Properties

Label 210.2.t
Level 210
Weight 2
Character orbit t
Rep. character \(\chi_{210}(59,\cdot)\)
Character field \(\Q(\zeta_{6})\)
Dimension 32
Newform subspaces 6
Sturm bound 96
Trace bound 3

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Defining parameters

Level: \( N \) = \( 210 = 2 \cdot 3 \cdot 5 \cdot 7 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 210.t (of order \(6\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) = \( 105 \)
Character field: \(\Q(\zeta_{6})\)
Newform subspaces: \( 6 \)
Sturm bound: \(96\)
Trace bound: \(3\)
Distinguishing \(T_p\): \(11\), \(13\), \(23\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(210, [\chi])\).

Total New Old
Modular forms 112 32 80
Cusp forms 80 32 48
Eisenstein series 32 0 32

Trace form

\( 32q - 16q^{4} + 6q^{9} + O(q^{10}) \) \( 32q - 16q^{4} + 6q^{9} - 6q^{10} + 20q^{15} - 16q^{16} - 24q^{19} + 10q^{21} - 6q^{24} + 6q^{25} - 10q^{30} - 12q^{31} - 12q^{36} - 12q^{39} + 6q^{40} + 6q^{45} - 4q^{46} + 40q^{49} - 4q^{51} - 10q^{60} - 60q^{61} + 32q^{64} + 48q^{66} - 10q^{70} - 96q^{75} - 52q^{79} + 22q^{81} + 16q^{84} - 88q^{85} + 16q^{91} + 96q^{94} + 6q^{96} + 32q^{99} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(210, [\chi])\) into newform subspaces

Label Dim. \(A\) Field CM Traces $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\)
210.2.t.a \(4\) \(1.677\) \(\Q(\sqrt{-3}, \sqrt{-11})\) None \(-2\) \(-2\) \(-6\) \(-10\) \(q+(-1+\beta _{2})q^{2}+(-\beta _{1}+\beta _{3})q^{3}-\beta _{2}q^{4}+\cdots\)
210.2.t.b \(4\) \(1.677\) \(\Q(\sqrt{-3}, \sqrt{-11})\) None \(-2\) \(-1\) \(-3\) \(10\) \(q+(-1+\beta _{2})q^{2}-\beta _{1}q^{3}-\beta _{2}q^{4}+(-2\beta _{2}+\cdots)q^{5}+\cdots\)
210.2.t.c \(4\) \(1.677\) \(\Q(\sqrt{-3}, \sqrt{-11})\) None \(2\) \(1\) \(6\) \(-10\) \(q+\beta _{2}q^{2}+(\beta _{2}+\beta _{3})q^{3}+(-1+\beta _{2}+\cdots)q^{4}+\cdots\)
210.2.t.d \(4\) \(1.677\) \(\Q(\sqrt{-3}, \sqrt{-11})\) None \(2\) \(2\) \(3\) \(10\) \(q+\beta _{2}q^{2}+(1-\beta _{1}+\beta _{3})q^{3}+(-1+\beta _{2}+\cdots)q^{4}+\cdots\)
210.2.t.e \(8\) \(1.677\) 8.0.3317760000.3 None \(-4\) \(0\) \(12\) \(0\) \(q-\beta _{4}q^{2}+(\beta _{5}-\beta _{6})q^{3}+(-1+\beta _{4}+\cdots)q^{4}+\cdots\)
210.2.t.f \(8\) \(1.677\) 8.0.3317760000.3 None \(4\) \(0\) \(-12\) \(0\) \(q+\beta _{4}q^{2}+(\beta _{5}-\beta _{6})q^{3}+(-1+\beta _{4}+\cdots)q^{4}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(210, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(210, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(105, [\chi])\)\(^{\oplus 2}\)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ (\( ( 1 + T + T^{2} )^{2} \))(\( ( 1 + T + T^{2} )^{2} \))(\( ( 1 - T + T^{2} )^{2} \))(\( ( 1 - T + T^{2} )^{2} \))(\( ( 1 + T + T^{2} )^{4} \))(\( ( 1 - T + T^{2} )^{4} \))
$3$ (\( ( 1 + T + 3 T^{2} )^{2} \))(\( 1 + T - 2 T^{2} + 3 T^{3} + 9 T^{4} \))(\( 1 - T - 2 T^{2} - 3 T^{3} + 9 T^{4} \))(\( ( 1 - T + 3 T^{2} )^{2} \))(\( 1 - 4 T^{2} + 7 T^{4} - 36 T^{6} + 81 T^{8} \))(\( 1 - 4 T^{2} + 7 T^{4} - 36 T^{6} + 81 T^{8} \))
$5$ (\( ( 1 + 3 T + 5 T^{2} )^{2} \))(\( 1 + 3 T + 4 T^{2} + 15 T^{3} + 25 T^{4} \))(\( ( 1 - 3 T + 5 T^{2} )^{2} \))(\( 1 - 3 T + 4 T^{2} - 15 T^{3} + 25 T^{4} \))(\( ( 1 - 6 T + 17 T^{2} - 30 T^{3} + 25 T^{4} )^{2} \))(\( ( 1 + 6 T + 17 T^{2} + 30 T^{3} + 25 T^{4} )^{2} \))
$7$ (\( ( 1 + 5 T + 7 T^{2} )^{2} \))(\( ( 1 - 5 T + 7 T^{2} )^{2} \))(\( ( 1 + 5 T + 7 T^{2} )^{2} \))(\( ( 1 - 5 T + 7 T^{2} )^{2} \))(\( ( 1 + 6 T^{2} + 49 T^{4} )^{2} \))(\( ( 1 + 6 T^{2} + 49 T^{4} )^{2} \))
$11$ (\( 1 + 9 T + 53 T^{2} + 234 T^{3} + 852 T^{4} + 2574 T^{5} + 6413 T^{6} + 11979 T^{7} + 14641 T^{8} \))(\( 1 - 9 T + 53 T^{2} - 234 T^{3} + 852 T^{4} - 2574 T^{5} + 6413 T^{6} - 11979 T^{7} + 14641 T^{8} \))(\( 1 - 9 T + 53 T^{2} - 234 T^{3} + 852 T^{4} - 2574 T^{5} + 6413 T^{6} - 11979 T^{7} + 14641 T^{8} \))(\( 1 + 9 T + 53 T^{2} + 234 T^{3} + 852 T^{4} + 2574 T^{5} + 6413 T^{6} + 11979 T^{7} + 14641 T^{8} \))(\( 1 + 22 T^{2} + 241 T^{4} + 22 T^{6} - 14156 T^{8} + 2662 T^{10} + 3528481 T^{12} + 38974342 T^{14} + 214358881 T^{16} \))(\( 1 + 22 T^{2} + 241 T^{4} + 22 T^{6} - 14156 T^{8} + 2662 T^{10} + 3528481 T^{12} + 38974342 T^{14} + 214358881 T^{16} \))
$13$ (\( ( 1 + 2 T + 13 T^{2} )^{4} \))(\( ( 1 - 2 T + 13 T^{2} )^{4} \))(\( ( 1 + 2 T + 13 T^{2} )^{4} \))(\( ( 1 - 2 T + 13 T^{2} )^{4} \))(\( ( 1 + 6 T^{2} - 133 T^{4} + 1014 T^{6} + 28561 T^{8} )^{2} \))(\( ( 1 + 6 T^{2} - 133 T^{4} + 1014 T^{6} + 28561 T^{8} )^{2} \))
$17$ (\( 1 - 10 T^{2} - 189 T^{4} - 2890 T^{6} + 83521 T^{8} \))(\( 1 - 10 T^{2} - 189 T^{4} - 2890 T^{6} + 83521 T^{8} \))(\( 1 - 10 T^{2} - 189 T^{4} - 2890 T^{6} + 83521 T^{8} \))(\( 1 - 10 T^{2} - 189 T^{4} - 2890 T^{6} + 83521 T^{8} \))(\( ( 1 + 24 T^{2} + 287 T^{4} + 6936 T^{6} + 83521 T^{8} )^{2} \))(\( ( 1 + 24 T^{2} + 287 T^{4} + 6936 T^{6} + 83521 T^{8} )^{2} \))
$19$ (\( ( 1 - T + 19 T^{2} )^{2}( 1 + 7 T + 19 T^{2} )^{2} \))(\( ( 1 - T + 19 T^{2} )^{2}( 1 + 7 T + 19 T^{2} )^{2} \))(\( ( 1 - T + 19 T^{2} )^{2}( 1 + 7 T + 19 T^{2} )^{2} \))(\( ( 1 - T + 19 T^{2} )^{2}( 1 + 7 T + 19 T^{2} )^{2} \))(\( ( 1 - 6 T + 43 T^{2} - 186 T^{3} + 828 T^{4} - 3534 T^{5} + 15523 T^{6} - 41154 T^{7} + 130321 T^{8} )^{2} \))(\( ( 1 - 6 T + 43 T^{2} - 186 T^{3} + 828 T^{4} - 3534 T^{5} + 15523 T^{6} - 41154 T^{7} + 130321 T^{8} )^{2} \))
$23$ (\( 1 + 3 T - 31 T^{2} - 18 T^{3} + 864 T^{4} - 414 T^{5} - 16399 T^{6} + 36501 T^{7} + 279841 T^{8} \))(\( 1 + 3 T - 31 T^{2} - 18 T^{3} + 864 T^{4} - 414 T^{5} - 16399 T^{6} + 36501 T^{7} + 279841 T^{8} \))(\( 1 - 3 T - 31 T^{2} + 18 T^{3} + 864 T^{4} + 414 T^{5} - 16399 T^{6} - 36501 T^{7} + 279841 T^{8} \))(\( 1 - 3 T - 31 T^{2} + 18 T^{3} + 864 T^{4} + 414 T^{5} - 16399 T^{6} - 36501 T^{7} + 279841 T^{8} \))(\( ( 1 - 2 T - 13 T^{2} + 58 T^{3} - 332 T^{4} + 1334 T^{5} - 6877 T^{6} - 24334 T^{7} + 279841 T^{8} )^{2} \))(\( ( 1 + 2 T - 13 T^{2} - 58 T^{3} - 332 T^{4} - 1334 T^{5} - 6877 T^{6} + 24334 T^{7} + 279841 T^{8} )^{2} \))
$29$ (\( ( 1 - 47 T^{2} + 841 T^{4} )^{2} \))(\( ( 1 - 47 T^{2} + 841 T^{4} )^{2} \))(\( ( 1 - 47 T^{2} + 841 T^{4} )^{2} \))(\( ( 1 - 47 T^{2} + 841 T^{4} )^{2} \))(\( ( 1 - 64 T^{2} + 2226 T^{4} - 53824 T^{6} + 707281 T^{8} )^{2} \))(\( ( 1 - 64 T^{2} + 2226 T^{4} - 53824 T^{6} + 707281 T^{8} )^{2} \))
$31$ (\( 1 - 9 T + 71 T^{2} - 396 T^{3} + 1812 T^{4} - 12276 T^{5} + 68231 T^{6} - 268119 T^{7} + 923521 T^{8} \))(\( 1 - 9 T + 71 T^{2} - 396 T^{3} + 1812 T^{4} - 12276 T^{5} + 68231 T^{6} - 268119 T^{7} + 923521 T^{8} \))(\( 1 - 9 T + 71 T^{2} - 396 T^{3} + 1812 T^{4} - 12276 T^{5} + 68231 T^{6} - 268119 T^{7} + 923521 T^{8} \))(\( 1 - 9 T + 71 T^{2} - 396 T^{3} + 1812 T^{4} - 12276 T^{5} + 68231 T^{6} - 268119 T^{7} + 923521 T^{8} \))(\( ( 1 + 12 T + 112 T^{2} + 768 T^{3} + 4623 T^{4} + 23808 T^{5} + 107632 T^{6} + 357492 T^{7} + 923521 T^{8} )^{2} \))(\( ( 1 + 12 T + 112 T^{2} + 768 T^{3} + 4623 T^{4} + 23808 T^{5} + 107632 T^{6} + 357492 T^{7} + 923521 T^{8} )^{2} \))
$37$ (\( 1 + 6 T - 10 T^{2} - 132 T^{3} - 441 T^{4} - 4884 T^{5} - 13690 T^{6} + 303918 T^{7} + 1874161 T^{8} \))(\( 1 - 6 T - 10 T^{2} + 132 T^{3} - 441 T^{4} + 4884 T^{5} - 13690 T^{6} - 303918 T^{7} + 1874161 T^{8} \))(\( 1 + 6 T - 10 T^{2} - 132 T^{3} - 441 T^{4} - 4884 T^{5} - 13690 T^{6} + 303918 T^{7} + 1874161 T^{8} \))(\( 1 - 6 T - 10 T^{2} + 132 T^{3} - 441 T^{4} + 4884 T^{5} - 13690 T^{6} - 303918 T^{7} + 1874161 T^{8} \))(\( 1 + 90 T^{2} + 3817 T^{4} + 139050 T^{6} + 5381028 T^{8} + 190359450 T^{10} + 7153672537 T^{12} + 230915376810 T^{14} + 3512479453921 T^{16} \))(\( 1 + 90 T^{2} + 3817 T^{4} + 139050 T^{6} + 5381028 T^{8} + 190359450 T^{10} + 7153672537 T^{12} + 230915376810 T^{14} + 3512479453921 T^{16} \))
$41$ (\( ( 1 + 9 T + 94 T^{2} + 369 T^{3} + 1681 T^{4} )^{2} \))(\( ( 1 - 9 T + 94 T^{2} - 369 T^{3} + 1681 T^{4} )^{2} \))(\( ( 1 - 9 T + 94 T^{2} - 369 T^{3} + 1681 T^{4} )^{2} \))(\( ( 1 + 9 T + 94 T^{2} + 369 T^{3} + 1681 T^{4} )^{2} \))(\( ( 1 + 118 T^{2} + 6363 T^{4} + 198358 T^{6} + 2825761 T^{8} )^{2} \))(\( ( 1 + 118 T^{2} + 6363 T^{4} + 198358 T^{6} + 2825761 T^{8} )^{2} \))
$43$ (\( 1 - 49 T^{2} + 660 T^{4} - 90601 T^{6} + 3418801 T^{8} \))(\( 1 - 49 T^{2} + 660 T^{4} - 90601 T^{6} + 3418801 T^{8} \))(\( 1 - 49 T^{2} + 660 T^{4} - 90601 T^{6} + 3418801 T^{8} \))(\( 1 - 49 T^{2} + 660 T^{4} - 90601 T^{6} + 3418801 T^{8} \))(\( ( 1 - 120 T^{2} + 6818 T^{4} - 221880 T^{6} + 3418801 T^{8} )^{2} \))(\( ( 1 - 120 T^{2} + 6818 T^{4} - 221880 T^{6} + 3418801 T^{8} )^{2} \))
$47$ (\( 1 + 18 T + 218 T^{2} + 1980 T^{3} + 14967 T^{4} + 93060 T^{5} + 481562 T^{6} + 1868814 T^{7} + 4879681 T^{8} \))(\( 1 + 18 T + 218 T^{2} + 1980 T^{3} + 14967 T^{4} + 93060 T^{5} + 481562 T^{6} + 1868814 T^{7} + 4879681 T^{8} \))(\( 1 - 18 T + 218 T^{2} - 1980 T^{3} + 14967 T^{4} - 93060 T^{5} + 481562 T^{6} - 1868814 T^{7} + 4879681 T^{8} \))(\( 1 - 18 T + 218 T^{2} - 1980 T^{3} + 14967 T^{4} - 93060 T^{5} + 481562 T^{6} - 1868814 T^{7} + 4879681 T^{8} \))(\( ( 1 + 6 T + 19 T^{2} + 42 T^{3} - 1596 T^{4} + 1974 T^{5} + 41971 T^{6} + 622938 T^{7} + 4879681 T^{8} )^{2} \))(\( ( 1 - 6 T + 19 T^{2} - 42 T^{3} - 1596 T^{4} - 1974 T^{5} + 41971 T^{6} - 622938 T^{7} + 4879681 T^{8} )^{2} \))
$53$ (\( 1 - 3 T - 91 T^{2} + 18 T^{3} + 6714 T^{4} + 954 T^{5} - 255619 T^{6} - 446631 T^{7} + 7890481 T^{8} \))(\( 1 - 3 T - 91 T^{2} + 18 T^{3} + 6714 T^{4} + 954 T^{5} - 255619 T^{6} - 446631 T^{7} + 7890481 T^{8} \))(\( 1 + 3 T - 91 T^{2} - 18 T^{3} + 6714 T^{4} - 954 T^{5} - 255619 T^{6} + 446631 T^{7} + 7890481 T^{8} \))(\( 1 + 3 T - 91 T^{2} - 18 T^{3} + 6714 T^{4} - 954 T^{5} - 255619 T^{6} + 446631 T^{7} + 7890481 T^{8} \))(\( ( 1 + 5 T - 28 T^{2} + 265 T^{3} + 2809 T^{4} )^{4} \))(\( ( 1 - 5 T - 28 T^{2} - 265 T^{3} + 2809 T^{4} )^{4} \))
$59$ (\( 1 - 9 T + 17 T^{2} + 486 T^{3} - 4164 T^{4} + 28674 T^{5} + 59177 T^{6} - 1848411 T^{7} + 12117361 T^{8} \))(\( 1 + 9 T + 17 T^{2} - 486 T^{3} - 4164 T^{4} - 28674 T^{5} + 59177 T^{6} + 1848411 T^{7} + 12117361 T^{8} \))(\( 1 + 9 T + 17 T^{2} - 486 T^{3} - 4164 T^{4} - 28674 T^{5} + 59177 T^{6} + 1848411 T^{7} + 12117361 T^{8} \))(\( 1 - 9 T + 17 T^{2} + 486 T^{3} - 4164 T^{4} + 28674 T^{5} + 59177 T^{6} - 1848411 T^{7} + 12117361 T^{8} \))(\( 1 - 100 T^{2} + 2458 T^{4} - 58000 T^{6} + 9954403 T^{8} - 201898000 T^{10} + 29784473338 T^{12} - 4218053364100 T^{14} + 146830437604321 T^{16} \))(\( 1 - 100 T^{2} + 2458 T^{4} - 58000 T^{6} + 9954403 T^{8} - 201898000 T^{10} + 29784473338 T^{12} - 4218053364100 T^{14} + 146830437604321 T^{16} \))
$61$ (\( 1 + 27 T + 401 T^{2} + 4266 T^{3} + 36066 T^{4} + 260226 T^{5} + 1492121 T^{6} + 6128487 T^{7} + 13845841 T^{8} \))(\( 1 + 27 T + 401 T^{2} + 4266 T^{3} + 36066 T^{4} + 260226 T^{5} + 1492121 T^{6} + 6128487 T^{7} + 13845841 T^{8} \))(\( 1 + 27 T + 401 T^{2} + 4266 T^{3} + 36066 T^{4} + 260226 T^{5} + 1492121 T^{6} + 6128487 T^{7} + 13845841 T^{8} \))(\( 1 + 27 T + 401 T^{2} + 4266 T^{3} + 36066 T^{4} + 260226 T^{5} + 1492121 T^{6} + 6128487 T^{7} + 13845841 T^{8} \))(\( ( 1 - 6 T + 73 T^{2} - 366 T^{3} + 3721 T^{4} )^{4} \))(\( ( 1 - 6 T + 73 T^{2} - 366 T^{3} + 3721 T^{4} )^{4} \))
$67$ (\( 1 - 9 T + 143 T^{2} - 1044 T^{3} + 10776 T^{4} - 69948 T^{5} + 641927 T^{6} - 2706867 T^{7} + 20151121 T^{8} \))(\( 1 + 9 T + 143 T^{2} + 1044 T^{3} + 10776 T^{4} + 69948 T^{5} + 641927 T^{6} + 2706867 T^{7} + 20151121 T^{8} \))(\( 1 - 9 T + 143 T^{2} - 1044 T^{3} + 10776 T^{4} - 69948 T^{5} + 641927 T^{6} - 2706867 T^{7} + 20151121 T^{8} \))(\( 1 + 9 T + 143 T^{2} + 1044 T^{3} + 10776 T^{4} + 69948 T^{5} + 641927 T^{6} + 2706867 T^{7} + 20151121 T^{8} \))(\( 1 + 36 T^{2} - 326 T^{4} - 264816 T^{6} - 23337981 T^{8} - 1188759024 T^{10} - 6569265446 T^{12} + 3256501758084 T^{14} + 406067677556641 T^{16} \))(\( 1 + 36 T^{2} - 326 T^{4} - 264816 T^{6} - 23337981 T^{8} - 1188759024 T^{10} - 6569265446 T^{12} + 3256501758084 T^{14} + 406067677556641 T^{16} \))
$71$ (\( 1 - 208 T^{2} + 19710 T^{4} - 1048528 T^{6} + 25411681 T^{8} \))(\( 1 - 208 T^{2} + 19710 T^{4} - 1048528 T^{6} + 25411681 T^{8} \))(\( 1 - 208 T^{2} + 19710 T^{4} - 1048528 T^{6} + 25411681 T^{8} \))(\( 1 - 208 T^{2} + 19710 T^{4} - 1048528 T^{6} + 25411681 T^{8} \))(\( ( 1 - 232 T^{2} + 23058 T^{4} - 1169512 T^{6} + 25411681 T^{8} )^{2} \))(\( ( 1 - 232 T^{2} + 23058 T^{4} - 1169512 T^{6} + 25411681 T^{8} )^{2} \))
$73$ (\( ( 1 - 2 T - 69 T^{2} - 146 T^{3} + 5329 T^{4} )^{2} \))(\( ( 1 + 2 T - 69 T^{2} + 146 T^{3} + 5329 T^{4} )^{2} \))(\( ( 1 - 2 T - 69 T^{2} - 146 T^{3} + 5329 T^{4} )^{2} \))(\( ( 1 + 2 T - 69 T^{2} + 146 T^{3} + 5329 T^{4} )^{2} \))(\( 1 - 136 T^{2} + 7534 T^{4} - 41344 T^{6} - 6797981 T^{8} - 220322176 T^{10} + 213952347694 T^{12} - 20581454775304 T^{14} + 806460091894081 T^{16} \))(\( 1 - 136 T^{2} + 7534 T^{4} - 41344 T^{6} - 6797981 T^{8} - 220322176 T^{10} + 213952347694 T^{12} - 20581454775304 T^{14} + 806460091894081 T^{16} \))
$79$ (\( 1 + T - 83 T^{2} - 74 T^{3} + 736 T^{4} - 5846 T^{5} - 518003 T^{6} + 493039 T^{7} + 38950081 T^{8} \))(\( 1 + T - 83 T^{2} - 74 T^{3} + 736 T^{4} - 5846 T^{5} - 518003 T^{6} + 493039 T^{7} + 38950081 T^{8} \))(\( 1 + T - 83 T^{2} - 74 T^{3} + 736 T^{4} - 5846 T^{5} - 518003 T^{6} + 493039 T^{7} + 38950081 T^{8} \))(\( 1 + T - 83 T^{2} - 74 T^{3} + 736 T^{4} - 5846 T^{5} - 518003 T^{6} + 493039 T^{7} + 38950081 T^{8} \))(\( ( 1 + 12 T - 20 T^{2} + 72 T^{3} + 9279 T^{4} + 5688 T^{5} - 124820 T^{6} + 5916468 T^{7} + 38950081 T^{8} )^{2} \))(\( ( 1 + 12 T - 20 T^{2} + 72 T^{3} + 9279 T^{4} + 5688 T^{5} - 124820 T^{6} + 5916468 T^{7} + 38950081 T^{8} )^{2} \))
$83$ (\( 1 - 190 T^{2} + 18051 T^{4} - 1308910 T^{6} + 47458321 T^{8} \))(\( 1 - 190 T^{2} + 18051 T^{4} - 1308910 T^{6} + 47458321 T^{8} \))(\( 1 - 190 T^{2} + 18051 T^{4} - 1308910 T^{6} + 47458321 T^{8} \))(\( 1 - 190 T^{2} + 18051 T^{4} - 1308910 T^{6} + 47458321 T^{8} \))(\( ( 1 - 36 T^{2} - 3178 T^{4} - 248004 T^{6} + 47458321 T^{8} )^{2} \))(\( ( 1 - 36 T^{2} - 3178 T^{4} - 248004 T^{6} + 47458321 T^{8} )^{2} \))
$89$ (\( 1 + 3 T - 163 T^{2} - 18 T^{3} + 20862 T^{4} - 1602 T^{5} - 1291123 T^{6} + 2114907 T^{7} + 62742241 T^{8} \))(\( 1 - 3 T - 163 T^{2} + 18 T^{3} + 20862 T^{4} + 1602 T^{5} - 1291123 T^{6} - 2114907 T^{7} + 62742241 T^{8} \))(\( 1 - 3 T - 163 T^{2} + 18 T^{3} + 20862 T^{4} + 1602 T^{5} - 1291123 T^{6} - 2114907 T^{7} + 62742241 T^{8} \))(\( 1 + 3 T - 163 T^{2} - 18 T^{3} + 20862 T^{4} - 1602 T^{5} - 1291123 T^{6} + 2114907 T^{7} + 62742241 T^{8} \))(\( 1 - 220 T^{2} + 22378 T^{4} - 2239600 T^{6} + 231025843 T^{8} - 17739871600 T^{10} + 1404045869098 T^{12} - 109335884011420 T^{14} + 3936588805702081 T^{16} \))(\( 1 - 220 T^{2} + 22378 T^{4} - 2239600 T^{6} + 231025843 T^{8} - 17739871600 T^{10} + 1404045869098 T^{12} - 109335884011420 T^{14} + 3936588805702081 T^{16} \))
$97$ (\( ( 1 + 13 T + 162 T^{2} + 1261 T^{3} + 9409 T^{4} )^{2} \))(\( ( 1 - 13 T + 162 T^{2} - 1261 T^{3} + 9409 T^{4} )^{2} \))(\( ( 1 + 13 T + 162 T^{2} + 1261 T^{3} + 9409 T^{4} )^{2} \))(\( ( 1 - 13 T + 162 T^{2} - 1261 T^{3} + 9409 T^{4} )^{2} \))(\( ( 1 + 232 T^{2} + 27954 T^{4} + 2182888 T^{6} + 88529281 T^{8} )^{2} \))(\( ( 1 + 232 T^{2} + 27954 T^{4} + 2182888 T^{6} + 88529281 T^{8} )^{2} \))
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