Properties

Label 2100.2.bo
Level 2100
Weight 2
Character orbit bo
Rep. character \(\chi_{2100}(1349,\cdot)\)
Character field \(\Q(\zeta_{6})\)
Dimension 96
Newform subspaces 9
Sturm bound 960
Trace bound 19

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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.bo (of order \(6\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 105 \)
Character field: \(\Q(\zeta_{6})\)
Newform subspaces: \( 9 \)
Sturm bound: \(960\)
Trace bound: \(19\)
Distinguishing \(T_p\): \(11\), \(13\), \(19\)

Dimensions

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

Total New Old
Modular forms 1032 96 936
Cusp forms 888 96 792
Eisenstein series 144 0 144

Trace form

\( 96q + O(q^{10}) \) \( 96q + 6q^{19} + 10q^{21} + 12q^{31} - 30q^{39} + 58q^{49} + 8q^{51} + 78q^{61} + 8q^{79} + 16q^{81} + 26q^{91} + 24q^{99} + O(q^{100}) \)

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.bo.a \(4\) \(16.769\) \(\Q(\zeta_{12})\) None \(0\) \(0\) \(0\) \(0\) \(q+(-\zeta_{12}-\zeta_{12}^{3})q^{3}+(2\zeta_{12}-3\zeta_{12}^{3})q^{7}+\cdots\)
2100.2.bo.b \(4\) \(16.769\) \(\Q(\zeta_{12})\) \(\Q(\sqrt{-3}) \) \(0\) \(0\) \(0\) \(0\) \(q+(\zeta_{12}+\zeta_{12}^{3})q^{3}+(\zeta_{12}-3\zeta_{12}^{3})q^{7}+\cdots\)
2100.2.bo.c \(4\) \(16.769\) \(\Q(\zeta_{12})\) \(\Q(\sqrt{-3}) \) \(0\) \(0\) \(0\) \(0\) \(q+(\zeta_{12}+\zeta_{12}^{3})q^{3}+(-3\zeta_{12}+\zeta_{12}^{3})q^{7}+\cdots\)
2100.2.bo.d \(4\) \(16.769\) \(\Q(\zeta_{12})\) \(\Q(\sqrt{-3}) \) \(0\) \(0\) \(0\) \(0\) \(q+(-\zeta_{12}-\zeta_{12}^{3})q^{3}+(-2\zeta_{12}-\zeta_{12}^{3})q^{7}+\cdots\)
2100.2.bo.e \(4\) \(16.769\) \(\Q(\zeta_{12})\) \(\Q(\sqrt{-3}) \) \(0\) \(0\) \(0\) \(0\) \(q+(-\zeta_{12}-\zeta_{12}^{3})q^{3}+(-\zeta_{12}-2\zeta_{12}^{3})q^{7}+\cdots\)
2100.2.bo.f \(4\) \(16.769\) \(\Q(\zeta_{12})\) None \(0\) \(0\) \(0\) \(0\) \(q+(2\zeta_{12}-\zeta_{12}^{3})q^{3}+(2\zeta_{12}+\zeta_{12}^{3})q^{7}+\cdots\)
2100.2.bo.g \(20\) \(16.769\) \(\mathbb{Q}[x]/(x^{20} - \cdots)\) None \(0\) \(0\) \(0\) \(0\) \(q+(\beta _{4}-\beta _{13})q^{3}+(-\beta _{1}+\beta _{2}+\beta _{18}+\cdots)q^{7}+\cdots\)
2100.2.bo.h \(20\) \(16.769\) \(\mathbb{Q}[x]/(x^{20} - \cdots)\) None \(0\) \(0\) \(0\) \(0\) \(q-\beta _{1}q^{3}+(\beta _{1}-\beta _{2}-\beta _{18})q^{7}+(-\beta _{7}+\cdots)q^{9}+\cdots\)
2100.2.bo.i \(32\) \(16.769\) None \(0\) \(0\) \(0\) \(0\)

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

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ (\( 1 + 3 T^{2} + 9 T^{4} \))(\( 1 + 3 T^{2} + 9 T^{4} \))(\( 1 + 3 T^{2} + 9 T^{4} \))(\( 1 + 3 T^{2} + 9 T^{4} \))(\( 1 + 3 T^{2} + 9 T^{4} \))(\( ( 1 - 3 T^{2} )^{2} \))(\( 1 + 3 T^{2} + 11 T^{4} + 54 T^{6} + 181 T^{8} + 345 T^{10} + 1629 T^{12} + 4374 T^{14} + 8019 T^{16} + 19683 T^{18} + 59049 T^{20} \))(\( 1 - 3 T^{2} - 16 T^{4} + 87 T^{6} + 91 T^{8} - 1104 T^{10} + 819 T^{12} + 7047 T^{14} - 11664 T^{16} - 19683 T^{18} + 59049 T^{20} \))
$5$ 1
$7$ (\( 1 + 2 T^{2} + 49 T^{4} \))(\( 1 + 11 T^{2} + 49 T^{4} \))(\( 1 - 13 T^{2} + 49 T^{4} \))(\( 1 + 2 T^{2} + 49 T^{4} \))(\( 1 + 11 T^{2} + 49 T^{4} \))(\( 1 + 2 T^{2} + 49 T^{4} \))(\( 1 - 13 T^{2} + 119 T^{4} - 562 T^{6} + 565 T^{8} + 10473 T^{10} + 27685 T^{12} - 1349362 T^{14} + 14000231 T^{16} - 74942413 T^{18} + 282475249 T^{20} \))(\( 1 - 13 T^{2} + 119 T^{4} - 562 T^{6} + 565 T^{8} + 10473 T^{10} + 27685 T^{12} - 1349362 T^{14} + 14000231 T^{16} - 74942413 T^{18} + 282475249 T^{20} \))
$11$ (\( ( 1 + 9 T + 38 T^{2} + 99 T^{3} + 121 T^{4} )^{2} \))(\( ( 1 + 11 T^{2} + 121 T^{4} )^{2} \))(\( ( 1 + 11 T^{2} + 121 T^{4} )^{2} \))(\( ( 1 + 11 T^{2} + 121 T^{4} )^{2} \))(\( ( 1 + 11 T^{2} + 121 T^{4} )^{2} \))(\( ( 1 - 9 T + 38 T^{2} - 99 T^{3} + 121 T^{4} )^{2} \))(\( ( 1 - 6 T + 51 T^{2} - 234 T^{3} + 1195 T^{4} - 3666 T^{5} + 12622 T^{6} - 19434 T^{7} + 25645 T^{8} + 184704 T^{9} - 509783 T^{10} + 2031744 T^{11} + 3103045 T^{12} - 25866654 T^{13} + 184798702 T^{14} - 590412966 T^{15} + 2117015395 T^{16} - 4559998014 T^{17} + 10932302931 T^{18} - 14147686146 T^{19} + 25937424601 T^{20} )^{2} \))(\( ( 1 + 6 T + 51 T^{2} + 234 T^{3} + 1195 T^{4} + 3666 T^{5} + 12622 T^{6} + 19434 T^{7} + 25645 T^{8} - 184704 T^{9} - 509783 T^{10} - 2031744 T^{11} + 3103045 T^{12} + 25866654 T^{13} + 184798702 T^{14} + 590412966 T^{15} + 2117015395 T^{16} + 4559998014 T^{17} + 10932302931 T^{18} + 14147686146 T^{19} + 25937424601 T^{20} )^{2} \))
$13$ (\( ( 1 + 13 T^{2} )^{4} \))(\( ( 1 - T^{2} + 169 T^{4} )^{2} \))(\( ( 1 + 23 T^{2} + 169 T^{4} )^{2} \))(\( ( 1 - 22 T^{2} + 169 T^{4} )^{2} \))(\( ( 1 - T^{2} + 169 T^{4} )^{2} \))(\( ( 1 + 13 T^{2} )^{4} \))(\( ( 1 + 49 T^{2} + 1364 T^{4} + 28371 T^{6} + 474775 T^{8} + 6687888 T^{10} + 80236975 T^{12} + 810304131 T^{14} + 6583767476 T^{16} + 39970805329 T^{18} + 137858491849 T^{20} )^{2} \))(\( ( 1 + 49 T^{2} + 1364 T^{4} + 28371 T^{6} + 474775 T^{8} + 6687888 T^{10} + 80236975 T^{12} + 810304131 T^{14} + 6583767476 T^{16} + 39970805329 T^{18} + 137858491849 T^{20} )^{2} \))
$17$ (\( 1 + 25 T^{2} + 336 T^{4} + 7225 T^{6} + 83521 T^{8} \))(\( ( 1 + 17 T^{2} + 289 T^{4} )^{2} \))(\( ( 1 + 17 T^{2} + 289 T^{4} )^{2} \))(\( ( 1 + 17 T^{2} + 289 T^{4} )^{2} \))(\( ( 1 + 17 T^{2} + 289 T^{4} )^{2} \))(\( 1 + 25 T^{2} + 336 T^{4} + 7225 T^{6} + 83521 T^{8} \))(\( 1 + 66 T^{2} + 2267 T^{4} + 48026 T^{6} + 572883 T^{8} - 801368 T^{10} - 179842402 T^{12} - 3509885976 T^{14} - 18012284507 T^{16} + 684073009458 T^{18} + 19619688493209 T^{20} + 197697099733362 T^{22} - 1504404014309147 T^{24} - 84720114927832344 T^{26} - 1254536973958813282 T^{28} - 1615553000015014232 T^{30} + \)\(33\!\cdots\!63\)\( T^{32} + \)\(80\!\cdots\!54\)\( T^{34} + \)\(11\!\cdots\!27\)\( T^{36} + \)\(92\!\cdots\!94\)\( T^{38} + \)\(40\!\cdots\!01\)\( T^{40} \))(\( 1 + 66 T^{2} + 2267 T^{4} + 48026 T^{6} + 572883 T^{8} - 801368 T^{10} - 179842402 T^{12} - 3509885976 T^{14} - 18012284507 T^{16} + 684073009458 T^{18} + 19619688493209 T^{20} + 197697099733362 T^{22} - 1504404014309147 T^{24} - 84720114927832344 T^{26} - 1254536973958813282 T^{28} - 1615553000015014232 T^{30} + \)\(33\!\cdots\!63\)\( T^{32} + \)\(80\!\cdots\!54\)\( T^{34} + \)\(11\!\cdots\!27\)\( T^{36} + \)\(92\!\cdots\!94\)\( T^{38} + \)\(40\!\cdots\!01\)\( T^{40} \))
$19$ (\( ( 1 + 3 T + 22 T^{2} + 57 T^{3} + 361 T^{4} )^{2} \))(\( ( 1 + 7 T + 19 T^{2} )^{2}( 1 + 8 T + 19 T^{2} )^{2} \))(\( ( 1 + T + 19 T^{2} )^{2}( 1 + 8 T + 19 T^{2} )^{2} \))(\( ( 1 - 7 T + 19 T^{2} )^{2}( 1 + T + 19 T^{2} )^{2} \))(\( ( 1 - 8 T + 19 T^{2} )^{2}( 1 - 7 T + 19 T^{2} )^{2} \))(\( ( 1 + 3 T + 22 T^{2} + 57 T^{3} + 361 T^{4} )^{2} \))(\( ( 1 + 3 T + 83 T^{2} + 240 T^{3} + 3800 T^{4} + 10380 T^{5} + 124845 T^{6} + 315315 T^{7} + 3213415 T^{8} + 7312140 T^{9} + 67225536 T^{10} + 138930660 T^{11} + 1160042815 T^{12} + 2162745585 T^{13} + 16269925245 T^{14} + 25701907620 T^{15} + 178774347800 T^{16} + 214529217360 T^{17} + 1409635732403 T^{18} + 968063093337 T^{19} + 6131066257801 T^{20} )^{2} \))(\( ( 1 + 3 T + 83 T^{2} + 240 T^{3} + 3800 T^{4} + 10380 T^{5} + 124845 T^{6} + 315315 T^{7} + 3213415 T^{8} + 7312140 T^{9} + 67225536 T^{10} + 138930660 T^{11} + 1160042815 T^{12} + 2162745585 T^{13} + 16269925245 T^{14} + 25701907620 T^{15} + 178774347800 T^{16} + 214529217360 T^{17} + 1409635732403 T^{18} + 968063093337 T^{19} + 6131066257801 T^{20} )^{2} \))
$23$ (\( 1 - 19 T^{2} - 168 T^{4} - 10051 T^{6} + 279841 T^{8} \))(\( ( 1 - 23 T^{2} + 529 T^{4} )^{2} \))(\( ( 1 - 23 T^{2} + 529 T^{4} )^{2} \))(\( ( 1 - 23 T^{2} + 529 T^{4} )^{2} \))(\( ( 1 - 23 T^{2} + 529 T^{4} )^{2} \))(\( 1 - 19 T^{2} - 168 T^{4} - 10051 T^{6} + 279841 T^{8} \))(\( 1 - 100 T^{2} + 4454 T^{4} - 135804 T^{6} + 3628499 T^{8} - 75944748 T^{10} + 808939708 T^{12} + 10318464096 T^{14} - 898818094003 T^{16} + 35402708232120 T^{18} - 973535494901246 T^{20} + 18728032654791480 T^{22} - 251526154243893523 T^{24} + 1527503005565941344 T^{26} + 63348865566404437948 T^{28} - \)\(31\!\cdots\!52\)\( T^{30} + \)\(79\!\cdots\!79\)\( T^{32} - \)\(15\!\cdots\!36\)\( T^{34} + \)\(27\!\cdots\!94\)\( T^{36} - \)\(32\!\cdots\!00\)\( T^{38} + \)\(17\!\cdots\!01\)\( T^{40} \))(\( 1 - 100 T^{2} + 4454 T^{4} - 135804 T^{6} + 3628499 T^{8} - 75944748 T^{10} + 808939708 T^{12} + 10318464096 T^{14} - 898818094003 T^{16} + 35402708232120 T^{18} - 973535494901246 T^{20} + 18728032654791480 T^{22} - 251526154243893523 T^{24} + 1527503005565941344 T^{26} + 63348865566404437948 T^{28} - \)\(31\!\cdots\!52\)\( T^{30} + \)\(79\!\cdots\!79\)\( T^{32} - \)\(15\!\cdots\!36\)\( T^{34} + \)\(27\!\cdots\!94\)\( T^{36} - \)\(32\!\cdots\!00\)\( T^{38} + \)\(17\!\cdots\!01\)\( T^{40} \))
$29$ (\( ( 1 - 29 T^{2} )^{4} \))(\( ( 1 - 29 T^{2} )^{4} \))(\( ( 1 - 29 T^{2} )^{4} \))(\( ( 1 - 29 T^{2} )^{4} \))(\( ( 1 - 29 T^{2} )^{4} \))(\( ( 1 - 29 T^{2} )^{4} \))(\( ( 1 - 148 T^{2} + 11438 T^{4} - 619594 T^{6} + 25499257 T^{8} - 826380936 T^{10} + 21444875137 T^{12} - 438227063914 T^{14} + 6803589145598 T^{16} - 74036469118228 T^{18} + 420707233300201 T^{20} )^{2} \))(\( ( 1 - 148 T^{2} + 11438 T^{4} - 619594 T^{6} + 25499257 T^{8} - 826380936 T^{10} + 21444875137 T^{12} - 438227063914 T^{14} + 6803589145598 T^{16} - 74036469118228 T^{18} + 420707233300201 T^{20} )^{2} \))
$31$ (\( ( 1 - 4 T + 31 T^{2} )^{2}( 1 + 7 T + 31 T^{2} )^{2} \))(\( ( 1 + 7 T + 31 T^{2} )^{2}( 1 + 11 T + 31 T^{2} )^{2} \))(\( ( 1 - 11 T + 31 T^{2} )^{2}( 1 - 7 T + 31 T^{2} )^{2} \))(\( ( 1 - 7 T + 31 T^{2} )^{2}( 1 + 4 T + 31 T^{2} )^{2} \))(\( ( 1 - 4 T + 31 T^{2} )^{2}( 1 + 7 T + 31 T^{2} )^{2} \))(\( ( 1 - 4 T + 31 T^{2} )^{2}( 1 + 7 T + 31 T^{2} )^{2} \))(\( ( 1 - 15 T + 167 T^{2} - 1380 T^{3} + 9668 T^{4} - 61404 T^{5} + 363093 T^{6} - 2129475 T^{7} + 12161347 T^{8} - 68552376 T^{9} + 387742776 T^{10} - 2125123656 T^{11} + 11687054467 T^{12} - 63439189725 T^{13} + 335324010453 T^{14} - 1757944388004 T^{15} + 8580385587908 T^{16} - 37967407473180 T^{17} + 142432803252647 T^{18} - 396594332410065 T^{19} + 819628286980801 T^{20} )^{2} \))(\( ( 1 - 15 T + 167 T^{2} - 1380 T^{3} + 9668 T^{4} - 61404 T^{5} + 363093 T^{6} - 2129475 T^{7} + 12161347 T^{8} - 68552376 T^{9} + 387742776 T^{10} - 2125123656 T^{11} + 11687054467 T^{12} - 63439189725 T^{13} + 335324010453 T^{14} - 1757944388004 T^{15} + 8580385587908 T^{16} - 37967407473180 T^{17} + 142432803252647 T^{18} - 396594332410065 T^{19} + 819628286980801 T^{20} )^{2} \))
$37$ (\( 1 + 25 T^{2} - 744 T^{4} + 34225 T^{6} + 1874161 T^{8} \))(\( ( 1 + 26 T^{2} + 1369 T^{4} )( 1 + 47 T^{2} + 1369 T^{4} ) \))(\( ( 1 - 73 T^{2} + 1369 T^{4} )( 1 + 26 T^{2} + 1369 T^{4} ) \))(\( ( 1 + 26 T^{2} + 1369 T^{4} )( 1 + 47 T^{2} + 1369 T^{4} ) \))(\( ( 1 - 73 T^{2} + 1369 T^{4} )( 1 + 26 T^{2} + 1369 T^{4} ) \))(\( 1 + 25 T^{2} - 744 T^{4} + 34225 T^{6} + 1874161 T^{8} \))(\( 1 + 235 T^{2} + 27377 T^{4} + 2212926 T^{6} + 144837806 T^{8} + 8250414270 T^{10} + 421139833759 T^{12} + 19588270538109 T^{14} + 844751235316463 T^{16} + 34241451973802580 T^{18} + 1307081650323006964 T^{20} + 46876547752135732020 T^{22} + \)\(15\!\cdots\!43\)\( T^{24} + \)\(50\!\cdots\!81\)\( T^{26} + \)\(14\!\cdots\!39\)\( T^{28} + \)\(39\!\cdots\!30\)\( T^{30} + \)\(95\!\cdots\!86\)\( T^{32} + \)\(19\!\cdots\!14\)\( T^{34} + \)\(33\!\cdots\!57\)\( T^{36} + \)\(39\!\cdots\!15\)\( T^{38} + \)\(23\!\cdots\!01\)\( T^{40} \))(\( 1 + 235 T^{2} + 27377 T^{4} + 2212926 T^{6} + 144837806 T^{8} + 8250414270 T^{10} + 421139833759 T^{12} + 19588270538109 T^{14} + 844751235316463 T^{16} + 34241451973802580 T^{18} + 1307081650323006964 T^{20} + 46876547752135732020 T^{22} + \)\(15\!\cdots\!43\)\( T^{24} + \)\(50\!\cdots\!81\)\( T^{26} + \)\(14\!\cdots\!39\)\( T^{28} + \)\(39\!\cdots\!30\)\( T^{30} + \)\(95\!\cdots\!86\)\( T^{32} + \)\(19\!\cdots\!14\)\( T^{34} + \)\(33\!\cdots\!57\)\( T^{36} + \)\(39\!\cdots\!15\)\( T^{38} + \)\(23\!\cdots\!01\)\( T^{40} \))
$41$ (\( ( 1 + 6 T + 41 T^{2} )^{4} \))(\( ( 1 + 41 T^{2} )^{4} \))(\( ( 1 + 41 T^{2} )^{4} \))(\( ( 1 + 41 T^{2} )^{4} \))(\( ( 1 + 41 T^{2} )^{4} \))(\( ( 1 - 6 T + 41 T^{2} )^{4} \))(\( ( 1 - 4 T + 90 T^{2} - 592 T^{3} + 3749 T^{4} - 36434 T^{5} + 153709 T^{6} - 995152 T^{7} + 6202890 T^{8} - 11303044 T^{9} + 115856201 T^{10} )^{4} \))(\( ( 1 + 4 T + 90 T^{2} + 592 T^{3} + 3749 T^{4} + 36434 T^{5} + 153709 T^{6} + 995152 T^{7} + 6202890 T^{8} + 11303044 T^{9} + 115856201 T^{10} )^{4} \))
$43$ (\( ( 1 - 70 T^{2} + 1849 T^{4} )^{2} \))(\( ( 1 - 22 T^{2} + 1849 T^{4} )^{2} \))(\( ( 1 - 22 T^{2} + 1849 T^{4} )^{2} \))(\( ( 1 + 83 T^{2} + 1849 T^{4} )^{2} \))(\( ( 1 + 83 T^{2} + 1849 T^{4} )^{2} \))(\( ( 1 - 70 T^{2} + 1849 T^{4} )^{2} \))(\( ( 1 - 177 T^{2} + 15651 T^{4} - 980958 T^{6} + 51911037 T^{8} - 2406825463 T^{10} + 95983507413 T^{12} - 3353700191358 T^{14} + 98935653079899 T^{16} - 2068811449135377 T^{18} + 21611482313284249 T^{20} )^{2} \))(\( ( 1 - 177 T^{2} + 15651 T^{4} - 980958 T^{6} + 51911037 T^{8} - 2406825463 T^{10} + 95983507413 T^{12} - 3353700191358 T^{14} + 98935653079899 T^{16} - 2068811449135377 T^{18} + 21611482313284249 T^{20} )^{2} \))
$47$ (\( 1 + 85 T^{2} + 5016 T^{4} + 187765 T^{6} + 4879681 T^{8} \))(\( ( 1 + 47 T^{2} + 2209 T^{4} )^{2} \))(\( ( 1 + 47 T^{2} + 2209 T^{4} )^{2} \))(\( ( 1 + 47 T^{2} + 2209 T^{4} )^{2} \))(\( ( 1 + 47 T^{2} + 2209 T^{4} )^{2} \))(\( 1 + 85 T^{2} + 5016 T^{4} + 187765 T^{6} + 4879681 T^{8} \))(\( 1 + 98 T^{2} + 1463 T^{4} - 172214 T^{6} - 7589337 T^{8} + 17111036 T^{10} + 16661205666 T^{12} + 813009774836 T^{14} - 550471722003 T^{16} - 837271397783898 T^{18} - 19354757954951431 T^{20} - 1849532517704630682 T^{22} - 2686126402895321043 T^{24} + \)\(87\!\cdots\!44\)\( T^{26} + \)\(39\!\cdots\!26\)\( T^{28} + \)\(90\!\cdots\!64\)\( T^{30} - \)\(88\!\cdots\!17\)\( T^{32} - \)\(44\!\cdots\!66\)\( T^{34} + \)\(82\!\cdots\!23\)\( T^{36} + \)\(12\!\cdots\!22\)\( T^{38} + \)\(27\!\cdots\!01\)\( T^{40} \))(\( 1 + 98 T^{2} + 1463 T^{4} - 172214 T^{6} - 7589337 T^{8} + 17111036 T^{10} + 16661205666 T^{12} + 813009774836 T^{14} - 550471722003 T^{16} - 837271397783898 T^{18} - 19354757954951431 T^{20} - 1849532517704630682 T^{22} - 2686126402895321043 T^{24} + \)\(87\!\cdots\!44\)\( T^{26} + \)\(39\!\cdots\!26\)\( T^{28} + \)\(90\!\cdots\!64\)\( T^{30} - \)\(88\!\cdots\!17\)\( T^{32} - \)\(44\!\cdots\!66\)\( T^{34} + \)\(82\!\cdots\!23\)\( T^{36} + \)\(12\!\cdots\!22\)\( T^{38} + \)\(27\!\cdots\!01\)\( T^{40} \))
$53$ (\( 1 - 79 T^{2} + 3432 T^{4} - 221911 T^{6} + 7890481 T^{8} \))(\( ( 1 - 53 T^{2} + 2809 T^{4} )^{2} \))(\( ( 1 - 53 T^{2} + 2809 T^{4} )^{2} \))(\( ( 1 - 53 T^{2} + 2809 T^{4} )^{2} \))(\( ( 1 - 53 T^{2} + 2809 T^{4} )^{2} \))(\( 1 - 79 T^{2} + 3432 T^{4} - 221911 T^{6} + 7890481 T^{8} \))(\( 1 - 98 T^{2} + 3839 T^{4} + 145862 T^{6} - 27406857 T^{8} + 1377609796 T^{10} - 37452311886 T^{12} - 261577435028 T^{14} + 89930621026509 T^{16} - 7083814035902694 T^{18} + 497135157539747921 T^{20} - 19898433626850667446 T^{22} + \)\(70\!\cdots\!29\)\( T^{24} - \)\(57\!\cdots\!12\)\( T^{26} - \)\(23\!\cdots\!46\)\( T^{28} + \)\(24\!\cdots\!04\)\( T^{30} - \)\(13\!\cdots\!37\)\( T^{32} + \)\(20\!\cdots\!78\)\( T^{34} + \)\(14\!\cdots\!19\)\( T^{36} - \)\(10\!\cdots\!22\)\( T^{38} + \)\(30\!\cdots\!01\)\( T^{40} \))(\( 1 - 98 T^{2} + 3839 T^{4} + 145862 T^{6} - 27406857 T^{8} + 1377609796 T^{10} - 37452311886 T^{12} - 261577435028 T^{14} + 89930621026509 T^{16} - 7083814035902694 T^{18} + 497135157539747921 T^{20} - 19898433626850667446 T^{22} + \)\(70\!\cdots\!29\)\( T^{24} - \)\(57\!\cdots\!12\)\( T^{26} - \)\(23\!\cdots\!46\)\( T^{28} + \)\(24\!\cdots\!04\)\( T^{30} - \)\(13\!\cdots\!37\)\( T^{32} + \)\(20\!\cdots\!78\)\( T^{34} + \)\(14\!\cdots\!19\)\( T^{36} - \)\(10\!\cdots\!22\)\( T^{38} + \)\(30\!\cdots\!01\)\( T^{40} \))
$59$ (\( ( 1 - 3 T - 50 T^{2} - 177 T^{3} + 3481 T^{4} )^{2} \))(\( ( 1 - 59 T^{2} + 3481 T^{4} )^{2} \))(\( ( 1 - 59 T^{2} + 3481 T^{4} )^{2} \))(\( ( 1 - 59 T^{2} + 3481 T^{4} )^{2} \))(\( ( 1 - 59 T^{2} + 3481 T^{4} )^{2} \))(\( ( 1 + 3 T - 50 T^{2} + 177 T^{3} + 3481 T^{4} )^{2} \))(\( ( 1 - 135 T^{2} - 756 T^{3} + 7351 T^{4} + 84450 T^{5} - 41974 T^{6} - 3597210 T^{7} - 8726459 T^{8} + 56020614 T^{9} + 414268763 T^{10} + 3305216226 T^{11} - 30376803779 T^{12} - 738791392590 T^{13} - 508614110614 T^{14} + 60375357050550 T^{15} + 310069102794991 T^{16} - 1881420522523164 T^{17} - 19822109076583335 T^{18} + 511116753300641401 T^{20} )^{2} \))(\( ( 1 - 135 T^{2} + 756 T^{3} + 7351 T^{4} - 84450 T^{5} - 41974 T^{6} + 3597210 T^{7} - 8726459 T^{8} - 56020614 T^{9} + 414268763 T^{10} - 3305216226 T^{11} - 30376803779 T^{12} + 738791392590 T^{13} - 508614110614 T^{14} - 60375357050550 T^{15} + 310069102794991 T^{16} + 1881420522523164 T^{17} - 19822109076583335 T^{18} + 511116753300641401 T^{20} )^{2} \))
$61$ (\( ( 1 + 21 T + 208 T^{2} + 1281 T^{3} + 3721 T^{4} )^{2} \))(\( ( 1 + 13 T + 61 T^{2} )^{2}( 1 + 14 T + 61 T^{2} )^{2} \))(\( ( 1 + T + 61 T^{2} )^{2}( 1 + 14 T + 61 T^{2} )^{2} \))(\( ( 1 - 14 T + 61 T^{2} )^{2}( 1 - T + 61 T^{2} )^{2} \))(\( ( 1 - T + 61 T^{2} )^{2}( 1 + 13 T + 61 T^{2} )^{2} \))(\( ( 1 + 21 T + 208 T^{2} + 1281 T^{3} + 3721 T^{4} )^{2} \))(\( ( 1 - 42 T + 1112 T^{2} - 22008 T^{3} + 361073 T^{4} - 5062200 T^{5} + 62481246 T^{6} - 687242394 T^{7} + 6821292673 T^{8} - 61325603712 T^{9} + 502052393310 T^{10} - 3740861826432 T^{11} + 25382030036233 T^{12} - 155990965832514 T^{13} + 865105397597886 T^{14} - 4275515394922200 T^{15} + 18602616131649353 T^{16} - 69165484335150168 T^{17} + 213178532052976472 T^{18} - 491154135899033922 T^{19} + 713342911662882601 T^{20} )^{2} \))(\( ( 1 - 42 T + 1112 T^{2} - 22008 T^{3} + 361073 T^{4} - 5062200 T^{5} + 62481246 T^{6} - 687242394 T^{7} + 6821292673 T^{8} - 61325603712 T^{9} + 502052393310 T^{10} - 3740861826432 T^{11} + 25382030036233 T^{12} - 155990965832514 T^{13} + 865105397597886 T^{14} - 4275515394922200 T^{15} + 18602616131649353 T^{16} - 69165484335150168 T^{17} + 213178532052976472 T^{18} - 491154135899033922 T^{19} + 713342911662882601 T^{20} )^{2} \))
$67$ (\( ( 1 - 13 T^{2} + 4489 T^{4} )( 1 + 122 T^{2} + 4489 T^{4} ) \))(\( ( 1 - 13 T^{2} + 4489 T^{4} )( 1 + 122 T^{2} + 4489 T^{4} ) \))(\( ( 1 - 109 T^{2} + 4489 T^{4} )( 1 + 122 T^{2} + 4489 T^{4} ) \))(\( ( 1 - 109 T^{2} + 4489 T^{4} )( 1 - 13 T^{2} + 4489 T^{4} ) \))(\( ( 1 - 13 T^{2} + 4489 T^{4} )( 1 + 122 T^{2} + 4489 T^{4} ) \))(\( ( 1 - 13 T^{2} + 4489 T^{4} )( 1 + 122 T^{2} + 4489 T^{4} ) \))(\( 1 + 237 T^{2} + 33122 T^{4} + 3625607 T^{6} + 334131306 T^{8} + 26888076115 T^{10} + 2089931081048 T^{12} + 163796249955747 T^{14} + 12948927080804797 T^{16} + 981044928724302918 T^{18} + 68872931546114565948 T^{20} + \)\(44\!\cdots\!02\)\( T^{22} + \)\(26\!\cdots\!37\)\( T^{24} + \)\(14\!\cdots\!43\)\( T^{26} + \)\(84\!\cdots\!68\)\( T^{28} + \)\(49\!\cdots\!35\)\( T^{30} + \)\(27\!\cdots\!66\)\( T^{32} + \)\(13\!\cdots\!03\)\( T^{34} + \)\(54\!\cdots\!82\)\( T^{36} + \)\(17\!\cdots\!33\)\( T^{38} + \)\(33\!\cdots\!01\)\( T^{40} \))(\( 1 + 237 T^{2} + 33122 T^{4} + 3625607 T^{6} + 334131306 T^{8} + 26888076115 T^{10} + 2089931081048 T^{12} + 163796249955747 T^{14} + 12948927080804797 T^{16} + 981044928724302918 T^{18} + 68872931546114565948 T^{20} + \)\(44\!\cdots\!02\)\( T^{22} + \)\(26\!\cdots\!37\)\( T^{24} + \)\(14\!\cdots\!43\)\( T^{26} + \)\(84\!\cdots\!68\)\( T^{28} + \)\(49\!\cdots\!35\)\( T^{30} + \)\(27\!\cdots\!66\)\( T^{32} + \)\(13\!\cdots\!03\)\( T^{34} + \)\(54\!\cdots\!82\)\( T^{36} + \)\(17\!\cdots\!33\)\( T^{38} + \)\(33\!\cdots\!01\)\( T^{40} \))
$71$ (\( ( 1 - 34 T^{2} + 5041 T^{4} )^{2} \))(\( ( 1 - 71 T^{2} )^{4} \))(\( ( 1 - 71 T^{2} )^{4} \))(\( ( 1 - 71 T^{2} )^{4} \))(\( ( 1 - 71 T^{2} )^{4} \))(\( ( 1 - 34 T^{2} + 5041 T^{4} )^{2} \))(\( ( 1 - 422 T^{2} + 92093 T^{4} - 13410012 T^{6} + 1429352290 T^{8} - 115763220252 T^{10} + 7205364893890 T^{12} - 340770947150172 T^{14} + 11797139447136653 T^{16} - 272507990185711142 T^{18} + 3255243551009881201 T^{20} )^{2} \))(\( ( 1 - 422 T^{2} + 92093 T^{4} - 13410012 T^{6} + 1429352290 T^{8} - 115763220252 T^{10} + 7205364893890 T^{12} - 340770947150172 T^{14} + 11797139447136653 T^{16} - 272507990185711142 T^{18} + 3255243551009881201 T^{20} )^{2} \))
$73$ (\( 1 + T^{2} - 5328 T^{4} + 5329 T^{6} + 28398241 T^{8} \))(\( ( 1 - 46 T^{2} + 5329 T^{4} )( 1 + 143 T^{2} + 5329 T^{4} ) \))(\( ( 1 - 97 T^{2} + 5329 T^{4} )( 1 - 46 T^{2} + 5329 T^{4} ) \))(\( ( 1 - 46 T^{2} + 5329 T^{4} )( 1 + 143 T^{2} + 5329 T^{4} ) \))(\( ( 1 - 97 T^{2} + 5329 T^{4} )( 1 - 46 T^{2} + 5329 T^{4} ) \))(\( 1 + T^{2} - 5328 T^{4} + 5329 T^{6} + 28398241 T^{8} \))(\( 1 - 221 T^{2} + 29033 T^{4} - 2168298 T^{6} + 83474342 T^{8} + 2057167494 T^{10} - 407779306505 T^{12} + 6444321727917 T^{14} + 3050137018094543 T^{16} - 468394461607084380 T^{18} + 40020648370576768996 T^{20} - \)\(24\!\cdots\!20\)\( T^{22} + \)\(86\!\cdots\!63\)\( T^{24} + \)\(97\!\cdots\!13\)\( T^{26} - \)\(32\!\cdots\!05\)\( T^{28} + \)\(88\!\cdots\!06\)\( T^{30} + \)\(19\!\cdots\!82\)\( T^{32} - \)\(26\!\cdots\!82\)\( T^{34} + \)\(18\!\cdots\!13\)\( T^{36} - \)\(76\!\cdots\!49\)\( T^{38} + \)\(18\!\cdots\!01\)\( T^{40} \))(\( 1 - 221 T^{2} + 29033 T^{4} - 2168298 T^{6} + 83474342 T^{8} + 2057167494 T^{10} - 407779306505 T^{12} + 6444321727917 T^{14} + 3050137018094543 T^{16} - 468394461607084380 T^{18} + 40020648370576768996 T^{20} - \)\(24\!\cdots\!20\)\( T^{22} + \)\(86\!\cdots\!63\)\( T^{24} + \)\(97\!\cdots\!13\)\( T^{26} - \)\(32\!\cdots\!05\)\( T^{28} + \)\(88\!\cdots\!06\)\( T^{30} + \)\(19\!\cdots\!82\)\( T^{32} - \)\(26\!\cdots\!82\)\( T^{34} + \)\(18\!\cdots\!13\)\( T^{36} - \)\(76\!\cdots\!49\)\( T^{38} + \)\(18\!\cdots\!01\)\( T^{40} \))
$79$ (\( ( 1 + T - 78 T^{2} + 79 T^{3} + 6241 T^{4} )^{2} \))(\( ( 1 - 4 T + 79 T^{2} )^{2}( 1 + 17 T + 79 T^{2} )^{2} \))(\( ( 1 - 13 T + 79 T^{2} )^{2}( 1 - 4 T + 79 T^{2} )^{2} \))(\( ( 1 - 4 T + 79 T^{2} )^{2}( 1 + 17 T + 79 T^{2} )^{2} \))(\( ( 1 - 13 T + 79 T^{2} )^{2}( 1 - 4 T + 79 T^{2} )^{2} \))(\( ( 1 + T - 78 T^{2} + 79 T^{3} + 6241 T^{4} )^{2} \))(\( ( 1 + T - 201 T^{2} - 740 T^{3} + 18128 T^{4} + 100900 T^{5} - 805347 T^{6} - 5115611 T^{7} + 21370651 T^{8} + 77003688 T^{9} - 699661200 T^{10} + 6083291352 T^{11} + 133374232891 T^{12} - 2522195731829 T^{13} - 31368330883107 T^{14} + 310474990659100 T^{15} + 4406689393684688 T^{16} - 14210892649757660 T^{17} - 304938870791218761 T^{18} + 119851595982618319 T^{19} + 9468276082626847201 T^{20} )^{2} \))(\( ( 1 + T - 201 T^{2} - 740 T^{3} + 18128 T^{4} + 100900 T^{5} - 805347 T^{6} - 5115611 T^{7} + 21370651 T^{8} + 77003688 T^{9} - 699661200 T^{10} + 6083291352 T^{11} + 133374232891 T^{12} - 2522195731829 T^{13} - 31368330883107 T^{14} + 310474990659100 T^{15} + 4406689393684688 T^{16} - 14210892649757660 T^{17} - 304938870791218761 T^{18} + 119851595982618319 T^{19} + 9468276082626847201 T^{20} )^{2} \))
$83$ (\( ( 1 - 22 T^{2} + 6889 T^{4} )^{2} \))(\( ( 1 - 83 T^{2} )^{4} \))(\( ( 1 - 83 T^{2} )^{4} \))(\( ( 1 - 83 T^{2} )^{4} \))(\( ( 1 - 83 T^{2} )^{4} \))(\( ( 1 - 22 T^{2} + 6889 T^{4} )^{2} \))(\( ( 1 - 56 T^{2} + 4922 T^{4} - 896022 T^{6} + 72023497 T^{8} - 6972835512 T^{10} + 496169870833 T^{12} - 42523699699062 T^{14} + 1609200517722218 T^{16} - 126128364999786296 T^{18} + 15516041187205853449 T^{20} )^{2} \))(\( ( 1 - 56 T^{2} + 4922 T^{4} - 896022 T^{6} + 72023497 T^{8} - 6972835512 T^{10} + 496169870833 T^{12} - 42523699699062 T^{14} + 1609200517722218 T^{16} - 126128364999786296 T^{18} + 15516041187205853449 T^{20} )^{2} \))
$89$ (\( ( 1 - 9 T - 8 T^{2} - 801 T^{3} + 7921 T^{4} )^{2} \))(\( ( 1 - 89 T^{2} + 7921 T^{4} )^{2} \))(\( ( 1 - 89 T^{2} + 7921 T^{4} )^{2} \))(\( ( 1 - 89 T^{2} + 7921 T^{4} )^{2} \))(\( ( 1 - 89 T^{2} + 7921 T^{4} )^{2} \))(\( ( 1 + 9 T - 8 T^{2} + 801 T^{3} + 7921 T^{4} )^{2} \))(\( ( 1 - 28 T + 118 T^{2} + 2240 T^{3} + 12999 T^{4} - 635638 T^{5} + 2053984 T^{6} + 15180164 T^{7} + 276922069 T^{8} - 3387791550 T^{9} + 10662822610 T^{10} - 301513447950 T^{11} + 2193499708549 T^{12} + 10701545034916 T^{13} + 128871559138144 T^{14} - 3549440380043462 T^{15} + 6460259801202039 T^{16} + 99078190165984960 T^{17} + 464517479072845558 T^{18} - 9809979303809585852 T^{19} + 31181719929966183601 T^{20} )^{2} \))(\( ( 1 + 28 T + 118 T^{2} - 2240 T^{3} + 12999 T^{4} + 635638 T^{5} + 2053984 T^{6} - 15180164 T^{7} + 276922069 T^{8} + 3387791550 T^{9} + 10662822610 T^{10} + 301513447950 T^{11} + 2193499708549 T^{12} - 10701545034916 T^{13} + 128871559138144 T^{14} + 3549440380043462 T^{15} + 6460259801202039 T^{16} - 99078190165984960 T^{17} + 464517479072845558 T^{18} + 9809979303809585852 T^{19} + 31181719929966183601 T^{20} )^{2} \))
$97$ (\( ( 1 + 146 T^{2} + 9409 T^{4} )^{2} \))(\( ( 1 + 167 T^{2} + 9409 T^{4} )^{2} \))(\( ( 1 - 169 T^{2} + 9409 T^{4} )^{2} \))(\( ( 1 - 169 T^{2} + 9409 T^{4} )^{2} \))(\( ( 1 + 2 T^{2} + 9409 T^{4} )^{2} \))(\( ( 1 + 146 T^{2} + 9409 T^{4} )^{2} \))(\( ( 1 + 758 T^{2} + 269933 T^{4} + 59888792 T^{6} + 9235543570 T^{8} + 1040392525668 T^{10} + 86897229450130 T^{12} + 5301911695718552 T^{14} + 224846632206499757 T^{16} + 5940774664537736438 T^{18} + 73742412689492826049 T^{20} )^{2} \))(\( ( 1 + 758 T^{2} + 269933 T^{4} + 59888792 T^{6} + 9235543570 T^{8} + 1040392525668 T^{10} + 86897229450130 T^{12} + 5301911695718552 T^{14} + 224846632206499757 T^{16} + 5940774664537736438 T^{18} + 73742412689492826049 T^{20} )^{2} \))
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