Properties

Label 1050.2.i
Level 1050
Weight 2
Character orbit i
Rep. character \(\chi_{1050}(151,\cdot)\)
Character field \(\Q(\zeta_{3})\)
Dimension 52
Newform subspaces 22
Sturm bound 480
Trace bound 13

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Defining parameters

Level: \( N \) = \( 1050 = 2 \cdot 3 \cdot 5^{2} \cdot 7 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 1050.i (of order \(3\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) = \( 7 \)
Character field: \(\Q(\zeta_{3})\)
Newform subspaces: \( 22 \)
Sturm bound: \(480\)
Trace bound: \(13\)
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 52 476
Cusp forms 432 52 380
Eisenstein series 96 0 96

Trace form

\( 52q - 26q^{4} + 4q^{6} - 6q^{7} - 26q^{9} + O(q^{10}) \) \( 52q - 26q^{4} + 4q^{6} - 6q^{7} - 26q^{9} + 8q^{11} - 8q^{13} - 26q^{16} - 12q^{17} - 8q^{19} + 8q^{21} + 12q^{22} - 4q^{23} - 2q^{24} + 4q^{26} + 6q^{28} + 8q^{29} + 26q^{31} + 6q^{33} + 24q^{34} + 52q^{36} + 12q^{37} + 4q^{38} - 10q^{42} - 32q^{43} + 8q^{44} + 4q^{46} - 28q^{47} + 22q^{49} + 12q^{51} + 4q^{52} - 36q^{53} - 2q^{54} + 12q^{56} - 8q^{57} - 14q^{58} + 24q^{59} - 36q^{61} - 40q^{62} + 52q^{64} - 8q^{66} - 4q^{67} - 12q^{68} + 8q^{69} + 56q^{71} + 4q^{73} - 28q^{74} + 16q^{76} + 52q^{77} - 24q^{78} - 18q^{79} - 26q^{81} + 16q^{82} + 16q^{83} - 16q^{84} - 4q^{86} + 14q^{87} - 6q^{88} + 16q^{89} - 32q^{91} + 8q^{92} - 4q^{93} - 8q^{94} - 2q^{96} + 52q^{97} - 24q^{98} - 16q^{99} + O(q^{100}) \)

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.i.a \(2\) \(8.384\) \(\Q(\sqrt{-3}) \) None \(-1\) \(-1\) \(0\) \(-5\) \(q-\zeta_{6}q^{2}+(-1+\zeta_{6})q^{3}+(-1+\zeta_{6})q^{4}+\cdots\)
1050.2.i.b \(2\) \(8.384\) \(\Q(\sqrt{-3}) \) None \(-1\) \(-1\) \(0\) \(-4\) \(q-\zeta_{6}q^{2}+(-1+\zeta_{6})q^{3}+(-1+\zeta_{6})q^{4}+\cdots\)
1050.2.i.c \(2\) \(8.384\) \(\Q(\sqrt{-3}) \) None \(-1\) \(-1\) \(0\) \(5\) \(q-\zeta_{6}q^{2}+(-1+\zeta_{6})q^{3}+(-1+\zeta_{6})q^{4}+\cdots\)
1050.2.i.d \(2\) \(8.384\) \(\Q(\sqrt{-3}) \) None \(-1\) \(-1\) \(0\) \(5\) \(q-\zeta_{6}q^{2}+(-1+\zeta_{6})q^{3}+(-1+\zeta_{6})q^{4}+\cdots\)
1050.2.i.e \(2\) \(8.384\) \(\Q(\sqrt{-3}) \) None \(-1\) \(1\) \(0\) \(-5\) \(q-\zeta_{6}q^{2}+(1-\zeta_{6})q^{3}+(-1+\zeta_{6})q^{4}+\cdots\)
1050.2.i.f \(2\) \(8.384\) \(\Q(\sqrt{-3}) \) None \(-1\) \(1\) \(0\) \(-4\) \(q-\zeta_{6}q^{2}+(1-\zeta_{6})q^{3}+(-1+\zeta_{6})q^{4}+\cdots\)
1050.2.i.g \(2\) \(8.384\) \(\Q(\sqrt{-3}) \) None \(-1\) \(1\) \(0\) \(1\) \(q-\zeta_{6}q^{2}+(1-\zeta_{6})q^{3}+(-1+\zeta_{6})q^{4}+\cdots\)
1050.2.i.h \(2\) \(8.384\) \(\Q(\sqrt{-3}) \) None \(-1\) \(1\) \(0\) \(1\) \(q-\zeta_{6}q^{2}+(1-\zeta_{6})q^{3}+(-1+\zeta_{6})q^{4}+\cdots\)
1050.2.i.i \(2\) \(8.384\) \(\Q(\sqrt{-3}) \) None \(-1\) \(1\) \(0\) \(4\) \(q-\zeta_{6}q^{2}+(1-\zeta_{6})q^{3}+(-1+\zeta_{6})q^{4}+\cdots\)
1050.2.i.j \(2\) \(8.384\) \(\Q(\sqrt{-3}) \) None \(-1\) \(1\) \(0\) \(5\) \(q-\zeta_{6}q^{2}+(1-\zeta_{6})q^{3}+(-1+\zeta_{6})q^{4}+\cdots\)
1050.2.i.k \(2\) \(8.384\) \(\Q(\sqrt{-3}) \) None \(1\) \(-1\) \(0\) \(-5\) \(q+\zeta_{6}q^{2}+(-1+\zeta_{6})q^{3}+(-1+\zeta_{6})q^{4}+\cdots\)
1050.2.i.l \(2\) \(8.384\) \(\Q(\sqrt{-3}) \) None \(1\) \(-1\) \(0\) \(-1\) \(q+\zeta_{6}q^{2}+(-1+\zeta_{6})q^{3}+(-1+\zeta_{6})q^{4}+\cdots\)
1050.2.i.m \(2\) \(8.384\) \(\Q(\sqrt{-3}) \) None \(1\) \(-1\) \(0\) \(-1\) \(q+\zeta_{6}q^{2}+(-1+\zeta_{6})q^{3}+(-1+\zeta_{6})q^{4}+\cdots\)
1050.2.i.n \(2\) \(8.384\) \(\Q(\sqrt{-3}) \) None \(1\) \(-1\) \(0\) \(-1\) \(q+\zeta_{6}q^{2}+(-1+\zeta_{6})q^{3}+(-1+\zeta_{6})q^{4}+\cdots\)
1050.2.i.o \(2\) \(8.384\) \(\Q(\sqrt{-3}) \) None \(1\) \(-1\) \(0\) \(4\) \(q+\zeta_{6}q^{2}+(-1+\zeta_{6})q^{3}+(-1+\zeta_{6})q^{4}+\cdots\)
1050.2.i.p \(2\) \(8.384\) \(\Q(\sqrt{-3}) \) None \(1\) \(-1\) \(0\) \(4\) \(q+\zeta_{6}q^{2}+(-1+\zeta_{6})q^{3}+(-1+\zeta_{6})q^{4}+\cdots\)
1050.2.i.q \(2\) \(8.384\) \(\Q(\sqrt{-3}) \) None \(1\) \(1\) \(0\) \(-5\) \(q+\zeta_{6}q^{2}+(1-\zeta_{6})q^{3}+(-1+\zeta_{6})q^{4}+\cdots\)
1050.2.i.r \(2\) \(8.384\) \(\Q(\sqrt{-3}) \) None \(1\) \(1\) \(0\) \(-5\) \(q+\zeta_{6}q^{2}+(1-\zeta_{6})q^{3}+(-1+\zeta_{6})q^{4}+\cdots\)
1050.2.i.s \(2\) \(8.384\) \(\Q(\sqrt{-3}) \) None \(1\) \(1\) \(0\) \(-4\) \(q+\zeta_{6}q^{2}+(1-\zeta_{6})q^{3}+(-1+\zeta_{6})q^{4}+\cdots\)
1050.2.i.t \(2\) \(8.384\) \(\Q(\sqrt{-3}) \) None \(1\) \(1\) \(0\) \(5\) \(q+\zeta_{6}q^{2}+(1-\zeta_{6})q^{3}+(-1+\zeta_{6})q^{4}+\cdots\)
1050.2.i.u \(6\) \(8.384\) 6.0.21870000.1 None \(-3\) \(-3\) \(0\) \(0\) \(q+\beta _{2}q^{2}+(-1-\beta _{2})q^{3}+(-1-\beta _{2}+\cdots)q^{4}+\cdots\)
1050.2.i.v \(6\) \(8.384\) 6.0.21870000.1 None \(3\) \(3\) \(0\) \(0\) \(q-\beta _{2}q^{2}+(1+\beta _{2})q^{3}+(-1-\beta _{2})q^{4}+\cdots\)

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

\( S_{2}^{\mathrm{old}}(1050, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(21, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(35, [\chi])\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(42, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(70, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(105, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(175, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(210, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(350, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(525, [\chi])\)\(^{\oplus 2}\)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ (\( 1 + T + T^{2} \))(\( 1 + T + T^{2} \))(\( 1 + T + T^{2} \))(\( 1 + T + T^{2} \))(\( 1 + T + T^{2} \))(\( 1 + T + T^{2} \))(\( 1 + T + T^{2} \))(\( 1 + T + T^{2} \))(\( 1 + T + T^{2} \))(\( 1 + T + T^{2} \))(\( 1 - T + T^{2} \))(\( 1 - T + T^{2} \))(\( 1 - T + T^{2} \))(\( 1 - T + T^{2} \))(\( 1 - T + T^{2} \))(\( 1 - T + T^{2} \))(\( 1 - T + T^{2} \))(\( 1 - T + T^{2} \))(\( 1 - T + T^{2} \))(\( 1 - T + T^{2} \))(\( ( 1 + T + T^{2} )^{3} \))(\( ( 1 - T + T^{2} )^{3} \))
$3$ (\( 1 + T + T^{2} \))(\( 1 + T + T^{2} \))(\( 1 + T + T^{2} \))(\( 1 + T + T^{2} \))(\( 1 - T + T^{2} \))(\( 1 - T + T^{2} \))(\( 1 - T + T^{2} \))(\( 1 - T + T^{2} \))(\( 1 - T + T^{2} \))(\( 1 - T + T^{2} \))(\( 1 + T + T^{2} \))(\( 1 + T + T^{2} \))(\( 1 + T + T^{2} \))(\( 1 + T + T^{2} \))(\( 1 + T + T^{2} \))(\( 1 + T + T^{2} \))(\( 1 - T + T^{2} \))(\( 1 - T + T^{2} \))(\( 1 - T + T^{2} \))(\( 1 - T + T^{2} \))(\( ( 1 + T + T^{2} )^{3} \))(\( ( 1 - T + T^{2} )^{3} \))
$5$ 1
$7$ (\( 1 + 5 T + 7 T^{2} \))(\( 1 + 4 T + 7 T^{2} \))(\( 1 - 5 T + 7 T^{2} \))(\( 1 - 5 T + 7 T^{2} \))(\( 1 + 5 T + 7 T^{2} \))(\( 1 + 4 T + 7 T^{2} \))(\( 1 - T + 7 T^{2} \))(\( 1 - T + 7 T^{2} \))(\( 1 - 4 T + 7 T^{2} \))(\( 1 - 5 T + 7 T^{2} \))(\( 1 + 5 T + 7 T^{2} \))(\( 1 + T + 7 T^{2} \))(\( 1 + T + 7 T^{2} \))(\( 1 + T + 7 T^{2} \))(\( 1 - 4 T + 7 T^{2} \))(\( 1 - 4 T + 7 T^{2} \))(\( 1 + 5 T + 7 T^{2} \))(\( 1 + 5 T + 7 T^{2} \))(\( 1 + 4 T + 7 T^{2} \))(\( 1 - 5 T + 7 T^{2} \))(\( 1 + 6 T^{2} + 20 T^{3} + 42 T^{4} + 343 T^{6} \))(\( 1 + 6 T^{2} - 20 T^{3} + 42 T^{4} + 343 T^{6} \))
$11$ (\( 1 - 4 T + 5 T^{2} - 44 T^{3} + 121 T^{4} \))(\( 1 - T - 10 T^{2} - 11 T^{3} + 121 T^{4} \))(\( 1 + 6 T + 25 T^{2} + 66 T^{3} + 121 T^{4} \))(\( 1 - 4 T + 5 T^{2} - 44 T^{3} + 121 T^{4} \))(\( 1 + 3 T - 2 T^{2} + 33 T^{3} + 121 T^{4} \))(\( 1 - 5 T + 14 T^{2} - 55 T^{3} + 121 T^{4} \))(\( 1 - 11 T^{2} + 121 T^{4} \))(\( 1 - 11 T^{2} + 121 T^{4} \))(\( 1 + 3 T - 2 T^{2} + 33 T^{3} + 121 T^{4} \))(\( 1 - 2 T - 7 T^{2} - 22 T^{3} + 121 T^{4} \))(\( 1 - 2 T - 7 T^{2} - 22 T^{3} + 121 T^{4} \))(\( 1 + 5 T + 14 T^{2} + 55 T^{3} + 121 T^{4} \))(\( 1 - 11 T^{2} + 121 T^{4} \))(\( 1 - 11 T^{2} + 121 T^{4} \))(\( 1 - 5 T + 14 T^{2} - 55 T^{3} + 121 T^{4} \))(\( 1 - 5 T + 14 T^{2} - 55 T^{3} + 121 T^{4} \))(\( 1 + 6 T + 25 T^{2} + 66 T^{3} + 121 T^{4} \))(\( 1 - 4 T + 5 T^{2} - 44 T^{3} + 121 T^{4} \))(\( 1 - T - 10 T^{2} - 11 T^{3} + 121 T^{4} \))(\( 1 - 4 T + 5 T^{2} - 44 T^{3} + 121 T^{4} \))(\( 1 + 3 T + 3 T^{2} - 16 T^{3} - 81 T^{4} + 69 T^{5} + 1466 T^{6} + 759 T^{7} - 9801 T^{8} - 21296 T^{9} + 43923 T^{10} + 483153 T^{11} + 1771561 T^{12} \))(\( 1 + 3 T + 3 T^{2} - 16 T^{3} - 81 T^{4} + 69 T^{5} + 1466 T^{6} + 759 T^{7} - 9801 T^{8} - 21296 T^{9} + 43923 T^{10} + 483153 T^{11} + 1771561 T^{12} \))
$13$ (\( ( 1 - T + 13 T^{2} )^{2} \))(\( ( 1 + T + 13 T^{2} )^{2} \))(\( ( 1 + 4 T + 13 T^{2} )^{2} \))(\( ( 1 + 4 T + 13 T^{2} )^{2} \))(\( ( 1 - 4 T + 13 T^{2} )^{2} \))(\( ( 1 + T + 13 T^{2} )^{2} \))(\( ( 1 + 5 T + 13 T^{2} )^{2} \))(\( ( 1 - 4 T + 13 T^{2} )^{2} \))(\( ( 1 + 5 T + 13 T^{2} )^{2} \))(\( ( 1 - 4 T + 13 T^{2} )^{2} \))(\( ( 1 + 4 T + 13 T^{2} )^{2} \))(\( ( 1 + 13 T^{2} )^{2} \))(\( ( 1 + 4 T + 13 T^{2} )^{2} \))(\( ( 1 - 5 T + 13 T^{2} )^{2} \))(\( ( 1 - T + 13 T^{2} )^{2} \))(\( ( 1 - 5 T + 13 T^{2} )^{2} \))(\( ( 1 - 4 T + 13 T^{2} )^{2} \))(\( ( 1 - 4 T + 13 T^{2} )^{2} \))(\( ( 1 + 7 T + 13 T^{2} )^{2} \))(\( ( 1 + T + 13 T^{2} )^{2} \))(\( ( 1 - 3 T + 27 T^{2} - 54 T^{3} + 351 T^{4} - 507 T^{5} + 2197 T^{6} )^{2} \))(\( ( 1 + 3 T + 27 T^{2} + 54 T^{3} + 351 T^{4} + 507 T^{5} + 2197 T^{6} )^{2} \))
$17$ (\( 1 + 2 T - 13 T^{2} + 34 T^{3} + 289 T^{4} \))(\( 1 - 17 T^{2} + 289 T^{4} \))(\( 1 - 3 T - 8 T^{2} - 51 T^{3} + 289 T^{4} \))(\( 1 - 3 T - 8 T^{2} - 51 T^{3} + 289 T^{4} \))(\( 1 - 17 T^{2} + 289 T^{4} \))(\( 1 + 2 T - 13 T^{2} + 34 T^{3} + 289 T^{4} \))(\( 1 + 6 T + 19 T^{2} + 102 T^{3} + 289 T^{4} \))(\( 1 - 3 T - 8 T^{2} - 51 T^{3} + 289 T^{4} \))(\( 1 - 17 T^{2} + 289 T^{4} \))(\( 1 + 5 T + 8 T^{2} + 85 T^{3} + 289 T^{4} \))(\( 1 - 5 T + 8 T^{2} - 85 T^{3} + 289 T^{4} \))(\( 1 + 4 T - T^{2} + 68 T^{3} + 289 T^{4} \))(\( 1 + 3 T - 8 T^{2} + 51 T^{3} + 289 T^{4} \))(\( 1 - 6 T + 19 T^{2} - 102 T^{3} + 289 T^{4} \))(\( 1 - 2 T - 13 T^{2} - 34 T^{3} + 289 T^{4} \))(\( 1 + 4 T - T^{2} + 68 T^{3} + 289 T^{4} \))(\( 1 + 3 T - 8 T^{2} + 51 T^{3} + 289 T^{4} \))(\( 1 + 3 T - 8 T^{2} + 51 T^{3} + 289 T^{4} \))(\( 1 + 4 T - T^{2} + 68 T^{3} + 289 T^{4} \))(\( 1 - 2 T - 13 T^{2} - 34 T^{3} + 289 T^{4} \))(\( 1 + 6 T + 33 T^{2} + 154 T^{3} + 366 T^{4} + 918 T^{5} + 3893 T^{6} + 15606 T^{7} + 105774 T^{8} + 756602 T^{9} + 2756193 T^{10} + 8519142 T^{11} + 24137569 T^{12} \))(\( 1 - 6 T + 33 T^{2} - 154 T^{3} + 366 T^{4} - 918 T^{5} + 3893 T^{6} - 15606 T^{7} + 105774 T^{8} - 756602 T^{9} + 2756193 T^{10} - 8519142 T^{11} + 24137569 T^{12} \))
$19$ (\( ( 1 - 7 T + 19 T^{2} )( 1 + 8 T + 19 T^{2} ) \))(\( 1 - 3 T - 10 T^{2} - 57 T^{3} + 361 T^{4} \))(\( 1 - 4 T - 3 T^{2} - 76 T^{3} + 361 T^{4} \))(\( 1 + 6 T + 17 T^{2} + 114 T^{3} + 361 T^{4} \))(\( 1 - 4 T - 3 T^{2} - 76 T^{3} + 361 T^{4} \))(\( ( 1 - T + 19 T^{2} )( 1 + 8 T + 19 T^{2} ) \))(\( ( 1 - 8 T + 19 T^{2} )( 1 + T + 19 T^{2} ) \))(\( 1 + 2 T - 15 T^{2} + 38 T^{3} + 361 T^{4} \))(\( 1 + 5 T + 6 T^{2} + 95 T^{3} + 361 T^{4} \))(\( 1 - 4 T - 3 T^{2} - 76 T^{3} + 361 T^{4} \))(\( 1 - 4 T - 3 T^{2} - 76 T^{3} + 361 T^{4} \))(\( ( 1 + T + 19 T^{2} )( 1 + 7 T + 19 T^{2} ) \))(\( 1 + 2 T - 15 T^{2} + 38 T^{3} + 361 T^{4} \))(\( ( 1 - 8 T + 19 T^{2} )( 1 + T + 19 T^{2} ) \))(\( ( 1 - T + 19 T^{2} )( 1 + 8 T + 19 T^{2} ) \))(\( ( 1 - 8 T + 19 T^{2} )( 1 + T + 19 T^{2} ) \))(\( 1 - 4 T - 3 T^{2} - 76 T^{3} + 361 T^{4} \))(\( 1 + 6 T + 17 T^{2} + 114 T^{3} + 361 T^{4} \))(\( ( 1 - 7 T + 19 T^{2} )( 1 + 8 T + 19 T^{2} ) \))(\( ( 1 - 7 T + 19 T^{2} )( 1 + 8 T + 19 T^{2} ) \))(\( 1 + 3 T - 36 T^{2} - 45 T^{3} + 900 T^{4} + 3 T^{5} - 19906 T^{6} + 57 T^{7} + 324900 T^{8} - 308655 T^{9} - 4691556 T^{10} + 7428297 T^{11} + 47045881 T^{12} \))(\( 1 + 3 T - 36 T^{2} - 45 T^{3} + 900 T^{4} + 3 T^{5} - 19906 T^{6} + 57 T^{7} + 324900 T^{8} - 308655 T^{9} - 4691556 T^{10} + 7428297 T^{11} + 47045881 T^{12} \))
$23$ (\( 1 - 2 T - 19 T^{2} - 46 T^{3} + 529 T^{4} \))(\( 1 - 7 T + 26 T^{2} - 161 T^{3} + 529 T^{4} \))(\( 1 + 3 T - 14 T^{2} + 69 T^{3} + 529 T^{4} \))(\( 1 - 7 T + 26 T^{2} - 161 T^{3} + 529 T^{4} \))(\( 1 - 23 T^{2} + 529 T^{4} \))(\( 1 - 3 T - 14 T^{2} - 69 T^{3} + 529 T^{4} \))(\( 1 + 6 T + 13 T^{2} + 138 T^{3} + 529 T^{4} \))(\( 1 - 3 T - 14 T^{2} - 69 T^{3} + 529 T^{4} \))(\( 1 + 9 T + 58 T^{2} + 207 T^{3} + 529 T^{4} \))(\( 1 - 5 T + 2 T^{2} - 115 T^{3} + 529 T^{4} \))(\( 1 + 5 T + 2 T^{2} + 115 T^{3} + 529 T^{4} \))(\( 1 + 4 T - 7 T^{2} + 92 T^{3} + 529 T^{4} \))(\( 1 + 3 T - 14 T^{2} + 69 T^{3} + 529 T^{4} \))(\( 1 - 6 T + 13 T^{2} - 138 T^{3} + 529 T^{4} \))(\( 1 + 3 T - 14 T^{2} + 69 T^{3} + 529 T^{4} \))(\( 1 - T - 22 T^{2} - 23 T^{3} + 529 T^{4} \))(\( 1 - 3 T - 14 T^{2} - 69 T^{3} + 529 T^{4} \))(\( 1 + 7 T + 26 T^{2} + 161 T^{3} + 529 T^{4} \))(\( 1 - T - 22 T^{2} - 23 T^{3} + 529 T^{4} \))(\( 1 + 2 T - 19 T^{2} + 46 T^{3} + 529 T^{4} \))(\( 1 + 9 T - 83 T^{3} + 1044 T^{4} + 4113 T^{5} - 4582 T^{6} + 94599 T^{7} + 552276 T^{8} - 1009861 T^{9} + 57927087 T^{11} + 148035889 T^{12} \))(\( 1 - 9 T + 83 T^{3} + 1044 T^{4} - 4113 T^{5} - 4582 T^{6} - 94599 T^{7} + 552276 T^{8} + 1009861 T^{9} - 57927087 T^{11} + 148035889 T^{12} \))
$29$ (\( ( 1 - 4 T + 29 T^{2} )^{2} \))(\( ( 1 + 8 T + 29 T^{2} )^{2} \))(\( ( 1 + 6 T + 29 T^{2} )^{2} \))(\( ( 1 - 4 T + 29 T^{2} )^{2} \))(\( ( 1 - 9 T + 29 T^{2} )^{2} \))(\( ( 1 + 29 T^{2} )^{2} \))(\( ( 1 + 29 T^{2} )^{2} \))(\( ( 1 + 29 T^{2} )^{2} \))(\( ( 1 + 29 T^{2} )^{2} \))(\( ( 1 + 6 T + 29 T^{2} )^{2} \))(\( ( 1 + 6 T + 29 T^{2} )^{2} \))(\( ( 1 + 5 T + 29 T^{2} )^{2} \))(\( ( 1 + 29 T^{2} )^{2} \))(\( ( 1 + 29 T^{2} )^{2} \))(\( ( 1 + 29 T^{2} )^{2} \))(\( ( 1 + 29 T^{2} )^{2} \))(\( ( 1 + 6 T + 29 T^{2} )^{2} \))(\( ( 1 - 4 T + 29 T^{2} )^{2} \))(\( ( 1 + 8 T + 29 T^{2} )^{2} \))(\( ( 1 - 4 T + 29 T^{2} )^{2} \))(\( ( 1 - 12 T + 120 T^{2} - 720 T^{3} + 3480 T^{4} - 10092 T^{5} + 24389 T^{6} )^{2} \))(\( ( 1 - 12 T + 120 T^{2} - 720 T^{3} + 3480 T^{4} - 10092 T^{5} + 24389 T^{6} )^{2} \))
$31$ (\( 1 - 31 T^{2} + 961 T^{4} \))(\( 1 - 2 T - 27 T^{2} - 62 T^{3} + 961 T^{4} \))(\( 1 + 5 T - 6 T^{2} + 155 T^{3} + 961 T^{4} \))(\( 1 - 5 T - 6 T^{2} - 155 T^{3} + 961 T^{4} \))(\( 1 - T - 30 T^{2} - 31 T^{3} + 961 T^{4} \))(\( 1 - 6 T + 5 T^{2} - 186 T^{3} + 961 T^{4} \))(\( 1 + 8 T + 33 T^{2} + 248 T^{3} + 961 T^{4} \))(\( 1 - T - 30 T^{2} - 31 T^{3} + 961 T^{4} \))(\( 1 - 10 T + 69 T^{2} - 310 T^{3} + 961 T^{4} \))(\( ( 1 - 7 T + 31 T^{2} )( 1 - 4 T + 31 T^{2} ) \))(\( ( 1 - 7 T + 31 T^{2} )( 1 - 4 T + 31 T^{2} ) \))(\( 1 + 3 T - 22 T^{2} + 93 T^{3} + 961 T^{4} \))(\( 1 - T - 30 T^{2} - 31 T^{3} + 961 T^{4} \))(\( 1 + 8 T + 33 T^{2} + 248 T^{3} + 961 T^{4} \))(\( 1 - 6 T + 5 T^{2} - 186 T^{3} + 961 T^{4} \))(\( 1 - 2 T - 27 T^{2} - 62 T^{3} + 961 T^{4} \))(\( 1 + 5 T - 6 T^{2} + 155 T^{3} + 961 T^{4} \))(\( 1 - 5 T - 6 T^{2} - 155 T^{3} + 961 T^{4} \))(\( 1 + 6 T + 5 T^{2} + 186 T^{3} + 961 T^{4} \))(\( 1 - 31 T^{2} + 961 T^{4} \))(\( 1 - 78 T^{2} - 20 T^{3} + 3666 T^{4} + 780 T^{5} - 128922 T^{6} + 24180 T^{7} + 3523026 T^{8} - 595820 T^{9} - 72034638 T^{10} + 887503681 T^{12} \))(\( 1 - 78 T^{2} - 20 T^{3} + 3666 T^{4} + 780 T^{5} - 128922 T^{6} + 24180 T^{7} + 3523026 T^{8} - 595820 T^{9} - 72034638 T^{10} + 887503681 T^{12} \))
$37$ (\( 1 + 3 T - 28 T^{2} + 111 T^{3} + 1369 T^{4} \))(\( ( 1 - 10 T + 37 T^{2} )( 1 - T + 37 T^{2} ) \))(\( 1 + 8 T + 27 T^{2} + 296 T^{3} + 1369 T^{4} \))(\( 1 - 2 T - 33 T^{2} - 74 T^{3} + 1369 T^{4} \))(\( 1 - 8 T + 27 T^{2} - 296 T^{3} + 1369 T^{4} \))(\( 1 + 5 T - 12 T^{2} + 185 T^{3} + 1369 T^{4} \))(\( ( 1 - 10 T + 37 T^{2} )( 1 + 11 T + 37 T^{2} ) \))(\( ( 1 - T + 37 T^{2} )( 1 + 11 T + 37 T^{2} ) \))(\( ( 1 - 10 T + 37 T^{2} )( 1 + 11 T + 37 T^{2} ) \))(\( 1 - 8 T + 27 T^{2} - 296 T^{3} + 1369 T^{4} \))(\( 1 + 8 T + 27 T^{2} + 296 T^{3} + 1369 T^{4} \))(\( 1 + 4 T - 21 T^{2} + 148 T^{3} + 1369 T^{4} \))(\( ( 1 - 11 T + 37 T^{2} )( 1 + T + 37 T^{2} ) \))(\( ( 1 - 11 T + 37 T^{2} )( 1 + 10 T + 37 T^{2} ) \))(\( 1 - 5 T - 12 T^{2} - 185 T^{3} + 1369 T^{4} \))(\( ( 1 - 11 T + 37 T^{2} )( 1 + 10 T + 37 T^{2} ) \))(\( 1 - 8 T + 27 T^{2} - 296 T^{3} + 1369 T^{4} \))(\( 1 + 2 T - 33 T^{2} + 74 T^{3} + 1369 T^{4} \))(\( 1 + 3 T - 28 T^{2} + 111 T^{3} + 1369 T^{4} \))(\( 1 - 3 T - 28 T^{2} - 111 T^{3} + 1369 T^{4} \))(\( 1 + 9 T - 42 T^{2} - 209 T^{3} + 4656 T^{4} + 11337 T^{5} - 136392 T^{6} + 419469 T^{7} + 6374064 T^{8} - 10586477 T^{9} - 78714762 T^{10} + 624095613 T^{11} + 2565726409 T^{12} \))(\( 1 - 9 T - 42 T^{2} + 209 T^{3} + 4656 T^{4} - 11337 T^{5} - 136392 T^{6} - 419469 T^{7} + 6374064 T^{8} + 10586477 T^{9} - 78714762 T^{10} - 624095613 T^{11} + 2565726409 T^{12} \))
$41$ (\( ( 1 - 12 T + 41 T^{2} )^{2} \))(\( ( 1 + 11 T + 41 T^{2} )^{2} \))(\( ( 1 + 3 T + 41 T^{2} )^{2} \))(\( ( 1 - 7 T + 41 T^{2} )^{2} \))(\( ( 1 + 41 T^{2} )^{2} \))(\( ( 1 + 9 T + 41 T^{2} )^{2} \))(\( ( 1 + 41 T^{2} )^{2} \))(\( ( 1 + 9 T + 41 T^{2} )^{2} \))(\( ( 1 - 9 T + 41 T^{2} )^{2} \))(\( ( 1 - 5 T + 41 T^{2} )^{2} \))(\( ( 1 - 5 T + 41 T^{2} )^{2} \))(\( ( 1 + 41 T^{2} )^{2} \))(\( ( 1 + 9 T + 41 T^{2} )^{2} \))(\( ( 1 + 41 T^{2} )^{2} \))(\( ( 1 + 9 T + 41 T^{2} )^{2} \))(\( ( 1 - 5 T + 41 T^{2} )^{2} \))(\( ( 1 + 3 T + 41 T^{2} )^{2} \))(\( ( 1 - 7 T + 41 T^{2} )^{2} \))(\( ( 1 - 9 T + 41 T^{2} )^{2} \))(\( ( 1 - 12 T + 41 T^{2} )^{2} \))(\( ( 1 + 9 T + 75 T^{2} + 610 T^{3} + 3075 T^{4} + 15129 T^{5} + 68921 T^{6} )^{2} \))(\( ( 1 + 9 T + 75 T^{2} + 610 T^{3} + 3075 T^{4} + 15129 T^{5} + 68921 T^{6} )^{2} \))
$43$ (\( ( 1 - 8 T + 43 T^{2} )^{2} \))(\( ( 1 + 8 T + 43 T^{2} )^{2} \))(\( ( 1 - 8 T + 43 T^{2} )^{2} \))(\( ( 1 + 2 T + 43 T^{2} )^{2} \))(\( ( 1 - 10 T + 43 T^{2} )^{2} \))(\( ( 1 + 10 T + 43 T^{2} )^{2} \))(\( ( 1 + 8 T + 43 T^{2} )^{2} \))(\( ( 1 - 10 T + 43 T^{2} )^{2} \))(\( ( 1 + 8 T + 43 T^{2} )^{2} \))(\( ( 1 + 43 T^{2} )^{2} \))(\( ( 1 + 43 T^{2} )^{2} \))(\( ( 1 + 2 T + 43 T^{2} )^{2} \))(\( ( 1 + 10 T + 43 T^{2} )^{2} \))(\( ( 1 - 8 T + 43 T^{2} )^{2} \))(\( ( 1 - 10 T + 43 T^{2} )^{2} \))(\( ( 1 + 12 T + 43 T^{2} )^{2} \))(\( ( 1 + 8 T + 43 T^{2} )^{2} \))(\( ( 1 - 2 T + 43 T^{2} )^{2} \))(\( ( 1 - 4 T + 43 T^{2} )^{2} \))(\( ( 1 + 8 T + 43 T^{2} )^{2} \))(\( ( 1 + 2 T + 43 T^{2} )^{6} \))(\( ( 1 - 2 T + 43 T^{2} )^{6} \))
$47$ (\( 1 - 6 T - 11 T^{2} - 282 T^{3} + 2209 T^{4} \))(\( 1 + 5 T - 22 T^{2} + 235 T^{3} + 2209 T^{4} \))(\( 1 + 9 T + 34 T^{2} + 423 T^{3} + 2209 T^{4} \))(\( 1 - T - 46 T^{2} - 47 T^{3} + 2209 T^{4} \))(\( 1 + 6 T - 11 T^{2} + 282 T^{3} + 2209 T^{4} \))(\( 1 + 13 T + 122 T^{2} + 611 T^{3} + 2209 T^{4} \))(\( 1 - 6 T - 11 T^{2} - 282 T^{3} + 2209 T^{4} \))(\( 1 + 3 T - 38 T^{2} + 141 T^{3} + 2209 T^{4} \))(\( 1 - 3 T - 38 T^{2} - 141 T^{3} + 2209 T^{4} \))(\( 1 + T - 46 T^{2} + 47 T^{3} + 2209 T^{4} \))(\( 1 - T - 46 T^{2} - 47 T^{3} + 2209 T^{4} \))(\( 1 + 6 T - 11 T^{2} + 282 T^{3} + 2209 T^{4} \))(\( 1 - 3 T - 38 T^{2} - 141 T^{3} + 2209 T^{4} \))(\( 1 + 6 T - 11 T^{2} + 282 T^{3} + 2209 T^{4} \))(\( 1 - 13 T + 122 T^{2} - 611 T^{3} + 2209 T^{4} \))(\( 1 + 11 T + 74 T^{2} + 517 T^{3} + 2209 T^{4} \))(\( 1 - 9 T + 34 T^{2} - 423 T^{3} + 2209 T^{4} \))(\( 1 + T - 46 T^{2} + 47 T^{3} + 2209 T^{4} \))(\( 1 + 3 T - 38 T^{2} + 141 T^{3} + 2209 T^{4} \))(\( 1 + 6 T - 11 T^{2} + 282 T^{3} + 2209 T^{4} \))(\( 1 - 3 T - 271 T^{3} + 24 T^{4} - 2991 T^{5} + 198842 T^{6} - 140577 T^{7} + 53016 T^{8} - 28136033 T^{9} - 688035021 T^{11} + 10779215329 T^{12} \))(\( 1 + 3 T + 271 T^{3} + 24 T^{4} + 2991 T^{5} + 198842 T^{6} + 140577 T^{7} + 53016 T^{8} + 28136033 T^{9} + 688035021 T^{11} + 10779215329 T^{12} \))
$53$ (\( 1 + 2 T - 49 T^{2} + 106 T^{3} + 2809 T^{4} \))(\( 1 + 11 T + 68 T^{2} + 583 T^{3} + 2809 T^{4} \))(\( 1 + 12 T + 91 T^{2} + 636 T^{3} + 2809 T^{4} \))(\( 1 + 2 T - 49 T^{2} + 106 T^{3} + 2809 T^{4} \))(\( 1 + 3 T - 44 T^{2} + 159 T^{3} + 2809 T^{4} \))(\( 1 + T - 52 T^{2} + 53 T^{3} + 2809 T^{4} \))(\( 1 + 6 T - 17 T^{2} + 318 T^{3} + 2809 T^{4} \))(\( 1 + 6 T - 17 T^{2} + 318 T^{3} + 2809 T^{4} \))(\( 1 + 3 T - 44 T^{2} + 159 T^{3} + 2809 T^{4} \))(\( 1 - 12 T + 91 T^{2} - 636 T^{3} + 2809 T^{4} \))(\( 1 + 12 T + 91 T^{2} + 636 T^{3} + 2809 T^{4} \))(\( 1 + 9 T + 28 T^{2} + 477 T^{3} + 2809 T^{4} \))(\( 1 - 6 T - 17 T^{2} - 318 T^{3} + 2809 T^{4} \))(\( 1 - 6 T - 17 T^{2} - 318 T^{3} + 2809 T^{4} \))(\( 1 - T - 52 T^{2} - 53 T^{3} + 2809 T^{4} \))(\( 1 + 9 T + 28 T^{2} + 477 T^{3} + 2809 T^{4} \))(\( 1 - 12 T + 91 T^{2} - 636 T^{3} + 2809 T^{4} \))(\( 1 - 2 T - 49 T^{2} - 106 T^{3} + 2809 T^{4} \))(\( 1 + T - 52 T^{2} + 53 T^{3} + 2809 T^{4} \))(\( 1 - 2 T - 49 T^{2} - 106 T^{3} + 2809 T^{4} \))(\( 1 - 9 T + 15 T^{2} - 312 T^{3} - 411 T^{4} + 28377 T^{5} - 140762 T^{6} + 1503981 T^{7} - 1154499 T^{8} - 46449624 T^{9} + 118357215 T^{10} - 3763759437 T^{11} + 22164361129 T^{12} \))(\( 1 + 9 T + 15 T^{2} + 312 T^{3} - 411 T^{4} - 28377 T^{5} - 140762 T^{6} - 1503981 T^{7} - 1154499 T^{8} + 46449624 T^{9} + 118357215 T^{10} + 3763759437 T^{11} + 22164361129 T^{12} \))
$59$ (\( 1 + 6 T - 23 T^{2} + 354 T^{3} + 3481 T^{4} \))(\( 1 + 4 T - 43 T^{2} + 236 T^{3} + 3481 T^{4} \))(\( 1 + 6 T - 23 T^{2} + 354 T^{3} + 3481 T^{4} \))(\( 1 - 14 T + 137 T^{2} - 826 T^{3} + 3481 T^{4} \))(\( 1 + 3 T - 50 T^{2} + 177 T^{3} + 3481 T^{4} \))(\( 1 + 4 T - 43 T^{2} + 236 T^{3} + 3481 T^{4} \))(\( 1 - 6 T - 23 T^{2} - 354 T^{3} + 3481 T^{4} \))(\( 1 - 6 T - 23 T^{2} - 354 T^{3} + 3481 T^{4} \))(\( 1 + 12 T + 85 T^{2} + 708 T^{3} + 3481 T^{4} \))(\( 1 - 2 T - 55 T^{2} - 118 T^{3} + 3481 T^{4} \))(\( 1 - 2 T - 55 T^{2} - 118 T^{3} + 3481 T^{4} \))(\( 1 - 11 T + 62 T^{2} - 649 T^{3} + 3481 T^{4} \))(\( 1 - 6 T - 23 T^{2} - 354 T^{3} + 3481 T^{4} \))(\( 1 - 6 T - 23 T^{2} - 354 T^{3} + 3481 T^{4} \))(\( 1 + 4 T - 43 T^{2} + 236 T^{3} + 3481 T^{4} \))(\( 1 + 4 T - 43 T^{2} + 236 T^{3} + 3481 T^{4} \))(\( 1 + 6 T - 23 T^{2} + 354 T^{3} + 3481 T^{4} \))(\( 1 - 14 T + 137 T^{2} - 826 T^{3} + 3481 T^{4} \))(\( 1 + 12 T + 85 T^{2} + 708 T^{3} + 3481 T^{4} \))(\( 1 + 6 T - 23 T^{2} + 354 T^{3} + 3481 T^{4} \))(\( 1 - 12 T - 6 T^{2} + 160 T^{3} + 1890 T^{4} + 23628 T^{5} - 457606 T^{6} + 1394052 T^{7} + 6579090 T^{8} + 32860640 T^{9} - 72704166 T^{10} - 8579091588 T^{11} + 42180533641 T^{12} \))(\( 1 - 12 T - 6 T^{2} + 160 T^{3} + 1890 T^{4} + 23628 T^{5} - 457606 T^{6} + 1394052 T^{7} + 6579090 T^{8} + 32860640 T^{9} - 72704166 T^{10} - 8579091588 T^{11} + 42180533641 T^{12} \))
$61$ (\( ( 1 - 14 T + 61 T^{2} )( 1 + T + 61 T^{2} ) \))(\( 1 - 61 T^{2} + 3721 T^{4} \))(\( 1 + 2 T - 57 T^{2} + 122 T^{3} + 3721 T^{4} \))(\( 1 + 12 T + 83 T^{2} + 732 T^{3} + 3721 T^{4} \))(\( 1 - 10 T + 39 T^{2} - 610 T^{3} + 3721 T^{4} \))(\( 1 - 2 T - 57 T^{2} - 122 T^{3} + 3721 T^{4} \))(\( ( 1 - 14 T + 61 T^{2} )( 1 + 13 T + 61 T^{2} ) \))(\( 1 + 8 T + 3 T^{2} + 488 T^{3} + 3721 T^{4} \))(\( 1 + 8 T + 3 T^{2} + 488 T^{3} + 3721 T^{4} \))(\( 1 + 10 T + 39 T^{2} + 610 T^{3} + 3721 T^{4} \))(\( 1 + 10 T + 39 T^{2} + 610 T^{3} + 3721 T^{4} \))(\( 1 - 6 T - 25 T^{2} - 366 T^{3} + 3721 T^{4} \))(\( 1 + 8 T + 3 T^{2} + 488 T^{3} + 3721 T^{4} \))(\( ( 1 - 14 T + 61 T^{2} )( 1 + 13 T + 61 T^{2} ) \))(\( 1 - 2 T - 57 T^{2} - 122 T^{3} + 3721 T^{4} \))(\( 1 + 4 T - 45 T^{2} + 244 T^{3} + 3721 T^{4} \))(\( 1 + 2 T - 57 T^{2} + 122 T^{3} + 3721 T^{4} \))(\( 1 + 12 T + 83 T^{2} + 732 T^{3} + 3721 T^{4} \))(\( 1 - 4 T - 45 T^{2} - 244 T^{3} + 3721 T^{4} \))(\( ( 1 - 14 T + 61 T^{2} )( 1 + T + 61 T^{2} ) \))(\( 1 + 6 T - 39 T^{2} + 410 T^{3} + 930 T^{4} - 31074 T^{5} - 22719 T^{6} - 1895514 T^{7} + 3460530 T^{8} + 93062210 T^{9} - 539987799 T^{10} + 5067577806 T^{11} + 51520374361 T^{12} \))(\( 1 + 6 T - 39 T^{2} + 410 T^{3} + 930 T^{4} - 31074 T^{5} - 22719 T^{6} - 1895514 T^{7} + 3460530 T^{8} + 93062210 T^{9} - 539987799 T^{10} + 5067577806 T^{11} + 51520374361 T^{12} \))
$67$ (\( 1 + 3 T - 58 T^{2} + 201 T^{3} + 4489 T^{4} \))(\( 1 - 67 T^{2} + 4489 T^{4} \))(\( 1 + 8 T - 3 T^{2} + 536 T^{3} + 4489 T^{4} \))(\( 1 - 12 T + 77 T^{2} - 804 T^{3} + 4489 T^{4} \))(\( 1 + 10 T + 33 T^{2} + 670 T^{3} + 4489 T^{4} \))(\( 1 - 6 T - 31 T^{2} - 402 T^{3} + 4489 T^{4} \))(\( 1 + 13 T + 102 T^{2} + 871 T^{3} + 4489 T^{4} \))(\( 1 + 4 T - 51 T^{2} + 268 T^{3} + 4489 T^{4} \))(\( 1 - 8 T - 3 T^{2} - 536 T^{3} + 4489 T^{4} \))(\( 1 - 67 T^{2} + 4489 T^{4} \))(\( 1 - 67 T^{2} + 4489 T^{4} \))(\( 1 + 2 T - 63 T^{2} + 134 T^{3} + 4489 T^{4} \))(\( 1 - 4 T - 51 T^{2} - 268 T^{3} + 4489 T^{4} \))(\( 1 - 13 T + 102 T^{2} - 871 T^{3} + 4489 T^{4} \))(\( 1 + 6 T - 31 T^{2} + 402 T^{3} + 4489 T^{4} \))(\( 1 + 12 T + 77 T^{2} + 804 T^{3} + 4489 T^{4} \))(\( 1 - 8 T - 3 T^{2} - 536 T^{3} + 4489 T^{4} \))(\( 1 + 12 T + 77 T^{2} + 804 T^{3} + 4489 T^{4} \))(\( 1 - 12 T + 77 T^{2} - 804 T^{3} + 4489 T^{4} \))(\( 1 - 3 T - 58 T^{2} - 201 T^{3} + 4489 T^{4} \))(\( 1 - 6 T - 57 T^{2} + 266 T^{3} - 54 T^{4} + 9522 T^{5} + 30183 T^{6} + 637974 T^{7} - 242406 T^{8} + 80002958 T^{9} - 1148613897 T^{10} - 8100750642 T^{11} + 90458382169 T^{12} \))(\( 1 + 6 T - 57 T^{2} - 266 T^{3} - 54 T^{4} - 9522 T^{5} + 30183 T^{6} - 637974 T^{7} - 242406 T^{8} - 80002958 T^{9} - 1148613897 T^{10} + 8100750642 T^{11} + 90458382169 T^{12} \))
$71$ (\( ( 1 - 16 T + 71 T^{2} )^{2} \))(\( ( 1 + 6 T + 71 T^{2} )^{2} \))(\( ( 1 + 9 T + 71 T^{2} )^{2} \))(\( ( 1 + 9 T + 71 T^{2} )^{2} \))(\( ( 1 + 6 T + 71 T^{2} )^{2} \))(\( ( 1 + 2 T + 71 T^{2} )^{2} \))(\( ( 1 - 12 T + 71 T^{2} )^{2} \))(\( ( 1 - 3 T + 71 T^{2} )^{2} \))(\( ( 1 + 6 T + 71 T^{2} )^{2} \))(\( ( 1 + T + 71 T^{2} )^{2} \))(\( ( 1 + T + 71 T^{2} )^{2} \))(\( ( 1 - 2 T + 71 T^{2} )^{2} \))(\( ( 1 - 3 T + 71 T^{2} )^{2} \))(\( ( 1 - 12 T + 71 T^{2} )^{2} \))(\( ( 1 + 2 T + 71 T^{2} )^{2} \))(\( ( 1 - 2 T + 71 T^{2} )^{2} \))(\( ( 1 + 9 T + 71 T^{2} )^{2} \))(\( ( 1 + 9 T + 71 T^{2} )^{2} \))(\( ( 1 + 14 T + 71 T^{2} )^{2} \))(\( ( 1 - 16 T + 71 T^{2} )^{2} \))(\( ( 1 - 6 T + 71 T^{2} )^{6} \))(\( ( 1 - 6 T + 71 T^{2} )^{6} \))
$73$ (\( 1 - 11 T + 48 T^{2} - 803 T^{3} + 5329 T^{4} \))(\( 1 + 6 T - 37 T^{2} + 438 T^{3} + 5329 T^{4} \))(\( 1 + 14 T + 123 T^{2} + 1022 T^{3} + 5329 T^{4} \))(\( 1 - 6 T - 37 T^{2} - 438 T^{3} + 5329 T^{4} \))(\( 1 - 2 T - 69 T^{2} - 146 T^{3} + 5329 T^{4} \))(\( 1 - 4 T - 57 T^{2} - 292 T^{3} + 5329 T^{4} \))(\( 1 - 5 T - 48 T^{2} - 365 T^{3} + 5329 T^{4} \))(\( 1 - 14 T + 123 T^{2} - 1022 T^{3} + 5329 T^{4} \))(\( 1 - 2 T - 69 T^{2} - 146 T^{3} + 5329 T^{4} \))(\( 1 - 2 T - 69 T^{2} - 146 T^{3} + 5329 T^{4} \))(\( 1 + 2 T - 69 T^{2} + 146 T^{3} + 5329 T^{4} \))(\( ( 1 - 17 T + 73 T^{2} )( 1 + 7 T + 73 T^{2} ) \))(\( 1 + 14 T + 123 T^{2} + 1022 T^{3} + 5329 T^{4} \))(\( 1 + 5 T - 48 T^{2} + 365 T^{3} + 5329 T^{4} \))(\( 1 + 4 T - 57 T^{2} + 292 T^{3} + 5329 T^{4} \))(\( ( 1 - 17 T + 73 T^{2} )( 1 + 7 T + 73 T^{2} ) \))(\( 1 - 14 T + 123 T^{2} - 1022 T^{3} + 5329 T^{4} \))(\( 1 + 6 T - 37 T^{2} + 438 T^{3} + 5329 T^{4} \))(\( 1 + 14 T + 123 T^{2} + 1022 T^{3} + 5329 T^{4} \))(\( 1 + 11 T + 48 T^{2} + 803 T^{3} + 5329 T^{4} \))(\( ( 1 + 4 T - 57 T^{2} + 292 T^{3} + 5329 T^{4} )^{3} \))(\( ( 1 - 4 T - 57 T^{2} - 292 T^{3} + 5329 T^{4} )^{3} \))
$79$ (\( ( 1 - 4 T + 79 T^{2} )( 1 + 17 T + 79 T^{2} ) \))(\( 1 - 8 T - 15 T^{2} - 632 T^{3} + 6241 T^{4} \))(\( 1 - 7 T - 30 T^{2} - 553 T^{3} + 6241 T^{4} \))(\( ( 1 - 13 T + 79 T^{2} )( 1 - 4 T + 79 T^{2} ) \))(\( 1 - T - 78 T^{2} - 79 T^{3} + 6241 T^{4} \))(\( 1 - 14 T + 117 T^{2} - 1106 T^{3} + 6241 T^{4} \))(\( 1 - 7 T - 30 T^{2} - 553 T^{3} + 6241 T^{4} \))(\( 1 + 11 T + 42 T^{2} + 869 T^{3} + 6241 T^{4} \))(\( 1 + 8 T - 15 T^{2} + 632 T^{3} + 6241 T^{4} \))(\( 1 + 9 T + 2 T^{2} + 711 T^{3} + 6241 T^{4} \))(\( 1 + 9 T + 2 T^{2} + 711 T^{3} + 6241 T^{4} \))(\( 1 + 3 T - 70 T^{2} + 237 T^{3} + 6241 T^{4} \))(\( 1 + 11 T + 42 T^{2} + 869 T^{3} + 6241 T^{4} \))(\( 1 - 7 T - 30 T^{2} - 553 T^{3} + 6241 T^{4} \))(\( 1 - 14 T + 117 T^{2} - 1106 T^{3} + 6241 T^{4} \))(\( 1 - 12 T + 65 T^{2} - 948 T^{3} + 6241 T^{4} \))(\( 1 - 7 T - 30 T^{2} - 553 T^{3} + 6241 T^{4} \))(\( ( 1 - 13 T + 79 T^{2} )( 1 - 4 T + 79 T^{2} ) \))(\( ( 1 - 13 T + 79 T^{2} )( 1 + 17 T + 79 T^{2} ) \))(\( ( 1 - 4 T + 79 T^{2} )( 1 + 17 T + 79 T^{2} ) \))(\( 1 + 24 T + 162 T^{2} + 1548 T^{3} + 38034 T^{4} + 316212 T^{5} + 1440614 T^{6} + 24980748 T^{7} + 237370194 T^{8} + 763224372 T^{9} + 6309913122 T^{10} + 73849353576 T^{11} + 243087455521 T^{12} \))(\( 1 + 24 T + 162 T^{2} + 1548 T^{3} + 38034 T^{4} + 316212 T^{5} + 1440614 T^{6} + 24980748 T^{7} + 237370194 T^{8} + 763224372 T^{9} + 6309913122 T^{10} + 73849353576 T^{11} + 243087455521 T^{12} \))
$83$ (\( ( 1 + 6 T + 83 T^{2} )^{2} \))(\( ( 1 + 8 T + 83 T^{2} )^{2} \))(\( ( 1 + 6 T + 83 T^{2} )^{2} \))(\( ( 1 - 4 T + 83 T^{2} )^{2} \))(\( ( 1 - 9 T + 83 T^{2} )^{2} \))(\( ( 1 - 10 T + 83 T^{2} )^{2} \))(\( ( 1 - 18 T + 83 T^{2} )^{2} \))(\( ( 1 + 83 T^{2} )^{2} \))(\( ( 1 + 83 T^{2} )^{2} \))(\( ( 1 + 6 T + 83 T^{2} )^{2} \))(\( ( 1 - 6 T + 83 T^{2} )^{2} \))(\( ( 1 - 7 T + 83 T^{2} )^{2} \))(\( ( 1 + 83 T^{2} )^{2} \))(\( ( 1 + 18 T + 83 T^{2} )^{2} \))(\( ( 1 + 10 T + 83 T^{2} )^{2} \))(\( ( 1 - 12 T + 83 T^{2} )^{2} \))(\( ( 1 - 6 T + 83 T^{2} )^{2} \))(\( ( 1 + 4 T + 83 T^{2} )^{2} \))(\( ( 1 + 12 T + 83 T^{2} )^{2} \))(\( ( 1 - 6 T + 83 T^{2} )^{2} \))(\( ( 1 - 12 T + 222 T^{2} - 1906 T^{3} + 18426 T^{4} - 82668 T^{5} + 571787 T^{6} )^{2} \))(\( ( 1 + 12 T + 222 T^{2} + 1906 T^{3} + 18426 T^{4} + 82668 T^{5} + 571787 T^{6} )^{2} \))
$89$ (\( 1 - 2 T - 85 T^{2} - 178 T^{3} + 7921 T^{4} \))(\( 1 - 10 T + 11 T^{2} - 890 T^{3} + 7921 T^{4} \))(\( 1 + 3 T - 80 T^{2} + 267 T^{3} + 7921 T^{4} \))(\( 1 - 7 T - 40 T^{2} - 623 T^{3} + 7921 T^{4} \))(\( 1 + 6 T - 53 T^{2} + 534 T^{3} + 7921 T^{4} \))(\( 1 + 10 T + 11 T^{2} + 890 T^{3} + 7921 T^{4} \))(\( 1 - 6 T - 53 T^{2} - 534 T^{3} + 7921 T^{4} \))(\( 1 - 15 T + 136 T^{2} - 1335 T^{3} + 7921 T^{4} \))(\( 1 + 6 T - 53 T^{2} + 534 T^{3} + 7921 T^{4} \))(\( 1 + 11 T + 32 T^{2} + 979 T^{3} + 7921 T^{4} \))(\( 1 + 11 T + 32 T^{2} + 979 T^{3} + 7921 T^{4} \))(\( 1 - 6 T - 53 T^{2} - 534 T^{3} + 7921 T^{4} \))(\( 1 - 15 T + 136 T^{2} - 1335 T^{3} + 7921 T^{4} \))(\( 1 - 6 T - 53 T^{2} - 534 T^{3} + 7921 T^{4} \))(\( 1 + 10 T + 11 T^{2} + 890 T^{3} + 7921 T^{4} \))(\( 1 + 14 T + 107 T^{2} + 1246 T^{3} + 7921 T^{4} \))(\( 1 + 3 T - 80 T^{2} + 267 T^{3} + 7921 T^{4} \))(\( 1 - 7 T - 40 T^{2} - 623 T^{3} + 7921 T^{4} \))(\( 1 - 2 T - 85 T^{2} - 178 T^{3} + 7921 T^{4} \))(\( 1 - 2 T - 85 T^{2} - 178 T^{3} + 7921 T^{4} \))(\( 1 - 6 T - 123 T^{2} + 398 T^{3} + 7074 T^{4} + 14802 T^{5} - 668251 T^{6} + 1317378 T^{7} + 56033154 T^{8} + 280577662 T^{9} - 7717295643 T^{10} - 33504356694 T^{11} + 496981290961 T^{12} \))(\( 1 - 6 T - 123 T^{2} + 398 T^{3} + 7074 T^{4} + 14802 T^{5} - 668251 T^{6} + 1317378 T^{7} + 56033154 T^{8} + 280577662 T^{9} - 7717295643 T^{10} - 33504356694 T^{11} + 496981290961 T^{12} \))
$97$ (\( ( 1 - 17 T + 97 T^{2} )^{2} \))(\( ( 1 - 16 T + 97 T^{2} )^{2} \))(\( ( 1 - 17 T + 97 T^{2} )^{2} \))(\( ( 1 - 7 T + 97 T^{2} )^{2} \))(\( ( 1 - T + 97 T^{2} )^{2} \))(\( ( 1 + 8 T + 97 T^{2} )^{2} \))(\( ( 1 - 7 T + 97 T^{2} )^{2} \))(\( ( 1 - 7 T + 97 T^{2} )^{2} \))(\( ( 1 + 8 T + 97 T^{2} )^{2} \))(\( ( 1 - T + 97 T^{2} )^{2} \))(\( ( 1 + T + 97 T^{2} )^{2} \))(\( ( 1 + 7 T + 97 T^{2} )^{2} \))(\( ( 1 + 7 T + 97 T^{2} )^{2} \))(\( ( 1 + 7 T + 97 T^{2} )^{2} \))(\( ( 1 - 8 T + 97 T^{2} )^{2} \))(\( ( 1 - 8 T + 97 T^{2} )^{2} \))(\( ( 1 + 17 T + 97 T^{2} )^{2} \))(\( ( 1 + 7 T + 97 T^{2} )^{2} \))(\( ( 1 - 16 T + 97 T^{2} )^{2} \))(\( ( 1 + 17 T + 97 T^{2} )^{2} \))(\( ( 1 + 24 T + 408 T^{2} + 4768 T^{3} + 39576 T^{4} + 225816 T^{5} + 912673 T^{6} )^{2} \))(\( ( 1 - 24 T + 408 T^{2} - 4768 T^{3} + 39576 T^{4} - 225816 T^{5} + 912673 T^{6} )^{2} \))
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