Properties

Label 1764.2.k
Level 1764
Weight 2
Character orbit k
Rep. character \(\chi_{1764}(361,\cdot)\)
Character field \(\Q(\zeta_{3})\)
Dimension 34
Newform subspaces 13
Sturm bound 672
Trace bound 13

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Defining parameters

Level: \( N \) \(=\) \( 1764 = 2^{2} \cdot 3^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1764.k (of order \(3\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 7 \)
Character field: \(\Q(\zeta_{3})\)
Newform subspaces: \( 13 \)
Sturm bound: \(672\)
Trace bound: \(13\)
Distinguishing \(T_p\): \(5\), \(11\), \(13\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(1764, [\chi])\).

Total New Old
Modular forms 768 34 734
Cusp forms 576 34 542
Eisenstein series 192 0 192

Trace form

\( 34q - q^{5} + O(q^{10}) \) \( 34q - q^{5} + 3q^{11} - 8q^{13} - 11q^{17} - 3q^{19} - 17q^{23} - 28q^{25} - 4q^{29} + 7q^{31} - 9q^{37} + 24q^{41} + 52q^{43} + 3q^{47} - 9q^{53} + 26q^{55} - 13q^{59} + 7q^{61} - 40q^{65} - 11q^{67} - 20q^{71} + 5q^{73} - 17q^{79} + 28q^{83} + 90q^{85} - 15q^{89} + 7q^{95} - 28q^{97} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(1764, [\chi])\) into newform subspaces

Label Dim. \(A\) Field CM Traces $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\)
1764.2.k.a \(2\) \(14.086\) \(\Q(\sqrt{-3}) \) None \(0\) \(0\) \(-4\) \(0\) \(q-4\zeta_{6}q^{5}+(2-2\zeta_{6})q^{11}+6q^{13}+\cdots\)
1764.2.k.b \(2\) \(14.086\) \(\Q(\sqrt{-3}) \) None \(0\) \(0\) \(-3\) \(0\) \(q-3\zeta_{6}q^{5}+(-3+3\zeta_{6})q^{11}-2q^{13}+\cdots\)
1764.2.k.c \(2\) \(14.086\) \(\Q(\sqrt{-3}) \) None \(0\) \(0\) \(-2\) \(0\) \(q-2\zeta_{6}q^{5}+(2-2\zeta_{6})q^{11}+4q^{13}+\cdots\)
1764.2.k.d \(2\) \(14.086\) \(\Q(\sqrt{-3}) \) None \(0\) \(0\) \(0\) \(0\) \(q+(-6+6\zeta_{6})q^{11}-2q^{13}-4\zeta_{6}q^{19}+\cdots\)
1764.2.k.e \(2\) \(14.086\) \(\Q(\sqrt{-3}) \) None \(0\) \(0\) \(0\) \(0\) \(q+(-6+6\zeta_{6})q^{11}+2q^{13}+4\zeta_{6}q^{19}+\cdots\)
1764.2.k.f \(2\) \(14.086\) \(\Q(\sqrt{-3}) \) \(\Q(\sqrt{-3}) \) \(0\) \(0\) \(0\) \(0\) \(q-5q^{13}-\zeta_{6}q^{19}+(5-5\zeta_{6})q^{25}+\cdots\)
1764.2.k.g \(2\) \(14.086\) \(\Q(\sqrt{-3}) \) \(\Q(\sqrt{-3}) \) \(0\) \(0\) \(0\) \(0\) \(q-2q^{13}+8\zeta_{6}q^{19}+(5-5\zeta_{6})q^{25}+\cdots\)
1764.2.k.h \(2\) \(14.086\) \(\Q(\sqrt{-3}) \) \(\Q(\sqrt{-3}) \) \(0\) \(0\) \(0\) \(0\) \(q+2q^{13}-8\zeta_{6}q^{19}+(5-5\zeta_{6})q^{25}+\cdots\)
1764.2.k.i \(2\) \(14.086\) \(\Q(\sqrt{-3}) \) None \(0\) \(0\) \(2\) \(0\) \(q+2\zeta_{6}q^{5}+(2-2\zeta_{6})q^{11}-4q^{13}+\cdots\)
1764.2.k.j \(2\) \(14.086\) \(\Q(\sqrt{-3}) \) None \(0\) \(0\) \(2\) \(0\) \(q+2\zeta_{6}q^{5}+(2-2\zeta_{6})q^{11}+3q^{13}+\cdots\)
1764.2.k.k \(2\) \(14.086\) \(\Q(\sqrt{-3}) \) None \(0\) \(0\) \(4\) \(0\) \(q+4\zeta_{6}q^{5}+(2-2\zeta_{6})q^{11}-6q^{13}+\cdots\)
1764.2.k.l \(4\) \(14.086\) \(\Q(\sqrt{2}, \sqrt{-3})\) None \(0\) \(0\) \(0\) \(0\) \(q+\beta _{1}q^{5}-4\beta _{2}q^{11}-3\beta _{3}q^{13}+(-\beta _{1}+\cdots)q^{17}+\cdots\)
1764.2.k.m \(8\) \(14.086\) 8.0.\(\cdots\).4 None \(0\) \(0\) \(0\) \(0\) \(q-\beta _{7}q^{5}-\beta _{5}q^{11}-3\beta _{6}q^{13}+\beta _{3}q^{17}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(1764, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(1764, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(21, [\chi])\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(28, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(42, [\chi])\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(49, [\chi])\)\(^{\oplus 9}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(63, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(84, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(98, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(126, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(147, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(196, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(252, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(294, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(441, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(588, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(882, [\chi])\)\(^{\oplus 2}\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ 1
$5$ (\( 1 + 4 T + 11 T^{2} + 20 T^{3} + 25 T^{4} \))(\( 1 + 3 T + 4 T^{2} + 15 T^{3} + 25 T^{4} \))(\( 1 + 2 T - T^{2} + 10 T^{3} + 25 T^{4} \))(\( 1 - 5 T^{2} + 25 T^{4} \))(\( 1 - 5 T^{2} + 25 T^{4} \))(\( 1 - 5 T^{2} + 25 T^{4} \))(\( 1 - 5 T^{2} + 25 T^{4} \))(\( 1 - 5 T^{2} + 25 T^{4} \))(\( 1 - 2 T - T^{2} - 10 T^{3} + 25 T^{4} \))(\( 1 - 2 T - T^{2} - 10 T^{3} + 25 T^{4} \))(\( 1 - 4 T + 11 T^{2} - 20 T^{3} + 25 T^{4} \))(\( 1 - 8 T^{2} + 39 T^{4} - 200 T^{6} + 625 T^{8} \))(\( ( 1 + 4 T^{2} - 9 T^{4} + 100 T^{6} + 625 T^{8} )^{2} \))
$7$ 1
$11$ (\( 1 - 2 T - 7 T^{2} - 22 T^{3} + 121 T^{4} \))(\( 1 + 3 T - 2 T^{2} + 33 T^{3} + 121 T^{4} \))(\( 1 - 2 T - 7 T^{2} - 22 T^{3} + 121 T^{4} \))(\( 1 + 6 T + 25 T^{2} + 66 T^{3} + 121 T^{4} \))(\( 1 + 6 T + 25 T^{2} + 66 T^{3} + 121 T^{4} \))(\( 1 - 11 T^{2} + 121 T^{4} \))(\( 1 - 11 T^{2} + 121 T^{4} \))(\( 1 - 11 T^{2} + 121 T^{4} \))(\( 1 - 2 T - 7 T^{2} - 22 T^{3} + 121 T^{4} \))(\( 1 - 2 T - 7 T^{2} - 22 T^{3} + 121 T^{4} \))(\( 1 - 2 T - 7 T^{2} - 22 T^{3} + 121 T^{4} \))(\( ( 1 - 4 T + 5 T^{2} - 44 T^{3} + 121 T^{4} )^{2} \))(\( ( 1 + 6 T^{2} - 85 T^{4} + 726 T^{6} + 14641 T^{8} )^{2} \))
$13$ (\( ( 1 - 6 T + 13 T^{2} )^{2} \))(\( ( 1 + 2 T + 13 T^{2} )^{2} \))(\( ( 1 - 4 T + 13 T^{2} )^{2} \))(\( ( 1 + 2 T + 13 T^{2} )^{2} \))(\( ( 1 - 2 T + 13 T^{2} )^{2} \))(\( ( 1 + 5 T + 13 T^{2} )^{2} \))(\( ( 1 + 2 T + 13 T^{2} )^{2} \))(\( ( 1 - 2 T + 13 T^{2} )^{2} \))(\( ( 1 + 4 T + 13 T^{2} )^{2} \))(\( ( 1 - 3 T + 13 T^{2} )^{2} \))(\( ( 1 + 6 T + 13 T^{2} )^{2} \))(\( ( 1 + 8 T^{2} + 169 T^{4} )^{2} \))(\( ( 1 + 8 T^{2} + 169 T^{4} )^{4} \))
$17$ (\( 1 - 4 T - T^{2} - 68 T^{3} + 289 T^{4} \))(\( 1 + 3 T - 8 T^{2} + 51 T^{3} + 289 T^{4} \))(\( 1 + 6 T + 19 T^{2} + 102 T^{3} + 289 T^{4} \))(\( 1 - 17 T^{2} + 289 T^{4} \))(\( 1 - 17 T^{2} + 289 T^{4} \))(\( 1 - 17 T^{2} + 289 T^{4} \))(\( 1 - 17 T^{2} + 289 T^{4} \))(\( 1 - 17 T^{2} + 289 T^{4} \))(\( 1 - 6 T + 19 T^{2} - 102 T^{3} + 289 T^{4} \))(\( 1 + 8 T + 47 T^{2} + 136 T^{3} + 289 T^{4} \))(\( 1 + 4 T - T^{2} + 68 T^{3} + 289 T^{4} \))(\( 1 - 32 T^{2} + 735 T^{4} - 9248 T^{6} + 83521 T^{8} \))(\( ( 1 - 20 T^{2} + 111 T^{4} - 5780 T^{6} + 83521 T^{8} )^{2} \))
$19$ (\( 1 + 4 T - 3 T^{2} + 76 T^{3} + 361 T^{4} \))(\( ( 1 - 7 T + 19 T^{2} )( 1 + 8 T + 19 T^{2} ) \))(\( ( 1 - 7 T + 19 T^{2} )( 1 - T + 19 T^{2} ) \))(\( 1 + 4 T - 3 T^{2} + 76 T^{3} + 361 T^{4} \))(\( 1 - 4 T - 3 T^{2} - 76 T^{3} + 361 T^{4} \))(\( ( 1 - 7 T + 19 T^{2} )( 1 + 8 T + 19 T^{2} ) \))(\( ( 1 - 7 T + 19 T^{2} )( 1 - T + 19 T^{2} ) \))(\( ( 1 + T + 19 T^{2} )( 1 + 7 T + 19 T^{2} ) \))(\( ( 1 + T + 19 T^{2} )( 1 + 7 T + 19 T^{2} ) \))(\( ( 1 - 7 T + 19 T^{2} )( 1 + 8 T + 19 T^{2} ) \))(\( 1 - 4 T - 3 T^{2} - 76 T^{3} + 361 T^{4} \))(\( 1 - 30 T^{2} + 539 T^{4} - 10830 T^{6} + 130321 T^{8} \))(\( ( 1 - 30 T^{2} + 539 T^{4} - 10830 T^{6} + 130321 T^{8} )^{2} \))
$23$ (\( 1 - 2 T - 19 T^{2} - 46 T^{3} + 529 T^{4} \))(\( 1 - 3 T - 14 T^{2} - 69 T^{3} + 529 T^{4} \))(\( 1 + 6 T + 13 T^{2} + 138 T^{3} + 529 T^{4} \))(\( 1 + 6 T + 13 T^{2} + 138 T^{3} + 529 T^{4} \))(\( 1 + 6 T + 13 T^{2} + 138 T^{3} + 529 T^{4} \))(\( 1 - 23 T^{2} + 529 T^{4} \))(\( 1 - 23 T^{2} + 529 T^{4} \))(\( 1 - 23 T^{2} + 529 T^{4} \))(\( 1 + 6 T + 13 T^{2} + 138 T^{3} + 529 T^{4} \))(\( 1 - 8 T + 41 T^{2} - 184 T^{3} + 529 T^{4} \))(\( 1 - 2 T - 19 T^{2} - 46 T^{3} + 529 T^{4} \))(\( ( 1 + 4 T - 7 T^{2} + 92 T^{3} + 529 T^{4} )^{2} \))(\( ( 1 - 18 T^{2} - 205 T^{4} - 9522 T^{6} + 279841 T^{8} )^{2} \))
$29$ (\( ( 1 - 2 T + 29 T^{2} )^{2} \))(\( ( 1 - 6 T + 29 T^{2} )^{2} \))(\( ( 1 - 10 T + 29 T^{2} )^{2} \))(\( ( 1 + 6 T + 29 T^{2} )^{2} \))(\( ( 1 + 6 T + 29 T^{2} )^{2} \))(\( ( 1 + 29 T^{2} )^{2} \))(\( ( 1 + 29 T^{2} )^{2} \))(\( ( 1 + 29 T^{2} )^{2} \))(\( ( 1 - 10 T + 29 T^{2} )^{2} \))(\( ( 1 + 4 T + 29 T^{2} )^{2} \))(\( ( 1 - 2 T + 29 T^{2} )^{2} \))(\( ( 1 + 8 T + 29 T^{2} )^{4} \))(\( ( 1 + 30 T^{2} + 841 T^{4} )^{4} \))
$31$ (\( 1 - 31 T^{2} + 961 T^{4} \))(\( ( 1 - 4 T + 31 T^{2} )( 1 + 11 T + 31 T^{2} ) \))(\( ( 1 - 11 T + 31 T^{2} )( 1 + 7 T + 31 T^{2} ) \))(\( 1 - 8 T + 33 T^{2} - 248 T^{3} + 961 T^{4} \))(\( 1 + 8 T + 33 T^{2} + 248 T^{3} + 961 T^{4} \))(\( ( 1 - 7 T + 31 T^{2} )( 1 - 4 T + 31 T^{2} ) \))(\( ( 1 - 7 T + 31 T^{2} )( 1 + 11 T + 31 T^{2} ) \))(\( ( 1 - 11 T + 31 T^{2} )( 1 + 7 T + 31 T^{2} ) \))(\( ( 1 - 7 T + 31 T^{2} )( 1 + 11 T + 31 T^{2} ) \))(\( 1 - 3 T - 22 T^{2} - 93 T^{3} + 961 T^{4} \))(\( 1 - 31 T^{2} + 961 T^{4} \))(\( ( 1 - 31 T^{2} + 961 T^{4} )^{2} \))(\( ( 1 + 10 T^{2} - 861 T^{4} + 9610 T^{6} + 923521 T^{8} )^{2} \))
$37$ (\( 1 + 2 T - 33 T^{2} + 74 T^{3} + 1369 T^{4} \))(\( ( 1 - 11 T + 37 T^{2} )( 1 + 10 T + 37 T^{2} ) \))(\( 1 + 6 T - T^{2} + 222 T^{3} + 1369 T^{4} \))(\( 1 + 2 T - 33 T^{2} + 74 T^{3} + 1369 T^{4} \))(\( 1 + 2 T - 33 T^{2} + 74 T^{3} + 1369 T^{4} \))(\( ( 1 + T + 37 T^{2} )( 1 + 10 T + 37 T^{2} ) \))(\( ( 1 - 11 T + 37 T^{2} )( 1 + T + 37 T^{2} ) \))(\( ( 1 - 11 T + 37 T^{2} )( 1 + T + 37 T^{2} ) \))(\( 1 + 6 T - T^{2} + 222 T^{3} + 1369 T^{4} \))(\( ( 1 - 11 T + 37 T^{2} )( 1 + 10 T + 37 T^{2} ) \))(\( 1 + 2 T - 33 T^{2} + 74 T^{3} + 1369 T^{4} \))(\( ( 1 - 8 T + 27 T^{2} - 296 T^{3} + 1369 T^{4} )^{2} \))(\( ( 1 + 4 T - 21 T^{2} + 148 T^{3} + 1369 T^{4} )^{4} \))
$41$ (\( ( 1 + 41 T^{2} )^{2} \))(\( ( 1 - 6 T + 41 T^{2} )^{2} \))(\( ( 1 + 6 T + 41 T^{2} )^{2} \))(\( ( 1 - 12 T + 41 T^{2} )^{2} \))(\( ( 1 + 12 T + 41 T^{2} )^{2} \))(\( ( 1 + 41 T^{2} )^{2} \))(\( ( 1 + 41 T^{2} )^{2} \))(\( ( 1 + 41 T^{2} )^{2} \))(\( ( 1 - 6 T + 41 T^{2} )^{2} \))(\( ( 1 - 6 T + 41 T^{2} )^{2} \))(\( ( 1 + 41 T^{2} )^{2} \))(\( ( 1 + 32 T^{2} + 1681 T^{4} )^{2} \))(\( ( 1 + 68 T^{2} + 1681 T^{4} )^{4} \))
$43$ (\( ( 1 + 4 T + 43 T^{2} )^{2} \))(\( ( 1 + 4 T + 43 T^{2} )^{2} \))(\( ( 1 - 4 T + 43 T^{2} )^{2} \))(\( ( 1 + 4 T + 43 T^{2} )^{2} \))(\( ( 1 + 4 T + 43 T^{2} )^{2} \))(\( ( 1 + 13 T + 43 T^{2} )^{2} \))(\( ( 1 - 8 T + 43 T^{2} )^{2} \))(\( ( 1 - 8 T + 43 T^{2} )^{2} \))(\( ( 1 - 4 T + 43 T^{2} )^{2} \))(\( ( 1 - 11 T + 43 T^{2} )^{2} \))(\( ( 1 + 4 T + 43 T^{2} )^{2} \))(\( ( 1 + 4 T + 43 T^{2} )^{4} \))(\( ( 1 - 8 T + 43 T^{2} )^{8} \))
$47$ (\( 1 + 12 T + 97 T^{2} + 564 T^{3} + 2209 T^{4} \))(\( 1 - 9 T + 34 T^{2} - 423 T^{3} + 2209 T^{4} \))(\( 1 + 8 T + 17 T^{2} + 376 T^{3} + 2209 T^{4} \))(\( 1 + 12 T + 97 T^{2} + 564 T^{3} + 2209 T^{4} \))(\( 1 - 12 T + 97 T^{2} - 564 T^{3} + 2209 T^{4} \))(\( 1 - 47 T^{2} + 2209 T^{4} \))(\( 1 - 47 T^{2} + 2209 T^{4} \))(\( 1 - 47 T^{2} + 2209 T^{4} \))(\( 1 - 8 T + 17 T^{2} - 376 T^{3} + 2209 T^{4} \))(\( 1 + 6 T - 11 T^{2} + 282 T^{3} + 2209 T^{4} \))(\( 1 - 12 T + 97 T^{2} - 564 T^{3} + 2209 T^{4} \))(\( 1 - 62 T^{2} + 1635 T^{4} - 136958 T^{6} + 4879681 T^{8} \))(\( ( 1 - 38 T^{2} - 765 T^{4} - 83942 T^{6} + 4879681 T^{8} )^{2} \))
$53$ (\( 1 + 6 T - 17 T^{2} + 318 T^{3} + 2809 T^{4} \))(\( 1 - 3 T - 44 T^{2} - 159 T^{3} + 2809 T^{4} \))(\( 1 - 2 T - 49 T^{2} - 106 T^{3} + 2809 T^{4} \))(\( 1 + 6 T - 17 T^{2} + 318 T^{3} + 2809 T^{4} \))(\( 1 + 6 T - 17 T^{2} + 318 T^{3} + 2809 T^{4} \))(\( 1 - 53 T^{2} + 2809 T^{4} \))(\( 1 - 53 T^{2} + 2809 T^{4} \))(\( 1 - 53 T^{2} + 2809 T^{4} \))(\( 1 - 2 T - 49 T^{2} - 106 T^{3} + 2809 T^{4} \))(\( 1 + 12 T + 91 T^{2} + 636 T^{3} + 2809 T^{4} \))(\( 1 + 6 T - 17 T^{2} + 318 T^{3} + 2809 T^{4} \))(\( ( 1 - 10 T + 47 T^{2} - 530 T^{3} + 2809 T^{4} )^{2} \))(\( ( 1 + 6 T^{2} - 2773 T^{4} + 16854 T^{6} + 7890481 T^{8} )^{2} \))
$59$ (\( 1 - 8 T + 5 T^{2} - 472 T^{3} + 3481 T^{4} \))(\( 1 + 9 T + 22 T^{2} + 531 T^{3} + 3481 T^{4} \))(\( 1 - 4 T - 43 T^{2} - 236 T^{3} + 3481 T^{4} \))(\( 1 - 59 T^{2} + 3481 T^{4} \))(\( 1 - 59 T^{2} + 3481 T^{4} \))(\( 1 - 59 T^{2} + 3481 T^{4} \))(\( 1 - 59 T^{2} + 3481 T^{4} \))(\( 1 - 59 T^{2} + 3481 T^{4} \))(\( 1 + 4 T - 43 T^{2} + 236 T^{3} + 3481 T^{4} \))(\( 1 + 4 T - 43 T^{2} + 236 T^{3} + 3481 T^{4} \))(\( 1 + 8 T + 5 T^{2} + 472 T^{3} + 3481 T^{4} \))(\( 1 + 82 T^{2} + 3243 T^{4} + 285442 T^{6} + 12117361 T^{8} \))(\( ( 1 - 62 T^{2} + 363 T^{4} - 215822 T^{6} + 12117361 T^{8} )^{2} \))
$61$ (\( 1 - 6 T - 25 T^{2} - 366 T^{3} + 3721 T^{4} \))(\( ( 1 - 13 T + 61 T^{2} )( 1 + 14 T + 61 T^{2} ) \))(\( 1 + 8 T + 3 T^{2} + 488 T^{3} + 3721 T^{4} \))(\( 1 + 10 T + 39 T^{2} + 610 T^{3} + 3721 T^{4} \))(\( 1 - 10 T + 39 T^{2} - 610 T^{3} + 3721 T^{4} \))(\( ( 1 - 13 T + 61 T^{2} )( 1 - T + 61 T^{2} ) \))(\( ( 1 - 13 T + 61 T^{2} )( 1 - T + 61 T^{2} ) \))(\( ( 1 + T + 61 T^{2} )( 1 + 13 T + 61 T^{2} ) \))(\( 1 - 8 T + 3 T^{2} - 488 T^{3} + 3721 T^{4} \))(\( 1 + 6 T - 25 T^{2} + 366 T^{3} + 3721 T^{4} \))(\( 1 + 6 T - 25 T^{2} + 366 T^{3} + 3721 T^{4} \))(\( 1 - 72 T^{2} + 1463 T^{4} - 267912 T^{6} + 13845841 T^{8} \))(\( ( 1 - 24 T^{2} - 3145 T^{4} - 89304 T^{6} + 13845841 T^{8} )^{2} \))
$67$ (\( 1 - 8 T - 3 T^{2} - 536 T^{3} + 4489 T^{4} \))(\( 1 - 7 T - 18 T^{2} - 469 T^{3} + 4489 T^{4} \))(\( 1 - 8 T - 3 T^{2} - 536 T^{3} + 4489 T^{4} \))(\( 1 + 8 T - 3 T^{2} + 536 T^{3} + 4489 T^{4} \))(\( 1 + 8 T - 3 T^{2} + 536 T^{3} + 4489 T^{4} \))(\( ( 1 - 11 T + 67 T^{2} )( 1 + 16 T + 67 T^{2} ) \))(\( ( 1 - 11 T + 67 T^{2} )( 1 - 5 T + 67 T^{2} ) \))(\( ( 1 - 11 T + 67 T^{2} )( 1 - 5 T + 67 T^{2} ) \))(\( 1 - 8 T - 3 T^{2} - 536 T^{3} + 4489 T^{4} \))(\( 1 + 13 T + 102 T^{2} + 871 T^{3} + 4489 T^{4} \))(\( 1 - 8 T - 3 T^{2} - 536 T^{3} + 4489 T^{4} \))(\( ( 1 - 67 T^{2} + 4489 T^{4} )^{2} \))(\( ( 1 + 12 T + 77 T^{2} + 804 T^{3} + 4489 T^{4} )^{4} \))
$71$ (\( ( 1 + 14 T + 71 T^{2} )^{2} \))(\( ( 1 + 71 T^{2} )^{2} \))(\( ( 1 - 10 T + 71 T^{2} )^{2} \))(\( ( 1 + 6 T + 71 T^{2} )^{2} \))(\( ( 1 + 6 T + 71 T^{2} )^{2} \))(\( ( 1 + 71 T^{2} )^{2} \))(\( ( 1 + 71 T^{2} )^{2} \))(\( ( 1 + 71 T^{2} )^{2} \))(\( ( 1 - 10 T + 71 T^{2} )^{2} \))(\( ( 1 - 10 T + 71 T^{2} )^{2} \))(\( ( 1 + 14 T + 71 T^{2} )^{2} \))(\( ( 1 + 71 T^{2} )^{4} \))(\( ( 1 - 110 T^{2} + 5041 T^{4} )^{4} \))
$73$ (\( 1 + 2 T - 69 T^{2} + 146 T^{3} + 5329 T^{4} \))(\( 1 + T - 72 T^{2} + 73 T^{3} + 5329 T^{4} \))(\( 1 - 4 T - 57 T^{2} - 292 T^{3} + 5329 T^{4} \))(\( ( 1 - 7 T + 73 T^{2} )( 1 + 17 T + 73 T^{2} ) \))(\( ( 1 - 17 T + 73 T^{2} )( 1 + 7 T + 73 T^{2} ) \))(\( ( 1 - 10 T + 73 T^{2} )( 1 - 7 T + 73 T^{2} ) \))(\( ( 1 - 7 T + 73 T^{2} )( 1 + 17 T + 73 T^{2} ) \))(\( ( 1 - 17 T + 73 T^{2} )( 1 + 7 T + 73 T^{2} ) \))(\( 1 + 4 T - 57 T^{2} + 292 T^{3} + 5329 T^{4} \))(\( 1 + 11 T + 48 T^{2} + 803 T^{3} + 5329 T^{4} \))(\( 1 - 2 T - 69 T^{2} - 146 T^{3} + 5329 T^{4} \))(\( 1 - 96 T^{2} + 3887 T^{4} - 511584 T^{6} + 28398241 T^{8} \))(\( ( 1 - 144 T^{2} + 15407 T^{4} - 767376 T^{6} + 28398241 T^{8} )^{2} \))
$79$ (\( 1 + 12 T + 65 T^{2} + 948 T^{3} + 6241 T^{4} \))(\( ( 1 - 17 T + 79 T^{2} )( 1 + 4 T + 79 T^{2} ) \))(\( ( 1 - 13 T + 79 T^{2} )( 1 + 17 T + 79 T^{2} ) \))(\( ( 1 - 17 T + 79 T^{2} )( 1 + 13 T + 79 T^{2} ) \))(\( ( 1 - 17 T + 79 T^{2} )( 1 + 13 T + 79 T^{2} ) \))(\( ( 1 + 4 T + 79 T^{2} )( 1 + 13 T + 79 T^{2} ) \))(\( ( 1 - 17 T + 79 T^{2} )( 1 + 13 T + 79 T^{2} ) \))(\( ( 1 - 17 T + 79 T^{2} )( 1 + 13 T + 79 T^{2} ) \))(\( ( 1 - 13 T + 79 T^{2} )( 1 + 17 T + 79 T^{2} ) \))(\( 1 - 3 T - 70 T^{2} - 237 T^{3} + 6241 T^{4} \))(\( 1 + 12 T + 65 T^{2} + 948 T^{3} + 6241 T^{4} \))(\( ( 1 + 8 T - 15 T^{2} + 632 T^{3} + 6241 T^{4} )^{2} \))(\( ( 1 - 17 T + 79 T^{2} )^{4}( 1 + 13 T + 79 T^{2} )^{4} \))
$83$ (\( ( 1 + 4 T + 83 T^{2} )^{2} \))(\( ( 1 - 12 T + 83 T^{2} )^{2} \))(\( ( 1 - 12 T + 83 T^{2} )^{2} \))(\( ( 1 + 12 T + 83 T^{2} )^{2} \))(\( ( 1 - 12 T + 83 T^{2} )^{2} \))(\( ( 1 + 83 T^{2} )^{2} \))(\( ( 1 + 83 T^{2} )^{2} \))(\( ( 1 + 83 T^{2} )^{2} \))(\( ( 1 + 12 T + 83 T^{2} )^{2} \))(\( ( 1 - 2 T + 83 T^{2} )^{2} \))(\( ( 1 - 4 T + 83 T^{2} )^{2} \))(\( ( 1 - 34 T^{2} + 6889 T^{4} )^{2} \))(\( ( 1 - 58 T^{2} + 6889 T^{4} )^{4} \))
$89$ (\( 1 - 89 T^{2} + 7921 T^{4} \))(\( 1 + 15 T + 136 T^{2} + 1335 T^{3} + 7921 T^{4} \))(\( 1 - 14 T + 107 T^{2} - 1246 T^{3} + 7921 T^{4} \))(\( 1 + 12 T + 55 T^{2} + 1068 T^{3} + 7921 T^{4} \))(\( 1 - 12 T + 55 T^{2} - 1068 T^{3} + 7921 T^{4} \))(\( 1 - 89 T^{2} + 7921 T^{4} \))(\( 1 - 89 T^{2} + 7921 T^{4} \))(\( 1 - 89 T^{2} + 7921 T^{4} \))(\( 1 + 14 T + 107 T^{2} + 1246 T^{3} + 7921 T^{4} \))(\( 1 - 89 T^{2} + 7921 T^{4} \))(\( 1 - 89 T^{2} + 7921 T^{4} \))(\( 1 - 128 T^{2} + 8463 T^{4} - 1013888 T^{6} + 62742241 T^{8} \))(\( ( 1 - 164 T^{2} + 18975 T^{4} - 1299044 T^{6} + 62742241 T^{8} )^{2} \))
$97$ (\( ( 1 - 2 T + 97 T^{2} )^{2} \))(\( ( 1 - 10 T + 97 T^{2} )^{2} \))(\( ( 1 + 4 T + 97 T^{2} )^{2} \))(\( ( 1 - 10 T + 97 T^{2} )^{2} \))(\( ( 1 + 10 T + 97 T^{2} )^{2} \))(\( ( 1 + 14 T + 97 T^{2} )^{2} \))(\( ( 1 + 14 T + 97 T^{2} )^{2} \))(\( ( 1 - 14 T + 97 T^{2} )^{2} \))(\( ( 1 - 4 T + 97 T^{2} )^{2} \))(\( ( 1 + 10 T + 97 T^{2} )^{2} \))(\( ( 1 + 2 T + 97 T^{2} )^{2} \))(\( ( 1 + 192 T^{2} + 9409 T^{4} )^{2} \))(\( ( 1 + 96 T^{2} + 9409 T^{4} )^{4} \))
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