Properties

Label 1050.2.o
Level 1050
Weight 2
Character orbit o
Rep. character \(\chi_{1050}(499,\cdot)\)
Character field \(\Q(\zeta_{6})\)
Dimension 48
Newform subspaces 12
Sturm bound 480
Trace bound 14

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Defining parameters

Level: \( N \) = \( 1050 = 2 \cdot 3 \cdot 5^{2} \cdot 7 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 1050.o (of order \(6\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) = \( 35 \)
Character field: \(\Q(\zeta_{6})\)
Newform subspaces: \( 12 \)
Sturm bound: \(480\)
Trace bound: \(14\)
Distinguishing \(T_p\): \(11\), \(13\)

Dimensions

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

Total New Old
Modular forms 528 48 480
Cusp forms 432 48 384
Eisenstein series 96 0 96

Trace form

\( 48q + 24q^{4} - 8q^{6} + 24q^{9} + O(q^{10}) \) \( 48q + 24q^{4} - 8q^{6} + 24q^{9} + 8q^{14} - 24q^{16} - 12q^{19} + 4q^{21} - 4q^{24} + 8q^{26} + 64q^{29} + 20q^{31} + 32q^{34} + 48q^{36} + 4q^{39} + 96q^{41} + 16q^{46} - 48q^{49} + 16q^{51} - 4q^{54} + 16q^{56} + 16q^{59} - 20q^{61} - 48q^{64} + 16q^{66} + 64q^{69} - 64q^{71} - 24q^{74} - 24q^{76} - 8q^{79} - 24q^{81} + 8q^{84} - 40q^{86} - 16q^{89} - 28q^{91} - 24q^{94} + 4q^{96} + O(q^{100}) \)

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.o.a \(4\) \(8.384\) \(\Q(\zeta_{12})\) None \(0\) \(0\) \(0\) \(0\) \(q+\zeta_{12}q^{2}+(-\zeta_{12}+\zeta_{12}^{3})q^{3}+\zeta_{12}^{2}q^{4}+\cdots\)
1050.2.o.b \(4\) \(8.384\) \(\Q(\zeta_{12})\) None \(0\) \(0\) \(0\) \(0\) \(q+(\zeta_{12}-\zeta_{12}^{3})q^{2}-\zeta_{12}q^{3}+(1-\zeta_{12}^{2}+\cdots)q^{4}+\cdots\)
1050.2.o.c \(4\) \(8.384\) \(\Q(\zeta_{12})\) None \(0\) \(0\) \(0\) \(0\) \(q+(-\zeta_{12}+\zeta_{12}^{3})q^{2}+\zeta_{12}q^{3}+(1+\cdots)q^{4}+\cdots\)
1050.2.o.d \(4\) \(8.384\) \(\Q(\zeta_{12})\) None \(0\) \(0\) \(0\) \(0\) \(q+(\zeta_{12}-\zeta_{12}^{3})q^{2}-\zeta_{12}q^{3}+(1-\zeta_{12}^{2}+\cdots)q^{4}+\cdots\)
1050.2.o.e \(4\) \(8.384\) \(\Q(\zeta_{12})\) None \(0\) \(0\) \(0\) \(0\) \(q+(-\zeta_{12}+\zeta_{12}^{3})q^{2}+\zeta_{12}q^{3}+(1+\cdots)q^{4}+\cdots\)
1050.2.o.f \(4\) \(8.384\) \(\Q(\zeta_{12})\) None \(0\) \(0\) \(0\) \(0\) \(q+(-\zeta_{12}+\zeta_{12}^{3})q^{2}+\zeta_{12}q^{3}+(1+\cdots)q^{4}+\cdots\)
1050.2.o.g \(4\) \(8.384\) \(\Q(\zeta_{12})\) None \(0\) \(0\) \(0\) \(0\) \(q+(-\zeta_{12}+\zeta_{12}^{3})q^{2}+\zeta_{12}q^{3}+(1+\cdots)q^{4}+\cdots\)
1050.2.o.h \(4\) \(8.384\) \(\Q(\zeta_{12})\) None \(0\) \(0\) \(0\) \(0\) \(q-\zeta_{12}q^{2}+(-\zeta_{12}+\zeta_{12}^{3})q^{3}+\zeta_{12}^{2}q^{4}+\cdots\)
1050.2.o.i \(4\) \(8.384\) \(\Q(\zeta_{12})\) None \(0\) \(0\) \(0\) \(0\) \(q+(\zeta_{12}-\zeta_{12}^{3})q^{2}+\zeta_{12}q^{3}+(1-\zeta_{12}^{2}+\cdots)q^{4}+\cdots\)
1050.2.o.j \(4\) \(8.384\) \(\Q(\zeta_{12})\) None \(0\) \(0\) \(0\) \(0\) \(q+(-\zeta_{12}+\zeta_{12}^{3})q^{2}-\zeta_{12}q^{3}+(1+\cdots)q^{4}+\cdots\)
1050.2.o.k \(4\) \(8.384\) \(\Q(\zeta_{12})\) None \(0\) \(0\) \(0\) \(0\) \(q+(-\zeta_{12}+\zeta_{12}^{3})q^{2}-\zeta_{12}q^{3}+(1+\cdots)q^{4}+\cdots\)
1050.2.o.l \(4\) \(8.384\) \(\Q(\zeta_{12})\) None \(0\) \(0\) \(0\) \(0\) \(q+(\zeta_{12}-\zeta_{12}^{3})q^{2}+\zeta_{12}q^{3}+(1-\zeta_{12}^{2}+\cdots)q^{4}+\cdots\)

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

\( S_{2}^{\mathrm{old}}(1050, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(35, [\chi])\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(70, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(105, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(175, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(210, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(350, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(525, [\chi])\)\(^{\oplus 2}\)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ (\( 1 - T^{2} + T^{4} \))(\( 1 - T^{2} + T^{4} \))(\( 1 - T^{2} + T^{4} \))(\( 1 - T^{2} + T^{4} \))(\( 1 - T^{2} + T^{4} \))(\( 1 - T^{2} + T^{4} \))(\( 1 - T^{2} + T^{4} \))(\( 1 - T^{2} + T^{4} \))(\( 1 - T^{2} + T^{4} \))(\( 1 - T^{2} + T^{4} \))(\( 1 - T^{2} + T^{4} \))(\( 1 - T^{2} + T^{4} \))
$3$ (\( 1 - T^{2} + T^{4} \))(\( 1 - T^{2} + T^{4} \))(\( 1 - T^{2} + T^{4} \))(\( 1 - T^{2} + T^{4} \))(\( 1 - T^{2} + T^{4} \))(\( 1 - T^{2} + T^{4} \))(\( 1 - T^{2} + T^{4} \))(\( 1 - T^{2} + T^{4} \))(\( 1 - T^{2} + T^{4} \))(\( 1 - T^{2} + T^{4} \))(\( 1 - T^{2} + T^{4} \))(\( 1 - T^{2} + T^{4} \))
$5$ 1
$7$ (\( 1 - 13 T^{2} + 49 T^{4} \))(\( 1 + 11 T^{2} + 49 T^{4} \))(\( 1 + 2 T^{2} + 49 T^{4} \))(\( 1 - 13 T^{2} + 49 T^{4} \))(\( 1 - 13 T^{2} + 49 T^{4} \))(\( 1 + 11 T^{2} + 49 T^{4} \))(\( 1 + 2 T^{2} + 49 T^{4} \))(\( 1 + 11 T^{2} + 49 T^{4} \))(\( 1 + 2 T^{2} + 49 T^{4} \))(\( 1 + 2 T^{2} + 49 T^{4} \))(\( 1 + 11 T^{2} + 49 T^{4} \))(\( 1 + 11 T^{2} + 49 T^{4} \))
$11$ (\( ( 1 + 5 T + 14 T^{2} + 55 T^{3} + 121 T^{4} )^{2} \))(\( ( 1 + 3 T - 2 T^{2} + 33 T^{3} + 121 T^{4} )^{2} \))(\( ( 1 + 3 T - 2 T^{2} + 33 T^{3} + 121 T^{4} )^{2} \))(\( ( 1 - 11 T^{2} + 121 T^{4} )^{2} \))(\( ( 1 - 11 T^{2} + 121 T^{4} )^{2} \))(\( ( 1 - 2 T - 7 T^{2} - 22 T^{3} + 121 T^{4} )^{2} \))(\( ( 1 - 5 T + 14 T^{2} - 55 T^{3} + 121 T^{4} )^{2} \))(\( ( 1 + 6 T + 25 T^{2} + 66 T^{3} + 121 T^{4} )^{2} \))(\( ( 1 - T - 10 T^{2} - 11 T^{3} + 121 T^{4} )^{2} \))(\( ( 1 - T - 10 T^{2} - 11 T^{3} + 121 T^{4} )^{2} \))(\( ( 1 - 4 T + 5 T^{2} - 44 T^{3} + 121 T^{4} )^{2} \))(\( ( 1 - 4 T + 5 T^{2} - 44 T^{3} + 121 T^{4} )^{2} \))
$13$ (\( ( 1 - 13 T^{2} )^{4} \))(\( ( 1 - 6 T + 13 T^{2} )^{2}( 1 + 6 T + 13 T^{2} )^{2} \))(\( ( 1 - T^{2} + 169 T^{4} )^{2} \))(\( ( 1 - T^{2} + 169 T^{4} )^{2} \))(\( ( 1 - 6 T + 13 T^{2} )^{2}( 1 + 6 T + 13 T^{2} )^{2} \))(\( ( 1 - 6 T + 13 T^{2} )^{2}( 1 + 6 T + 13 T^{2} )^{2} \))(\( ( 1 - T^{2} + 169 T^{4} )^{2} \))(\( ( 1 - 6 T + 13 T^{2} )^{2}( 1 + 6 T + 13 T^{2} )^{2} \))(\( ( 1 - 25 T^{2} + 169 T^{4} )^{2} \))(\( ( 1 + 23 T^{2} + 169 T^{4} )^{2} \))(\( ( 1 - 25 T^{2} + 169 T^{4} )^{2} \))(\( ( 1 - 6 T + 13 T^{2} )^{2}( 1 + 6 T + 13 T^{2} )^{2} \))
$17$ (\( 1 + 18 T^{2} + 35 T^{4} + 5202 T^{6} + 83521 T^{8} \))(\( ( 1 + 17 T^{2} + 289 T^{4} )^{2} \))(\( ( 1 + 17 T^{2} + 289 T^{4} )^{2} \))(\( 1 - 2 T^{2} - 285 T^{4} - 578 T^{6} + 83521 T^{8} \))(\( 1 + 25 T^{2} + 336 T^{4} + 7225 T^{6} + 83521 T^{8} \))(\( 1 + 9 T^{2} - 208 T^{4} + 2601 T^{6} + 83521 T^{8} \))(\( 1 + 18 T^{2} + 35 T^{4} + 5202 T^{6} + 83521 T^{8} \))(\( 1 + 25 T^{2} + 336 T^{4} + 7225 T^{6} + 83521 T^{8} \))(\( ( 1 + 17 T^{2} + 289 T^{4} )^{2} \))(\( 1 + 18 T^{2} + 35 T^{4} + 5202 T^{6} + 83521 T^{8} \))(\( ( 1 - 8 T + 47 T^{2} - 136 T^{3} + 289 T^{4} )( 1 + 8 T + 47 T^{2} + 136 T^{3} + 289 T^{4} ) \))(\( 1 + 25 T^{2} + 336 T^{4} + 7225 T^{6} + 83521 T^{8} \))
$19$ (\( ( 1 - 7 T + 19 T^{2} )^{2}( 1 - T + 19 T^{2} )^{2} \))(\( ( 1 + 4 T - 3 T^{2} + 76 T^{3} + 361 T^{4} )^{2} \))(\( ( 1 - 5 T + 6 T^{2} - 95 T^{3} + 361 T^{4} )^{2} \))(\( ( 1 - T + 19 T^{2} )^{2}( 1 + 8 T + 19 T^{2} )^{2} \))(\( ( 1 - 2 T - 15 T^{2} - 38 T^{3} + 361 T^{4} )^{2} \))(\( ( 1 + 4 T - 3 T^{2} + 76 T^{3} + 361 T^{4} )^{2} \))(\( ( 1 - T + 19 T^{2} )^{2}( 1 + 8 T + 19 T^{2} )^{2} \))(\( ( 1 + 4 T - 3 T^{2} + 76 T^{3} + 361 T^{4} )^{2} \))(\( ( 1 + 3 T - 10 T^{2} + 57 T^{3} + 361 T^{4} )^{2} \))(\( ( 1 - 8 T + 19 T^{2} )^{2}( 1 + 7 T + 19 T^{2} )^{2} \))(\( ( 1 - 8 T + 19 T^{2} )^{2}( 1 + 7 T + 19 T^{2} )^{2} \))(\( ( 1 - 6 T + 17 T^{2} - 114 T^{3} + 361 T^{4} )^{2} \))
$23$ (\( 1 + 30 T^{2} + 371 T^{4} + 15870 T^{6} + 279841 T^{8} \))(\( ( 1 + 23 T^{2} + 529 T^{4} )^{2} \))(\( 1 - 35 T^{2} + 696 T^{4} - 18515 T^{6} + 279841 T^{8} \))(\( 1 + 10 T^{2} - 429 T^{4} + 5290 T^{6} + 279841 T^{8} \))(\( 1 + 37 T^{2} + 840 T^{4} + 19573 T^{6} + 279841 T^{8} \))(\( 1 + 21 T^{2} - 88 T^{4} + 11109 T^{6} + 279841 T^{8} \))(\( 1 + 45 T^{2} + 1496 T^{4} + 23805 T^{6} + 279841 T^{8} \))(\( 1 + 37 T^{2} + 840 T^{4} + 19573 T^{6} + 279841 T^{8} \))(\( 1 - 3 T^{2} - 520 T^{4} - 1587 T^{6} + 279841 T^{8} \))(\( 1 + 45 T^{2} + 1496 T^{4} + 23805 T^{6} + 279841 T^{8} \))(\( 1 + 42 T^{2} + 1235 T^{4} + 22218 T^{6} + 279841 T^{8} \))(\( 1 - 3 T^{2} - 520 T^{4} - 1587 T^{6} + 279841 T^{8} \))
$29$ (\( ( 1 - 5 T + 29 T^{2} )^{4} \))(\( ( 1 + 9 T + 29 T^{2} )^{4} \))(\( ( 1 + 29 T^{2} )^{4} \))(\( ( 1 + 29 T^{2} )^{4} \))(\( ( 1 + 29 T^{2} )^{4} \))(\( ( 1 - 6 T + 29 T^{2} )^{4} \))(\( ( 1 + 29 T^{2} )^{4} \))(\( ( 1 - 6 T + 29 T^{2} )^{4} \))(\( ( 1 - 8 T + 29 T^{2} )^{4} \))(\( ( 1 - 8 T + 29 T^{2} )^{4} \))(\( ( 1 + 4 T + 29 T^{2} )^{4} \))(\( ( 1 + 4 T + 29 T^{2} )^{4} \))
$31$ (\( ( 1 + 3 T - 22 T^{2} + 93 T^{3} + 961 T^{4} )^{2} \))(\( ( 1 - T - 30 T^{2} - 31 T^{3} + 961 T^{4} )^{2} \))(\( ( 1 - 10 T + 69 T^{2} - 310 T^{3} + 961 T^{4} )^{2} \))(\( ( 1 + 8 T + 33 T^{2} + 248 T^{3} + 961 T^{4} )^{2} \))(\( ( 1 - T - 30 T^{2} - 31 T^{3} + 961 T^{4} )^{2} \))(\( ( 1 - 7 T + 31 T^{2} )^{2}( 1 - 4 T + 31 T^{2} )^{2} \))(\( ( 1 - 2 T - 27 T^{2} - 62 T^{3} + 961 T^{4} )^{2} \))(\( ( 1 + 5 T - 6 T^{2} + 155 T^{3} + 961 T^{4} )^{2} \))(\( ( 1 - 2 T - 27 T^{2} - 62 T^{3} + 961 T^{4} )^{2} \))(\( ( 1 + 6 T + 5 T^{2} + 186 T^{3} + 961 T^{4} )^{2} \))(\( ( 1 - 31 T^{2} + 961 T^{4} )^{2} \))(\( ( 1 - 5 T - 6 T^{2} - 155 T^{3} + 961 T^{4} )^{2} \))
$37$ (\( 1 + 58 T^{2} + 1995 T^{4} + 79402 T^{6} + 1874161 T^{8} \))(\( 1 + 10 T^{2} - 1269 T^{4} + 13690 T^{6} + 1874161 T^{8} \))(\( ( 1 + 26 T^{2} + 1369 T^{4} )( 1 + 47 T^{2} + 1369 T^{4} ) \))(\( ( 1 + 26 T^{2} + 1369 T^{4} )( 1 + 47 T^{2} + 1369 T^{4} ) \))(\( ( 1 - 73 T^{2} + 1369 T^{4} )( 1 + 47 T^{2} + 1369 T^{4} ) \))(\( 1 + 10 T^{2} - 1269 T^{4} + 13690 T^{6} + 1874161 T^{8} \))(\( ( 1 + 26 T^{2} + 1369 T^{4} )( 1 + 47 T^{2} + 1369 T^{4} ) \))(\( 1 + 10 T^{2} - 1269 T^{4} + 13690 T^{6} + 1874161 T^{8} \))(\( ( 1 - 73 T^{2} + 1369 T^{4} )( 1 + 26 T^{2} + 1369 T^{4} ) \))(\( 1 + 65 T^{2} + 2856 T^{4} + 88985 T^{6} + 1874161 T^{8} \))(\( 1 + 65 T^{2} + 2856 T^{4} + 88985 T^{6} + 1874161 T^{8} \))(\( ( 1 - 12 T + 107 T^{2} - 444 T^{3} + 1369 T^{4} )( 1 + 12 T + 107 T^{2} + 444 T^{3} + 1369 T^{4} ) \))
$41$ (\( ( 1 + 41 T^{2} )^{4} \))(\( ( 1 + 41 T^{2} )^{4} \))(\( ( 1 - 9 T + 41 T^{2} )^{4} \))(\( ( 1 + 41 T^{2} )^{4} \))(\( ( 1 + 9 T + 41 T^{2} )^{4} \))(\( ( 1 - 5 T + 41 T^{2} )^{4} \))(\( ( 1 - 5 T + 41 T^{2} )^{4} \))(\( ( 1 + 3 T + 41 T^{2} )^{4} \))(\( ( 1 + 11 T + 41 T^{2} )^{4} \))(\( ( 1 - 9 T + 41 T^{2} )^{4} \))(\( ( 1 - 12 T + 41 T^{2} )^{4} \))(\( ( 1 - 7 T + 41 T^{2} )^{4} \))
$43$ (\( ( 1 - 82 T^{2} + 1849 T^{4} )^{2} \))(\( ( 1 + 14 T^{2} + 1849 T^{4} )^{2} \))(\( ( 1 - 22 T^{2} + 1849 T^{4} )^{2} \))(\( ( 1 - 22 T^{2} + 1849 T^{4} )^{2} \))(\( ( 1 + 14 T^{2} + 1849 T^{4} )^{2} \))(\( ( 1 - 43 T^{2} )^{4} \))(\( ( 1 + 58 T^{2} + 1849 T^{4} )^{2} \))(\( ( 1 - 22 T^{2} + 1849 T^{4} )^{2} \))(\( ( 1 - 22 T^{2} + 1849 T^{4} )^{2} \))(\( ( 1 - 70 T^{2} + 1849 T^{4} )^{2} \))(\( ( 1 - 22 T^{2} + 1849 T^{4} )^{2} \))(\( ( 1 - 82 T^{2} + 1849 T^{4} )^{2} \))
$47$ (\( 1 + 58 T^{2} + 1155 T^{4} + 128122 T^{6} + 4879681 T^{8} \))(\( 1 + 58 T^{2} + 1155 T^{4} + 128122 T^{6} + 4879681 T^{8} \))(\( 1 + 85 T^{2} + 5016 T^{4} + 187765 T^{6} + 4879681 T^{8} \))(\( 1 + 58 T^{2} + 1155 T^{4} + 128122 T^{6} + 4879681 T^{8} \))(\( 1 + 85 T^{2} + 5016 T^{4} + 187765 T^{6} + 4879681 T^{8} \))(\( 1 + 93 T^{2} + 6440 T^{4} + 205437 T^{6} + 4879681 T^{8} \))(\( 1 - 27 T^{2} - 1480 T^{4} - 59643 T^{6} + 4879681 T^{8} \))(\( 1 + 13 T^{2} - 2040 T^{4} + 28717 T^{6} + 4879681 T^{8} \))(\( 1 + 69 T^{2} + 2552 T^{4} + 152421 T^{6} + 4879681 T^{8} \))(\( 1 + 85 T^{2} + 5016 T^{4} + 187765 T^{6} + 4879681 T^{8} \))(\( 1 + 58 T^{2} + 1155 T^{4} + 128122 T^{6} + 4879681 T^{8} \))(\( 1 + 93 T^{2} + 6440 T^{4} + 205437 T^{6} + 4879681 T^{8} \))
$53$ (\( 1 + 25 T^{2} - 2184 T^{4} + 70225 T^{6} + 7890481 T^{8} \))(\( 1 + 97 T^{2} + 6600 T^{4} + 272473 T^{6} + 7890481 T^{8} \))(\( 1 + 97 T^{2} + 6600 T^{4} + 272473 T^{6} + 7890481 T^{8} \))(\( 1 + 70 T^{2} + 2091 T^{4} + 196630 T^{6} + 7890481 T^{8} \))(\( 1 + 70 T^{2} + 2091 T^{4} + 196630 T^{6} + 7890481 T^{8} \))(\( 1 - 38 T^{2} - 1365 T^{4} - 106742 T^{6} + 7890481 T^{8} \))(\( 1 + 25 T^{2} - 2184 T^{4} + 70225 T^{6} + 7890481 T^{8} \))(\( 1 - 38 T^{2} - 1365 T^{4} - 106742 T^{6} + 7890481 T^{8} \))(\( 1 - 15 T^{2} - 2584 T^{4} - 42135 T^{6} + 7890481 T^{8} \))(\( 1 + 105 T^{2} + 8216 T^{4} + 294945 T^{6} + 7890481 T^{8} \))(\( 1 + 102 T^{2} + 7595 T^{4} + 286518 T^{6} + 7890481 T^{8} \))(\( 1 + 102 T^{2} + 7595 T^{4} + 286518 T^{6} + 7890481 T^{8} \))
$59$ (\( ( 1 + 11 T + 62 T^{2} + 649 T^{3} + 3481 T^{4} )^{2} \))(\( ( 1 - 3 T - 50 T^{2} - 177 T^{3} + 3481 T^{4} )^{2} \))(\( ( 1 - 12 T + 85 T^{2} - 708 T^{3} + 3481 T^{4} )^{2} \))(\( ( 1 + 6 T - 23 T^{2} + 354 T^{3} + 3481 T^{4} )^{2} \))(\( ( 1 + 6 T - 23 T^{2} + 354 T^{3} + 3481 T^{4} )^{2} \))(\( ( 1 + 2 T - 55 T^{2} + 118 T^{3} + 3481 T^{4} )^{2} \))(\( ( 1 - 4 T - 43 T^{2} - 236 T^{3} + 3481 T^{4} )^{2} \))(\( ( 1 - 6 T - 23 T^{2} - 354 T^{3} + 3481 T^{4} )^{2} \))(\( ( 1 - 4 T - 43 T^{2} - 236 T^{3} + 3481 T^{4} )^{2} \))(\( ( 1 - 12 T + 85 T^{2} - 708 T^{3} + 3481 T^{4} )^{2} \))(\( ( 1 - 6 T - 23 T^{2} - 354 T^{3} + 3481 T^{4} )^{2} \))(\( ( 1 + 14 T + 137 T^{2} + 826 T^{3} + 3481 T^{4} )^{2} \))
$61$ (\( ( 1 - 6 T - 25 T^{2} - 366 T^{3} + 3721 T^{4} )^{2} \))(\( ( 1 - 10 T + 39 T^{2} - 610 T^{3} + 3721 T^{4} )^{2} \))(\( ( 1 + 8 T + 3 T^{2} + 488 T^{3} + 3721 T^{4} )^{2} \))(\( ( 1 - 14 T + 61 T^{2} )^{2}( 1 + 13 T + 61 T^{2} )^{2} \))(\( ( 1 + 8 T + 3 T^{2} + 488 T^{3} + 3721 T^{4} )^{2} \))(\( ( 1 + 10 T + 39 T^{2} + 610 T^{3} + 3721 T^{4} )^{2} \))(\( ( 1 + 4 T - 45 T^{2} + 244 T^{3} + 3721 T^{4} )^{2} \))(\( ( 1 + 2 T - 57 T^{2} + 122 T^{3} + 3721 T^{4} )^{2} \))(\( ( 1 - 61 T^{2} + 3721 T^{4} )^{2} \))(\( ( 1 - 4 T - 45 T^{2} - 244 T^{3} + 3721 T^{4} )^{2} \))(\( ( 1 - 14 T + 61 T^{2} )^{2}( 1 + T + 61 T^{2} )^{2} \))(\( ( 1 + 12 T + 83 T^{2} + 732 T^{3} + 3721 T^{4} )^{2} \))
$67$ (\( 1 + 130 T^{2} + 12411 T^{4} + 583570 T^{6} + 20151121 T^{8} \))(\( 1 + 34 T^{2} - 3333 T^{4} + 152626 T^{6} + 20151121 T^{8} \))(\( 1 + 70 T^{2} + 411 T^{4} + 314230 T^{6} + 20151121 T^{8} \))(\( 1 - 35 T^{2} - 3264 T^{4} - 157115 T^{6} + 20151121 T^{8} \))(\( 1 + 118 T^{2} + 9435 T^{4} + 529702 T^{6} + 20151121 T^{8} \))(\( ( 1 + 67 T^{2} + 4489 T^{4} )^{2} \))(\( 1 - 10 T^{2} - 4389 T^{4} - 44890 T^{6} + 20151121 T^{8} \))(\( 1 + 70 T^{2} + 411 T^{4} + 314230 T^{6} + 20151121 T^{8} \))(\( ( 1 + 67 T^{2} + 4489 T^{4} )^{2} \))(\( 1 - 10 T^{2} - 4389 T^{4} - 44890 T^{6} + 20151121 T^{8} \))(\( 1 + 125 T^{2} + 11136 T^{4} + 561125 T^{6} + 20151121 T^{8} \))(\( 1 - 10 T^{2} - 4389 T^{4} - 44890 T^{6} + 20151121 T^{8} \))
$71$ (\( ( 1 - 2 T + 71 T^{2} )^{4} \))(\( ( 1 + 6 T + 71 T^{2} )^{4} \))(\( ( 1 + 6 T + 71 T^{2} )^{4} \))(\( ( 1 - 12 T + 71 T^{2} )^{4} \))(\( ( 1 - 3 T + 71 T^{2} )^{4} \))(\( ( 1 + T + 71 T^{2} )^{4} \))(\( ( 1 - 2 T + 71 T^{2} )^{4} \))(\( ( 1 + 9 T + 71 T^{2} )^{4} \))(\( ( 1 + 6 T + 71 T^{2} )^{4} \))(\( ( 1 + 14 T + 71 T^{2} )^{4} \))(\( ( 1 - 16 T + 71 T^{2} )^{4} \))(\( ( 1 + 9 T + 71 T^{2} )^{4} \))
$73$ (\( ( 1 - 97 T^{2} + 5329 T^{4} )( 1 + 143 T^{2} + 5329 T^{4} ) \))(\( 1 + 142 T^{2} + 14835 T^{4} + 756718 T^{6} + 28398241 T^{8} \))(\( 1 + 142 T^{2} + 14835 T^{4} + 756718 T^{6} + 28398241 T^{8} \))(\( 1 + 121 T^{2} + 9312 T^{4} + 644809 T^{6} + 28398241 T^{8} \))(\( 1 - 50 T^{2} - 2829 T^{4} - 266450 T^{6} + 28398241 T^{8} \))(\( 1 + 142 T^{2} + 14835 T^{4} + 756718 T^{6} + 28398241 T^{8} \))(\( ( 1 - 97 T^{2} + 5329 T^{4} )( 1 + 143 T^{2} + 5329 T^{4} ) \))(\( 1 - 50 T^{2} - 2829 T^{4} - 266450 T^{6} + 28398241 T^{8} \))(\( ( 1 - 16 T + 183 T^{2} - 1168 T^{3} + 5329 T^{4} )( 1 + 16 T + 183 T^{2} + 1168 T^{3} + 5329 T^{4} ) \))(\( 1 - 50 T^{2} - 2829 T^{4} - 266450 T^{6} + 28398241 T^{8} \))(\( 1 + 25 T^{2} - 4704 T^{4} + 133225 T^{6} + 28398241 T^{8} \))(\( ( 1 - 16 T + 183 T^{2} - 1168 T^{3} + 5329 T^{4} )( 1 + 16 T + 183 T^{2} + 1168 T^{3} + 5329 T^{4} ) \))
$79$ (\( ( 1 - 3 T - 70 T^{2} - 237 T^{3} + 6241 T^{4} )^{2} \))(\( ( 1 + T - 78 T^{2} + 79 T^{3} + 6241 T^{4} )^{2} \))(\( ( 1 - 8 T - 15 T^{2} - 632 T^{3} + 6241 T^{4} )^{2} \))(\( ( 1 + 7 T - 30 T^{2} + 553 T^{3} + 6241 T^{4} )^{2} \))(\( ( 1 - 11 T + 42 T^{2} - 869 T^{3} + 6241 T^{4} )^{2} \))(\( ( 1 - 9 T + 2 T^{2} - 711 T^{3} + 6241 T^{4} )^{2} \))(\( ( 1 + 12 T + 65 T^{2} + 948 T^{3} + 6241 T^{4} )^{2} \))(\( ( 1 + 7 T - 30 T^{2} + 553 T^{3} + 6241 T^{4} )^{2} \))(\( ( 1 + 8 T - 15 T^{2} + 632 T^{3} + 6241 T^{4} )^{2} \))(\( ( 1 - 17 T + 79 T^{2} )^{2}( 1 + 13 T + 79 T^{2} )^{2} \))(\( ( 1 - 17 T + 79 T^{2} )^{2}( 1 + 4 T + 79 T^{2} )^{2} \))(\( ( 1 + 4 T + 79 T^{2} )^{2}( 1 + 13 T + 79 T^{2} )^{2} \))
$83$ (\( ( 1 - 117 T^{2} + 6889 T^{4} )^{2} \))(\( ( 1 - 85 T^{2} + 6889 T^{4} )^{2} \))(\( ( 1 - 83 T^{2} )^{4} \))(\( ( 1 + 158 T^{2} + 6889 T^{4} )^{2} \))(\( ( 1 - 83 T^{2} )^{4} \))(\( ( 1 - 130 T^{2} + 6889 T^{4} )^{2} \))(\( ( 1 - 22 T^{2} + 6889 T^{4} )^{2} \))(\( ( 1 - 130 T^{2} + 6889 T^{4} )^{2} \))(\( ( 1 - 102 T^{2} + 6889 T^{4} )^{2} \))(\( ( 1 - 22 T^{2} + 6889 T^{4} )^{2} \))(\( ( 1 - 130 T^{2} + 6889 T^{4} )^{2} \))(\( ( 1 - 150 T^{2} + 6889 T^{4} )^{2} \))
$89$ (\( ( 1 + 6 T - 53 T^{2} + 534 T^{3} + 7921 T^{4} )^{2} \))(\( ( 1 - 6 T - 53 T^{2} - 534 T^{3} + 7921 T^{4} )^{2} \))(\( ( 1 - 6 T - 53 T^{2} - 534 T^{3} + 7921 T^{4} )^{2} \))(\( ( 1 + 6 T - 53 T^{2} + 534 T^{3} + 7921 T^{4} )^{2} \))(\( ( 1 + 15 T + 136 T^{2} + 1335 T^{3} + 7921 T^{4} )^{2} \))(\( ( 1 - 11 T + 32 T^{2} - 979 T^{3} + 7921 T^{4} )^{2} \))(\( ( 1 - 14 T + 107 T^{2} - 1246 T^{3} + 7921 T^{4} )^{2} \))(\( ( 1 - 3 T - 80 T^{2} - 267 T^{3} + 7921 T^{4} )^{2} \))(\( ( 1 + 10 T + 11 T^{2} + 890 T^{3} + 7921 T^{4} )^{2} \))(\( ( 1 + 2 T - 85 T^{2} + 178 T^{3} + 7921 T^{4} )^{2} \))(\( ( 1 + 2 T - 85 T^{2} + 178 T^{3} + 7921 T^{4} )^{2} \))(\( ( 1 + 7 T - 40 T^{2} + 623 T^{3} + 7921 T^{4} )^{2} \))
$97$ (\( ( 1 - 145 T^{2} + 9409 T^{4} )^{2} \))(\( ( 1 - 193 T^{2} + 9409 T^{4} )^{2} \))(\( ( 1 - 18 T + 97 T^{2} )^{2}( 1 + 18 T + 97 T^{2} )^{2} \))(\( ( 1 - 145 T^{2} + 9409 T^{4} )^{2} \))(\( ( 1 - 145 T^{2} + 9409 T^{4} )^{2} \))(\( ( 1 - 193 T^{2} + 9409 T^{4} )^{2} \))(\( ( 1 - 18 T + 97 T^{2} )^{2}( 1 + 18 T + 97 T^{2} )^{2} \))(\( ( 1 + 95 T^{2} + 9409 T^{4} )^{2} \))(\( ( 1 + 62 T^{2} + 9409 T^{4} )^{2} \))(\( ( 1 + 62 T^{2} + 9409 T^{4} )^{2} \))(\( ( 1 + 95 T^{2} + 9409 T^{4} )^{2} \))(\( ( 1 - 145 T^{2} + 9409 T^{4} )^{2} \))
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