16} - 1665093009287864 T^{18} + 23654175392055803 T^{20} + 1640998185078527882 T^{22} - \)\(21\!\cdots\!63\)\( T^{24} + \)\(85\!\cdots\!20\)\( T^{26} - \)\(11\!\cdots\!25\)\( T^{28} - \)\(87\!\cdots\!46\)\( T^{30} + \)\(56\!\cdots\!21\)\( T^{32} \))(\( 1 - 34 T^{2} - 1025 T^{4} + 161680 T^{6} - 9103783 T^{8} + 152237258 T^{10} + 4847483963 T^{12} - 753776826296 T^{14} + 44233333645801 T^{16} - 1665093009287864 T^{18} + 23654175392055803 T^{20} + 1640998185078527882 T^{22} - \)\(21\!\cdots\!63\)\( T^{24} + \)\(85\!\cdots\!20\)\( T^{26} - \)\(11\!\cdots\!25\)\( T^{28} - \)\(87\!\cdots\!46\)\( T^{30} + \)\(56\!\cdots\!21\)\( T^{32} \))
$53$ (\( 1 + 53 T^{2} + 2809 T^{4} \))(\( 1 + 53 T^{2} + 2809 T^{4} \))(\( 1 + 53 T^{2} + 2809 T^{4} \))(\( 1 + 53 T^{2} + 2809 T^{4} \))(\( 1 - 9 T + 80 T^{2} - 477 T^{3} + 2809 T^{4} \))(\( 1 + 9 T + 80 T^{2} + 477 T^{3} + 2809 T^{4} \))(\( 1 + 53 T^{2} + 2809 T^{4} \))(\( 1 + 53 T^{2} + 2809 T^{4} \))(\( 1 + 53 T^{2} + 2809 T^{4} \))(\( 1 + 24 T + 337 T^{2} + 3480 T^{3} + 28655 T^{4} + 224976 T^{5} + 1724958 T^{6} + 14318448 T^{7} + 120990445 T^{8} + 964554888 T^{9} + 7393360287 T^{10} + 51121409064 T^{11} + 339862160005 T^{12} + 2131687582896 T^{13} + 13610748324798 T^{14} + 94083949233168 T^{15} + 635119768151495 T^{16} + 4087994766632760 T^{17} + 20981515668628657 T^{18} + 79194326203251192 T^{19} + 174887470365513049 T^{20} \))(\( 1 - 24 T + 337 T^{2} - 3480 T^{3} + 28655 T^{4} - 224976 T^{5} + 1724958 T^{6} - 14318448 T^{7} + 120990445 T^{8} - 964554888 T^{9} + 7393360287 T^{10} - 51121409064 T^{11} + 339862160005 T^{12} - 2131687582896 T^{13} + 13610748324798 T^{14} - 94083949233168 T^{15} + 635119768151495 T^{16} - 4087994766632760 T^{17} + 20981515668628657 T^{18} - 79194326203251192 T^{19} + 174887470365513049 T^{20} \))(\( 1 + 262 T^{2} + 35395 T^{4} + 3231368 T^{6} + 221761433 T^{8} + 11879858878 T^{10} + 500403165131 T^{12} + 17219962264736 T^{14} + 683285105805889 T^{16} + 48370874001643424 T^{18} + 3948421666806018011 T^{20} + \)\(26\!\cdots\!62\)\( T^{22} + \)\(13\!\cdots\!13\)\( T^{24} + \)\(56\!\cdots\!32\)\( T^{26} + \)\(17\!\cdots\!95\)\( T^{28} + \)\(36\!\cdots\!78\)\( T^{30} + \)\(38\!\cdots\!21\)\( T^{32} \))(\( 1 + 262 T^{2} + 35395 T^{4} + 3231368 T^{6} + 221761433 T^{8} + 11879858878 T^{10} + 500403165131 T^{12} + 17219962264736 T^{14} + 683285105805889 T^{16} + 48370874001643424 T^{18} + 3948421666806018011 T^{20} + \)\(26\!\cdots\!62\)\( T^{22} + \)\(13\!\cdots\!13\)\( T^{24} + \)\(56\!\cdots\!32\)\( T^{26} + \)\(17\!\cdots\!95\)\( T^{28} + \)\(36\!\cdots\!78\)\( T^{30} + \)\(38\!\cdots\!21\)\( T^{32} \))
$59$ (\( 1 - 59 T^{2} + 3481 T^{4} \))(\( 1 - 59 T^{2} + 3481 T^{4} \))(\( 1 - 59 T^{2} + 3481 T^{4} \))(\( 1 - 59 T^{2} + 3481 T^{4} \))(\( 1 - 3 T - 50 T^{2} - 177 T^{3} + 3481 T^{4} \))(\( 1 + 3 T - 50 T^{2} + 177 T^{3} + 3481 T^{4} \))(\( 1 - 59 T^{2} + 3481 T^{4} \))(\( 1 - 59 T^{2} + 3481 T^{4} \))(\( 1 - 59 T^{2} + 3481 T^{4} \))(\( 1 - 135 T^{2} + 756 T^{3} + 7351 T^{4} - 84450 T^{5} - 41974 T^{6} + 3597210 T^{7} - 8726459 T^{8} - 56020614 T^{9} + 414268763 T^{10} - 3305216226 T^{11} - 30376803779 T^{12} + 738791392590 T^{13} - 508614110614 T^{14} - 60375357050550 T^{15} + 310069102794991 T^{16} + 1881420522523164 T^{17} - 19822109076583335 T^{18} + 511116753300641401 T^{20} \))(\( 1 - 135 T^{2} - 756 T^{3} + 7351 T^{4} + 84450 T^{5} - 41974 T^{6} - 3597210 T^{7} - 8726459 T^{8} + 56020614 T^{9} + 414268763 T^{10} + 3305216226 T^{11} - 30376803779 T^{12} - 738791392590 T^{13} - 508614110614 T^{14} + 60375357050550 T^{15} + 310069102794991 T^{16} - 1881420522523164 T^{17} - 19822109076583335 T^{18} + 511116753300641401 T^{20} \))(\( 1 - 179 T^{2} + 16761 T^{4} - 465136 T^{6} - 46752400 T^{8} + 6296484144 T^{10} - 226286292944 T^{12} - 7423721157083 T^{14} + 1182150698554881 T^{16} - 25841973347805923 T^{18} - 2741992700954200784 T^{20} + \)\(26\!\cdots\!04\)\( T^{22} - \)\(68\!\cdots\!00\)\( T^{24} - \)\(23\!\cdots\!36\)\( T^{26} + \)\(29\!\cdots\!41\)\( T^{28} - \)\(11\!\cdots\!19\)\( T^{30} + \)\(21\!\cdots\!41\)\( T^{32} \))(\( 1 - 179 T^{2} + 16761 T^{4} - 465136 T^{6} - 46752400 T^{8} + 6296484144 T^{10} - 226286292944 T^{12} - 7423721157083 T^{14} + 1182150698554881 T^{16} - 25841973347805923 T^{18} - 2741992700954200784 T^{20} + \)\(26\!\cdots\!04\)\( T^{22} - \)\(68\!\cdots\!00\)\( T^{24} - \)\(23\!\cdots\!36\)\( T^{26} + \)\(29\!\cdots\!41\)\( T^{28} - \)\(11\!\cdots\!19\)\( T^{30} + \)\(21\!\cdots\!41\)\( T^{32} \))
$61$ (\( ( 1 + 13 T + 61 T^{2} )( 1 + 14 T + 61 T^{2} ) \))(\( ( 1 + T + 61 T^{2} )( 1 + 14 T + 61 T^{2} ) \))(\( ( 1 - 14 T + 61 T^{2} )( 1 - T + 61 T^{2} ) \))(\( ( 1 - T + 61 T^{2} )( 1 + 13 T + 61 T^{2} ) \))(\( 1 + 21 T + 208 T^{2} + 1281 T^{3} + 3721 T^{4} \))(\( 1 + 21 T + 208 T^{2} + 1281 T^{3} + 3721 T^{4} \))(\( ( 1 - 14 T + 61 T^{2} )( 1 - T + 61 T^{2} ) \))(\( ( 1 + T + 61 T^{2} )( 1 + 14 T + 61 T^{2} ) \))(\( ( 1 + 13 T + 61 T^{2} )( 1 + 14 T + 61 T^{2} ) \))(\( 1 - 42 T + 1112 T^{2} - 22008 T^{3} + 361073 T^{4} - 5062200 T^{5} + 62481246 T^{6} - 687242394 T^{7} + 6821292673 T^{8} - 61325603712 T^{9} + 502052393310 T^{10} - 3740861826432 T^{11} + 25382030036233 T^{12} - 155990965832514 T^{13} + 865105397597886 T^{14} - 4275515394922200 T^{15} + 18602616131649353 T^{16} - 69165484335150168 T^{17} + 213178532052976472 T^{18} - 491154135899033922 T^{19} + 713342911662882601 T^{20} \))(\( 1 - 42 T + 1112 T^{2} - 22008 T^{3} + 361073 T^{4} - 5062200 T^{5} + 62481246 T^{6} - 687242394 T^{7} + 6821292673 T^{8} - 61325603712 T^{9} + 502052393310 T^{10} - 3740861826432 T^{11} + 25382030036233 T^{12} - 155990965832514 T^{13} + 865105397597886 T^{14} - 4275515394922200 T^{15} + 18602616131649353 T^{16} - 69165484335150168 T^{17} + 213178532052976472 T^{18} - 491154135899033922 T^{19} + 713342911662882601 T^{20} \))(\( ( 1 - 18 T + 325 T^{2} - 3906 T^{3} + 44196 T^{4} - 437580 T^{5} + 4261022 T^{6} - 36703131 T^{7} + 310887998 T^{8} - 2238890991 T^{9} + 15855262862 T^{10} - 99322345980 T^{11} + 611930788836 T^{12} - 3298993151706 T^{13} + 16744121667325 T^{14} - 56569371048378 T^{15} + 191707312997281 T^{16} )^{2} \))(\( ( 1 - 18 T + 325 T^{2} - 3906 T^{3} + 44196 T^{4} - 437580 T^{5} + 4261022 T^{6} - 36703131 T^{7} + 310887998 T^{8} - 2238890991 T^{9} + 15855262862 T^{10} - 99322345980 T^{11} + 611930788836 T^{12} - 3298993151706 T^{13} + 16744121667325 T^{14} - 56569371048378 T^{15} + 191707312997281 T^{16} )^{2} \))
$67$ (\( ( 1 - 11 T + 67 T^{2} )( 1 + 16 T + 67 T^{2} ) \))(\( ( 1 - 16 T + 67 T^{2} )( 1 + 5 T + 67 T^{2} ) \))(\( ( 1 - 11 T + 67 T^{2} )( 1 - 5 T + 67 T^{2} ) \))(\( ( 1 - 16 T + 67 T^{2} )( 1 + 11 T + 67 T^{2} ) \))(\( ( 1 - 16 T + 67 T^{2} )( 1 + 11 T + 67 T^{2} ) \))(\( ( 1 - 16 T + 67 T^{2} )( 1 + 11 T + 67 T^{2} ) \))(\( ( 1 + 5 T + 67 T^{2} )( 1 + 11 T + 67 T^{2} ) \))(\( ( 1 - 5 T + 67 T^{2} )( 1 + 16 T + 67 T^{2} ) \))(\( ( 1 - 16 T + 67 T^{2} )( 1 + 11 T + 67 T^{2} ) \))(\( 1 + 7 T - 94 T^{2} - 1731 T^{3} + 26 T^{4} + 170241 T^{5} + 879508 T^{6} - 10247793 T^{7} - 116682655 T^{8} + 286659444 T^{9} + 9801469996 T^{10} + 19206182748 T^{11} - 523788438295 T^{12} - 3082156966059 T^{13} + 17723072128468 T^{14} + 229846648340787 T^{15} + 2351917936394 T^{16} - 10491091788814113 T^{17} - 38170361690324254 T^{18} + 190445740774064629 T^{19} + 1822837804551761449 T^{20} \))(\( 1 + 7 T - 94 T^{2} - 1731 T^{3} + 26 T^{4} + 170241 T^{5} + 879508 T^{6} - 10247793 T^{7} - 116682655 T^{8} + 286659444 T^{9} + 9801469996 T^{10} + 19206182748 T^{11} - 523788438295 T^{12} - 3082156966059 T^{13} + 17723072128468 T^{14} + 229846648340787 T^{15} + 2351917936394 T^{16} - 10491091788814113 T^{17} - 38170361690324254 T^{18} + 190445740774064629 T^{19} + 1822837804551761449 T^{20} \))(\( ( 1 + 15 T - 19 T^{2} - 1578 T^{3} - 3827 T^{4} + 92169 T^{5} + 542666 T^{6} - 1629357 T^{7} - 31878578 T^{8} - 109166919 T^{9} + 2436027674 T^{10} + 27721024947 T^{11} - 77118340067 T^{12} - 2130497418846 T^{13} - 1718709261211 T^{14} + 90910674079845 T^{15} + 406067677556641 T^{16} )^{2} \))(\( ( 1 - 15 T - 19 T^{2} + 1578 T^{3} - 3827 T^{4} - 92169 T^{5} + 542666 T^{6} + 1629357 T^{7} - 31878578 T^{8} + 109166919 T^{9} + 2436027674 T^{10} - 27721024947 T^{11} - 77118340067 T^{12} + 2130497418846 T^{13} - 1718709261211 T^{14} - 90910674079845 T^{15} + 406067677556641 T^{16} )^{2} \))
$71$ (\( ( 1 - 71 T^{2} )^{2} \))(\( ( 1 - 71 T^{2} )^{2} \))(\( ( 1 - 71 T^{2} )^{2} \))(\( ( 1 - 71 T^{2} )^{2} \))(\( 1 - 34 T^{2} + 5041 T^{4} \))(\( 1 - 34 T^{2} + 5041 T^{4} \))(\( ( 1 - 71 T^{2} )^{2} \))(\( ( 1 - 71 T^{2} )^{2} \))(\( ( 1 - 71 T^{2} )^{2} \))(\( 1 - 422 T^{2} + 92093 T^{4} - 13410012 T^{6} + 1429352290 T^{8} - 115763220252 T^{10} + 7205364893890 T^{12} - 340770947150172 T^{14} + 11797139447136653 T^{16} - 272507990185711142 T^{18} + 3255243551009881201 T^{20} \))(\( 1 - 422 T^{2} + 92093 T^{4} - 13410012 T^{6} + 1429352290 T^{8} - 115763220252 T^{10} + 7205364893890 T^{12} - 340770947150172 T^{14} + 11797139447136653 T^{16} - 272507990185711142 T^{18} + 3255243551009881201 T^{20} \))(\( ( 1 - 139 T^{2} + 17517 T^{4} - 1749386 T^{6} + 123496010 T^{8} - 8818654826 T^{10} + 445136416077 T^{12} - 17805939465019 T^{14} + 645753531245761 T^{16} )^{2} \))(\( ( 1 - 139 T^{2} + 17517 T^{4} - 1749386 T^{6} + 123496010 T^{8} - 8818654826 T^{10} + 445136416077 T^{12} - 17805939465019 T^{14} + 645753531245761 T^{16} )^{2} \))
$73$ (\( ( 1 + 10 T + 73 T^{2} )( 1 + 17 T + 73 T^{2} ) \))(\( ( 1 - 10 T + 73 T^{2} )( 1 + 7 T + 73 T^{2} ) \))(\( ( 1 + 10 T + 73 T^{2} )( 1 + 17 T + 73 T^{2} ) \))(\( ( 1 - 10 T + 73 T^{2} )( 1 + 7 T + 73 T^{2} ) \))(\( 1 - 21 T + 220 T^{2} - 1533 T^{3} + 5329 T^{4} \))(\( 1 - 21 T + 220 T^{2} - 1533 T^{3} + 5329 T^{4} \))(\( ( 1 - 17 T + 73 T^{2} )( 1 - 10 T + 73 T^{2} ) \))(\( ( 1 - 7 T + 73 T^{2} )( 1 + 10 T + 73 T^{2} ) \))(\( ( 1 - 17 T + 73 T^{2} )( 1 - 10 T + 73 T^{2} ) \))(\( 1 - 3 T + 115 T^{2} - 336 T^{3} + 8912 T^{4} - 58260 T^{5} + 290497 T^{6} - 1508853 T^{7} - 7279937 T^{8} + 55999116 T^{9} - 744209196 T^{10} + 4087935468 T^{11} - 38794784273 T^{12} - 586969467501 T^{13} + 8249603815777 T^{14} - 120777151008180 T^{15} + 1348690624687568 T^{16} - 3711925902416592 T^{17} + 92742910567819315 T^{18} - 176614760124803739 T^{19} + 4297625829703557649 T^{20} \))(\( 1 - 3 T + 115 T^{2} - 336 T^{3} + 8912 T^{4} - 58260 T^{5} + 290497 T^{6} - 1508853 T^{7} - 7279937 T^{8} + 55999116 T^{9} - 744209196 T^{10} + 4087935468 T^{11} - 38794784273 T^{12} - 586969467501 T^{13} + 8249603815777 T^{14} - 120777151008180 T^{15} + 1348690624687568 T^{16} - 3711925902416592 T^{17} + 92742910567819315 T^{18} - 176614760124803739 T^{19} + 4297625829703557649 T^{20} \))(\( ( 1 + 27 T + 513 T^{2} + 7290 T^{3} + 85003 T^{4} + 866997 T^{5} + 8113770 T^{6} + 71702037 T^{7} + 621942924 T^{8} + 5234248701 T^{9} + 43238280330 T^{10} + 337276571949 T^{11} + 2413935679723 T^{12} + 15112691912970 T^{13} + 77634458086257 T^{14} + 298279760015619 T^{15} + 806460091894081 T^{16} )^{2} \))(\( ( 1 - 27 T + 513 T^{2} - 7290 T^{3} + 85003 T^{4} - 866997 T^{5} + 8113770 T^{6} - 71702037 T^{7} + 621942924 T^{8} - 5234248701 T^{9} + 43238280330 T^{10} - 337276571949 T^{11} + 2413935679723 T^{12} - 15112691912970 T^{13} + 77634458086257 T^{14} - 298279760015619 T^{15} + 806460091894081 T^{16} )^{2} \))
$79$ (\( ( 1 - 17 T + 79 T^{2} )( 1 + 4 T + 79 T^{2} ) \))(\( ( 1 + 4 T + 79 T^{2} )( 1 + 13 T + 79 T^{2} ) \))(\( ( 1 - 17 T + 79 T^{2} )( 1 + 4 T + 79 T^{2} ) \))(\( ( 1 + 4 T + 79 T^{2} )( 1 + 13 T + 79 T^{2} ) \))(\( 1 - T - 78 T^{2} - 79 T^{3} + 6241 T^{4} \))(\( 1 - T - 78 T^{2} - 79 T^{3} + 6241 T^{4} \))(\( ( 1 - 17 T + 79 T^{2} )( 1 + 4 T + 79 T^{2} ) \))(\( ( 1 + 4 T + 79 T^{2} )( 1 + 13 T + 79 T^{2} ) \))(\( ( 1 - 17 T + 79 T^{2} )( 1 + 4 T + 79 T^{2} ) \))(\( 1 - T - 201 T^{2} + 740 T^{3} + 18128 T^{4} - 100900 T^{5} - 805347 T^{6} + 5115611 T^{7} + 21370651 T^{8} - 77003688 T^{9} - 699661200 T^{10} - 6083291352 T^{11} + 133374232891 T^{12} + 2522195731829 T^{13} - 31368330883107 T^{14} - 310474990659100 T^{15} + 4406689393684688 T^{16} + 14210892649757660 T^{17} - 304938870791218761 T^{18} - 119851595982618319 T^{19} + 9468276082626847201 T^{20} \))(\( 1 - T - 201 T^{2} + 740 T^{3} + 18128 T^{4} - 100900 T^{5} - 805347 T^{6} + 5115611 T^{7} + 21370651 T^{8} - 77003688 T^{9} - 699661200 T^{10} - 6083291352 T^{11} + 133374232891 T^{12} + 2522195731829 T^{13} - 31368330883107 T^{14} - 310474990659100 T^{15} + 4406689393684688 T^{16} + 14210892649757660 T^{17} - 304938870791218761 T^{18} - 119851595982618319 T^{19} + 9468276082626847201 T^{20} \))(\( ( 1 - 195 T^{2} - 582 T^{3} + 18208 T^{4} + 79734 T^{5} - 1345644 T^{6} - 3198963 T^{7} + 109759584 T^{8} - 252718077 T^{9} - 8398164204 T^{10} + 39311971626 T^{11} + 709203074848 T^{12} - 1790846824218 T^{13} - 47402053826595 T^{14} + 1517108809906561 T^{16} )^{2} \))(\( ( 1 - 195 T^{2} - 582 T^{3} + 18208 T^{4} + 79734 T^{5} - 1345644 T^{6} - 3198963 T^{7} + 109759584 T^{8} - 252718077 T^{9} - 8398164204 T^{10} + 39311971626 T^{11} + 709203074848 T^{12} - 1790846824218 T^{13} - 47402053826595 T^{14} + 1517108809906561 T^{16} )^{2} \))
$83$ (\( ( 1 + 83 T^{2} )^{2} \))(\( ( 1 + 83 T^{2} )^{2} \))(\( ( 1 + 83 T^{2} )^{2} \))(\( ( 1 + 83 T^{2} )^{2} \))(\( ( 1 - 12 T + 83 T^{2} )^{2} \))(\( ( 1 + 12 T + 83 T^{2} )^{2} \))(\( ( 1 + 83 T^{2} )^{2} \))(\( ( 1 + 83 T^{2} )^{2} \))(\( ( 1 + 83 T^{2} )^{2} \))(\( ( 1 + 4 T + 36 T^{2} + 304 T^{3} + 3029 T^{4} - 33442 T^{5} + 251407 T^{6} + 2094256 T^{7} + 20584332 T^{8} + 189833284 T^{9} + 3939040643 T^{10} )^{2} \))(\( ( 1 - 4 T + 36 T^{2} - 304 T^{3} + 3029 T^{4} + 33442 T^{5} + 251407 T^{6} - 2094256 T^{7} + 20584332 T^{8} - 189833284 T^{9} + 3939040643 T^{10} )^{2} \))(\( ( 1 + 406 T^{2} + 83877 T^{4} + 11459777 T^{6} + 1116323102 T^{8} + 78946403753 T^{10} + 3980661590517 T^{12} + 132737791587814 T^{14} + 2252292232139041 T^{16} )^{2} \))(\( ( 1 + 406 T^{2} + 83877 T^{4} + 11459777 T^{6} + 1116323102 T^{8} + 78946403753 T^{10} + 3980661590517 T^{12} + 132737791587814 T^{14} + 2252292232139041 T^{16} )^{2} \))
$89$ (\( 1 - 89 T^{2} + 7921 T^{4} \))(\( 1 - 89 T^{2} + 7921 T^{4} \))(\( 1 - 89 T^{2} + 7921 T^{4} \))(\( 1 - 89 T^{2} + 7921 T^{4} \))(\( 1 - 9 T - 8 T^{2} - 801 T^{3} + 7921 T^{4} \))(\( 1 + 9 T - 8 T^{2} + 801 T^{3} + 7921 T^{4} \))(\( 1 - 89 T^{2} + 7921 T^{4} \))(\( 1 - 89 T^{2} + 7921 T^{4} \))(\( 1 - 89 T^{2} + 7921 T^{4} \))(\( 1 + 28 T + 118 T^{2} - 2240 T^{3} + 12999 T^{4} + 635638 T^{5} + 2053984 T^{6} - 15180164 T^{7} + 276922069 T^{8} + 3387791550 T^{9} + 10662822610 T^{10} + 301513447950 T^{11} + 2193499708549 T^{12} - 10701545034916 T^{13} + 128871559138144 T^{14} + 3549440380043462 T^{15} + 6460259801202039 T^{16} - 99078190165984960 T^{17} + 464517479072845558 T^{18} + 9809979303809585852 T^{19} + 31181719929966183601 T^{20} \))(\( 1 - 28 T + 118 T^{2} + 2240 T^{3} + 12999 T^{4} - 635638 T^{5} + 2053984 T^{6} + 15180164 T^{7} + 276922069 T^{8} - 3387791550 T^{9} + 10662822610 T^{10} - 301513447950 T^{11} + 2193499708549 T^{12} + 10701545034916 T^{13} + 128871559138144 T^{14} - 3549440380043462 T^{15} + 6460259801202039 T^{16} + 99078190165984960 T^{17} + 464517479072845558 T^{18} - 9809979303809585852 T^{19} + 31181719929966183601 T^{20} \))(\( 1 - 118 T^{2} + 3595 T^{4} + 1013020 T^{6} - 177688255 T^{8} + 15671126798 T^{10} - 78833677741 T^{12} - 122913712595300 T^{14} + 16662250146905929 T^{16} - 973599517467371300 T^{18} - 4946201607742157581 T^{20} + \)\(77\!\cdots\!78\)\( T^{22} - \)\(69\!\cdots\!55\)\( T^{24} + \)\(31\!\cdots\!20\)\( T^{26} + \)\(88\!\cdots\!95\)\( T^{28} - \)\(23\!\cdots\!38\)\( T^{30} + \)\(15\!\cdots\!61\)\( T^{32} \))(\( 1 - 118 T^{2} + 3595 T^{4} + 1013020 T^{6} - 177688255 T^{8} + 15671126798 T^{10} - 78833677741 T^{12} - 122913712595300 T^{14} + 16662250146905929 T^{16} - 973599517467371300 T^{18} - 4946201607742157581 T^{20} + \)\(77\!\cdots\!78\)\( T^{22} - \)\(69\!\cdots\!55\)\( T^{24} + \)\(31\!\cdots\!20\)\( T^{26} + \)\(88\!\cdots\!95\)\( T^{28} - \)\(23\!\cdots\!38\)\( T^{30} + \)\(15\!\cdots\!61\)\( T^{32} \))
$97$ (\( ( 1 - 19 T + 97 T^{2} )( 1 + 19 T + 97 T^{2} ) \))(\( ( 1 - 5 T + 97 T^{2} )( 1 + 5 T + 97 T^{2} ) \))(\( ( 1 - 5 T + 97 T^{2} )( 1 + 5 T + 97 T^{2} ) \))(\( ( 1 - 14 T + 97 T^{2} )( 1 + 14 T + 97 T^{2} ) \))(\( 1 - 146 T^{2} + 9409 T^{4} \))(\( 1 - 146 T^{2} + 9409 T^{4} \))(\( ( 1 - 5 T + 97 T^{2} )( 1 + 5 T + 97 T^{2} ) \))(\( ( 1 - 5 T + 97 T^{2} )( 1 + 5 T + 97 T^{2} ) \))(\( ( 1 - 19 T + 97 T^{2} )( 1 + 19 T + 97 T^{2} ) \))(\( 1 - 758 T^{2} + 269933 T^{4} - 59888792 T^{6} + 9235543570 T^{8} - 1040392525668 T^{10} + 86897229450130 T^{12} - 5301911695718552 T^{14} + 224846632206499757 T^{16} - 5940774664537736438 T^{18} + 73742412689492826049 T^{20} \))(\( 1 - 758 T^{2} + 269933 T^{4} - 59888792 T^{6} + 9235543570 T^{8} - 1040392525668 T^{10} + 86897229450130 T^{12} - 5301911695718552 T^{14} + 224846632206499757 T^{16} - 5940774664537736438 T^{18} + 73742412689492826049 T^{20} \))(\( ( 1 - 402 T^{2} + 92431 T^{4} - 14249709 T^{6} + 1608196287 T^{8} - 134075511981 T^{10} + 8182849972111 T^{12} - 334854745981458 T^{14} + 7837433594376961 T^{16} )^{2} \))(\( ( 1 - 402 T^{2} + 92431 T^{4} - 14249709 T^{6} + 1608196287 T^{8} - 134075511981 T^{10} + 8182849972111 T^{12} - 334854745981458 T^{14} + 7837433594376961 T^{16} )^{2} \))
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