Properties

Label 1050.2.d
Level 1050
Weight 2
Character orbit d
Rep. character \(\chi_{1050}(1049,\cdot)\)
Character field \(\Q\)
Dimension 48
Newform subspaces 8
Sturm bound 480
Trace bound 3

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Defining parameters

Level: \( N \) = \( 1050 = 2 \cdot 3 \cdot 5^{2} \cdot 7 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 1050.d (of order \(2\) and degree \(1\))
Character conductor: \(\operatorname{cond}(\chi)\) = \( 105 \)
Character field: \(\Q\)
Newform subspaces: \( 8 \)
Sturm bound: \(480\)
Trace bound: \(3\)
Distinguishing \(T_p\): \(11\), \(13\), \(23\)

Dimensions

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

Total New Old
Modular forms 264 48 216
Cusp forms 216 48 168
Eisenstein series 48 0 48

Trace form

\( 48q + 48q^{4} + O(q^{10}) \) \( 48q + 48q^{4} + 48q^{16} + 20q^{21} + 48q^{39} + 56q^{46} - 16q^{49} + 40q^{51} + 48q^{64} - 32q^{79} - 88q^{81} + 20q^{84} - 56q^{91} - 96q^{99} + O(q^{100}) \)

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.d.a \(4\) \(8.384\) \(\Q(i, \sqrt{5})\) None \(-4\) \(-2\) \(0\) \(6\) \(q-q^{2}+(-\beta _{2}-\beta _{3})q^{3}+q^{4}+(\beta _{2}+\beta _{3})q^{6}+\cdots\)
1050.2.d.b \(4\) \(8.384\) \(\Q(i, \sqrt{6})\) None \(-4\) \(0\) \(0\) \(0\) \(q-q^{2}-\beta _{3}q^{3}+q^{4}+\beta _{3}q^{6}+(\beta _{1}+\beta _{2}+\cdots)q^{7}+\cdots\)
1050.2.d.c \(4\) \(8.384\) \(\Q(i, \sqrt{5})\) None \(-4\) \(2\) \(0\) \(-6\) \(q-q^{2}+(\beta _{2}+\beta _{3})q^{3}+q^{4}+(-\beta _{2}-\beta _{3})q^{6}+\cdots\)
1050.2.d.d \(4\) \(8.384\) \(\Q(i, \sqrt{5})\) None \(4\) \(-2\) \(0\) \(6\) \(q+q^{2}+(-\beta _{2}-\beta _{3})q^{3}+q^{4}+(-\beta _{2}+\cdots)q^{6}+\cdots\)
1050.2.d.e \(4\) \(8.384\) \(\Q(i, \sqrt{6})\) None \(4\) \(0\) \(0\) \(0\) \(q+q^{2}+\beta _{3}q^{3}+q^{4}+\beta _{3}q^{6}+(-\beta _{1}+\cdots)q^{7}+\cdots\)
1050.2.d.f \(4\) \(8.384\) \(\Q(i, \sqrt{5})\) None \(4\) \(2\) \(0\) \(-6\) \(q+q^{2}+(\beta _{2}+\beta _{3})q^{3}+q^{4}+(\beta _{2}+\beta _{3})q^{6}+\cdots\)
1050.2.d.g \(12\) \(8.384\) \(\mathbb{Q}[x]/(x^{12} + \cdots)\) None \(-12\) \(0\) \(0\) \(0\) \(q-q^{2}+\beta _{10}q^{3}+q^{4}-\beta _{10}q^{6}+(-\beta _{1}+\cdots)q^{7}+\cdots\)
1050.2.d.h \(12\) \(8.384\) \(\mathbb{Q}[x]/(x^{12} + \cdots)\) None \(12\) \(0\) \(0\) \(0\) \(q+q^{2}-\beta _{10}q^{3}+q^{4}-\beta _{10}q^{6}+(\beta _{1}+\cdots)q^{7}+\cdots\)

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

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

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ (\( ( 1 + T )^{4} \))(\( ( 1 + T )^{4} \))(\( ( 1 + T )^{4} \))(\( ( 1 - T )^{4} \))(\( ( 1 - T )^{4} \))(\( ( 1 - T )^{4} \))(\( ( 1 + T )^{12} \))(\( ( 1 - T )^{12} \))
$3$ (\( 1 + 2 T + 2 T^{2} + 6 T^{3} + 9 T^{4} \))(\( 1 + 9 T^{4} \))(\( 1 - 2 T + 2 T^{2} - 6 T^{3} + 9 T^{4} \))(\( 1 + 2 T + 2 T^{2} + 6 T^{3} + 9 T^{4} \))(\( 1 + 9 T^{4} \))(\( 1 - 2 T + 2 T^{2} - 6 T^{3} + 9 T^{4} \))(\( 1 + 4 T^{4} + 30 T^{6} + 36 T^{8} + 729 T^{12} \))(\( 1 + 4 T^{4} + 30 T^{6} + 36 T^{8} + 729 T^{12} \))
$5$ 1
$7$ (\( 1 - 6 T + 18 T^{2} - 42 T^{3} + 49 T^{4} \))(\( 1 - 10 T^{2} + 49 T^{4} \))(\( 1 + 6 T + 18 T^{2} + 42 T^{3} + 49 T^{4} \))(\( 1 - 6 T + 18 T^{2} - 42 T^{3} + 49 T^{4} \))(\( 1 - 10 T^{2} + 49 T^{4} \))(\( 1 + 6 T + 18 T^{2} + 42 T^{3} + 49 T^{4} \))(\( 1 + 14 T^{2} + 99 T^{4} + 652 T^{6} + 4851 T^{8} + 33614 T^{10} + 117649 T^{12} \))(\( 1 + 14 T^{2} + 99 T^{4} + 652 T^{6} + 4851 T^{8} + 33614 T^{10} + 117649 T^{12} \))
$11$ (\( ( 1 - 2 T^{2} + 121 T^{4} )^{2} \))(\( ( 1 - 11 T^{2} )^{4} \))(\( ( 1 - 2 T^{2} + 121 T^{4} )^{2} \))(\( ( 1 - 2 T^{2} + 121 T^{4} )^{2} \))(\( ( 1 - 11 T^{2} )^{4} \))(\( ( 1 - 2 T^{2} + 121 T^{4} )^{2} \))(\( ( 1 - 20 T^{2} + 320 T^{4} - 3962 T^{6} + 38720 T^{8} - 292820 T^{10} + 1771561 T^{12} )^{2} \))(\( ( 1 - 20 T^{2} + 320 T^{4} - 3962 T^{6} + 38720 T^{8} - 292820 T^{10} + 1771561 T^{12} )^{2} \))
$13$ (\( ( 1 + 2 T + 22 T^{2} + 26 T^{3} + 169 T^{4} )^{2} \))(\( ( 1 + 20 T^{2} + 169 T^{4} )^{2} \))(\( ( 1 - 2 T + 22 T^{2} - 26 T^{3} + 169 T^{4} )^{2} \))(\( ( 1 + 2 T + 22 T^{2} + 26 T^{3} + 169 T^{4} )^{2} \))(\( ( 1 + 20 T^{2} + 169 T^{4} )^{2} \))(\( ( 1 - 2 T + 22 T^{2} - 26 T^{3} + 169 T^{4} )^{2} \))(\( ( 1 + 15 T^{2} + 559 T^{4} + 5110 T^{6} + 94471 T^{8} + 428415 T^{10} + 4826809 T^{12} )^{2} \))(\( ( 1 + 15 T^{2} + 559 T^{4} + 5110 T^{6} + 94471 T^{8} + 428415 T^{10} + 4826809 T^{12} )^{2} \))
$17$ (\( 1 - 40 T^{2} + 798 T^{4} - 11560 T^{6} + 83521 T^{8} \))(\( ( 1 - 10 T^{2} + 289 T^{4} )^{2} \))(\( 1 - 40 T^{2} + 798 T^{4} - 11560 T^{6} + 83521 T^{8} \))(\( 1 - 40 T^{2} + 798 T^{4} - 11560 T^{6} + 83521 T^{8} \))(\( ( 1 - 10 T^{2} + 289 T^{4} )^{2} \))(\( 1 - 40 T^{2} + 798 T^{4} - 11560 T^{6} + 83521 T^{8} \))(\( ( 1 - 29 T^{2} + 494 T^{4} - 5297 T^{6} + 142766 T^{8} - 2422109 T^{10} + 24137569 T^{12} )^{2} \))(\( ( 1 - 29 T^{2} + 494 T^{4} - 5297 T^{6} + 142766 T^{8} - 2422109 T^{10} + 24137569 T^{12} )^{2} \))
$19$ (\( 1 - 4 T^{2} - 554 T^{4} - 1444 T^{6} + 130321 T^{8} \))(\( ( 1 - 32 T^{2} + 361 T^{4} )^{2} \))(\( 1 - 4 T^{2} - 554 T^{4} - 1444 T^{6} + 130321 T^{8} \))(\( 1 - 4 T^{2} - 554 T^{4} - 1444 T^{6} + 130321 T^{8} \))(\( ( 1 - 32 T^{2} + 361 T^{4} )^{2} \))(\( 1 - 4 T^{2} - 554 T^{4} - 1444 T^{6} + 130321 T^{8} \))(\( ( 1 - 96 T^{2} + 4132 T^{4} - 101374 T^{6} + 1491652 T^{8} - 12510816 T^{10} + 47045881 T^{12} )^{2} \))(\( ( 1 - 96 T^{2} + 4132 T^{4} - 101374 T^{6} + 1491652 T^{8} - 12510816 T^{10} + 47045881 T^{12} )^{2} \))
$23$ (\( ( 1 + 4 T + 23 T^{2} )^{4} \))(\( ( 1 - 6 T + 23 T^{2} )^{4} \))(\( ( 1 + 4 T + 23 T^{2} )^{4} \))(\( ( 1 - 4 T + 23 T^{2} )^{4} \))(\( ( 1 + 6 T + 23 T^{2} )^{4} \))(\( ( 1 - 4 T + 23 T^{2} )^{4} \))(\( ( 1 + 5 T + 49 T^{2} + 224 T^{3} + 1127 T^{4} + 2645 T^{5} + 12167 T^{6} )^{4} \))(\( ( 1 - 5 T + 49 T^{2} - 224 T^{3} + 1127 T^{4} - 2645 T^{5} + 12167 T^{6} )^{4} \))
$29$ (\( 1 - 24 T^{2} + 1646 T^{4} - 20184 T^{6} + 707281 T^{8} \))(\( ( 1 - 22 T^{2} + 841 T^{4} )^{2} \))(\( 1 - 24 T^{2} + 1646 T^{4} - 20184 T^{6} + 707281 T^{8} \))(\( 1 - 24 T^{2} + 1646 T^{4} - 20184 T^{6} + 707281 T^{8} \))(\( ( 1 - 22 T^{2} + 841 T^{4} )^{2} \))(\( 1 - 24 T^{2} + 1646 T^{4} - 20184 T^{6} + 707281 T^{8} \))(\( ( 1 - 109 T^{2} + 5535 T^{4} - 186434 T^{6} + 4654935 T^{8} - 77093629 T^{10} + 594823321 T^{12} )^{2} \))(\( ( 1 - 109 T^{2} + 5535 T^{4} - 186434 T^{6} + 4654935 T^{8} - 77093629 T^{10} + 594823321 T^{12} )^{2} \))
$31$ (\( 1 - 64 T^{2} + 2446 T^{4} - 61504 T^{6} + 923521 T^{8} \))(\( ( 1 - 31 T^{2} )^{4} \))(\( 1 - 64 T^{2} + 2446 T^{4} - 61504 T^{6} + 923521 T^{8} \))(\( 1 - 64 T^{2} + 2446 T^{4} - 61504 T^{6} + 923521 T^{8} \))(\( ( 1 - 31 T^{2} )^{4} \))(\( 1 - 64 T^{2} + 2446 T^{4} - 61504 T^{6} + 923521 T^{8} \))(\( ( 1 - 51 T^{2} + 2623 T^{4} - 73486 T^{6} + 2520703 T^{8} - 47099571 T^{10} + 887503681 T^{12} )^{2} \))(\( ( 1 - 51 T^{2} + 2623 T^{4} - 73486 T^{6} + 2520703 T^{8} - 47099571 T^{10} + 887503681 T^{12} )^{2} \))
$37$ (\( 1 - 120 T^{2} + 6158 T^{4} - 164280 T^{6} + 1874161 T^{8} \))(\( ( 1 - 12 T + 37 T^{2} )^{2}( 1 + 12 T + 37 T^{2} )^{2} \))(\( 1 - 120 T^{2} + 6158 T^{4} - 164280 T^{6} + 1874161 T^{8} \))(\( 1 - 120 T^{2} + 6158 T^{4} - 164280 T^{6} + 1874161 T^{8} \))(\( ( 1 - 12 T + 37 T^{2} )^{2}( 1 + 12 T + 37 T^{2} )^{2} \))(\( 1 - 120 T^{2} + 6158 T^{4} - 164280 T^{6} + 1874161 T^{8} \))(\( ( 1 - 70 T^{2} + 2907 T^{4} - 101660 T^{6} + 3979683 T^{8} - 131191270 T^{10} + 2565726409 T^{12} )^{2} \))(\( ( 1 - 70 T^{2} + 2907 T^{4} - 101660 T^{6} + 3979683 T^{8} - 131191270 T^{10} + 2565726409 T^{12} )^{2} \))
$41$ (\( ( 1 + 4 T + 66 T^{2} + 164 T^{3} + 1681 T^{4} )^{2} \))(\( ( 1 + 58 T^{2} + 1681 T^{4} )^{2} \))(\( ( 1 - 4 T + 66 T^{2} - 164 T^{3} + 1681 T^{4} )^{2} \))(\( ( 1 - 4 T + 66 T^{2} - 164 T^{3} + 1681 T^{4} )^{2} \))(\( ( 1 + 58 T^{2} + 1681 T^{4} )^{2} \))(\( ( 1 + 4 T + 66 T^{2} + 164 T^{3} + 1681 T^{4} )^{2} \))(\( ( 1 - 3 T^{2} + 3166 T^{4} - 943 T^{6} + 5322046 T^{8} - 8477283 T^{10} + 4750104241 T^{12} )^{2} \))(\( ( 1 - 3 T^{2} + 3166 T^{4} - 943 T^{6} + 5322046 T^{8} - 8477283 T^{10} + 4750104241 T^{12} )^{2} \))
$43$ (\( 1 + 20 T^{2} - 1322 T^{4} + 36980 T^{6} + 3418801 T^{8} \))(\( ( 1 - 70 T^{2} + 1849 T^{4} )^{2} \))(\( 1 + 20 T^{2} - 1322 T^{4} + 36980 T^{6} + 3418801 T^{8} \))(\( 1 + 20 T^{2} - 1322 T^{4} + 36980 T^{6} + 3418801 T^{8} \))(\( ( 1 - 70 T^{2} + 1849 T^{4} )^{2} \))(\( 1 + 20 T^{2} - 1322 T^{4} + 36980 T^{6} + 3418801 T^{8} \))(\( ( 1 - 209 T^{2} + 19659 T^{4} - 1076742 T^{6} + 36349491 T^{8} - 714529409 T^{10} + 6321363049 T^{12} )^{2} \))(\( ( 1 - 209 T^{2} + 19659 T^{4} - 1076742 T^{6} + 36349491 T^{8} - 714529409 T^{10} + 6321363049 T^{12} )^{2} \))
$47$ (\( 1 - 140 T^{2} + 8998 T^{4} - 309260 T^{6} + 4879681 T^{8} \))(\( ( 1 - 70 T^{2} + 2209 T^{4} )^{2} \))(\( 1 - 140 T^{2} + 8998 T^{4} - 309260 T^{6} + 4879681 T^{8} \))(\( 1 - 140 T^{2} + 8998 T^{4} - 309260 T^{6} + 4879681 T^{8} \))(\( ( 1 - 70 T^{2} + 2209 T^{4} )^{2} \))(\( 1 - 140 T^{2} + 8998 T^{4} - 309260 T^{6} + 4879681 T^{8} \))(\( ( 1 - 78 T^{2} + 7315 T^{4} - 322124 T^{6} + 16158835 T^{8} - 380615118 T^{10} + 10779215329 T^{12} )^{2} \))(\( ( 1 - 78 T^{2} + 7315 T^{4} - 322124 T^{6} + 16158835 T^{8} - 380615118 T^{10} + 10779215329 T^{12} )^{2} \))
$53$ (\( ( 1 + 8 T + 102 T^{2} + 424 T^{3} + 2809 T^{4} )^{2} \))(\( ( 1 - 6 T + 53 T^{2} )^{4} \))(\( ( 1 + 8 T + 102 T^{2} + 424 T^{3} + 2809 T^{4} )^{2} \))(\( ( 1 - 8 T + 102 T^{2} - 424 T^{3} + 2809 T^{4} )^{2} \))(\( ( 1 + 6 T + 53 T^{2} )^{4} \))(\( ( 1 - 8 T + 102 T^{2} - 424 T^{3} + 2809 T^{4} )^{2} \))(\( ( 1 + 5 T + 95 T^{2} + 548 T^{3} + 5035 T^{4} + 14045 T^{5} + 148877 T^{6} )^{4} \))(\( ( 1 - 5 T + 95 T^{2} - 548 T^{3} + 5035 T^{4} - 14045 T^{5} + 148877 T^{6} )^{4} \))
$59$ (\( ( 1 + 98 T^{2} + 3481 T^{4} )^{2} \))(\( ( 1 - 32 T^{2} + 3481 T^{4} )^{2} \))(\( ( 1 + 98 T^{2} + 3481 T^{4} )^{2} \))(\( ( 1 + 98 T^{2} + 3481 T^{4} )^{2} \))(\( ( 1 - 32 T^{2} + 3481 T^{4} )^{2} \))(\( ( 1 + 98 T^{2} + 3481 T^{4} )^{2} \))(\( ( 1 + 179 T^{2} + 18587 T^{4} + 1260626 T^{6} + 64701347 T^{8} + 2169007619 T^{10} + 42180533641 T^{12} )^{2} \))(\( ( 1 + 179 T^{2} + 18587 T^{4} + 1260626 T^{6} + 64701347 T^{8} + 2169007619 T^{10} + 42180533641 T^{12} )^{2} \))
$61$ (\( 1 - 184 T^{2} + 15406 T^{4} - 684664 T^{6} + 13845841 T^{8} \))(\( ( 1 + 28 T^{2} + 3721 T^{4} )^{2} \))(\( 1 - 184 T^{2} + 15406 T^{4} - 684664 T^{6} + 13845841 T^{8} \))(\( 1 - 184 T^{2} + 15406 T^{4} - 684664 T^{6} + 13845841 T^{8} \))(\( ( 1 + 28 T^{2} + 3721 T^{4} )^{2} \))(\( 1 - 184 T^{2} + 15406 T^{4} - 684664 T^{6} + 13845841 T^{8} \))(\( ( 1 - 183 T^{2} + 12451 T^{4} - 609898 T^{6} + 46330171 T^{8} - 2533788903 T^{10} + 51520374361 T^{12} )^{2} \))(\( ( 1 - 183 T^{2} + 12451 T^{4} - 609898 T^{6} + 46330171 T^{8} - 2533788903 T^{10} + 51520374361 T^{12} )^{2} \))
$67$ (\( ( 1 + 10 T^{2} + 4489 T^{4} )^{2} \))(\( ( 1 - 70 T^{2} + 4489 T^{4} )^{2} \))(\( ( 1 + 10 T^{2} + 4489 T^{4} )^{2} \))(\( ( 1 + 10 T^{2} + 4489 T^{4} )^{2} \))(\( ( 1 - 70 T^{2} + 4489 T^{4} )^{2} \))(\( ( 1 + 10 T^{2} + 4489 T^{4} )^{2} \))(\( ( 1 - 260 T^{2} + 33680 T^{4} - 2779070 T^{6} + 151189520 T^{8} - 5239291460 T^{10} + 90458382169 T^{12} )^{2} \))(\( ( 1 - 260 T^{2} + 33680 T^{4} - 2779070 T^{6} + 151189520 T^{8} - 5239291460 T^{10} + 90458382169 T^{12} )^{2} \))
$71$ (\( 1 - 224 T^{2} + 22126 T^{4} - 1129184 T^{6} + 25411681 T^{8} \))(\( ( 1 - 71 T^{2} )^{4} \))(\( 1 - 224 T^{2} + 22126 T^{4} - 1129184 T^{6} + 25411681 T^{8} \))(\( 1 - 224 T^{2} + 22126 T^{4} - 1129184 T^{6} + 25411681 T^{8} \))(\( ( 1 - 71 T^{2} )^{4} \))(\( 1 - 224 T^{2} + 22126 T^{4} - 1129184 T^{6} + 25411681 T^{8} \))(\( ( 1 - 210 T^{2} + 21667 T^{4} - 1617716 T^{6} + 109223347 T^{8} - 5336453010 T^{10} + 128100283921 T^{12} )^{2} \))(\( ( 1 - 210 T^{2} + 21667 T^{4} - 1617716 T^{6} + 109223347 T^{8} - 5336453010 T^{10} + 128100283921 T^{12} )^{2} \))
$73$ (\( ( 1 - 18 T + 222 T^{2} - 1314 T^{3} + 5329 T^{4} )^{2} \))(\( ( 1 + 50 T^{2} + 5329 T^{4} )^{2} \))(\( ( 1 + 18 T + 222 T^{2} + 1314 T^{3} + 5329 T^{4} )^{2} \))(\( ( 1 - 18 T + 222 T^{2} - 1314 T^{3} + 5329 T^{4} )^{2} \))(\( ( 1 + 50 T^{2} + 5329 T^{4} )^{2} \))(\( ( 1 + 18 T + 222 T^{2} + 1314 T^{3} + 5329 T^{4} )^{2} \))(\( ( 1 + 332 T^{2} + 52356 T^{4} + 4849846 T^{6} + 279005124 T^{8} + 9428216012 T^{10} + 151334226289 T^{12} )^{2} \))(\( ( 1 + 332 T^{2} + 52356 T^{4} + 4849846 T^{6} + 279005124 T^{8} + 9428216012 T^{10} + 151334226289 T^{12} )^{2} \))
$79$ (\( ( 1 + 78 T^{2} + 6241 T^{4} )^{2} \))(\( ( 1 - 10 T + 79 T^{2} )^{4} \))(\( ( 1 + 78 T^{2} + 6241 T^{4} )^{2} \))(\( ( 1 + 78 T^{2} + 6241 T^{4} )^{2} \))(\( ( 1 - 10 T + 79 T^{2} )^{4} \))(\( ( 1 + 78 T^{2} + 6241 T^{4} )^{2} \))(\( ( 1 + 14 T + 193 T^{2} + 2172 T^{3} + 15247 T^{4} + 87374 T^{5} + 493039 T^{6} )^{4} \))(\( ( 1 + 14 T + 193 T^{2} + 2172 T^{3} + 15247 T^{4} + 87374 T^{5} + 493039 T^{6} )^{4} \))
$83$ (\( 1 + 160 T^{2} + 19198 T^{4} + 1102240 T^{6} + 47458321 T^{8} \))(\( ( 1 - 160 T^{2} + 6889 T^{4} )^{2} \))(\( 1 + 160 T^{2} + 19198 T^{4} + 1102240 T^{6} + 47458321 T^{8} \))(\( 1 + 160 T^{2} + 19198 T^{4} + 1102240 T^{6} + 47458321 T^{8} \))(\( ( 1 - 160 T^{2} + 6889 T^{4} )^{2} \))(\( 1 + 160 T^{2} + 19198 T^{4} + 1102240 T^{6} + 47458321 T^{8} \))(\( ( 1 - 309 T^{2} + 44338 T^{4} - 4238477 T^{6} + 305444482 T^{8} - 14664621189 T^{10} + 326940373369 T^{12} )^{2} \))(\( ( 1 - 309 T^{2} + 44338 T^{4} - 4238477 T^{6} + 305444482 T^{8} - 14664621189 T^{10} + 326940373369 T^{12} )^{2} \))
$89$ (\( ( 1 + 20 T + 258 T^{2} + 1780 T^{3} + 7921 T^{4} )^{2} \))(\( ( 1 + 89 T^{2} )^{4} \))(\( ( 1 - 20 T + 258 T^{2} - 1780 T^{3} + 7921 T^{4} )^{2} \))(\( ( 1 - 20 T + 258 T^{2} - 1780 T^{3} + 7921 T^{4} )^{2} \))(\( ( 1 + 89 T^{2} )^{4} \))(\( ( 1 + 20 T + 258 T^{2} + 1780 T^{3} + 7921 T^{4} )^{2} \))(\( ( 1 + 348 T^{2} + 58036 T^{4} + 6184130 T^{6} + 459703156 T^{8} + 21834299868 T^{10} + 496981290961 T^{12} )^{2} \))(\( ( 1 + 348 T^{2} + 58036 T^{4} + 6184130 T^{6} + 459703156 T^{8} + 21834299868 T^{10} + 496981290961 T^{12} )^{2} \))
$97$ (\( ( 1 - 6 T + 198 T^{2} - 582 T^{3} + 9409 T^{4} )^{2} \))(\( ( 1 + 170 T^{2} + 9409 T^{4} )^{2} \))(\( ( 1 + 6 T + 198 T^{2} + 582 T^{3} + 9409 T^{4} )^{2} \))(\( ( 1 - 6 T + 198 T^{2} - 582 T^{3} + 9409 T^{4} )^{2} \))(\( ( 1 + 170 T^{2} + 9409 T^{4} )^{2} \))(\( ( 1 + 6 T + 198 T^{2} + 582 T^{3} + 9409 T^{4} )^{2} \))(\( ( 1 + 158 T^{2} + 21375 T^{4} + 2986564 T^{6} + 201117375 T^{8} + 13987626398 T^{10} + 832972004929 T^{12} )^{2} \))(\( ( 1 + 158 T^{2} + 21375 T^{4} + 2986564 T^{6} + 201117375 T^{8} + 13987626398 T^{10} + 832972004929 T^{12} )^{2} \))
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