Properties

Label 1050.2.s
Level 1050
Weight 2
Character orbit s
Rep. character \(\chi_{1050}(101,\cdot)\)
Character field \(\Q(\zeta_{6})\)
Dimension 100
Newform subspaces 10
Sturm bound 480
Trace bound 6

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Defining parameters

Level: \( N \) = \( 1050 = 2 \cdot 3 \cdot 5^{2} \cdot 7 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 1050.s (of order \(6\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) = \( 21 \)
Character field: \(\Q(\zeta_{6})\)
Newform subspaces: \( 10 \)
Sturm bound: \(480\)
Trace bound: \(6\)
Distinguishing \(T_p\): \(11\), \(37\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(1050, [\chi])\).

Total New Old
Modular forms 528 100 428
Cusp forms 432 100 332
Eisenstein series 96 0 96

Trace form

\( 100q + 50q^{4} - 6q^{7} + 6q^{9} + O(q^{10}) \) \( 100q + 50q^{4} - 6q^{7} + 6q^{9} - 50q^{16} + 8q^{18} + 24q^{19} + 16q^{21} - 12q^{22} - 6q^{24} - 6q^{28} - 18q^{31} + 42q^{33} + 12q^{36} + 12q^{37} + 12q^{39} + 2q^{42} + 32q^{43} + 20q^{46} - 2q^{49} - 16q^{51} + 12q^{52} + 18q^{54} + 72q^{57} - 22q^{58} + 60q^{61} - 40q^{63} - 100q^{64} + 48q^{66} + 12q^{67} - 8q^{72} - 24q^{73} - 40q^{78} + 2q^{79} + 10q^{81} + 24q^{82} - 4q^{84} - 54q^{87} - 6q^{88} - 8q^{91} + 28q^{93} + 24q^{94} - 6q^{96} - 128q^{99} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(1050, [\chi])\) into newform subspaces

Label Dim. \(A\) Field CM Traces $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\)
1050.2.s.a \(4\) \(8.384\) \(\Q(\zeta_{12})\) None \(0\) \(-6\) \(0\) \(10\) \(q+(-\zeta_{12}+\zeta_{12}^{3})q^{2}+(-1-\zeta_{12}^{2}+\cdots)q^{3}+\cdots\)
1050.2.s.b \(4\) \(8.384\) \(\Q(\zeta_{12})\) None \(0\) \(0\) \(0\) \(10\) \(q+(-\zeta_{12}+\zeta_{12}^{3})q^{2}+(\zeta_{12}-2\zeta_{12}^{3})q^{3}+\cdots\)
1050.2.s.c \(4\) \(8.384\) \(\Q(\zeta_{12})\) None \(0\) \(6\) \(0\) \(-10\) \(q+(\zeta_{12}-\zeta_{12}^{3})q^{2}+(1+\zeta_{12}^{2})q^{3}+\cdots\)
1050.2.s.d \(8\) \(8.384\) 8.0.303595776.1 None \(0\) \(0\) \(0\) \(0\) \(q-\beta _{5}q^{2}+(-\beta _{1}+\beta _{5})q^{3}+(1+\beta _{4}+\cdots)q^{4}+\cdots\)
1050.2.s.e \(8\) \(8.384\) 8.0.303595776.1 None \(0\) \(0\) \(0\) \(0\) \(q+\beta _{5}q^{2}+(-\beta _{1}+\beta _{7})q^{3}+(1+\beta _{4}+\cdots)q^{4}+\cdots\)
1050.2.s.f \(12\) \(8.384\) \(\mathbb{Q}[x]/(x^{12} - \cdots)\) None \(0\) \(-2\) \(0\) \(-8\) \(q+\beta _{4}q^{2}+\beta _{7}q^{3}-\beta _{6}q^{4}+(\beta _{2}+\beta _{10}+\cdots)q^{6}+\cdots\)
1050.2.s.g \(12\) \(8.384\) \(\mathbb{Q}[x]/(x^{12} - \cdots)\) None \(0\) \(2\) \(0\) \(-8\) \(q-\beta _{4}q^{2}+(\beta _{3}-\beta _{9})q^{3}-\beta _{6}q^{4}+(-\beta _{1}+\cdots)q^{6}+\cdots\)
1050.2.s.h \(16\) \(8.384\) 16.0.\(\cdots\).8 None \(0\) \(-6\) \(0\) \(0\) \(q+\beta _{7}q^{2}+\beta _{1}q^{3}+(1+\beta _{12})q^{4}+(-\beta _{11}+\cdots)q^{6}+\cdots\)
1050.2.s.i \(16\) \(8.384\) 16.0.\(\cdots\).1 None \(0\) \(0\) \(0\) \(0\) \(q-\beta _{1}q^{2}+(\beta _{8}+\beta _{12}+\beta _{15})q^{3}+(1+\cdots)q^{4}+\cdots\)
1050.2.s.j \(16\) \(8.384\) 16.0.\(\cdots\).8 None \(0\) \(6\) \(0\) \(0\) \(q-\beta _{7}q^{2}-\beta _{1}q^{3}+(1+\beta _{12})q^{4}+(-\beta _{11}+\cdots)q^{6}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(1050, [\chi])\) into lower level spaces

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

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ (\( 1 - T^{2} + T^{4} \))(\( 1 - T^{2} + T^{4} \))(\( 1 - T^{2} + T^{4} \))(\( ( 1 - T^{2} + T^{4} )^{2} \))(\( ( 1 - T^{2} + T^{4} )^{2} \))(\( ( 1 - T^{2} + T^{4} )^{3} \))(\( ( 1 - T^{2} + T^{4} )^{3} \))(\( ( 1 - T^{2} + T^{4} )^{4} \))(\( ( 1 - T^{2} + T^{4} )^{4} \))(\( ( 1 - T^{2} + T^{4} )^{4} \))
$3$ (\( ( 1 + 3 T + 3 T^{2} )^{2} \))(\( 1 + 3 T^{2} + 9 T^{4} \))(\( ( 1 - 3 T + 3 T^{2} )^{2} \))(\( 1 + 5 T^{2} + 16 T^{4} + 45 T^{6} + 81 T^{8} \))(\( ( 1 - 5 T^{2} + 9 T^{4} )^{2} \))(\( 1 + 2 T + 2 T^{2} + 4 T^{3} + 16 T^{4} + 18 T^{5} + 6 T^{6} + 54 T^{7} + 144 T^{8} + 108 T^{9} + 162 T^{10} + 486 T^{11} + 729 T^{12} \))(\( 1 - 2 T - T^{2} + 2 T^{3} - 8 T^{4} + 6 T^{5} + 33 T^{6} + 18 T^{7} - 72 T^{8} + 54 T^{9} - 81 T^{10} - 486 T^{11} + 729 T^{12} \))(\( 1 + 6 T + 18 T^{2} + 36 T^{3} + 50 T^{4} + 18 T^{5} - 144 T^{6} - 498 T^{7} - 1013 T^{8} - 1494 T^{9} - 1296 T^{10} + 486 T^{11} + 4050 T^{12} + 8748 T^{13} + 13122 T^{14} + 13122 T^{15} + 6561 T^{16} \))(\( ( 1 + 4 T^{2} + 7 T^{4} + 36 T^{6} + 81 T^{8} )^{2} \))(\( 1 - 6 T + 18 T^{2} - 36 T^{3} + 50 T^{4} - 18 T^{5} - 144 T^{6} + 498 T^{7} - 1013 T^{8} + 1494 T^{9} - 1296 T^{10} - 486 T^{11} + 4050 T^{12} - 8748 T^{13} + 13122 T^{14} - 13122 T^{15} + 6561 T^{16} \))
$5$ 1
$7$ (\( ( 1 - 5 T + 7 T^{2} )^{2} \))(\( ( 1 - 5 T + 7 T^{2} )^{2} \))(\( ( 1 + 5 T + 7 T^{2} )^{2} \))(\( ( 1 + 11 T^{2} + 49 T^{4} )^{2} \))(\( ( 1 + 11 T^{2} + 49 T^{4} )^{2} \))(\( 1 + 8 T + 39 T^{2} + 152 T^{3} + 483 T^{4} + 1360 T^{5} + 3642 T^{6} + 9520 T^{7} + 23667 T^{8} + 52136 T^{9} + 93639 T^{10} + 134456 T^{11} + 117649 T^{12} \))(\( 1 + 8 T + 39 T^{2} + 152 T^{3} + 483 T^{4} + 1360 T^{5} + 3642 T^{6} + 9520 T^{7} + 23667 T^{8} + 52136 T^{9} + 93639 T^{10} + 134456 T^{11} + 117649 T^{12} \))(\( ( 1 + 77 T^{4} + 2401 T^{8} )^{2} \))(\( ( 1 - 6 T^{2} + 49 T^{4} )^{4} \))(\( ( 1 + 77 T^{4} + 2401 T^{8} )^{2} \))
$11$ (\( 1 - 14 T^{2} + 75 T^{4} - 1694 T^{6} + 14641 T^{8} \))(\( 1 + 13 T^{2} + 48 T^{4} + 1573 T^{6} + 14641 T^{8} \))(\( 1 - 14 T^{2} + 75 T^{4} - 1694 T^{6} + 14641 T^{8} \))(\( ( 1 - 9 T + 53 T^{2} - 234 T^{3} + 852 T^{4} - 2574 T^{5} + 6413 T^{6} - 11979 T^{7} + 14641 T^{8} )^{2} \))(\( ( 1 + 9 T + 53 T^{2} + 234 T^{3} + 852 T^{4} + 2574 T^{5} + 6413 T^{6} + 11979 T^{7} + 14641 T^{8} )^{2} \))(\( 1 - 12 T + 104 T^{2} - 672 T^{3} + 3656 T^{4} - 16524 T^{5} + 63276 T^{6} - 198252 T^{7} + 468664 T^{8} - 516768 T^{9} - 2129576 T^{10} + 16995156 T^{11} - 68897930 T^{12} + 186946716 T^{13} - 257678696 T^{14} - 687818208 T^{15} + 6861709624 T^{16} - 31928682852 T^{17} + 112097293836 T^{18} - 322006013604 T^{19} + 783696068936 T^{20} - 1584540848352 T^{21} + 2697492158504 T^{22} - 3423740047332 T^{23} + 3138428376721 T^{24} \))(\( 1 + 12 T + 104 T^{2} + 672 T^{3} + 3656 T^{4} + 16524 T^{5} + 63276 T^{6} + 198252 T^{7} + 468664 T^{8} + 516768 T^{9} - 2129576 T^{10} - 16995156 T^{11} - 68897930 T^{12} - 186946716 T^{13} - 257678696 T^{14} + 687818208 T^{15} + 6861709624 T^{16} + 31928682852 T^{17} + 112097293836 T^{18} + 322006013604 T^{19} + 783696068936 T^{20} + 1584540848352 T^{21} + 2697492158504 T^{22} + 3423740047332 T^{23} + 3138428376721 T^{24} \))(\( 1 + 44 T^{2} + 852 T^{4} + 11656 T^{6} + 155594 T^{8} + 1774884 T^{10} + 14908720 T^{12} + 132122900 T^{14} + 1489364739 T^{16} + 15986870900 T^{18} + 218278569520 T^{20} + 3144315273924 T^{22} + 33352955730314 T^{24} + 302326621149256 T^{26} + 2673940976966292 T^{28} + 16708992677662604 T^{30} + 45949729863572161 T^{32} \))(\( ( 1 + 22 T^{2} + 241 T^{4} + 22 T^{6} - 14156 T^{8} + 2662 T^{10} + 3528481 T^{12} + 38974342 T^{14} + 214358881 T^{16} )^{2} \))(\( 1 + 44 T^{2} + 852 T^{4} + 11656 T^{6} + 155594 T^{8} + 1774884 T^{10} + 14908720 T^{12} + 132122900 T^{14} + 1489364739 T^{16} + 15986870900 T^{18} + 218278569520 T^{20} + 3144315273924 T^{22} + 33352955730314 T^{24} + 302326621149256 T^{26} + 2673940976966292 T^{28} + 16708992677662604 T^{30} + 45949729863572161 T^{32} \))
$13$ (\( ( 1 - 7 T + 13 T^{2} )^{2}( 1 + 7 T + 13 T^{2} )^{2} \))(\( ( 1 - 14 T^{2} + 169 T^{4} )^{2} \))(\( ( 1 - 7 T + 13 T^{2} )^{2}( 1 + 7 T + 13 T^{2} )^{2} \))(\( ( 1 - 22 T^{2} + 169 T^{4} )^{4} \))(\( ( 1 - 22 T^{2} + 169 T^{4} )^{4} \))(\( 1 - 24 T^{2} + 632 T^{4} - 13172 T^{6} + 225768 T^{8} - 3379208 T^{10} + 49928150 T^{12} - 571086152 T^{14} + 6448159848 T^{16} - 63578728148 T^{18} + 515541815672 T^{20} - 3308603804376 T^{22} + 23298085122481 T^{24} \))(\( 1 - 24 T^{2} + 632 T^{4} - 13172 T^{6} + 225768 T^{8} - 3379208 T^{10} + 49928150 T^{12} - 571086152 T^{14} + 6448159848 T^{16} - 63578728148 T^{18} + 515541815672 T^{20} - 3308603804376 T^{22} + 23298085122481 T^{24} \))(\( ( 1 - 56 T^{2} + 1320 T^{4} - 18088 T^{6} + 213086 T^{8} - 3056872 T^{10} + 37700520 T^{12} - 270301304 T^{14} + 815730721 T^{16} )^{2} \))(\( ( 1 - 6 T^{2} - 133 T^{4} - 1014 T^{6} + 28561 T^{8} )^{4} \))(\( ( 1 - 56 T^{2} + 1320 T^{4} - 18088 T^{6} + 213086 T^{8} - 3056872 T^{10} + 37700520 T^{12} - 270301304 T^{14} + 815730721 T^{16} )^{2} \))
$17$ (\( ( 1 - 17 T^{2} + 289 T^{4} )^{2} \))(\( 1 - 22 T^{2} + 195 T^{4} - 6358 T^{6} + 83521 T^{8} \))(\( ( 1 - 17 T^{2} + 289 T^{4} )^{2} \))(\( ( 1 + 10 T^{2} - 189 T^{4} + 2890 T^{6} + 83521 T^{8} )^{2} \))(\( ( 1 + 10 T^{2} - 189 T^{4} + 2890 T^{6} + 83521 T^{8} )^{2} \))(\( 1 - 12 T + 10 T^{2} + 240 T^{3} + 929 T^{4} - 10248 T^{5} - 21642 T^{6} + 175188 T^{7} + 650254 T^{8} - 1893108 T^{9} - 17461318 T^{10} + 24663192 T^{11} + 233538997 T^{12} + 419274264 T^{13} - 5046320902 T^{14} - 9300839604 T^{15} + 54309864334 T^{16} + 248741908116 T^{17} - 522385268298 T^{18} - 4205150720904 T^{19} + 6480478662689 T^{20} + 28461090359280 T^{21} + 20159939004490 T^{22} - 411262755691596 T^{23} + 582622237229761 T^{24} \))(\( 1 + 12 T + 10 T^{2} - 240 T^{3} + 929 T^{4} + 10248 T^{5} - 21642 T^{6} - 175188 T^{7} + 650254 T^{8} + 1893108 T^{9} - 17461318 T^{10} - 24663192 T^{11} + 233538997 T^{12} - 419274264 T^{13} - 5046320902 T^{14} + 9300839604 T^{15} + 54309864334 T^{16} - 248741908116 T^{17} - 522385268298 T^{18} + 4205150720904 T^{19} + 6480478662689 T^{20} - 28461090359280 T^{21} + 20159939004490 T^{22} + 411262755691596 T^{23} + 582622237229761 T^{24} \))(\( 1 - 56 T^{2} + 1182 T^{4} - 9520 T^{6} + 61409 T^{8} - 4590096 T^{10} + 138521374 T^{12} - 1998216584 T^{14} + 24847509060 T^{16} - 577484592776 T^{18} + 11569443677854 T^{20} - 110793758916624 T^{22} + 428374288694369 T^{24} - 19192261932274480 T^{26} + 688659484405577502 T^{28} - 9429158287326452024 T^{30} + 48661191875666868481 T^{32} \))(\( ( 1 - 24 T^{2} + 287 T^{4} - 6936 T^{6} + 83521 T^{8} )^{4} \))(\( 1 - 56 T^{2} + 1182 T^{4} - 9520 T^{6} + 61409 T^{8} - 4590096 T^{10} + 138521374 T^{12} - 1998216584 T^{14} + 24847509060 T^{16} - 577484592776 T^{18} + 11569443677854 T^{20} - 110793758916624 T^{22} + 428374288694369 T^{24} - 19192261932274480 T^{26} + 688659484405577502 T^{28} - 9429158287326452024 T^{30} + 48661191875666868481 T^{32} \))
$19$ (\( ( 1 + 3 T + 22 T^{2} + 57 T^{3} + 361 T^{4} )^{2} \))(\( ( 1 - 7 T + 19 T^{2} )^{2}( 1 + T + 19 T^{2} )^{2} \))(\( ( 1 + 3 T + 22 T^{2} + 57 T^{3} + 361 T^{4} )^{2} \))(\( ( 1 - 7 T + 19 T^{2} )^{4}( 1 + T + 19 T^{2} )^{4} \))(\( ( 1 - 7 T + 19 T^{2} )^{4}( 1 + T + 19 T^{2} )^{4} \))(\( 1 + 24 T^{2} - 316 T^{4} + 144 T^{5} - 5636 T^{6} - 33888 T^{7} + 181380 T^{8} - 941760 T^{9} + 1187824 T^{10} + 9129072 T^{11} - 54892570 T^{12} + 173452368 T^{13} + 428804464 T^{14} - 6459531840 T^{15} + 23637622980 T^{16} - 83910042912 T^{17} - 265150585316 T^{18} + 128717530416 T^{19} - 5366805920956 T^{20} + 147145590187224 T^{22} + 2213314919066161 T^{24} \))(\( 1 + 24 T^{2} - 316 T^{4} + 144 T^{5} - 5636 T^{6} - 33888 T^{7} + 181380 T^{8} - 941760 T^{9} + 1187824 T^{10} + 9129072 T^{11} - 54892570 T^{12} + 173452368 T^{13} + 428804464 T^{14} - 6459531840 T^{15} + 23637622980 T^{16} - 83910042912 T^{17} - 265150585316 T^{18} + 128717530416 T^{19} - 5366805920956 T^{20} + 147145590187224 T^{22} + 2213314919066161 T^{24} \))(\( ( 1 + 34 T^{2} + 558 T^{4} - 1260 T^{5} - 1864 T^{6} - 50232 T^{7} - 145345 T^{8} - 954408 T^{9} - 672904 T^{10} - 8642340 T^{11} + 72719118 T^{12} + 1599559954 T^{14} + 16983563041 T^{16} )^{2} \))(\( ( 1 + 6 T + 43 T^{2} + 186 T^{3} + 828 T^{4} + 3534 T^{5} + 15523 T^{6} + 41154 T^{7} + 130321 T^{8} )^{4} \))(\( ( 1 + 34 T^{2} + 558 T^{4} - 1260 T^{5} - 1864 T^{6} - 50232 T^{7} - 145345 T^{8} - 954408 T^{9} - 672904 T^{10} - 8642340 T^{11} + 72719118 T^{12} + 1599559954 T^{14} + 16983563041 T^{16} )^{2} \))
$23$ (\( 1 + 10 T^{2} - 429 T^{4} + 5290 T^{6} + 279841 T^{8} \))(\( 1 + 10 T^{2} - 429 T^{4} + 5290 T^{6} + 279841 T^{8} \))(\( 1 + 10 T^{2} - 429 T^{4} + 5290 T^{6} + 279841 T^{8} \))(\( 1 + 71 T^{2} + 2797 T^{4} + 84206 T^{6} + 2089006 T^{8} + 44544974 T^{10} + 782715277 T^{12} + 10510548119 T^{14} + 78310985281 T^{16} \))(\( 1 + 71 T^{2} + 2797 T^{4} + 84206 T^{6} + 2089006 T^{8} + 44544974 T^{10} + 782715277 T^{12} + 10510548119 T^{14} + 78310985281 T^{16} \))(\( 1 + 24 T + 367 T^{2} + 4200 T^{3} + 39535 T^{4} + 321816 T^{5} + 2347952 T^{6} + 15699792 T^{7} + 97859137 T^{8} + 573159912 T^{9} + 3169289665 T^{10} + 16552915872 T^{11} + 81697151198 T^{12} + 380717065056 T^{13} + 1676554232785 T^{14} + 6973636649304 T^{15} + 27384998757217 T^{16} + 101049246340656 T^{17} + 347581161649328 T^{18} + 1095727306051752 T^{19} + 3096024803084335 T^{20} + 7564841178144600 T^{21} + 15203529615409183 T^{22} + 22867434189934248 T^{23} + 21914624432020321 T^{24} \))(\( 1 - 24 T + 367 T^{2} - 4200 T^{3} + 39535 T^{4} - 321816 T^{5} + 2347952 T^{6} - 15699792 T^{7} + 97859137 T^{8} - 573159912 T^{9} + 3169289665 T^{10} - 16552915872 T^{11} + 81697151198 T^{12} - 380717065056 T^{13} + 1676554232785 T^{14} - 6973636649304 T^{15} + 27384998757217 T^{16} - 101049246340656 T^{17} + 347581161649328 T^{18} - 1095727306051752 T^{19} + 3096024803084335 T^{20} - 7564841178144600 T^{21} + 15203529615409183 T^{22} - 22867434189934248 T^{23} + 21914624432020321 T^{24} \))(\( 1 + 36 T^{2} + 374 T^{4} - 2232 T^{6} + 34033 T^{8} + 12853944 T^{10} + 327932102 T^{12} + 500300892 T^{14} - 66612183452 T^{16} + 264659171868 T^{18} + 91768847355782 T^{20} + 1902845027196216 T^{22} + 2665157762068273 T^{24} - 92463973028864568 T^{26} + 8196069537575600054 T^{28} + \)\(41\!\cdots\!24\)\( T^{30} + \)\(61\!\cdots\!61\)\( T^{32} \))(\( ( 1 + 30 T^{2} - 263 T^{4} + 3150 T^{6} + 596628 T^{8} + 1666350 T^{10} - 73598183 T^{12} + 4441076670 T^{14} + 78310985281 T^{16} )^{2} \))(\( 1 + 36 T^{2} + 374 T^{4} - 2232 T^{6} + 34033 T^{8} + 12853944 T^{10} + 327932102 T^{12} + 500300892 T^{14} - 66612183452 T^{16} + 264659171868 T^{18} + 91768847355782 T^{20} + 1902845027196216 T^{22} + 2665157762068273 T^{24} - 92463973028864568 T^{26} + 8196069537575600054 T^{28} + \)\(41\!\cdots\!24\)\( T^{30} + \)\(61\!\cdots\!61\)\( T^{32} \))
$29$ (\( ( 1 - 22 T^{2} + 841 T^{4} )^{2} \))(\( ( 1 - 49 T^{2} + 841 T^{4} )^{2} \))(\( ( 1 - 22 T^{2} + 841 T^{4} )^{2} \))(\( ( 1 - 47 T^{2} + 841 T^{4} )^{4} \))(\( ( 1 - 47 T^{2} + 841 T^{4} )^{4} \))(\( 1 - 158 T^{2} + 12415 T^{4} - 625726 T^{6} + 22774147 T^{8} - 673287380 T^{10} + 19008498890 T^{12} - 566234686580 T^{14} + 16107721464307 T^{16} - 372196417356046 T^{18} + 6210559216910815 T^{20} - 66471742861431758 T^{22} + 353814783205469041 T^{24} \))(\( 1 - 158 T^{2} + 12415 T^{4} - 625726 T^{6} + 22774147 T^{8} - 673287380 T^{10} + 19008498890 T^{12} - 566234686580 T^{14} + 16107721464307 T^{16} - 372196417356046 T^{18} + 6210559216910815 T^{20} - 66471742861431758 T^{22} + 353814783205469041 T^{24} \))(\( ( 1 - 60 T^{2} + 4000 T^{4} - 140436 T^{6} + 5155470 T^{8} - 118106676 T^{10} + 2829124000 T^{12} - 35689399260 T^{14} + 500246412961 T^{16} )^{2} \))(\( ( 1 - 64 T^{2} + 2226 T^{4} - 53824 T^{6} + 707281 T^{8} )^{4} \))(\( ( 1 - 60 T^{2} + 4000 T^{4} - 140436 T^{6} + 5155470 T^{8} - 118106676 T^{10} + 2829124000 T^{12} - 35689399260 T^{14} + 500246412961 T^{16} )^{2} \))
$31$ (\( ( 1 + 6 T + 43 T^{2} + 186 T^{3} + 961 T^{4} )^{2} \))(\( ( 1 - 4 T + 31 T^{2} )^{2}( 1 + 7 T + 31 T^{2} )^{2} \))(\( ( 1 + 6 T + 43 T^{2} + 186 T^{3} + 961 T^{4} )^{2} \))(\( ( 1 - 9 T + 71 T^{2} - 396 T^{3} + 1812 T^{4} - 12276 T^{5} + 68231 T^{6} - 268119 T^{7} + 923521 T^{8} )^{2} \))(\( ( 1 - 9 T + 71 T^{2} - 396 T^{3} + 1812 T^{4} - 12276 T^{5} + 68231 T^{6} - 268119 T^{7} + 923521 T^{8} )^{2} \))(\( 1 - 12 T + 162 T^{2} - 1368 T^{3} + 10769 T^{4} - 77664 T^{5} + 554878 T^{6} - 3968772 T^{7} + 27631230 T^{8} - 178188132 T^{9} + 1059815698 T^{10} - 6109630032 T^{11} + 33269854325 T^{12} - 189398530992 T^{13} + 1018482885778 T^{14} - 5308402640412 T^{15} + 25518021160830 T^{16} - 113622572872572 T^{17} + 492456267505918 T^{18} - 2136739662316704 T^{19} + 9184783582202129 T^{20} - 36169403115797928 T^{21} + 132779782490889762 T^{22} - 304901722756857972 T^{23} + 787662783788549761 T^{24} \))(\( 1 - 12 T + 162 T^{2} - 1368 T^{3} + 10769 T^{4} - 77664 T^{5} + 554878 T^{6} - 3968772 T^{7} + 27631230 T^{8} - 178188132 T^{9} + 1059815698 T^{10} - 6109630032 T^{11} + 33269854325 T^{12} - 189398530992 T^{13} + 1018482885778 T^{14} - 5308402640412 T^{15} + 25518021160830 T^{16} - 113622572872572 T^{17} + 492456267505918 T^{18} - 2136739662316704 T^{19} + 9184783582202129 T^{20} - 36169403115797928 T^{21} + 132779782490889762 T^{22} - 304901722756857972 T^{23} + 787662783788549761 T^{24} \))(\( ( 1 + 40 T^{2} + 1035 T^{4} + 4536 T^{5} + 30392 T^{6} + 284760 T^{7} + 482936 T^{8} + 8827560 T^{9} + 29206712 T^{10} + 135131976 T^{11} + 955844235 T^{12} + 35500147240 T^{14} + 852891037441 T^{16} )^{2} \))(\( ( 1 + 12 T + 112 T^{2} + 768 T^{3} + 4623 T^{4} + 23808 T^{5} + 107632 T^{6} + 357492 T^{7} + 923521 T^{8} )^{4} \))(\( ( 1 + 40 T^{2} + 1035 T^{4} + 4536 T^{5} + 30392 T^{6} + 284760 T^{7} + 482936 T^{8} + 8827560 T^{9} + 29206712 T^{10} + 135131976 T^{11} + 955844235 T^{12} + 35500147240 T^{14} + 852891037441 T^{16} )^{2} \))
$37$ (\( ( 1 - 7 T + 12 T^{2} - 259 T^{3} + 1369 T^{4} )^{2} \))(\( ( 1 + 2 T - 33 T^{2} + 74 T^{3} + 1369 T^{4} )^{2} \))(\( ( 1 + 7 T + 12 T^{2} + 259 T^{3} + 1369 T^{4} )^{2} \))(\( 1 + 56 T^{2} + 802 T^{4} - 22624 T^{6} - 719789 T^{8} - 30972256 T^{10} + 1503077122 T^{12} + 143680678904 T^{14} + 3512479453921 T^{16} \))(\( 1 + 56 T^{2} + 802 T^{4} - 22624 T^{6} - 719789 T^{8} - 30972256 T^{10} + 1503077122 T^{12} + 143680678904 T^{14} + 3512479453921 T^{16} \))(\( 1 - 8 T - 104 T^{2} + 832 T^{3} + 6788 T^{4} - 46312 T^{5} - 326044 T^{6} + 1583912 T^{7} + 13285604 T^{8} - 33469952 T^{9} - 514631840 T^{10} + 373683016 T^{11} + 19047573830 T^{12} + 13826271592 T^{13} - 704530988960 T^{14} - 1695353478656 T^{15} + 24899360878244 T^{16} + 109834725619784 T^{17} - 836539701295996 T^{18} - 4396485093783496 T^{19} + 23842710533215748 T^{20} + 108128167509504064 T^{21} - 500092774731456296 T^{22} - 1423340974235683304 T^{23} + 6582952005840035281 T^{24} \))(\( 1 - 8 T - 104 T^{2} + 832 T^{3} + 6788 T^{4} - 46312 T^{5} - 326044 T^{6} + 1583912 T^{7} + 13285604 T^{8} - 33469952 T^{9} - 514631840 T^{10} + 373683016 T^{11} + 19047573830 T^{12} + 13826271592 T^{13} - 704530988960 T^{14} - 1695353478656 T^{15} + 24899360878244 T^{16} + 109834725619784 T^{17} - 836539701295996 T^{18} - 4396485093783496 T^{19} + 23842710533215748 T^{20} + 108128167509504064 T^{21} - 500092774731456296 T^{22} - 1423340974235683304 T^{23} + 6582952005840035281 T^{24} \))(\( ( 1 - 12 T - 30 T^{2} + 456 T^{3} + 4870 T^{4} - 24984 T^{5} - 235080 T^{6} + 11340 T^{7} + 14260815 T^{8} + 419580 T^{9} - 321824520 T^{10} - 1265514552 T^{11} + 9127164070 T^{12} + 31620844392 T^{13} - 76971792270 T^{14} - 1139182525596 T^{15} + 3512479453921 T^{16} )^{2} \))(\( ( 1 - 90 T^{2} + 3817 T^{4} - 139050 T^{6} + 5381028 T^{8} - 190359450 T^{10} + 7153672537 T^{12} - 230915376810 T^{14} + 3512479453921 T^{16} )^{2} \))(\( ( 1 + 12 T - 30 T^{2} - 456 T^{3} + 4870 T^{4} + 24984 T^{5} - 235080 T^{6} - 11340 T^{7} + 14260815 T^{8} - 419580 T^{9} - 321824520 T^{10} + 1265514552 T^{11} + 9127164070 T^{12} - 31620844392 T^{13} - 76971792270 T^{14} + 1139182525596 T^{15} + 3512479453921 T^{16} )^{2} \))
$41$ (\( ( 1 - 26 T^{2} + 1681 T^{4} )^{2} \))(\( ( 1 + 34 T^{2} + 1681 T^{4} )^{2} \))(\( ( 1 - 26 T^{2} + 1681 T^{4} )^{2} \))(\( ( 1 - 9 T + 94 T^{2} - 369 T^{3} + 1681 T^{4} )^{4} \))(\( ( 1 + 9 T + 94 T^{2} + 369 T^{3} + 1681 T^{4} )^{4} \))(\( ( 1 - 2 T + 101 T^{2} - 150 T^{3} + 6062 T^{4} - 7442 T^{5} + 295913 T^{6} - 305122 T^{7} + 10190222 T^{8} - 10338150 T^{9} + 285401861 T^{10} - 231712402 T^{11} + 4750104241 T^{12} )^{2} \))(\( ( 1 + 2 T + 101 T^{2} + 150 T^{3} + 6062 T^{4} + 7442 T^{5} + 295913 T^{6} + 305122 T^{7} + 10190222 T^{8} + 10338150 T^{9} + 285401861 T^{10} + 231712402 T^{11} + 4750104241 T^{12} )^{2} \))(\( ( 1 + 280 T^{2} + 35970 T^{4} + 2762144 T^{6} + 138512387 T^{8} + 4643164064 T^{10} + 101642623170 T^{12} + 1330029187480 T^{14} + 7984925229121 T^{16} )^{2} \))(\( ( 1 + 118 T^{2} + 6363 T^{4} + 198358 T^{6} + 2825761 T^{8} )^{4} \))(\( ( 1 + 280 T^{2} + 35970 T^{4} + 2762144 T^{6} + 138512387 T^{8} + 4643164064 T^{10} + 101642623170 T^{12} + 1330029187480 T^{14} + 7984925229121 T^{16} )^{2} \))
$43$ (\( ( 1 + 4 T + 43 T^{2} )^{4} \))(\( ( 1 - 8 T + 43 T^{2} )^{4} \))(\( ( 1 - 4 T + 43 T^{2} )^{4} \))(\( ( 1 + 49 T^{2} + 660 T^{4} + 90601 T^{6} + 3418801 T^{8} )^{2} \))(\( ( 1 + 49 T^{2} + 660 T^{4} + 90601 T^{6} + 3418801 T^{8} )^{2} \))(\( ( 1 + 131 T^{2} + 196 T^{3} + 7479 T^{4} + 24892 T^{5} + 314798 T^{6} + 1070356 T^{7} + 13828671 T^{8} + 15583372 T^{9} + 447862931 T^{10} + 6321363049 T^{12} )^{2} \))(\( ( 1 + 131 T^{2} + 196 T^{3} + 7479 T^{4} + 24892 T^{5} + 314798 T^{6} + 1070356 T^{7} + 13828671 T^{8} + 15583372 T^{9} + 447862931 T^{10} + 6321363049 T^{12} )^{2} \))(\( ( 1 - 10 T + 108 T^{2} - 782 T^{3} + 6746 T^{4} - 33626 T^{5} + 199692 T^{6} - 795070 T^{7} + 3418801 T^{8} )^{4} \))(\( ( 1 + 120 T^{2} + 6818 T^{4} + 221880 T^{6} + 3418801 T^{8} )^{4} \))(\( ( 1 + 10 T + 108 T^{2} + 782 T^{3} + 6746 T^{4} + 33626 T^{5} + 199692 T^{6} + 795070 T^{7} + 3418801 T^{8} )^{4} \))
$47$ (\( 1 + 14 T^{2} - 2013 T^{4} + 30926 T^{6} + 4879681 T^{8} \))(\( 1 - 46 T^{2} - 93 T^{4} - 101614 T^{6} + 4879681 T^{8} \))(\( 1 + 14 T^{2} - 2013 T^{4} + 30926 T^{6} + 4879681 T^{8} \))(\( 1 - 112 T^{2} + 6178 T^{4} - 218176 T^{6} + 7936579 T^{8} - 481950784 T^{10} + 30146669218 T^{12} - 1207272116848 T^{14} + 23811286661761 T^{16} \))(\( 1 - 112 T^{2} + 6178 T^{4} - 218176 T^{6} + 7936579 T^{8} - 481950784 T^{10} + 30146669218 T^{12} - 1207272116848 T^{14} + 23811286661761 T^{16} \))(\( 1 - 16 T + 32 T^{2} + 544 T^{3} + 428 T^{4} - 39000 T^{5} + 312596 T^{6} - 2377608 T^{7} + 2268284 T^{8} + 67449856 T^{9} + 407948184 T^{10} - 5931851872 T^{11} + 27380961750 T^{12} - 278797037984 T^{13} + 901157538456 T^{14} + 7002846399488 T^{15} + 11068502337404 T^{16} - 545292523403256 T^{17} + 3369539594984084 T^{18} - 19758301698057000 T^{19} + 10191230691233708 T^{20} + 608806977367905248 T^{21} + 1683172231546561568 T^{22} - 39554547441344196848 T^{23} + \)\(11\!\cdots\!41\)\( T^{24} \))(\( 1 + 16 T + 32 T^{2} - 544 T^{3} + 428 T^{4} + 39000 T^{5} + 312596 T^{6} + 2377608 T^{7} + 2268284 T^{8} - 67449856 T^{9} + 407948184 T^{10} + 5931851872 T^{11} + 27380961750 T^{12} + 278797037984 T^{13} + 901157538456 T^{14} - 7002846399488 T^{15} + 11068502337404 T^{16} + 545292523403256 T^{17} + 3369539594984084 T^{18} + 19758301698057000 T^{19} + 10191230691233708 T^{20} - 608806977367905248 T^{21} + 1683172231546561568 T^{22} + 39554547441344196848 T^{23} + \)\(11\!\cdots\!41\)\( T^{24} \))(\( 1 - 124 T^{2} + 6262 T^{4} + 71176 T^{6} - 24294991 T^{8} + 1352675576 T^{10} - 14450785594 T^{12} - 2414274589508 T^{14} + 186365750498596 T^{16} - 5333132568223172 T^{18} - 70515223898115514 T^{20} + 14580781303983104504 T^{22} - \)\(57\!\cdots\!51\)\( T^{24} + \)\(37\!\cdots\!24\)\( T^{26} + \)\(72\!\cdots\!42\)\( T^{28} - \)\(31\!\cdots\!56\)\( T^{30} + \)\(56\!\cdots\!21\)\( T^{32} \))(\( ( 1 - 2 T^{2} - 3335 T^{4} + 2158 T^{6} + 6260404 T^{8} + 4767022 T^{10} - 16273736135 T^{12} - 21558430658 T^{14} + 23811286661761 T^{16} )^{2} \))(\( 1 - 124 T^{2} + 6262 T^{4} + 71176 T^{6} - 24294991 T^{8} + 1352675576 T^{10} - 14450785594 T^{12} - 2414274589508 T^{14} + 186365750498596 T^{16} - 5333132568223172 T^{18} - 70515223898115514 T^{20} + 14580781303983104504 T^{22} - \)\(57\!\cdots\!51\)\( T^{24} + \)\(37\!\cdots\!24\)\( T^{26} + \)\(72\!\cdots\!42\)\( T^{28} - \)\(31\!\cdots\!56\)\( T^{30} + \)\(56\!\cdots\!21\)\( T^{32} \))
$53$ (\( ( 1 + 53 T^{2} + 2809 T^{4} )^{2} \))(\( 1 + 25 T^{2} - 2184 T^{4} + 70225 T^{6} + 7890481 T^{8} \))(\( ( 1 + 53 T^{2} + 2809 T^{4} )^{2} \))(\( 1 + 191 T^{2} + 21817 T^{4} + 1727786 T^{6} + 104667286 T^{8} + 4853350874 T^{10} + 172146623977 T^{12} + 4233392975639 T^{14} + 62259690411361 T^{16} \))(\( 1 + 191 T^{2} + 21817 T^{4} + 1727786 T^{6} + 104667286 T^{8} + 4853350874 T^{10} + 172146623977 T^{12} + 4233392975639 T^{14} + 62259690411361 T^{16} \))(\( 1 + 48 T + 1272 T^{2} + 24192 T^{3} + 372104 T^{4} + 4941024 T^{5} + 58639988 T^{6} + 632702016 T^{7} + 6267491832 T^{8} + 57492176256 T^{9} + 491936676296 T^{10} + 3943130126256 T^{11} + 29642342340470 T^{12} + 208985896691568 T^{13} + 1381850123715464 T^{14} + 8559262724464512 T^{15} + 49453525218051192 T^{16} + 264593131503213888 T^{17} + 1299717870632226452 T^{18} + 5804275935001973088 T^{19} + 23167079840829073544 T^{20} + 79827880812877201536 T^{21} + \)\(22\!\cdots\!28\)\( T^{22} + \)\(44\!\cdots\!56\)\( T^{23} + \)\(49\!\cdots\!41\)\( T^{24} \))(\( 1 - 48 T + 1272 T^{2} - 24192 T^{3} + 372104 T^{4} - 4941024 T^{5} + 58639988 T^{6} - 632702016 T^{7} + 6267491832 T^{8} - 57492176256 T^{9} + 491936676296 T^{10} - 3943130126256 T^{11} + 29642342340470 T^{12} - 208985896691568 T^{13} + 1381850123715464 T^{14} - 8559262724464512 T^{15} + 49453525218051192 T^{16} - 264593131503213888 T^{17} + 1299717870632226452 T^{18} - 5804275935001973088 T^{19} + 23167079840829073544 T^{20} - 79827880812877201536 T^{21} + \)\(22\!\cdots\!28\)\( T^{22} - \)\(44\!\cdots\!56\)\( T^{23} + \)\(49\!\cdots\!41\)\( T^{24} \))(\( 1 + 172 T^{2} + 15556 T^{4} + 929096 T^{6} + 33434282 T^{8} + 192496612 T^{10} - 88205745040 T^{12} - 9092994584236 T^{14} - 571411990249469 T^{16} - 25542221787118924 T^{18} - 695985755328964240 T^{20} + 4266564424476994948 T^{22} + \)\(20\!\cdots\!02\)\( T^{24} + \)\(16\!\cdots\!04\)\( T^{26} + \)\(76\!\cdots\!96\)\( T^{28} + \)\(23\!\cdots\!68\)\( T^{30} + \)\(38\!\cdots\!21\)\( T^{32} \))(\( ( 1 + 81 T^{2} + 3752 T^{4} + 227529 T^{6} + 7890481 T^{8} )^{4} \))(\( 1 + 172 T^{2} + 15556 T^{4} + 929096 T^{6} + 33434282 T^{8} + 192496612 T^{10} - 88205745040 T^{12} - 9092994584236 T^{14} - 571411990249469 T^{16} - 25542221787118924 T^{18} - 695985755328964240 T^{20} + 4266564424476994948 T^{22} + \)\(20\!\cdots\!02\)\( T^{24} + \)\(16\!\cdots\!04\)\( T^{26} + \)\(76\!\cdots\!96\)\( T^{28} + \)\(23\!\cdots\!68\)\( T^{30} + \)\(38\!\cdots\!21\)\( T^{32} \))
$59$ (\( 1 - 10 T^{2} - 3381 T^{4} - 34810 T^{6} + 12117361 T^{8} \))(\( 1 - 115 T^{2} + 9744 T^{4} - 400315 T^{6} + 12117361 T^{8} \))(\( 1 - 10 T^{2} - 3381 T^{4} - 34810 T^{6} + 12117361 T^{8} \))(\( ( 1 - 9 T + 17 T^{2} + 486 T^{3} - 4164 T^{4} + 28674 T^{5} + 59177 T^{6} - 1848411 T^{7} + 12117361 T^{8} )^{2} \))(\( ( 1 + 9 T + 17 T^{2} - 486 T^{3} - 4164 T^{4} - 28674 T^{5} + 59177 T^{6} + 1848411 T^{7} + 12117361 T^{8} )^{2} \))(\( 1 + 12 T - 62 T^{2} - 192 T^{3} + 13661 T^{4} + 4488 T^{5} - 291210 T^{6} + 10035204 T^{7} + 17708314 T^{8} - 439140108 T^{9} + 4538367434 T^{10} + 27175198584 T^{11} - 197381258843 T^{12} + 1603336716456 T^{13} + 15798057037754 T^{14} - 90190156240932 T^{15} + 214578033439354 T^{16} + 7174411185021996 T^{17} - 12283393201595610 T^{18} + 11169067863867672 T^{19} + 2005850608112629181 T^{20} - 1663295197181748288 T^{21} - 31689238704639766862 T^{22} + \)\(36\!\cdots\!08\)\( T^{23} + \)\(17\!\cdots\!81\)\( T^{24} \))(\( 1 - 12 T - 62 T^{2} + 192 T^{3} + 13661 T^{4} - 4488 T^{5} - 291210 T^{6} - 10035204 T^{7} + 17708314 T^{8} + 439140108 T^{9} + 4538367434 T^{10} - 27175198584 T^{11} - 197381258843 T^{12} - 1603336716456 T^{13} + 15798057037754 T^{14} + 90190156240932 T^{15} + 214578033439354 T^{16} - 7174411185021996 T^{17} - 12283393201595610 T^{18} - 11169067863867672 T^{19} + 2005850608112629181 T^{20} + 1663295197181748288 T^{21} - 31689238704639766862 T^{22} - \)\(36\!\cdots\!08\)\( T^{23} + \)\(17\!\cdots\!81\)\( T^{24} \))(\( 1 - 392 T^{2} + 82872 T^{4} - 12348112 T^{6} + 1439957090 T^{8} - 138681126360 T^{10} + 11389108537504 T^{12} - 813599166279704 T^{14} + 51103980716496291 T^{16} - 2832138697819649624 T^{18} + \)\(13\!\cdots\!44\)\( T^{20} - \)\(58\!\cdots\!60\)\( T^{22} + \)\(21\!\cdots\!90\)\( T^{24} - \)\(63\!\cdots\!12\)\( T^{26} + \)\(14\!\cdots\!32\)\( T^{28} - \)\(24\!\cdots\!12\)\( T^{30} + \)\(21\!\cdots\!41\)\( T^{32} \))(\( ( 1 - 100 T^{2} + 2458 T^{4} - 58000 T^{6} + 9954403 T^{8} - 201898000 T^{10} + 29784473338 T^{12} - 4218053364100 T^{14} + 146830437604321 T^{16} )^{2} \))(\( 1 - 392 T^{2} + 82872 T^{4} - 12348112 T^{6} + 1439957090 T^{8} - 138681126360 T^{10} + 11389108537504 T^{12} - 813599166279704 T^{14} + 51103980716496291 T^{16} - 2832138697819649624 T^{18} + \)\(13\!\cdots\!44\)\( T^{20} - \)\(58\!\cdots\!60\)\( T^{22} + \)\(21\!\cdots\!90\)\( T^{24} - \)\(63\!\cdots\!12\)\( T^{26} + \)\(14\!\cdots\!32\)\( T^{28} - \)\(24\!\cdots\!12\)\( T^{30} + \)\(21\!\cdots\!41\)\( T^{32} \))
$61$ (\( ( 1 + 9 T + 88 T^{2} + 549 T^{3} + 3721 T^{4} )^{2} \))(\( ( 1 + 61 T^{2} + 3721 T^{4} )^{2} \))(\( ( 1 + 9 T + 88 T^{2} + 549 T^{3} + 3721 T^{4} )^{2} \))(\( ( 1 + 27 T + 401 T^{2} + 4266 T^{3} + 36066 T^{4} + 260226 T^{5} + 1492121 T^{6} + 6128487 T^{7} + 13845841 T^{8} )^{2} \))(\( ( 1 + 27 T + 401 T^{2} + 4266 T^{3} + 36066 T^{4} + 260226 T^{5} + 1492121 T^{6} + 6128487 T^{7} + 13845841 T^{8} )^{2} \))(\( 1 + 30 T + 633 T^{2} + 9990 T^{3} + 134066 T^{4} + 1612650 T^{5} + 18045535 T^{6} + 190552410 T^{7} + 1899056448 T^{8} + 17814782910 T^{9} + 157346567941 T^{10} + 1317498445782 T^{11} + 10515537066752 T^{12} + 80367405192702 T^{13} + 585486579308461 T^{14} + 4043617239694710 T^{15} + 26294033629032768 T^{16} + 160939860632635410 T^{17} + 929712718744528135 T^{18} + 5068144234509265650 T^{19} + 25701432624293474546 T^{20} + \)\(11\!\cdots\!90\)\( T^{21} + \)\(45\!\cdots\!33\)\( T^{22} + \)\(13\!\cdots\!30\)\( T^{23} + \)\(26\!\cdots\!21\)\( T^{24} \))(\( 1 + 30 T + 633 T^{2} + 9990 T^{3} + 134066 T^{4} + 1612650 T^{5} + 18045535 T^{6} + 190552410 T^{7} + 1899056448 T^{8} + 17814782910 T^{9} + 157346567941 T^{10} + 1317498445782 T^{11} + 10515537066752 T^{12} + 80367405192702 T^{13} + 585486579308461 T^{14} + 4043617239694710 T^{15} + 26294033629032768 T^{16} + 160939860632635410 T^{17} + 929712718744528135 T^{18} + 5068144234509265650 T^{19} + 25701432624293474546 T^{20} + \)\(11\!\cdots\!90\)\( T^{21} + \)\(45\!\cdots\!33\)\( T^{22} + \)\(13\!\cdots\!30\)\( T^{23} + \)\(26\!\cdots\!21\)\( T^{24} \))(\( ( 1 - 54 T + 1528 T^{2} - 30024 T^{3} + 457410 T^{4} - 5741010 T^{5} + 61597952 T^{6} - 577586742 T^{7} + 4788758087 T^{8} - 35232791262 T^{9} + 229205979392 T^{10} - 1303100190810 T^{11} + 6333226131810 T^{12} - 25358159341224 T^{13} + 78723132023608 T^{14} - 169708113145134 T^{15} + 191707312997281 T^{16} )^{2} \))(\( ( 1 - 6 T + 73 T^{2} - 366 T^{3} + 3721 T^{4} )^{8} \))(\( ( 1 - 54 T + 1528 T^{2} - 30024 T^{3} + 457410 T^{4} - 5741010 T^{5} + 61597952 T^{6} - 577586742 T^{7} + 4788758087 T^{8} - 35232791262 T^{9} + 229205979392 T^{10} - 1303100190810 T^{11} + 6333226131810 T^{12} - 25358159341224 T^{13} + 78723132023608 T^{14} - 169708113145134 T^{15} + 191707312997281 T^{16} )^{2} \))
$67$ (\( ( 1 - 16 T + 67 T^{2} )^{2}( 1 + 5 T + 67 T^{2} )^{2} \))(\( ( 1 - 2 T - 63 T^{2} - 134 T^{3} + 4489 T^{4} )^{2} \))(\( ( 1 - 5 T + 67 T^{2} )^{2}( 1 + 16 T + 67 T^{2} )^{2} \))(\( 1 - 205 T^{2} + 23209 T^{4} - 2016790 T^{6} + 145240510 T^{8} - 9053370310 T^{10} + 467687367289 T^{12} - 18543968344645 T^{14} + 406067677556641 T^{16} \))(\( 1 - 205 T^{2} + 23209 T^{4} - 2016790 T^{6} + 145240510 T^{8} - 9053370310 T^{10} + 467687367289 T^{12} - 18543968344645 T^{14} + 406067677556641 T^{16} \))(\( 1 - 4 T - 339 T^{2} + 1212 T^{3} + 65398 T^{4} - 197860 T^{5} - 8949237 T^{6} + 19422588 T^{7} + 965230072 T^{8} - 1219402116 T^{9} - 85058357911 T^{10} + 32989889052 T^{11} + 6246904001980 T^{12} + 2210322566484 T^{13} - 381826968662479 T^{14} - 366751038614508 T^{15} + 19450467973710712 T^{16} + 26222923701716916 T^{17} - 809533500666955053 T^{18} - 1199172398229208780 T^{19} + 26556013976849208118 T^{20} + 32974319688309475764 T^{21} - \)\(61\!\cdots\!11\)\( T^{22} - \)\(48\!\cdots\!32\)\( T^{23} + \)\(81\!\cdots\!61\)\( T^{24} \))(\( 1 - 4 T - 339 T^{2} + 1212 T^{3} + 65398 T^{4} - 197860 T^{5} - 8949237 T^{6} + 19422588 T^{7} + 965230072 T^{8} - 1219402116 T^{9} - 85058357911 T^{10} + 32989889052 T^{11} + 6246904001980 T^{12} + 2210322566484 T^{13} - 381826968662479 T^{14} - 366751038614508 T^{15} + 19450467973710712 T^{16} + 26222923701716916 T^{17} - 809533500666955053 T^{18} - 1199172398229208780 T^{19} + 26556013976849208118 T^{20} + 32974319688309475764 T^{21} - \)\(61\!\cdots\!11\)\( T^{22} - \)\(48\!\cdots\!32\)\( T^{23} + \)\(81\!\cdots\!61\)\( T^{24} \))(\( ( 1 + 6 T - 46 T^{2} + 384 T^{3} + 1546 T^{4} - 50202 T^{5} + 319376 T^{6} + 3681930 T^{7} - 5314277 T^{8} + 246689310 T^{9} + 1433678864 T^{10} - 15098904126 T^{11} + 31153633066 T^{12} + 518448041088 T^{13} - 4161085579774 T^{14} + 36364269631938 T^{15} + 406067677556641 T^{16} )^{2} \))(\( ( 1 - 36 T^{2} - 326 T^{4} + 264816 T^{6} - 23337981 T^{8} + 1188759024 T^{10} - 6569265446 T^{12} - 3256501758084 T^{14} + 406067677556641 T^{16} )^{2} \))(\( ( 1 - 6 T - 46 T^{2} - 384 T^{3} + 1546 T^{4} + 50202 T^{5} + 319376 T^{6} - 3681930 T^{7} - 5314277 T^{8} - 246689310 T^{9} + 1433678864 T^{10} + 15098904126 T^{11} + 31153633066 T^{12} - 518448041088 T^{13} - 4161085579774 T^{14} - 36364269631938 T^{15} + 406067677556641 T^{16} )^{2} \))
$71$ (\( ( 1 - 106 T^{2} + 5041 T^{4} )^{2} \))(\( ( 1 + 2 T^{2} + 5041 T^{4} )^{2} \))(\( ( 1 - 106 T^{2} + 5041 T^{4} )^{2} \))(\( ( 1 - 208 T^{2} + 19710 T^{4} - 1048528 T^{6} + 25411681 T^{8} )^{2} \))(\( ( 1 - 208 T^{2} + 19710 T^{4} - 1048528 T^{6} + 25411681 T^{8} )^{2} \))(\( 1 - 300 T^{2} + 49610 T^{4} - 6300412 T^{6} + 667955583 T^{8} - 59469293656 T^{10} + 4533955360172 T^{12} - 299784709319896 T^{14} + 16973874197365023 T^{16} - 807084566019275452 T^{18} + 32035832685102203210 T^{20} - \)\(97\!\cdots\!00\)\( T^{22} + \)\(16\!\cdots\!41\)\( T^{24} \))(\( 1 - 300 T^{2} + 49610 T^{4} - 6300412 T^{6} + 667955583 T^{8} - 59469293656 T^{10} + 4533955360172 T^{12} - 299784709319896 T^{14} + 16973874197365023 T^{16} - 807084566019275452 T^{18} + 32035832685102203210 T^{20} - \)\(97\!\cdots\!00\)\( T^{22} + \)\(16\!\cdots\!41\)\( T^{24} \))(\( ( 1 - 172 T^{2} + 12162 T^{4} - 750752 T^{6} + 58967291 T^{8} - 3784540832 T^{10} + 309056864322 T^{12} - 22033248834412 T^{14} + 645753531245761 T^{16} )^{2} \))(\( ( 1 - 232 T^{2} + 23058 T^{4} - 1169512 T^{6} + 25411681 T^{8} )^{4} \))(\( ( 1 - 172 T^{2} + 12162 T^{4} - 750752 T^{6} + 58967291 T^{8} - 3784540832 T^{10} + 309056864322 T^{12} - 22033248834412 T^{14} + 645753531245761 T^{16} )^{2} \))
$73$ (\( ( 1 - 10 T + 73 T^{2} )^{2}( 1 + 7 T + 73 T^{2} )^{2} \))(\( ( 1 + 12 T + 121 T^{2} + 876 T^{3} + 5329 T^{4} )^{2} \))(\( ( 1 - 7 T + 73 T^{2} )^{2}( 1 + 10 T + 73 T^{2} )^{2} \))(\( ( 1 + 142 T^{2} + 14835 T^{4} + 756718 T^{6} + 28398241 T^{8} )^{2} \))(\( ( 1 + 142 T^{2} + 14835 T^{4} + 756718 T^{6} + 28398241 T^{8} )^{2} \))(\( 1 + 158 T^{2} + 8161 T^{4} + 137736 T^{5} + 413890 T^{6} + 21056400 T^{7} + 34702222 T^{8} + 1012533192 T^{9} + 9774206798 T^{10} + 39652148664 T^{11} + 1268896234901 T^{12} + 2894606852472 T^{13} + 52086748026542 T^{14} + 393892624752264 T^{15} + 985482063591502 T^{16} + 43651424690845200 T^{17} + 62635722918754210 T^{18} + 1521624482426344392 T^{19} + 6581520809947595041 T^{20} + \)\(67\!\cdots\!42\)\( T^{22} + \)\(22\!\cdots\!21\)\( T^{24} \))(\( 1 + 158 T^{2} + 8161 T^{4} + 137736 T^{5} + 413890 T^{6} + 21056400 T^{7} + 34702222 T^{8} + 1012533192 T^{9} + 9774206798 T^{10} + 39652148664 T^{11} + 1268896234901 T^{12} + 2894606852472 T^{13} + 52086748026542 T^{14} + 393892624752264 T^{15} + 985482063591502 T^{16} + 43651424690845200 T^{17} + 62635722918754210 T^{18} + 1521624482426344392 T^{19} + 6581520809947595041 T^{20} + \)\(67\!\cdots\!42\)\( T^{22} + \)\(22\!\cdots\!21\)\( T^{24} \))(\( ( 1 + 180 T^{2} + 16666 T^{4} + 913680 T^{6} + 46500915 T^{8} + 4869000720 T^{10} + 473285084506 T^{12} + 27240160732020 T^{14} + 806460091894081 T^{16} )^{2} \))(\( ( 1 + 136 T^{2} + 7534 T^{4} + 41344 T^{6} - 6797981 T^{8} + 220322176 T^{10} + 213952347694 T^{12} + 20581454775304 T^{14} + 806460091894081 T^{16} )^{2} \))(\( ( 1 + 180 T^{2} + 16666 T^{4} + 913680 T^{6} + 46500915 T^{8} + 4869000720 T^{10} + 473285084506 T^{12} + 27240160732020 T^{14} + 806460091894081 T^{16} )^{2} \))
$79$ (\( ( 1 - T - 78 T^{2} - 79 T^{3} + 6241 T^{4} )^{2} \))(\( ( 1 - T - 78 T^{2} - 79 T^{3} + 6241 T^{4} )^{2} \))(\( ( 1 - T - 78 T^{2} - 79 T^{3} + 6241 T^{4} )^{2} \))(\( ( 1 - T - 83 T^{2} + 74 T^{3} + 736 T^{4} + 5846 T^{5} - 518003 T^{6} - 493039 T^{7} + 38950081 T^{8} )^{2} \))(\( ( 1 - T - 83 T^{2} + 74 T^{3} + 736 T^{4} + 5846 T^{5} - 518003 T^{6} - 493039 T^{7} + 38950081 T^{8} )^{2} \))(\( 1 + 4 T - 254 T^{2} + 1984 T^{3} + 51521 T^{4} - 543784 T^{5} - 2546434 T^{6} + 103559324 T^{7} - 209408530 T^{8} - 8333233172 T^{9} + 86160957682 T^{10} + 365142325960 T^{11} - 8321189358475 T^{12} + 28846243750840 T^{13} + 537730536893362 T^{14} - 4108608949889708 T^{15} - 8156479205590930 T^{16} + 318657880590314276 T^{17} - 619006161712162114 T^{18} - 10442778444129485656 T^{19} + 78162962995195929281 T^{20} + \)\(23\!\cdots\!96\)\( T^{21} - \)\(24\!\cdots\!54\)\( T^{22} + \)\(29\!\cdots\!16\)\( T^{23} + \)\(59\!\cdots\!41\)\( T^{24} \))(\( 1 + 4 T - 254 T^{2} + 1984 T^{3} + 51521 T^{4} - 543784 T^{5} - 2546434 T^{6} + 103559324 T^{7} - 209408530 T^{8} - 8333233172 T^{9} + 86160957682 T^{10} + 365142325960 T^{11} - 8321189358475 T^{12} + 28846243750840 T^{13} + 537730536893362 T^{14} - 4108608949889708 T^{15} - 8156479205590930 T^{16} + 318657880590314276 T^{17} - 619006161712162114 T^{18} - 10442778444129485656 T^{19} + 78162962995195929281 T^{20} + \)\(23\!\cdots\!96\)\( T^{21} - \)\(24\!\cdots\!54\)\( T^{22} + \)\(29\!\cdots\!16\)\( T^{23} + \)\(59\!\cdots\!41\)\( T^{24} \))(\( ( 1 + 12 T - 156 T^{2} - 1632 T^{3} + 23959 T^{4} + 159888 T^{5} - 2286612 T^{6} - 4972548 T^{7} + 201275328 T^{8} - 392831292 T^{9} - 14270745492 T^{10} + 78831019632 T^{11} + 933204990679 T^{12} - 5021756043168 T^{13} - 37921643061276 T^{14} + 230446907833908 T^{15} + 1517108809906561 T^{16} )^{2} \))(\( ( 1 - 12 T - 20 T^{2} - 72 T^{3} + 9279 T^{4} - 5688 T^{5} - 124820 T^{6} - 5916468 T^{7} + 38950081 T^{8} )^{4} \))(\( ( 1 + 12 T - 156 T^{2} - 1632 T^{3} + 23959 T^{4} + 159888 T^{5} - 2286612 T^{6} - 4972548 T^{7} + 201275328 T^{8} - 392831292 T^{9} - 14270745492 T^{10} + 78831019632 T^{11} + 933204990679 T^{12} - 5021756043168 T^{13} - 37921643061276 T^{14} + 230446907833908 T^{15} + 1517108809906561 T^{16} )^{2} \))
$83$ (\( ( 1 + 58 T^{2} + 6889 T^{4} )^{2} \))(\( ( 1 + 91 T^{2} + 6889 T^{4} )^{2} \))(\( ( 1 + 58 T^{2} + 6889 T^{4} )^{2} \))(\( ( 1 + 190 T^{2} + 18051 T^{4} + 1308910 T^{6} + 47458321 T^{8} )^{2} \))(\( ( 1 + 190 T^{2} + 18051 T^{4} + 1308910 T^{6} + 47458321 T^{8} )^{2} \))(\( ( 1 + 20 T + 635 T^{2} + 8628 T^{3} + 148691 T^{4} + 1458128 T^{5} + 17076098 T^{6} + 121024624 T^{7} + 1024332299 T^{8} + 4933378236 T^{9} + 30136033835 T^{10} + 78780812860 T^{11} + 326940373369 T^{12} )^{2} \))(\( ( 1 - 20 T + 635 T^{2} - 8628 T^{3} + 148691 T^{4} - 1458128 T^{5} + 17076098 T^{6} - 121024624 T^{7} + 1024332299 T^{8} - 4933378236 T^{9} + 30136033835 T^{10} - 78780812860 T^{11} + 326940373369 T^{12} )^{2} \))(\( ( 1 + 244 T^{2} + 29232 T^{4} + 2582300 T^{6} + 210508718 T^{8} + 17789464700 T^{10} + 1387301639472 T^{12} + 79773451102036 T^{14} + 2252292232139041 T^{16} )^{2} \))(\( ( 1 + 36 T^{2} - 3178 T^{4} + 248004 T^{6} + 47458321 T^{8} )^{4} \))(\( ( 1 + 244 T^{2} + 29232 T^{4} + 2582300 T^{6} + 210508718 T^{8} + 17789464700 T^{10} + 1387301639472 T^{12} + 79773451102036 T^{14} + 2252292232139041 T^{16} )^{2} \))
$89$ (\( 1 - 70 T^{2} - 3021 T^{4} - 554470 T^{6} + 62742241 T^{8} \))(\( 1 - 70 T^{2} - 3021 T^{4} - 554470 T^{6} + 62742241 T^{8} \))(\( 1 - 70 T^{2} - 3021 T^{4} - 554470 T^{6} + 62742241 T^{8} \))(\( ( 1 + 3 T - 163 T^{2} - 18 T^{3} + 20862 T^{4} - 1602 T^{5} - 1291123 T^{6} + 2114907 T^{7} + 62742241 T^{8} )^{2} \))(\( ( 1 - 3 T - 163 T^{2} + 18 T^{3} + 20862 T^{4} + 1602 T^{5} - 1291123 T^{6} - 2114907 T^{7} + 62742241 T^{8} )^{2} \))(\( 1 + 26 T + 101 T^{2} - 1838 T^{3} + 1058 T^{4} + 234822 T^{5} - 191269 T^{6} - 17352042 T^{7} - 36283792 T^{8} + 94067578 T^{9} - 6641487 T^{10} + 7507089026 T^{11} - 13384168368 T^{12} + 668130923314 T^{13} - 52607218527 T^{14} + 66314726395082 T^{15} - 2276526422057872 T^{16} - 96894834089544858 T^{17} - 95057114540819509 T^{18} + 10386490522837910838 T^{19} + 4164910956432801698 T^{20} - \)\(64\!\cdots\!42\)\( T^{21} + \)\(31\!\cdots\!01\)\( T^{22} + \)\(72\!\cdots\!14\)\( T^{23} + \)\(24\!\cdots\!21\)\( T^{24} \))(\( 1 - 26 T + 101 T^{2} + 1838 T^{3} + 1058 T^{4} - 234822 T^{5} - 191269 T^{6} + 17352042 T^{7} - 36283792 T^{8} - 94067578 T^{9} - 6641487 T^{10} - 7507089026 T^{11} - 13384168368 T^{12} - 668130923314 T^{13} - 52607218527 T^{14} - 66314726395082 T^{15} - 2276526422057872 T^{16} + 96894834089544858 T^{17} - 95057114540819509 T^{18} - 10386490522837910838 T^{19} + 4164910956432801698 T^{20} + \)\(64\!\cdots\!42\)\( T^{21} + \)\(31\!\cdots\!01\)\( T^{22} - \)\(72\!\cdots\!14\)\( T^{23} + \)\(24\!\cdots\!21\)\( T^{24} \))(\( 1 - 280 T^{2} + 33934 T^{4} - 2359280 T^{6} + 86826641 T^{8} + 3149241200 T^{10} - 1390981471378 T^{12} + 224015580284440 T^{14} - 23587997383426652 T^{16} + 1774427411433049240 T^{18} - 87273294703733078098 T^{20} + \)\(15\!\cdots\!00\)\( T^{22} + \)\(34\!\cdots\!21\)\( T^{24} - \)\(73\!\cdots\!80\)\( T^{26} + \)\(83\!\cdots\!14\)\( T^{28} - \)\(54\!\cdots\!80\)\( T^{30} + \)\(15\!\cdots\!61\)\( T^{32} \))(\( ( 1 - 220 T^{2} + 22378 T^{4} - 2239600 T^{6} + 231025843 T^{8} - 17739871600 T^{10} + 1404045869098 T^{12} - 109335884011420 T^{14} + 3936588805702081 T^{16} )^{2} \))(\( 1 - 280 T^{2} + 33934 T^{4} - 2359280 T^{6} + 86826641 T^{8} + 3149241200 T^{10} - 1390981471378 T^{12} + 224015580284440 T^{14} - 23587997383426652 T^{16} + 1774427411433049240 T^{18} - 87273294703733078098 T^{20} + \)\(15\!\cdots\!00\)\( T^{22} + \)\(34\!\cdots\!21\)\( T^{24} - \)\(73\!\cdots\!80\)\( T^{26} + \)\(83\!\cdots\!14\)\( T^{28} - \)\(54\!\cdots\!80\)\( T^{30} + \)\(15\!\cdots\!61\)\( T^{32} \))
$97$ (\( ( 1 + 49 T^{2} + 9409 T^{4} )^{2} \))(\( ( 1 - 19 T + 97 T^{2} )^{2}( 1 + 19 T + 97 T^{2} )^{2} \))(\( ( 1 + 49 T^{2} + 9409 T^{4} )^{2} \))(\( ( 1 - 155 T^{2} + 12276 T^{4} - 1458395 T^{6} + 88529281 T^{8} )^{2} \))(\( ( 1 - 155 T^{2} + 12276 T^{4} - 1458395 T^{6} + 88529281 T^{8} )^{2} \))(\( 1 - 484 T^{2} + 116170 T^{4} - 18891668 T^{6} + 2452131295 T^{8} - 279695423560 T^{10} + 28680693351980 T^{12} - 2631654240276040 T^{14} + 217085420463948895 T^{16} - 15736230570413031572 T^{18} + \)\(91\!\cdots\!70\)\( T^{20} - \)\(35\!\cdots\!16\)\( T^{22} + \)\(69\!\cdots\!41\)\( T^{24} \))(\( 1 - 484 T^{2} + 116170 T^{4} - 18891668 T^{6} + 2452131295 T^{8} - 279695423560 T^{10} + 28680693351980 T^{12} - 2631654240276040 T^{14} + 217085420463948895 T^{16} - 15736230570413031572 T^{18} + \)\(91\!\cdots\!70\)\( T^{20} - \)\(35\!\cdots\!16\)\( T^{22} + \)\(69\!\cdots\!41\)\( T^{24} \))(\( ( 1 - 468 T^{2} + 112546 T^{4} - 17841600 T^{6} + 2025951627 T^{8} - 167871614400 T^{10} + 9963616459426 T^{12} - 389830898306772 T^{14} + 7837433594376961 T^{16} )^{2} \))(\( ( 1 - 232 T^{2} + 27954 T^{4} - 2182888 T^{6} + 88529281 T^{8} )^{4} \))(\( ( 1 - 468 T^{2} + 112546 T^{4} - 17841600 T^{6} + 2025951627 T^{8} - 167871614400 T^{10} + 9963616459426 T^{12} - 389830898306772 T^{14} + 7837433594376961 T^{16} )^{2} \))
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