Properties

Label 1008.2.cs
Level 1008
Weight 2
Character orbit cs
Rep. character \(\chi_{1008}(271,\cdot)\)
Character field \(\Q(\zeta_{6})\)
Dimension 40
Newform subspaces 17
Sturm bound 384
Trace bound 7

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Defining parameters

Level: \( N \) = \( 1008 = 2^{4} \cdot 3^{2} \cdot 7 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 1008.cs (of order \(6\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) = \( 28 \)
Character field: \(\Q(\zeta_{6})\)
Newform subspaces: \( 17 \)
Sturm bound: \(384\)
Trace bound: \(7\)
Distinguishing \(T_p\): \(5\), \(11\), \(13\), \(17\), \(19\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(1008, [\chi])\).

Total New Old
Modular forms 432 40 392
Cusp forms 336 40 296
Eisenstein series 96 0 96

Trace form

\( 40q + O(q^{10}) \) \( 40q + 32q^{25} + 24q^{29} + 4q^{37} + 16q^{49} + 12q^{53} + 36q^{61} + 36q^{65} + 12q^{73} + 48q^{77} + 24q^{85} + 36q^{89} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(1008, [\chi])\) into newform subspaces

Label Dim. \(A\) Field CM Traces $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\)
1008.2.cs.a \(2\) \(8.049\) \(\Q(\sqrt{-3}) \) None \(0\) \(0\) \(-6\) \(-1\) \(q+(-4+2\zeta_{6})q^{5}+(-2+3\zeta_{6})q^{7}+\cdots\)
1008.2.cs.b \(2\) \(8.049\) \(\Q(\sqrt{-3}) \) None \(0\) \(0\) \(-6\) \(1\) \(q+(-4+2\zeta_{6})q^{5}+(2-3\zeta_{6})q^{7}+(2+\cdots)q^{11}+\cdots\)
1008.2.cs.c \(2\) \(8.049\) \(\Q(\sqrt{-3}) \) None \(0\) \(0\) \(-3\) \(-4\) \(q+(-2+\zeta_{6})q^{5}+(-1-2\zeta_{6})q^{7}+(-1+\cdots)q^{11}+\cdots\)
1008.2.cs.d \(2\) \(8.049\) \(\Q(\sqrt{-3}) \) None \(0\) \(0\) \(-3\) \(-1\) \(q+(-2+\zeta_{6})q^{5}+(1-3\zeta_{6})q^{7}+(1+\zeta_{6})q^{11}+\cdots\)
1008.2.cs.e \(2\) \(8.049\) \(\Q(\sqrt{-3}) \) None \(0\) \(0\) \(-3\) \(1\) \(q+(-2+\zeta_{6})q^{5}+(-1+3\zeta_{6})q^{7}+(-1+\cdots)q^{11}+\cdots\)
1008.2.cs.f \(2\) \(8.049\) \(\Q(\sqrt{-3}) \) None \(0\) \(0\) \(-3\) \(4\) \(q+(-2+\zeta_{6})q^{5}+(1+2\zeta_{6})q^{7}+(1+\zeta_{6})q^{11}+\cdots\)
1008.2.cs.g \(2\) \(8.049\) \(\Q(\sqrt{-3}) \) \(\Q(\sqrt{-3}) \) \(0\) \(0\) \(0\) \(-5\) \(q+(-2-\zeta_{6})q^{7}+(-3+6\zeta_{6})q^{13}+\cdots\)
1008.2.cs.h \(2\) \(8.049\) \(\Q(\sqrt{-3}) \) \(\Q(\sqrt{-3}) \) \(0\) \(0\) \(0\) \(-1\) \(q+(-2+3\zeta_{6})q^{7}+(1-2\zeta_{6})q^{13}+7\zeta_{6}q^{19}+\cdots\)
1008.2.cs.i \(2\) \(8.049\) \(\Q(\sqrt{-3}) \) \(\Q(\sqrt{-3}) \) \(0\) \(0\) \(0\) \(1\) \(q+(2-3\zeta_{6})q^{7}+(1-2\zeta_{6})q^{13}-7\zeta_{6}q^{19}+\cdots\)
1008.2.cs.j \(2\) \(8.049\) \(\Q(\sqrt{-3}) \) \(\Q(\sqrt{-3}) \) \(0\) \(0\) \(0\) \(5\) \(q+(2+\zeta_{6})q^{7}+(-3+6\zeta_{6})q^{13}+\zeta_{6}q^{19}+\cdots\)
1008.2.cs.k \(2\) \(8.049\) \(\Q(\sqrt{-3}) \) None \(0\) \(0\) \(3\) \(-5\) \(q+(2-\zeta_{6})q^{5}+(-3+\zeta_{6})q^{7}+(3+3\zeta_{6})q^{11}+\cdots\)
1008.2.cs.l \(2\) \(8.049\) \(\Q(\sqrt{-3}) \) None \(0\) \(0\) \(3\) \(5\) \(q+(2-\zeta_{6})q^{5}+(3-\zeta_{6})q^{7}+(-3-3\zeta_{6})q^{11}+\cdots\)
1008.2.cs.m \(2\) \(8.049\) \(\Q(\sqrt{-3}) \) None \(0\) \(0\) \(6\) \(-1\) \(q+(4-2\zeta_{6})q^{5}+(-2+3\zeta_{6})q^{7}+(-2+\cdots)q^{11}+\cdots\)
1008.2.cs.n \(2\) \(8.049\) \(\Q(\sqrt{-3}) \) None \(0\) \(0\) \(6\) \(1\) \(q+(4-2\zeta_{6})q^{5}+(2-3\zeta_{6})q^{7}+(2+2\zeta_{6})q^{11}+\cdots\)
1008.2.cs.o \(4\) \(8.049\) \(\Q(\sqrt{-3}, \sqrt{5})\) None \(0\) \(0\) \(0\) \(-10\) \(q-\beta _{2}q^{5}+(-3-\beta _{1})q^{7}+(\beta _{2}-\beta _{3})q^{11}+\cdots\)
1008.2.cs.p \(4\) \(8.049\) \(\Q(\sqrt{-3}, \sqrt{5})\) None \(0\) \(0\) \(0\) \(10\) \(q-\beta _{2}q^{5}+(3+\beta _{1})q^{7}+(-\beta _{2}+\beta _{3})q^{11}+\cdots\)
1008.2.cs.q \(4\) \(8.049\) \(\Q(\sqrt{-3}, \sqrt{7})\) None \(0\) \(0\) \(6\) \(0\) \(q+(1-\beta _{2})q^{5}+\beta _{3}q^{7}+(-\beta _{1}+\beta _{3})q^{11}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(1008, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(1008, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(28, [\chi])\)\(^{\oplus 9}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(84, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(112, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(252, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(336, [\chi])\)\(^{\oplus 2}\)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ 1
$3$ 1
$5$ (\( 1 + 6 T + 17 T^{2} + 30 T^{3} + 25 T^{4} \))(\( 1 + 6 T + 17 T^{2} + 30 T^{3} + 25 T^{4} \))(\( 1 + 3 T + 8 T^{2} + 15 T^{3} + 25 T^{4} \))(\( 1 + 3 T + 8 T^{2} + 15 T^{3} + 25 T^{4} \))(\( 1 + 3 T + 8 T^{2} + 15 T^{3} + 25 T^{4} \))(\( 1 + 3 T + 8 T^{2} + 15 T^{3} + 25 T^{4} \))(\( 1 + 5 T^{2} + 25 T^{4} \))(\( 1 + 5 T^{2} + 25 T^{4} \))(\( 1 + 5 T^{2} + 25 T^{4} \))(\( 1 + 5 T^{2} + 25 T^{4} \))(\( 1 - 3 T + 8 T^{2} - 15 T^{3} + 25 T^{4} \))(\( 1 - 3 T + 8 T^{2} - 15 T^{3} + 25 T^{4} \))(\( 1 - 6 T + 17 T^{2} - 30 T^{3} + 25 T^{4} \))(\( 1 - 6 T + 17 T^{2} - 30 T^{3} + 25 T^{4} \))(\( ( 1 - 5 T^{2} )^{2}( 1 + 5 T^{2} + 25 T^{4} ) \))(\( ( 1 - 5 T^{2} )^{2}( 1 + 5 T^{2} + 25 T^{4} ) \))(\( ( 1 - 3 T + 8 T^{2} - 15 T^{3} + 25 T^{4} )^{2} \))
$7$ (\( 1 + T + 7 T^{2} \))(\( 1 - T + 7 T^{2} \))(\( 1 + 4 T + 7 T^{2} \))(\( 1 + T + 7 T^{2} \))(\( 1 - T + 7 T^{2} \))(\( 1 - 4 T + 7 T^{2} \))(\( 1 + 5 T + 7 T^{2} \))(\( 1 + T + 7 T^{2} \))(\( 1 - T + 7 T^{2} \))(\( 1 - 5 T + 7 T^{2} \))(\( 1 + 5 T + 7 T^{2} \))(\( 1 - 5 T + 7 T^{2} \))(\( 1 + T + 7 T^{2} \))(\( 1 - T + 7 T^{2} \))(\( ( 1 + 5 T + 7 T^{2} )^{2} \))(\( ( 1 - 5 T + 7 T^{2} )^{2} \))(\( ( 1 - 7 T^{2} )^{2} \))
$11$ (\( 1 + 6 T + 23 T^{2} + 66 T^{3} + 121 T^{4} \))(\( 1 - 6 T + 23 T^{2} - 66 T^{3} + 121 T^{4} \))(\( 1 + 3 T + 14 T^{2} + 33 T^{3} + 121 T^{4} \))(\( 1 - 3 T + 14 T^{2} - 33 T^{3} + 121 T^{4} \))(\( 1 + 3 T + 14 T^{2} + 33 T^{3} + 121 T^{4} \))(\( 1 - 3 T + 14 T^{2} - 33 T^{3} + 121 T^{4} \))(\( 1 + 11 T^{2} + 121 T^{4} \))(\( 1 + 11 T^{2} + 121 T^{4} \))(\( 1 + 11 T^{2} + 121 T^{4} \))(\( 1 + 11 T^{2} + 121 T^{4} \))(\( 1 - 9 T + 38 T^{2} - 99 T^{3} + 121 T^{4} \))(\( 1 + 9 T + 38 T^{2} + 99 T^{3} + 121 T^{4} \))(\( 1 + 6 T + 23 T^{2} + 66 T^{3} + 121 T^{4} \))(\( 1 - 6 T + 23 T^{2} - 66 T^{3} + 121 T^{4} \))(\( 1 + 7 T^{2} - 72 T^{4} + 847 T^{6} + 14641 T^{8} \))(\( 1 + 7 T^{2} - 72 T^{4} + 847 T^{6} + 14641 T^{8} \))(\( 1 + T^{2} - 120 T^{4} + 121 T^{6} + 14641 T^{8} \))
$13$ (\( ( 1 - 7 T + 13 T^{2} )( 1 + 7 T + 13 T^{2} ) \))(\( ( 1 - 7 T + 13 T^{2} )( 1 + 7 T + 13 T^{2} ) \))(\( ( 1 - 13 T^{2} )^{2} \))(\( ( 1 - 13 T^{2} )^{2} \))(\( ( 1 - 13 T^{2} )^{2} \))(\( ( 1 - 13 T^{2} )^{2} \))(\( ( 1 - 5 T + 13 T^{2} )( 1 + 5 T + 13 T^{2} ) \))(\( ( 1 - 7 T + 13 T^{2} )( 1 + 7 T + 13 T^{2} ) \))(\( ( 1 - 7 T + 13 T^{2} )( 1 + 7 T + 13 T^{2} ) \))(\( ( 1 - 5 T + 13 T^{2} )( 1 + 5 T + 13 T^{2} ) \))(\( ( 1 - 2 T + 13 T^{2} )( 1 + 2 T + 13 T^{2} ) \))(\( ( 1 - 2 T + 13 T^{2} )( 1 + 2 T + 13 T^{2} ) \))(\( ( 1 - 5 T + 13 T^{2} )( 1 + 5 T + 13 T^{2} ) \))(\( ( 1 - 5 T + 13 T^{2} )( 1 + 5 T + 13 T^{2} ) \))(\( ( 1 - 14 T^{2} + 169 T^{4} )^{2} \))(\( ( 1 - 14 T^{2} + 169 T^{4} )^{2} \))(\( ( 1 - 14 T^{2} + 169 T^{4} )^{2} \))
$17$ (\( 1 + 17 T^{2} + 289 T^{4} \))(\( 1 + 17 T^{2} + 289 T^{4} \))(\( 1 - 9 T + 44 T^{2} - 153 T^{3} + 289 T^{4} \))(\( 1 + 6 T + 29 T^{2} + 102 T^{3} + 289 T^{4} \))(\( 1 + 6 T + 29 T^{2} + 102 T^{3} + 289 T^{4} \))(\( 1 - 9 T + 44 T^{2} - 153 T^{3} + 289 T^{4} \))(\( 1 + 17 T^{2} + 289 T^{4} \))(\( 1 + 17 T^{2} + 289 T^{4} \))(\( 1 + 17 T^{2} + 289 T^{4} \))(\( 1 + 17 T^{2} + 289 T^{4} \))(\( 1 + 6 T + 29 T^{2} + 102 T^{3} + 289 T^{4} \))(\( 1 + 6 T + 29 T^{2} + 102 T^{3} + 289 T^{4} \))(\( 1 - 12 T + 65 T^{2} - 204 T^{3} + 289 T^{4} \))(\( 1 - 12 T + 65 T^{2} - 204 T^{3} + 289 T^{4} \))(\( ( 1 + 17 T^{2} + 289 T^{4} )^{2} \))(\( ( 1 + 17 T^{2} + 289 T^{4} )^{2} \))(\( ( 1 + 9 T + 44 T^{2} + 153 T^{3} + 289 T^{4} )^{2} \))
$19$ (\( 1 + 5 T + 6 T^{2} + 95 T^{3} + 361 T^{4} \))(\( 1 - 5 T + 6 T^{2} - 95 T^{3} + 361 T^{4} \))(\( ( 1 - 8 T + 19 T^{2} )( 1 + T + 19 T^{2} ) \))(\( 1 + 2 T - 15 T^{2} + 38 T^{3} + 361 T^{4} \))(\( 1 - 2 T - 15 T^{2} - 38 T^{3} + 361 T^{4} \))(\( ( 1 - T + 19 T^{2} )( 1 + 8 T + 19 T^{2} ) \))(\( ( 1 - 7 T + 19 T^{2} )( 1 + 8 T + 19 T^{2} ) \))(\( ( 1 - 8 T + 19 T^{2} )( 1 + T + 19 T^{2} ) \))(\( ( 1 - T + 19 T^{2} )( 1 + 8 T + 19 T^{2} ) \))(\( ( 1 - 8 T + 19 T^{2} )( 1 + 7 T + 19 T^{2} ) \))(\( 1 - 2 T - 15 T^{2} - 38 T^{3} + 361 T^{4} \))(\( 1 + 2 T - 15 T^{2} + 38 T^{3} + 361 T^{4} \))(\( ( 1 - 8 T + 19 T^{2} )( 1 + T + 19 T^{2} ) \))(\( ( 1 - T + 19 T^{2} )( 1 + 8 T + 19 T^{2} ) \))(\( ( 1 + 4 T - 3 T^{2} + 76 T^{3} + 361 T^{4} )^{2} \))(\( ( 1 - 4 T - 3 T^{2} - 76 T^{3} + 361 T^{4} )^{2} \))(\( 1 - 31 T^{2} + 600 T^{4} - 11191 T^{6} + 130321 T^{8} \))
$23$ (\( 1 - 12 T + 71 T^{2} - 276 T^{3} + 529 T^{4} \))(\( 1 + 12 T + 71 T^{2} + 276 T^{3} + 529 T^{4} \))(\( 1 - 15 T + 98 T^{2} - 345 T^{3} + 529 T^{4} \))(\( 1 + 23 T^{2} + 529 T^{4} \))(\( 1 + 23 T^{2} + 529 T^{4} \))(\( 1 + 15 T + 98 T^{2} + 345 T^{3} + 529 T^{4} \))(\( 1 + 23 T^{2} + 529 T^{4} \))(\( 1 + 23 T^{2} + 529 T^{4} \))(\( 1 + 23 T^{2} + 529 T^{4} \))(\( 1 + 23 T^{2} + 529 T^{4} \))(\( 1 - 12 T + 71 T^{2} - 276 T^{3} + 529 T^{4} \))(\( 1 + 12 T + 71 T^{2} + 276 T^{3} + 529 T^{4} \))(\( 1 + 23 T^{2} + 529 T^{4} \))(\( 1 + 23 T^{2} + 529 T^{4} \))(\( 1 - 14 T^{2} - 333 T^{4} - 7406 T^{6} + 279841 T^{8} \))(\( 1 - 14 T^{2} - 333 T^{4} - 7406 T^{6} + 279841 T^{8} \))(\( 1 + 25 T^{2} + 96 T^{4} + 13225 T^{6} + 279841 T^{8} \))
$29$ (\( ( 1 + 29 T^{2} )^{2} \))(\( ( 1 + 29 T^{2} )^{2} \))(\( ( 1 - 6 T + 29 T^{2} )^{2} \))(\( ( 1 + 9 T + 29 T^{2} )^{2} \))(\( ( 1 + 9 T + 29 T^{2} )^{2} \))(\( ( 1 - 6 T + 29 T^{2} )^{2} \))(\( ( 1 + 29 T^{2} )^{2} \))(\( ( 1 + 29 T^{2} )^{2} \))(\( ( 1 + 29 T^{2} )^{2} \))(\( ( 1 + 29 T^{2} )^{2} \))(\( ( 1 - 9 T + 29 T^{2} )^{2} \))(\( ( 1 - 9 T + 29 T^{2} )^{2} \))(\( ( 1 + 29 T^{2} )^{2} \))(\( ( 1 + 29 T^{2} )^{2} \))(\( ( 1 + 13 T^{2} + 841 T^{4} )^{2} \))(\( ( 1 + 13 T^{2} + 841 T^{4} )^{2} \))(\( ( 1 + 29 T^{2} )^{4} \))
$31$ (\( 1 - 5 T - 6 T^{2} - 155 T^{3} + 961 T^{4} \))(\( 1 + 5 T - 6 T^{2} + 155 T^{3} + 961 T^{4} \))(\( 1 - 5 T - 6 T^{2} - 155 T^{3} + 961 T^{4} \))(\( 1 - 5 T - 6 T^{2} - 155 T^{3} + 961 T^{4} \))(\( 1 + 5 T - 6 T^{2} + 155 T^{3} + 961 T^{4} \))(\( 1 + 5 T - 6 T^{2} + 155 T^{3} + 961 T^{4} \))(\( ( 1 + 4 T + 31 T^{2} )( 1 + 7 T + 31 T^{2} ) \))(\( ( 1 - 4 T + 31 T^{2} )( 1 + 11 T + 31 T^{2} ) \))(\( ( 1 - 11 T + 31 T^{2} )( 1 + 4 T + 31 T^{2} ) \))(\( ( 1 - 7 T + 31 T^{2} )( 1 - 4 T + 31 T^{2} ) \))(\( 1 - T - 30 T^{2} - 31 T^{3} + 961 T^{4} \))(\( 1 + T - 30 T^{2} + 31 T^{3} + 961 T^{4} \))(\( 1 - 5 T - 6 T^{2} - 155 T^{3} + 961 T^{4} \))(\( 1 + 5 T - 6 T^{2} + 155 T^{3} + 961 T^{4} \))(\( ( 1 - T - 30 T^{2} - 31 T^{3} + 961 T^{4} )^{2} \))(\( ( 1 + T - 30 T^{2} + 31 T^{3} + 961 T^{4} )^{2} \))(\( 1 - 55 T^{2} + 2064 T^{4} - 52855 T^{6} + 923521 T^{8} \))
$37$ (\( ( 1 - 10 T + 37 T^{2} )( 1 - T + 37 T^{2} ) \))(\( ( 1 - 10 T + 37 T^{2} )( 1 - T + 37 T^{2} ) \))(\( 1 - 5 T - 12 T^{2} - 185 T^{3} + 1369 T^{4} \))(\( ( 1 - T + 37 T^{2} )( 1 + 11 T + 37 T^{2} ) \))(\( ( 1 - T + 37 T^{2} )( 1 + 11 T + 37 T^{2} ) \))(\( 1 - 5 T - 12 T^{2} - 185 T^{3} + 1369 T^{4} \))(\( ( 1 - 10 T + 37 T^{2} )( 1 - T + 37 T^{2} ) \))(\( ( 1 - 10 T + 37 T^{2} )( 1 + 11 T + 37 T^{2} ) \))(\( ( 1 - 10 T + 37 T^{2} )( 1 + 11 T + 37 T^{2} ) \))(\( ( 1 - 10 T + 37 T^{2} )( 1 - T + 37 T^{2} ) \))(\( 1 - 2 T - 33 T^{2} - 74 T^{3} + 1369 T^{4} \))(\( 1 - 2 T - 33 T^{2} - 74 T^{3} + 1369 T^{4} \))(\( ( 1 - 10 T + 37 T^{2} )( 1 + 11 T + 37 T^{2} ) \))(\( ( 1 - 10 T + 37 T^{2} )( 1 + 11 T + 37 T^{2} ) \))(\( ( 1 + 4 T - 21 T^{2} + 148 T^{3} + 1369 T^{4} )^{2} \))(\( ( 1 + 4 T - 21 T^{2} + 148 T^{3} + 1369 T^{4} )^{2} \))(\( ( 1 + 7 T + 12 T^{2} + 259 T^{3} + 1369 T^{4} )^{2} \))
$41$ (\( 1 - 70 T^{2} + 1681 T^{4} \))(\( 1 - 70 T^{2} + 1681 T^{4} \))(\( 1 - 34 T^{2} + 1681 T^{4} \))(\( 1 + 26 T^{2} + 1681 T^{4} \))(\( 1 + 26 T^{2} + 1681 T^{4} \))(\( 1 - 34 T^{2} + 1681 T^{4} \))(\( ( 1 - 41 T^{2} )^{2} \))(\( ( 1 - 41 T^{2} )^{2} \))(\( ( 1 - 41 T^{2} )^{2} \))(\( ( 1 - 41 T^{2} )^{2} \))(\( 1 - 70 T^{2} + 1681 T^{4} \))(\( 1 - 70 T^{2} + 1681 T^{4} \))(\( 1 + 26 T^{2} + 1681 T^{4} \))(\( 1 + 26 T^{2} + 1681 T^{4} \))(\( ( 1 - 22 T^{2} + 1681 T^{4} )^{2} \))(\( ( 1 - 22 T^{2} + 1681 T^{4} )^{2} \))(\( ( 1 - 70 T^{2} + 1681 T^{4} )^{2} \))
$43$ (\( 1 - 11 T^{2} + 1849 T^{4} \))(\( 1 - 11 T^{2} + 1849 T^{4} \))(\( 1 - 74 T^{2} + 1849 T^{4} \))(\( 1 - 74 T^{2} + 1849 T^{4} \))(\( 1 - 74 T^{2} + 1849 T^{4} \))(\( 1 - 74 T^{2} + 1849 T^{4} \))(\( ( 1 - 13 T + 43 T^{2} )( 1 + 13 T + 43 T^{2} ) \))(\( ( 1 - 5 T + 43 T^{2} )( 1 + 5 T + 43 T^{2} ) \))(\( ( 1 - 5 T + 43 T^{2} )( 1 + 5 T + 43 T^{2} ) \))(\( ( 1 - 13 T + 43 T^{2} )( 1 + 13 T + 43 T^{2} ) \))(\( 1 - 74 T^{2} + 1849 T^{4} \))(\( 1 - 74 T^{2} + 1849 T^{4} \))(\( ( 1 - 13 T + 43 T^{2} )( 1 + 13 T + 43 T^{2} ) \))(\( ( 1 - 13 T + 43 T^{2} )( 1 + 13 T + 43 T^{2} ) \))(\( ( 1 - 38 T^{2} + 1849 T^{4} )^{2} \))(\( ( 1 - 38 T^{2} + 1849 T^{4} )^{2} \))(\( ( 1 - 2 T^{2} + 1849 T^{4} )^{2} \))
$47$ (\( 1 + 6 T - 11 T^{2} + 282 T^{3} + 2209 T^{4} \))(\( 1 - 6 T - 11 T^{2} - 282 T^{3} + 2209 T^{4} \))(\( 1 + 3 T - 38 T^{2} + 141 T^{3} + 2209 T^{4} \))(\( 1 + 12 T + 97 T^{2} + 564 T^{3} + 2209 T^{4} \))(\( 1 - 12 T + 97 T^{2} - 564 T^{3} + 2209 T^{4} \))(\( 1 - 3 T - 38 T^{2} - 141 T^{3} + 2209 T^{4} \))(\( 1 - 47 T^{2} + 2209 T^{4} \))(\( 1 - 47 T^{2} + 2209 T^{4} \))(\( 1 - 47 T^{2} + 2209 T^{4} \))(\( 1 - 47 T^{2} + 2209 T^{4} \))(\( 1 - 47 T^{2} + 2209 T^{4} \))(\( 1 - 47 T^{2} + 2209 T^{4} \))(\( 1 - 6 T - 11 T^{2} - 282 T^{3} + 2209 T^{4} \))(\( 1 + 6 T - 11 T^{2} + 282 T^{3} + 2209 T^{4} \))(\( 1 + 86 T^{2} + 5187 T^{4} + 189974 T^{6} + 4879681 T^{8} \))(\( 1 + 86 T^{2} + 5187 T^{4} + 189974 T^{6} + 4879681 T^{8} \))(\( 1 - 31 T^{2} - 1248 T^{4} - 68479 T^{6} + 4879681 T^{8} \))
$53$ (\( 1 - 12 T + 91 T^{2} - 636 T^{3} + 2809 T^{4} \))(\( 1 - 12 T + 91 T^{2} - 636 T^{3} + 2809 T^{4} \))(\( 1 + 9 T + 28 T^{2} + 477 T^{3} + 2809 T^{4} \))(\( 1 + 9 T + 28 T^{2} + 477 T^{3} + 2809 T^{4} \))(\( 1 + 9 T + 28 T^{2} + 477 T^{3} + 2809 T^{4} \))(\( 1 + 9 T + 28 T^{2} + 477 T^{3} + 2809 T^{4} \))(\( 1 - 53 T^{2} + 2809 T^{4} \))(\( 1 - 53 T^{2} + 2809 T^{4} \))(\( 1 - 53 T^{2} + 2809 T^{4} \))(\( 1 - 53 T^{2} + 2809 T^{4} \))(\( 1 - 9 T + 28 T^{2} - 477 T^{3} + 2809 T^{4} \))(\( 1 - 9 T + 28 T^{2} - 477 T^{3} + 2809 T^{4} \))(\( 1 - 53 T^{2} + 2809 T^{4} \))(\( 1 - 53 T^{2} + 2809 T^{4} \))(\( 1 - 61 T^{2} + 912 T^{4} - 171349 T^{6} + 7890481 T^{8} \))(\( 1 - 61 T^{2} + 912 T^{4} - 171349 T^{6} + 7890481 T^{8} \))(\( ( 1 - 3 T - 44 T^{2} - 159 T^{3} + 2809 T^{4} )^{2} \))
$59$ (\( 1 - 12 T + 85 T^{2} - 708 T^{3} + 3481 T^{4} \))(\( 1 + 12 T + 85 T^{2} + 708 T^{3} + 3481 T^{4} \))(\( 1 + 9 T + 22 T^{2} + 531 T^{3} + 3481 T^{4} \))(\( 1 - 9 T + 22 T^{2} - 531 T^{3} + 3481 T^{4} \))(\( 1 + 9 T + 22 T^{2} + 531 T^{3} + 3481 T^{4} \))(\( 1 - 9 T + 22 T^{2} - 531 T^{3} + 3481 T^{4} \))(\( 1 - 59 T^{2} + 3481 T^{4} \))(\( 1 - 59 T^{2} + 3481 T^{4} \))(\( 1 - 59 T^{2} + 3481 T^{4} \))(\( 1 - 59 T^{2} + 3481 T^{4} \))(\( 1 - 3 T - 50 T^{2} - 177 T^{3} + 3481 T^{4} \))(\( 1 + 3 T - 50 T^{2} + 177 T^{3} + 3481 T^{4} \))(\( 1 - 59 T^{2} + 3481 T^{4} \))(\( 1 - 59 T^{2} + 3481 T^{4} \))(\( 1 - 73 T^{2} + 1848 T^{4} - 254113 T^{6} + 12117361 T^{8} \))(\( 1 - 73 T^{2} + 1848 T^{4} - 254113 T^{6} + 12117361 T^{8} \))(\( 1 - 55 T^{2} - 456 T^{4} - 191455 T^{6} + 12117361 T^{8} \))
$61$ (\( 1 + 24 T + 253 T^{2} + 1464 T^{3} + 3721 T^{4} \))(\( 1 + 24 T + 253 T^{2} + 1464 T^{3} + 3721 T^{4} \))(\( ( 1 - 14 T + 61 T^{2} )( 1 - T + 61 T^{2} ) \))(\( 1 + 61 T^{2} + 3721 T^{4} \))(\( 1 + 61 T^{2} + 3721 T^{4} \))(\( ( 1 - 14 T + 61 T^{2} )( 1 - T + 61 T^{2} ) \))(\( ( 1 - T + 61 T^{2} )( 1 + 13 T + 61 T^{2} ) \))(\( ( 1 - 13 T + 61 T^{2} )( 1 + T + 61 T^{2} ) \))(\( ( 1 - 13 T + 61 T^{2} )( 1 + T + 61 T^{2} ) \))(\( ( 1 - T + 61 T^{2} )( 1 + 13 T + 61 T^{2} ) \))(\( ( 1 - T + 61 T^{2} )( 1 + 13 T + 61 T^{2} ) \))(\( ( 1 - T + 61 T^{2} )( 1 + 13 T + 61 T^{2} ) \))(\( 1 + 61 T^{2} + 3721 T^{4} \))(\( 1 + 61 T^{2} + 3721 T^{4} \))(\( ( 1 - 18 T + 169 T^{2} - 1098 T^{3} + 3721 T^{4} )^{2} \))(\( ( 1 - 18 T + 169 T^{2} - 1098 T^{3} + 3721 T^{4} )^{2} \))(\( ( 1 - 3 T + 64 T^{2} - 183 T^{3} + 3721 T^{4} )^{2} \))
$67$ (\( 1 + 15 T + 142 T^{2} + 1005 T^{3} + 4489 T^{4} \))(\( 1 - 15 T + 142 T^{2} - 1005 T^{3} + 4489 T^{4} \))(\( 1 + 9 T + 94 T^{2} + 603 T^{3} + 4489 T^{4} \))(\( 1 - 24 T + 259 T^{2} - 1608 T^{3} + 4489 T^{4} \))(\( 1 + 24 T + 259 T^{2} + 1608 T^{3} + 4489 T^{4} \))(\( 1 - 9 T + 94 T^{2} - 603 T^{3} + 4489 T^{4} \))(\( ( 1 + 11 T + 67 T^{2} )( 1 + 16 T + 67 T^{2} ) \))(\( ( 1 - 16 T + 67 T^{2} )( 1 - 5 T + 67 T^{2} ) \))(\( ( 1 + 5 T + 67 T^{2} )( 1 + 16 T + 67 T^{2} ) \))(\( ( 1 - 16 T + 67 T^{2} )( 1 - 11 T + 67 T^{2} ) \))(\( 1 + 67 T^{2} + 4489 T^{4} \))(\( 1 + 67 T^{2} + 4489 T^{4} \))(\( 1 + 3 T + 70 T^{2} + 201 T^{3} + 4489 T^{4} \))(\( 1 - 3 T + 70 T^{2} - 201 T^{3} + 4489 T^{4} \))(\( ( 1 - 12 T + 115 T^{2} - 804 T^{3} + 4489 T^{4} )^{2} \))(\( ( 1 + 12 T + 115 T^{2} + 804 T^{3} + 4489 T^{4} )^{2} \))(\( 1 + 113 T^{2} + 8280 T^{4} + 507257 T^{6} + 20151121 T^{8} \))
$71$ (\( 1 - 130 T^{2} + 5041 T^{4} \))(\( 1 - 130 T^{2} + 5041 T^{4} \))(\( 1 - 130 T^{2} + 5041 T^{4} \))(\( 1 + 50 T^{2} + 5041 T^{4} \))(\( 1 + 50 T^{2} + 5041 T^{4} \))(\( 1 - 130 T^{2} + 5041 T^{4} \))(\( ( 1 - 71 T^{2} )^{2} \))(\( ( 1 - 71 T^{2} )^{2} \))(\( ( 1 - 71 T^{2} )^{2} \))(\( ( 1 - 71 T^{2} )^{2} \))(\( 1 - 94 T^{2} + 5041 T^{4} \))(\( 1 - 94 T^{2} + 5041 T^{4} \))(\( 1 - 130 T^{2} + 5041 T^{4} \))(\( 1 - 130 T^{2} + 5041 T^{4} \))(\( ( 1 - 82 T^{2} + 5041 T^{4} )^{2} \))(\( ( 1 - 82 T^{2} + 5041 T^{4} )^{2} \))(\( ( 1 - 58 T^{2} + 5041 T^{4} )^{2} \))
$73$ (\( 1 + 9 T + 100 T^{2} + 657 T^{3} + 5329 T^{4} \))(\( 1 + 9 T + 100 T^{2} + 657 T^{3} + 5329 T^{4} \))(\( ( 1 - 10 T + 73 T^{2} )( 1 + 7 T + 73 T^{2} ) \))(\( 1 + 12 T + 121 T^{2} + 876 T^{3} + 5329 T^{4} \))(\( 1 + 12 T + 121 T^{2} + 876 T^{3} + 5329 T^{4} \))(\( ( 1 - 10 T + 73 T^{2} )( 1 + 7 T + 73 T^{2} ) \))(\( ( 1 - 10 T + 73 T^{2} )( 1 + 7 T + 73 T^{2} ) \))(\( ( 1 - 17 T + 73 T^{2} )( 1 - 10 T + 73 T^{2} ) \))(\( ( 1 - 17 T + 73 T^{2} )( 1 - 10 T + 73 T^{2} ) \))(\( ( 1 - 10 T + 73 T^{2} )( 1 + 7 T + 73 T^{2} ) \))(\( 1 - 12 T + 121 T^{2} - 876 T^{3} + 5329 T^{4} \))(\( 1 - 12 T + 121 T^{2} - 876 T^{3} + 5329 T^{4} \))(\( 1 - 15 T + 148 T^{2} - 1095 T^{3} + 5329 T^{4} \))(\( 1 - 15 T + 148 T^{2} - 1095 T^{3} + 5329 T^{4} \))(\( ( 1 + 12 T + 121 T^{2} + 876 T^{3} + 5329 T^{4} )^{2} \))(\( ( 1 + 12 T + 121 T^{2} + 876 T^{3} + 5329 T^{4} )^{2} \))(\( ( 1 + 9 T + 100 T^{2} + 657 T^{3} + 5329 T^{4} )^{2} \))
$79$ (\( ( 1 + 4 T + 79 T^{2} )( 1 + 17 T + 79 T^{2} ) \))(\( ( 1 - 17 T + 79 T^{2} )( 1 - 4 T + 79 T^{2} ) \))(\( ( 1 - 13 T + 79 T^{2} )( 1 + 4 T + 79 T^{2} ) \))(\( ( 1 - 4 T + 79 T^{2} )( 1 + 13 T + 79 T^{2} ) \))(\( ( 1 - 13 T + 79 T^{2} )( 1 + 4 T + 79 T^{2} ) \))(\( ( 1 - 4 T + 79 T^{2} )( 1 + 13 T + 79 T^{2} ) \))(\( ( 1 - 4 T + 79 T^{2} )( 1 + 13 T + 79 T^{2} ) \))(\( ( 1 + 4 T + 79 T^{2} )( 1 + 17 T + 79 T^{2} ) \))(\( ( 1 - 17 T + 79 T^{2} )( 1 - 4 T + 79 T^{2} ) \))(\( ( 1 - 13 T + 79 T^{2} )( 1 + 4 T + 79 T^{2} ) \))(\( 1 - 3 T + 82 T^{2} - 237 T^{3} + 6241 T^{4} \))(\( 1 + 3 T + 82 T^{2} + 237 T^{3} + 6241 T^{4} \))(\( 1 - 27 T + 322 T^{2} - 2133 T^{3} + 6241 T^{4} \))(\( 1 + 27 T + 322 T^{2} + 2133 T^{3} + 6241 T^{4} \))(\( ( 1 + 4 T + 79 T^{2} )^{2}( 1 + 17 T + 79 T^{2} )^{2} \))(\( ( 1 - 17 T + 79 T^{2} )^{2}( 1 - 4 T + 79 T^{2} )^{2} \))(\( 1 + 137 T^{2} + 12528 T^{4} + 855017 T^{6} + 38950081 T^{8} \))
$83$ (\( ( 1 + 18 T + 83 T^{2} )^{2} \))(\( ( 1 - 18 T + 83 T^{2} )^{2} \))(\( ( 1 - 12 T + 83 T^{2} )^{2} \))(\( ( 1 - 3 T + 83 T^{2} )^{2} \))(\( ( 1 + 3 T + 83 T^{2} )^{2} \))(\( ( 1 + 12 T + 83 T^{2} )^{2} \))(\( ( 1 + 83 T^{2} )^{2} \))(\( ( 1 + 83 T^{2} )^{2} \))(\( ( 1 + 83 T^{2} )^{2} \))(\( ( 1 + 83 T^{2} )^{2} \))(\( ( 1 + 15 T + 83 T^{2} )^{2} \))(\( ( 1 - 15 T + 83 T^{2} )^{2} \))(\( ( 1 + 6 T + 83 T^{2} )^{2} \))(\( ( 1 - 6 T + 83 T^{2} )^{2} \))(\( ( 1 + 121 T^{2} + 6889 T^{4} )^{2} \))(\( ( 1 + 121 T^{2} + 6889 T^{4} )^{2} \))(\( ( 1 + 83 T^{2} )^{4} \))
$89$ (\( 1 + 12 T + 137 T^{2} + 1068 T^{3} + 7921 T^{4} \))(\( 1 + 12 T + 137 T^{2} + 1068 T^{3} + 7921 T^{4} \))(\( 1 - 21 T + 236 T^{2} - 1869 T^{3} + 7921 T^{4} \))(\( 1 - 6 T + 101 T^{2} - 534 T^{3} + 7921 T^{4} \))(\( 1 - 6 T + 101 T^{2} - 534 T^{3} + 7921 T^{4} \))(\( 1 - 21 T + 236 T^{2} - 1869 T^{3} + 7921 T^{4} \))(\( 1 + 89 T^{2} + 7921 T^{4} \))(\( 1 + 89 T^{2} + 7921 T^{4} \))(\( 1 + 89 T^{2} + 7921 T^{4} \))(\( 1 + 89 T^{2} + 7921 T^{4} \))(\( 1 - 18 T + 197 T^{2} - 1602 T^{3} + 7921 T^{4} \))(\( 1 - 18 T + 197 T^{2} - 1602 T^{3} + 7921 T^{4} \))(\( 1 + 12 T + 137 T^{2} + 1068 T^{3} + 7921 T^{4} \))(\( 1 + 12 T + 137 T^{2} + 1068 T^{3} + 7921 T^{4} \))(\( 1 + 118 T^{2} + 6003 T^{4} + 934678 T^{6} + 62742241 T^{8} \))(\( 1 + 118 T^{2} + 6003 T^{4} + 934678 T^{6} + 62742241 T^{8} \))(\( ( 1 + 3 T + 92 T^{2} + 267 T^{3} + 7921 T^{4} )^{2} \))
$97$ (\( 1 - 146 T^{2} + 9409 T^{4} \))(\( 1 - 146 T^{2} + 9409 T^{4} \))(\( 1 - 146 T^{2} + 9409 T^{4} \))(\( ( 1 - 5 T + 97 T^{2} )( 1 + 5 T + 97 T^{2} ) \))(\( ( 1 - 5 T + 97 T^{2} )( 1 + 5 T + 97 T^{2} ) \))(\( 1 - 146 T^{2} + 9409 T^{4} \))(\( ( 1 - 14 T + 97 T^{2} )( 1 + 14 T + 97 T^{2} ) \))(\( ( 1 - 14 T + 97 T^{2} )( 1 + 14 T + 97 T^{2} ) \))(\( ( 1 - 14 T + 97 T^{2} )( 1 + 14 T + 97 T^{2} ) \))(\( ( 1 - 14 T + 97 T^{2} )( 1 + 14 T + 97 T^{2} ) \))(\( 1 - 119 T^{2} + 9409 T^{4} \))(\( 1 - 119 T^{2} + 9409 T^{4} \))(\( 1 - 146 T^{2} + 9409 T^{4} \))(\( 1 - 146 T^{2} + 9409 T^{4} \))(\( ( 1 - 19 T + 97 T^{2} )^{2}( 1 + 19 T + 97 T^{2} )^{2} \))(\( ( 1 - 19 T + 97 T^{2} )^{2}( 1 + 19 T + 97 T^{2} )^{2} \))(\( ( 1 - 182 T^{2} + 9409 T^{4} )^{2} \))
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