Properties

Label 1620.2.i
Level 1620
Weight 2
Character orbit i
Rep. character \(\chi_{1620}(541,\cdot)\)
Character field \(\Q(\zeta_{3})\)
Dimension 32
Newform subspaces 14
Sturm bound 648
Trace bound 17

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Defining parameters

Level: \( N \) \(=\) \( 1620 = 2^{2} \cdot 3^{4} \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1620.i (of order \(3\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 9 \)
Character field: \(\Q(\zeta_{3})\)
Newform subspaces: \( 14 \)
Sturm bound: \(648\)
Trace bound: \(17\)
Distinguishing \(T_p\): \(7\), \(11\), \(17\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(1620, [\chi])\).

Total New Old
Modular forms 720 32 688
Cusp forms 576 32 544
Eisenstein series 144 0 144

Trace form

\( 32q + 10q^{7} + O(q^{10}) \) \( 32q + 10q^{7} + 10q^{13} - 44q^{19} - 16q^{25} - 20q^{31} - 20q^{37} - 8q^{43} + 6q^{49} + 10q^{61} + 10q^{67} - 20q^{73} - 2q^{79} - 12q^{85} + 68q^{91} + 22q^{97} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(1620, [\chi])\) into newform subspaces

Label Dim. \(A\) Field CM Traces $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\)
1620.2.i.a \(2\) \(12.936\) \(\Q(\sqrt{-3}) \) None \(0\) \(0\) \(-1\) \(-2\) \(q-\zeta_{6}q^{5}+(-2+2\zeta_{6})q^{7}-2\zeta_{6}q^{13}+\cdots\)
1620.2.i.b \(2\) \(12.936\) \(\Q(\sqrt{-3}) \) None \(0\) \(0\) \(-1\) \(-2\) \(q-\zeta_{6}q^{5}+(-2+2\zeta_{6})q^{7}-2\zeta_{6}q^{13}+\cdots\)
1620.2.i.c \(2\) \(12.936\) \(\Q(\sqrt{-3}) \) None \(0\) \(0\) \(-1\) \(-2\) \(q-\zeta_{6}q^{5}+(-2+2\zeta_{6})q^{7}+(3-3\zeta_{6})q^{11}+\cdots\)
1620.2.i.d \(2\) \(12.936\) \(\Q(\sqrt{-3}) \) None \(0\) \(0\) \(-1\) \(1\) \(q-\zeta_{6}q^{5}+(1-\zeta_{6})q^{7}+(-6+6\zeta_{6})q^{11}+\cdots\)
1620.2.i.e \(2\) \(12.936\) \(\Q(\sqrt{-3}) \) None \(0\) \(0\) \(-1\) \(4\) \(q-\zeta_{6}q^{5}+(4-4\zeta_{6})q^{7}+(-3+3\zeta_{6})q^{11}+\cdots\)
1620.2.i.f \(2\) \(12.936\) \(\Q(\sqrt{-3}) \) None \(0\) \(0\) \(-1\) \(4\) \(q-\zeta_{6}q^{5}+(4-4\zeta_{6})q^{7}+(6-6\zeta_{6})q^{11}+\cdots\)
1620.2.i.g \(2\) \(12.936\) \(\Q(\sqrt{-3}) \) None \(0\) \(0\) \(1\) \(-2\) \(q+\zeta_{6}q^{5}+(-2+2\zeta_{6})q^{7}+(-3+3\zeta_{6})q^{11}+\cdots\)
1620.2.i.h \(2\) \(12.936\) \(\Q(\sqrt{-3}) \) None \(0\) \(0\) \(1\) \(-2\) \(q+\zeta_{6}q^{5}+(-2+2\zeta_{6})q^{7}-2\zeta_{6}q^{13}+\cdots\)
1620.2.i.i \(2\) \(12.936\) \(\Q(\sqrt{-3}) \) None \(0\) \(0\) \(1\) \(-2\) \(q+\zeta_{6}q^{5}+(-2+2\zeta_{6})q^{7}-2\zeta_{6}q^{13}+\cdots\)
1620.2.i.j \(2\) \(12.936\) \(\Q(\sqrt{-3}) \) None \(0\) \(0\) \(1\) \(1\) \(q+\zeta_{6}q^{5}+(1-\zeta_{6})q^{7}+(6-6\zeta_{6})q^{11}+\cdots\)
1620.2.i.k \(2\) \(12.936\) \(\Q(\sqrt{-3}) \) None \(0\) \(0\) \(1\) \(4\) \(q+\zeta_{6}q^{5}+(4-4\zeta_{6})q^{7}+(-6+6\zeta_{6})q^{11}+\cdots\)
1620.2.i.l \(2\) \(12.936\) \(\Q(\sqrt{-3}) \) None \(0\) \(0\) \(1\) \(4\) \(q+\zeta_{6}q^{5}+(4-4\zeta_{6})q^{7}+(3-3\zeta_{6})q^{11}+\cdots\)
1620.2.i.m \(4\) \(12.936\) \(\Q(\zeta_{12})\) None \(0\) \(0\) \(-2\) \(2\) \(q+(-1+\zeta_{12})q^{5}+(\zeta_{12}-\zeta_{12}^{2})q^{7}+\cdots\)
1620.2.i.n \(4\) \(12.936\) \(\Q(\zeta_{12})\) None \(0\) \(0\) \(2\) \(2\) \(q+(1-\zeta_{12})q^{5}+(\zeta_{12}-\zeta_{12}^{2})q^{7}+\zeta_{12}^{2}q^{11}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(1620, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(1620, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(18, [\chi])\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(36, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(45, [\chi])\)\(^{\oplus 9}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(54, [\chi])\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(81, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(90, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(108, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(135, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(162, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(180, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(270, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(324, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(405, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(540, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(810, [\chi])\)\(^{\oplus 2}\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ 1
$5$ (\( 1 + T + T^{2} \))(\( 1 + T + T^{2} \))(\( 1 + T + T^{2} \))(\( 1 + T + T^{2} \))(\( 1 + T + T^{2} \))(\( 1 + T + T^{2} \))(\( 1 - T + T^{2} \))(\( 1 - T + T^{2} \))(\( 1 - T + T^{2} \))(\( 1 - T + T^{2} \))(\( 1 - T + T^{2} \))(\( 1 - T + T^{2} \))(\( ( 1 + T + T^{2} )^{2} \))(\( ( 1 - T + T^{2} )^{2} \))
$7$ (\( 1 + 2 T - 3 T^{2} + 14 T^{3} + 49 T^{4} \))(\( 1 + 2 T - 3 T^{2} + 14 T^{3} + 49 T^{4} \))(\( 1 + 2 T - 3 T^{2} + 14 T^{3} + 49 T^{4} \))(\( ( 1 - 5 T + 7 T^{2} )( 1 + 4 T + 7 T^{2} ) \))(\( ( 1 - 5 T + 7 T^{2} )( 1 + T + 7 T^{2} ) \))(\( ( 1 - 5 T + 7 T^{2} )( 1 + T + 7 T^{2} ) \))(\( 1 + 2 T - 3 T^{2} + 14 T^{3} + 49 T^{4} \))(\( 1 + 2 T - 3 T^{2} + 14 T^{3} + 49 T^{4} \))(\( 1 + 2 T - 3 T^{2} + 14 T^{3} + 49 T^{4} \))(\( ( 1 - 5 T + 7 T^{2} )( 1 + 4 T + 7 T^{2} ) \))(\( ( 1 - 5 T + 7 T^{2} )( 1 + T + 7 T^{2} ) \))(\( ( 1 - 5 T + 7 T^{2} )( 1 + T + 7 T^{2} ) \))(\( 1 - 2 T - 8 T^{2} + 4 T^{3} + 67 T^{4} + 28 T^{5} - 392 T^{6} - 686 T^{7} + 2401 T^{8} \))(\( 1 - 2 T - 8 T^{2} + 4 T^{3} + 67 T^{4} + 28 T^{5} - 392 T^{6} - 686 T^{7} + 2401 T^{8} \))
$11$ (\( 1 - 11 T^{2} + 121 T^{4} \))(\( 1 - 11 T^{2} + 121 T^{4} \))(\( 1 - 3 T - 2 T^{2} - 33 T^{3} + 121 T^{4} \))(\( 1 + 6 T + 25 T^{2} + 66 T^{3} + 121 T^{4} \))(\( 1 + 3 T - 2 T^{2} + 33 T^{3} + 121 T^{4} \))(\( 1 - 6 T + 25 T^{2} - 66 T^{3} + 121 T^{4} \))(\( 1 + 3 T - 2 T^{2} + 33 T^{3} + 121 T^{4} \))(\( 1 - 11 T^{2} + 121 T^{4} \))(\( 1 - 11 T^{2} + 121 T^{4} \))(\( 1 - 6 T + 25 T^{2} - 66 T^{3} + 121 T^{4} \))(\( 1 + 6 T + 25 T^{2} + 66 T^{3} + 121 T^{4} \))(\( 1 - 3 T - 2 T^{2} - 33 T^{3} + 121 T^{4} \))(\( 1 - 19 T^{2} + 240 T^{4} - 2299 T^{6} + 14641 T^{8} \))(\( 1 - 19 T^{2} + 240 T^{4} - 2299 T^{6} + 14641 T^{8} \))
$13$ (\( ( 1 - 5 T + 13 T^{2} )( 1 + 7 T + 13 T^{2} ) \))(\( ( 1 - 5 T + 13 T^{2} )( 1 + 7 T + 13 T^{2} ) \))(\( 1 - 4 T + 3 T^{2} - 52 T^{3} + 169 T^{4} \))(\( 1 - T - 12 T^{2} - 13 T^{3} + 169 T^{4} \))(\( 1 - 4 T + 3 T^{2} - 52 T^{3} + 169 T^{4} \))(\( 1 - 4 T + 3 T^{2} - 52 T^{3} + 169 T^{4} \))(\( 1 - 4 T + 3 T^{2} - 52 T^{3} + 169 T^{4} \))(\( ( 1 - 5 T + 13 T^{2} )( 1 + 7 T + 13 T^{2} ) \))(\( ( 1 - 5 T + 13 T^{2} )( 1 + 7 T + 13 T^{2} ) \))(\( 1 - T - 12 T^{2} - 13 T^{3} + 169 T^{4} \))(\( 1 - 4 T + 3 T^{2} - 52 T^{3} + 169 T^{4} \))(\( 1 - 4 T + 3 T^{2} - 52 T^{3} + 169 T^{4} \))(\( 1 + 4 T - 2 T^{2} - 32 T^{3} - 53 T^{4} - 416 T^{5} - 338 T^{6} + 8788 T^{7} + 28561 T^{8} \))(\( 1 + 4 T - 2 T^{2} - 32 T^{3} - 53 T^{4} - 416 T^{5} - 338 T^{6} + 8788 T^{7} + 28561 T^{8} \))
$17$ (\( ( 1 + 3 T + 17 T^{2} )^{2} \))(\( ( 1 - 6 T + 17 T^{2} )^{2} \))(\( ( 1 + 6 T + 17 T^{2} )^{2} \))(\( ( 1 + 17 T^{2} )^{2} \))(\( ( 1 + 17 T^{2} )^{2} \))(\( ( 1 - 3 T + 17 T^{2} )^{2} \))(\( ( 1 - 6 T + 17 T^{2} )^{2} \))(\( ( 1 + 6 T + 17 T^{2} )^{2} \))(\( ( 1 - 3 T + 17 T^{2} )^{2} \))(\( ( 1 + 17 T^{2} )^{2} \))(\( ( 1 + 3 T + 17 T^{2} )^{2} \))(\( ( 1 + 17 T^{2} )^{2} \))(\( ( 1 - 6 T + 40 T^{2} - 102 T^{3} + 289 T^{4} )^{2} \))(\( ( 1 + 6 T + 40 T^{2} + 102 T^{3} + 289 T^{4} )^{2} \))
$19$ (\( ( 1 - 5 T + 19 T^{2} )^{2} \))(\( ( 1 + 4 T + 19 T^{2} )^{2} \))(\( ( 1 + 7 T + 19 T^{2} )^{2} \))(\( ( 1 + T + 19 T^{2} )^{2} \))(\( ( 1 - 5 T + 19 T^{2} )^{2} \))(\( ( 1 + 7 T + 19 T^{2} )^{2} \))(\( ( 1 + 7 T + 19 T^{2} )^{2} \))(\( ( 1 + 4 T + 19 T^{2} )^{2} \))(\( ( 1 - 5 T + 19 T^{2} )^{2} \))(\( ( 1 + T + 19 T^{2} )^{2} \))(\( ( 1 + 7 T + 19 T^{2} )^{2} \))(\( ( 1 - 5 T + 19 T^{2} )^{2} \))(\( ( 1 + 2 T + 27 T^{2} + 38 T^{3} + 361 T^{4} )^{2} \))(\( ( 1 + 2 T + 27 T^{2} + 38 T^{3} + 361 T^{4} )^{2} \))
$23$ (\( 1 + 3 T - 14 T^{2} + 69 T^{3} + 529 T^{4} \))(\( 1 - 6 T + 13 T^{2} - 138 T^{3} + 529 T^{4} \))(\( 1 - 6 T + 13 T^{2} - 138 T^{3} + 529 T^{4} \))(\( 1 + 6 T + 13 T^{2} + 138 T^{3} + 529 T^{4} \))(\( 1 - 6 T + 13 T^{2} - 138 T^{3} + 529 T^{4} \))(\( 1 + 9 T + 58 T^{2} + 207 T^{3} + 529 T^{4} \))(\( 1 + 6 T + 13 T^{2} + 138 T^{3} + 529 T^{4} \))(\( 1 + 6 T + 13 T^{2} + 138 T^{3} + 529 T^{4} \))(\( 1 - 3 T - 14 T^{2} - 69 T^{3} + 529 T^{4} \))(\( 1 - 6 T + 13 T^{2} - 138 T^{3} + 529 T^{4} \))(\( 1 - 9 T + 58 T^{2} - 207 T^{3} + 529 T^{4} \))(\( 1 + 6 T + 13 T^{2} + 138 T^{3} + 529 T^{4} \))(\( 1 - 34 T^{2} + 627 T^{4} - 17986 T^{6} + 279841 T^{8} \))(\( 1 - 34 T^{2} + 627 T^{4} - 17986 T^{6} + 279841 T^{8} \))
$29$ (\( 1 - 6 T + 7 T^{2} - 174 T^{3} + 841 T^{4} \))(\( 1 - 6 T + 7 T^{2} - 174 T^{3} + 841 T^{4} \))(\( 1 - 3 T - 20 T^{2} - 87 T^{3} + 841 T^{4} \))(\( 1 + 6 T + 7 T^{2} + 174 T^{3} + 841 T^{4} \))(\( 1 - 9 T + 52 T^{2} - 261 T^{3} + 841 T^{4} \))(\( 1 - 29 T^{2} + 841 T^{4} \))(\( 1 + 3 T - 20 T^{2} + 87 T^{3} + 841 T^{4} \))(\( 1 + 6 T + 7 T^{2} + 174 T^{3} + 841 T^{4} \))(\( 1 + 6 T + 7 T^{2} + 174 T^{3} + 841 T^{4} \))(\( 1 - 6 T + 7 T^{2} - 174 T^{3} + 841 T^{4} \))(\( 1 - 29 T^{2} + 841 T^{4} \))(\( 1 + 9 T + 52 T^{2} + 261 T^{3} + 841 T^{4} \))(\( 1 + 12 T + 53 T^{2} + 396 T^{3} + 3264 T^{4} + 11484 T^{5} + 44573 T^{6} + 292668 T^{7} + 707281 T^{8} \))(\( 1 - 12 T + 53 T^{2} - 396 T^{3} + 3264 T^{4} - 11484 T^{5} + 44573 T^{6} - 292668 T^{7} + 707281 T^{8} \))
$31$ (\( 1 + 5 T - 6 T^{2} + 155 T^{3} + 961 T^{4} \))(\( ( 1 - 11 T + 31 T^{2} )( 1 + 7 T + 31 T^{2} ) \))(\( 1 + 5 T - 6 T^{2} + 155 T^{3} + 961 T^{4} \))(\( 1 + 8 T + 33 T^{2} + 248 T^{3} + 961 T^{4} \))(\( 1 + 5 T - 6 T^{2} + 155 T^{3} + 961 T^{4} \))(\( ( 1 - 11 T + 31 T^{2} )( 1 + 4 T + 31 T^{2} ) \))(\( 1 + 5 T - 6 T^{2} + 155 T^{3} + 961 T^{4} \))(\( ( 1 - 11 T + 31 T^{2} )( 1 + 7 T + 31 T^{2} ) \))(\( 1 + 5 T - 6 T^{2} + 155 T^{3} + 961 T^{4} \))(\( 1 + 8 T + 33 T^{2} + 248 T^{3} + 961 T^{4} \))(\( ( 1 - 11 T + 31 T^{2} )( 1 + 4 T + 31 T^{2} ) \))(\( 1 + 5 T - 6 T^{2} + 155 T^{3} + 961 T^{4} \))(\( 1 - 2 T - 11 T^{2} + 94 T^{3} - 860 T^{4} + 2914 T^{5} - 10571 T^{6} - 59582 T^{7} + 923521 T^{8} \))(\( 1 - 2 T - 11 T^{2} + 94 T^{3} - 860 T^{4} + 2914 T^{5} - 10571 T^{6} - 59582 T^{7} + 923521 T^{8} \))
$37$ (\( ( 1 - 2 T + 37 T^{2} )^{2} \))(\( ( 1 - 2 T + 37 T^{2} )^{2} \))(\( ( 1 + 4 T + 37 T^{2} )^{2} \))(\( ( 1 + 7 T + 37 T^{2} )^{2} \))(\( ( 1 - 2 T + 37 T^{2} )^{2} \))(\( ( 1 - 2 T + 37 T^{2} )^{2} \))(\( ( 1 + 4 T + 37 T^{2} )^{2} \))(\( ( 1 - 2 T + 37 T^{2} )^{2} \))(\( ( 1 - 2 T + 37 T^{2} )^{2} \))(\( ( 1 + 7 T + 37 T^{2} )^{2} \))(\( ( 1 - 2 T + 37 T^{2} )^{2} \))(\( ( 1 - 2 T + 37 T^{2} )^{2} \))(\( ( 1 + 2 T + 48 T^{2} + 74 T^{3} + 1369 T^{4} )^{2} \))(\( ( 1 + 2 T + 48 T^{2} + 74 T^{3} + 1369 T^{4} )^{2} \))
$41$ (\( 1 + 12 T + 103 T^{2} + 492 T^{3} + 1681 T^{4} \))(\( 1 - 6 T - 5 T^{2} - 246 T^{3} + 1681 T^{4} \))(\( 1 - 3 T - 32 T^{2} - 123 T^{3} + 1681 T^{4} \))(\( 1 - 6 T - 5 T^{2} - 246 T^{3} + 1681 T^{4} \))(\( 1 - 9 T + 40 T^{2} - 369 T^{3} + 1681 T^{4} \))(\( 1 - 6 T - 5 T^{2} - 246 T^{3} + 1681 T^{4} \))(\( 1 + 3 T - 32 T^{2} + 123 T^{3} + 1681 T^{4} \))(\( 1 + 6 T - 5 T^{2} + 246 T^{3} + 1681 T^{4} \))(\( 1 - 12 T + 103 T^{2} - 492 T^{3} + 1681 T^{4} \))(\( 1 + 6 T - 5 T^{2} + 246 T^{3} + 1681 T^{4} \))(\( 1 + 6 T - 5 T^{2} + 246 T^{3} + 1681 T^{4} \))(\( 1 + 9 T + 40 T^{2} + 369 T^{3} + 1681 T^{4} \))(\( 1 + 12 T + 53 T^{2} + 108 T^{3} + 696 T^{4} + 4428 T^{5} + 89093 T^{6} + 827052 T^{7} + 2825761 T^{8} \))(\( 1 - 12 T + 53 T^{2} - 108 T^{3} + 696 T^{4} - 4428 T^{5} + 89093 T^{6} - 827052 T^{7} + 2825761 T^{8} \))
$43$ (\( ( 1 - 5 T + 43 T^{2} )( 1 + 13 T + 43 T^{2} ) \))(\( 1 - 10 T + 57 T^{2} - 430 T^{3} + 1849 T^{4} \))(\( ( 1 - 5 T + 43 T^{2} )( 1 + 13 T + 43 T^{2} ) \))(\( 1 - 4 T - 27 T^{2} - 172 T^{3} + 1849 T^{4} \))(\( 1 - 10 T + 57 T^{2} - 430 T^{3} + 1849 T^{4} \))(\( 1 + 2 T - 39 T^{2} + 86 T^{3} + 1849 T^{4} \))(\( ( 1 - 5 T + 43 T^{2} )( 1 + 13 T + 43 T^{2} ) \))(\( 1 - 10 T + 57 T^{2} - 430 T^{3} + 1849 T^{4} \))(\( ( 1 - 5 T + 43 T^{2} )( 1 + 13 T + 43 T^{2} ) \))(\( 1 - 4 T - 27 T^{2} - 172 T^{3} + 1849 T^{4} \))(\( 1 + 2 T - 39 T^{2} + 86 T^{3} + 1849 T^{4} \))(\( 1 - 10 T + 57 T^{2} - 430 T^{3} + 1849 T^{4} \))(\( 1 + 10 T - 8 T^{2} + 220 T^{3} + 5515 T^{4} + 9460 T^{5} - 14792 T^{6} + 795070 T^{7} + 3418801 T^{8} \))(\( 1 + 10 T - 8 T^{2} + 220 T^{3} + 5515 T^{4} + 9460 T^{5} - 14792 T^{6} + 795070 T^{7} + 3418801 T^{8} \))
$47$ (\( 1 - 12 T + 97 T^{2} - 564 T^{3} + 2209 T^{4} \))(\( 1 + 6 T - 11 T^{2} + 282 T^{3} + 2209 T^{4} \))(\( 1 - 47 T^{2} + 2209 T^{4} \))(\( 1 + 12 T + 97 T^{2} + 564 T^{3} + 2209 T^{4} \))(\( 1 - 6 T - 11 T^{2} - 282 T^{3} + 2209 T^{4} \))(\( 1 - 47 T^{2} + 2209 T^{4} \))(\( 1 - 47 T^{2} + 2209 T^{4} \))(\( 1 - 6 T - 11 T^{2} - 282 T^{3} + 2209 T^{4} \))(\( 1 + 12 T + 97 T^{2} + 564 T^{3} + 2209 T^{4} \))(\( 1 - 12 T + 97 T^{2} - 564 T^{3} + 2209 T^{4} \))(\( 1 - 47 T^{2} + 2209 T^{4} \))(\( 1 + 6 T - 11 T^{2} + 282 T^{3} + 2209 T^{4} \))(\( 1 + 6 T - 64 T^{2} + 36 T^{3} + 6099 T^{4} + 1692 T^{5} - 141376 T^{6} + 622938 T^{7} + 4879681 T^{8} \))(\( 1 - 6 T - 64 T^{2} - 36 T^{3} + 6099 T^{4} - 1692 T^{5} - 141376 T^{6} - 622938 T^{7} + 4879681 T^{8} \))
$53$ (\( ( 1 + 3 T + 53 T^{2} )^{2} \))(\( ( 1 - 6 T + 53 T^{2} )^{2} \))(\( ( 1 - 6 T + 53 T^{2} )^{2} \))(\( ( 1 + 6 T + 53 T^{2} )^{2} \))(\( ( 1 + 12 T + 53 T^{2} )^{2} \))(\( ( 1 + 9 T + 53 T^{2} )^{2} \))(\( ( 1 + 6 T + 53 T^{2} )^{2} \))(\( ( 1 + 6 T + 53 T^{2} )^{2} \))(\( ( 1 - 3 T + 53 T^{2} )^{2} \))(\( ( 1 - 6 T + 53 T^{2} )^{2} \))(\( ( 1 - 9 T + 53 T^{2} )^{2} \))(\( ( 1 - 12 T + 53 T^{2} )^{2} \))(\( ( 1 - 18 T + 184 T^{2} - 954 T^{3} + 2809 T^{4} )^{2} \))(\( ( 1 + 18 T + 184 T^{2} + 954 T^{3} + 2809 T^{4} )^{2} \))
$59$ (\( 1 + 6 T - 23 T^{2} + 354 T^{3} + 3481 T^{4} \))(\( 1 - 12 T + 85 T^{2} - 708 T^{3} + 3481 T^{4} \))(\( 1 + 3 T - 50 T^{2} + 177 T^{3} + 3481 T^{4} \))(\( 1 - 59 T^{2} + 3481 T^{4} \))(\( 1 + 9 T + 22 T^{2} + 531 T^{3} + 3481 T^{4} \))(\( 1 + 12 T + 85 T^{2} + 708 T^{3} + 3481 T^{4} \))(\( 1 - 3 T - 50 T^{2} - 177 T^{3} + 3481 T^{4} \))(\( 1 + 12 T + 85 T^{2} + 708 T^{3} + 3481 T^{4} \))(\( 1 - 6 T - 23 T^{2} - 354 T^{3} + 3481 T^{4} \))(\( 1 - 59 T^{2} + 3481 T^{4} \))(\( 1 - 12 T + 85 T^{2} - 708 T^{3} + 3481 T^{4} \))(\( 1 - 9 T + 22 T^{2} - 531 T^{3} + 3481 T^{4} \))(\( 1 + 12 T - 7 T^{2} + 396 T^{3} + 10824 T^{4} + 23364 T^{5} - 24367 T^{6} + 2464548 T^{7} + 12117361 T^{8} \))(\( 1 - 12 T - 7 T^{2} - 396 T^{3} + 10824 T^{4} - 23364 T^{5} - 24367 T^{6} - 2464548 T^{7} + 12117361 T^{8} \))
$61$ (\( 1 - 7 T - 12 T^{2} - 427 T^{3} + 3721 T^{4} \))(\( 1 + 2 T - 57 T^{2} + 122 T^{3} + 3721 T^{4} \))(\( ( 1 + T + 61 T^{2} )( 1 + 13 T + 61 T^{2} ) \))(\( 1 + 11 T + 60 T^{2} + 671 T^{3} + 3721 T^{4} \))(\( 1 - 10 T + 39 T^{2} - 610 T^{3} + 3721 T^{4} \))(\( 1 - 7 T - 12 T^{2} - 427 T^{3} + 3721 T^{4} \))(\( ( 1 + T + 61 T^{2} )( 1 + 13 T + 61 T^{2} ) \))(\( 1 + 2 T - 57 T^{2} + 122 T^{3} + 3721 T^{4} \))(\( 1 - 7 T - 12 T^{2} - 427 T^{3} + 3721 T^{4} \))(\( 1 + 11 T + 60 T^{2} + 671 T^{3} + 3721 T^{4} \))(\( 1 - 7 T - 12 T^{2} - 427 T^{3} + 3721 T^{4} \))(\( 1 - 10 T + 39 T^{2} - 610 T^{3} + 3721 T^{4} \))(\( ( 1 - 4 T - 45 T^{2} - 244 T^{3} + 3721 T^{4} )^{2} \))(\( ( 1 - 4 T - 45 T^{2} - 244 T^{3} + 3721 T^{4} )^{2} \))
$67$ (\( 1 + 2 T - 63 T^{2} + 134 T^{3} + 4489 T^{4} \))(\( 1 + 2 T - 63 T^{2} + 134 T^{3} + 4489 T^{4} \))(\( 1 + 2 T - 63 T^{2} + 134 T^{3} + 4489 T^{4} \))(\( 1 - 7 T - 18 T^{2} - 469 T^{3} + 4489 T^{4} \))(\( 1 + 2 T - 63 T^{2} + 134 T^{3} + 4489 T^{4} \))(\( 1 + 2 T - 63 T^{2} + 134 T^{3} + 4489 T^{4} \))(\( 1 + 2 T - 63 T^{2} + 134 T^{3} + 4489 T^{4} \))(\( 1 + 2 T - 63 T^{2} + 134 T^{3} + 4489 T^{4} \))(\( 1 + 2 T - 63 T^{2} + 134 T^{3} + 4489 T^{4} \))(\( 1 - 7 T - 18 T^{2} - 469 T^{3} + 4489 T^{4} \))(\( 1 + 2 T - 63 T^{2} + 134 T^{3} + 4489 T^{4} \))(\( 1 + 2 T - 63 T^{2} + 134 T^{3} + 4489 T^{4} \))(\( 1 - 8 T + 22 T^{2} + 736 T^{3} - 7013 T^{4} + 49312 T^{5} + 98758 T^{6} - 2406104 T^{7} + 20151121 T^{8} \))(\( 1 - 8 T + 22 T^{2} + 736 T^{3} - 7013 T^{4} + 49312 T^{5} + 98758 T^{6} - 2406104 T^{7} + 20151121 T^{8} \))
$71$ (\( ( 1 - 12 T + 71 T^{2} )^{2} \))(\( ( 1 - 12 T + 71 T^{2} )^{2} \))(\( ( 1 + 15 T + 71 T^{2} )^{2} \))(\( ( 1 + 6 T + 71 T^{2} )^{2} \))(\( ( 1 - 3 T + 71 T^{2} )^{2} \))(\( ( 1 + 6 T + 71 T^{2} )^{2} \))(\( ( 1 - 15 T + 71 T^{2} )^{2} \))(\( ( 1 + 12 T + 71 T^{2} )^{2} \))(\( ( 1 + 12 T + 71 T^{2} )^{2} \))(\( ( 1 - 6 T + 71 T^{2} )^{2} \))(\( ( 1 - 6 T + 71 T^{2} )^{2} \))(\( ( 1 + 3 T + 71 T^{2} )^{2} \))(\( ( 1 - 12 T + 151 T^{2} - 852 T^{3} + 5041 T^{4} )^{2} \))(\( ( 1 + 12 T + 151 T^{2} + 852 T^{3} + 5041 T^{4} )^{2} \))
$73$ (\( ( 1 + 16 T + 73 T^{2} )^{2} \))(\( ( 1 - 2 T + 73 T^{2} )^{2} \))(\( ( 1 + 10 T + 73 T^{2} )^{2} \))(\( ( 1 - 11 T + 73 T^{2} )^{2} \))(\( ( 1 + 4 T + 73 T^{2} )^{2} \))(\( ( 1 - 2 T + 73 T^{2} )^{2} \))(\( ( 1 + 10 T + 73 T^{2} )^{2} \))(\( ( 1 - 2 T + 73 T^{2} )^{2} \))(\( ( 1 + 16 T + 73 T^{2} )^{2} \))(\( ( 1 - 11 T + 73 T^{2} )^{2} \))(\( ( 1 - 2 T + 73 T^{2} )^{2} \))(\( ( 1 + 4 T + 73 T^{2} )^{2} \))(\( ( 1 - 10 T + 144 T^{2} - 730 T^{3} + 5329 T^{4} )^{2} \))(\( ( 1 - 10 T + 144 T^{2} - 730 T^{3} + 5329 T^{4} )^{2} \))
$79$ (\( 1 - T - 78 T^{2} - 79 T^{3} + 6241 T^{4} \))(\( 1 + 8 T - 15 T^{2} + 632 T^{3} + 6241 T^{4} \))(\( 1 + 8 T - 15 T^{2} + 632 T^{3} + 6241 T^{4} \))(\( 1 - T - 78 T^{2} - 79 T^{3} + 6241 T^{4} \))(\( ( 1 - 17 T + 79 T^{2} )( 1 + 13 T + 79 T^{2} ) \))(\( 1 - T - 78 T^{2} - 79 T^{3} + 6241 T^{4} \))(\( 1 + 8 T - 15 T^{2} + 632 T^{3} + 6241 T^{4} \))(\( 1 + 8 T - 15 T^{2} + 632 T^{3} + 6241 T^{4} \))(\( 1 - T - 78 T^{2} - 79 T^{3} + 6241 T^{4} \))(\( 1 - T - 78 T^{2} - 79 T^{3} + 6241 T^{4} \))(\( 1 - T - 78 T^{2} - 79 T^{3} + 6241 T^{4} \))(\( ( 1 - 17 T + 79 T^{2} )( 1 + 13 T + 79 T^{2} ) \))(\( 1 - 8 T - 2 T^{2} + 736 T^{3} - 6941 T^{4} + 58144 T^{5} - 12482 T^{6} - 3944312 T^{7} + 38950081 T^{8} \))(\( 1 - 8 T - 2 T^{2} + 736 T^{3} - 6941 T^{4} + 58144 T^{5} - 12482 T^{6} - 3944312 T^{7} + 38950081 T^{8} \))
$83$ (\( 1 - 15 T + 142 T^{2} - 1245 T^{3} + 6889 T^{4} \))(\( 1 - 6 T - 47 T^{2} - 498 T^{3} + 6889 T^{4} \))(\( 1 - 83 T^{2} + 6889 T^{4} \))(\( 1 + 6 T - 47 T^{2} + 498 T^{3} + 6889 T^{4} \))(\( 1 + 6 T - 47 T^{2} + 498 T^{3} + 6889 T^{4} \))(\( 1 - 9 T - 2 T^{2} - 747 T^{3} + 6889 T^{4} \))(\( 1 - 83 T^{2} + 6889 T^{4} \))(\( 1 + 6 T - 47 T^{2} + 498 T^{3} + 6889 T^{4} \))(\( 1 + 15 T + 142 T^{2} + 1245 T^{3} + 6889 T^{4} \))(\( 1 - 6 T - 47 T^{2} - 498 T^{3} + 6889 T^{4} \))(\( 1 + 9 T - 2 T^{2} + 747 T^{3} + 6889 T^{4} \))(\( 1 - 6 T - 47 T^{2} - 498 T^{3} + 6889 T^{4} \))(\( 1 + 6 T + 8 T^{2} - 828 T^{3} - 9093 T^{4} - 68724 T^{5} + 55112 T^{6} + 3430722 T^{7} + 47458321 T^{8} \))(\( 1 - 6 T + 8 T^{2} + 828 T^{3} - 9093 T^{4} + 68724 T^{5} + 55112 T^{6} - 3430722 T^{7} + 47458321 T^{8} \))
$89$ (\( ( 1 + 12 T + 89 T^{2} )^{2} \))(\( ( 1 - 6 T + 89 T^{2} )^{2} \))(\( ( 1 + 15 T + 89 T^{2} )^{2} \))(\( ( 1 + 12 T + 89 T^{2} )^{2} \))(\( ( 1 + 9 T + 89 T^{2} )^{2} \))(\( ( 1 - 6 T + 89 T^{2} )^{2} \))(\( ( 1 - 15 T + 89 T^{2} )^{2} \))(\( ( 1 + 6 T + 89 T^{2} )^{2} \))(\( ( 1 - 12 T + 89 T^{2} )^{2} \))(\( ( 1 - 12 T + 89 T^{2} )^{2} \))(\( ( 1 + 6 T + 89 T^{2} )^{2} \))(\( ( 1 - 9 T + 89 T^{2} )^{2} \))(\( ( 1 + 151 T^{2} + 7921 T^{4} )^{2} \))(\( ( 1 + 151 T^{2} + 7921 T^{4} )^{2} \))
$97$ (\( 1 - 16 T + 159 T^{2} - 1552 T^{3} + 9409 T^{4} \))(\( 1 + 2 T - 93 T^{2} + 194 T^{3} + 9409 T^{4} \))(\( 1 + 8 T - 33 T^{2} + 776 T^{3} + 9409 T^{4} \))(\( 1 - 13 T + 72 T^{2} - 1261 T^{3} + 9409 T^{4} \))(\( 1 + 2 T - 93 T^{2} + 194 T^{3} + 9409 T^{4} \))(\( 1 + 8 T - 33 T^{2} + 776 T^{3} + 9409 T^{4} \))(\( 1 + 8 T - 33 T^{2} + 776 T^{3} + 9409 T^{4} \))(\( 1 + 2 T - 93 T^{2} + 194 T^{3} + 9409 T^{4} \))(\( 1 - 16 T + 159 T^{2} - 1552 T^{3} + 9409 T^{4} \))(\( 1 - 13 T + 72 T^{2} - 1261 T^{3} + 9409 T^{4} \))(\( 1 + 8 T - 33 T^{2} + 776 T^{3} + 9409 T^{4} \))(\( 1 + 2 T - 93 T^{2} + 194 T^{3} + 9409 T^{4} \))(\( 1 - 2 T - 188 T^{2} + 4 T^{3} + 27067 T^{4} + 388 T^{5} - 1768892 T^{6} - 1825346 T^{7} + 88529281 T^{8} \))(\( 1 - 2 T - 188 T^{2} + 4 T^{3} + 27067 T^{4} + 388 T^{5} - 1768892 T^{6} - 1825346 T^{7} + 88529281 T^{8} \))
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