Properties

Label 1694.2.a
Level 1694
Weight 2
Character orbit a
Rep. character \(\chi_{1694}(1,\cdot)\)
Character field \(\Q\)
Dimension 54
Newform subspaces 27
Sturm bound 528
Trace bound 5

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Defining parameters

Level: \( N \) = \( 1694 = 2 \cdot 7 \cdot 11^{2} \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 1694.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 27 \)
Sturm bound: \(528\)
Trace bound: \(5\)
Distinguishing \(T_p\): \(3\), \(5\), \(13\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(1694))\).

Total New Old
Modular forms 288 54 234
Cusp forms 241 54 187
Eisenstein series 47 0 47

The following table gives the dimensions of the cuspidal new subspaces with specified eigenvalues for the Atkin-Lehner operators and the Fricke involution.

\(2\)\(7\)\(11\)FrickeDim.
\(+\)\(+\)\(+\)\(+\)\(4\)
\(+\)\(+\)\(-\)\(-\)\(9\)
\(+\)\(-\)\(+\)\(-\)\(10\)
\(+\)\(-\)\(-\)\(+\)\(4\)
\(-\)\(+\)\(+\)\(-\)\(8\)
\(-\)\(+\)\(-\)\(+\)\(6\)
\(-\)\(-\)\(+\)\(+\)\(2\)
\(-\)\(-\)\(-\)\(-\)\(11\)
Plus space\(+\)\(16\)
Minus space\(-\)\(38\)

Trace form

\( 54q + 6q^{3} + 54q^{4} + 2q^{5} + 2q^{6} + 62q^{9} + O(q^{10}) \) \( 54q + 6q^{3} + 54q^{4} + 2q^{5} + 2q^{6} + 62q^{9} - 6q^{10} + 6q^{12} + 6q^{13} - 2q^{14} + 16q^{15} + 54q^{16} - 4q^{18} + 10q^{19} + 2q^{20} + 6q^{21} + 2q^{24} + 66q^{25} - 2q^{26} + 36q^{27} + 8q^{29} + 16q^{30} + 36q^{31} + 4q^{34} - 2q^{35} + 62q^{36} + 12q^{37} + 14q^{38} + 8q^{39} - 6q^{40} - 24q^{41} - 2q^{42} + 12q^{43} - 14q^{45} + 8q^{46} + 4q^{47} + 6q^{48} + 54q^{49} + 4q^{50} - 12q^{51} + 6q^{52} - 8q^{53} + 20q^{54} - 2q^{56} - 4q^{57} + 2q^{59} + 16q^{60} + 10q^{61} - 12q^{62} - 12q^{63} + 54q^{64} + 4q^{65} + 36q^{67} + 24q^{69} + 2q^{70} + 16q^{71} - 4q^{72} - 4q^{73} + 12q^{74} + 42q^{75} + 10q^{76} + 16q^{78} - 24q^{79} + 2q^{80} + 82q^{81} - 4q^{82} + 10q^{83} + 6q^{84} + 4q^{85} + 16q^{86} - 36q^{87} + 12q^{89} - 6q^{90} + 6q^{91} + 32q^{93} - 4q^{94} - 32q^{95} + 2q^{96} + 8q^{97} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(1694))\) into newform subspaces

Label Dim. \(A\) Field CM Traces A-L signs $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\) 2 7 11
1694.2.a.a \(1\) \(13.527\) \(\Q\) None \(-1\) \(-1\) \(2\) \(-1\) \(+\) \(+\) \(-\) \(q-q^{2}-q^{3}+q^{4}+2q^{5}+q^{6}-q^{7}+\cdots\)
1694.2.a.b \(1\) \(13.527\) \(\Q\) None \(-1\) \(0\) \(-1\) \(1\) \(+\) \(-\) \(-\) \(q-q^{2}+q^{4}-q^{5}+q^{7}-q^{8}-3q^{9}+\cdots\)
1694.2.a.c \(1\) \(13.527\) \(\Q\) None \(-1\) \(0\) \(2\) \(1\) \(+\) \(-\) \(-\) \(q-q^{2}+q^{4}+2q^{5}+q^{7}-q^{8}-3q^{9}+\cdots\)
1694.2.a.d \(1\) \(13.527\) \(\Q\) None \(-1\) \(3\) \(2\) \(-1\) \(+\) \(+\) \(-\) \(q-q^{2}+3q^{3}+q^{4}+2q^{5}-3q^{6}-q^{7}+\cdots\)
1694.2.a.e \(1\) \(13.527\) \(\Q\) None \(1\) \(-2\) \(0\) \(-1\) \(-\) \(+\) \(-\) \(q+q^{2}-2q^{3}+q^{4}-2q^{6}-q^{7}+q^{8}+\cdots\)
1694.2.a.f \(1\) \(13.527\) \(\Q\) None \(1\) \(-1\) \(2\) \(1\) \(-\) \(-\) \(-\) \(q+q^{2}-q^{3}+q^{4}+2q^{5}-q^{6}+q^{7}+\cdots\)
1694.2.a.g \(1\) \(13.527\) \(\Q\) None \(1\) \(0\) \(-4\) \(1\) \(-\) \(-\) \(-\) \(q+q^{2}+q^{4}-4q^{5}+q^{7}+q^{8}-3q^{9}+\cdots\)
1694.2.a.h \(1\) \(13.527\) \(\Q\) None \(1\) \(0\) \(-1\) \(-1\) \(-\) \(+\) \(-\) \(q+q^{2}+q^{4}-q^{5}-q^{7}+q^{8}-3q^{9}+\cdots\)
1694.2.a.i \(1\) \(13.527\) \(\Q\) None \(1\) \(2\) \(2\) \(1\) \(-\) \(-\) \(-\) \(q+q^{2}+2q^{3}+q^{4}+2q^{5}+2q^{6}+q^{7}+\cdots\)
1694.2.a.j \(1\) \(13.527\) \(\Q\) None \(1\) \(3\) \(2\) \(1\) \(-\) \(-\) \(-\) \(q+q^{2}+3q^{3}+q^{4}+2q^{5}+3q^{6}+q^{7}+\cdots\)
1694.2.a.k \(2\) \(13.527\) \(\Q(\sqrt{5}) \) None \(-2\) \(-2\) \(-3\) \(-2\) \(+\) \(+\) \(-\) \(q-q^{2}-q^{3}+q^{4}+(-1-\beta )q^{5}+q^{6}+\cdots\)
1694.2.a.l \(2\) \(13.527\) \(\Q(\sqrt{5}) \) None \(-2\) \(-2\) \(2\) \(-2\) \(+\) \(+\) \(-\) \(q-q^{2}+(-1-\beta )q^{3}+q^{4}+(1+\beta )q^{5}+\cdots\)
1694.2.a.m \(2\) \(13.527\) \(\Q(\sqrt{5}) \) None \(-2\) \(-1\) \(4\) \(-2\) \(+\) \(+\) \(+\) \(q-q^{2}-\beta q^{3}+q^{4}+2q^{5}+\beta q^{6}-q^{7}+\cdots\)
1694.2.a.n \(2\) \(13.527\) \(\Q(\sqrt{5}) \) None \(-2\) \(0\) \(-3\) \(-2\) \(+\) \(+\) \(+\) \(q-q^{2}+(1-2\beta )q^{3}+q^{4}+(-1-\beta )q^{5}+\cdots\)
1694.2.a.o \(2\) \(13.527\) \(\Q(\sqrt{5}) \) None \(-2\) \(0\) \(-1\) \(2\) \(+\) \(-\) \(-\) \(q-q^{2}+(1-2\beta )q^{3}+q^{4}+(-1+\beta )q^{5}+\cdots\)
1694.2.a.p \(2\) \(13.527\) \(\Q(\sqrt{3}) \) None \(-2\) \(2\) \(2\) \(2\) \(+\) \(-\) \(+\) \(q-q^{2}+(1+\beta )q^{3}+q^{4}+(1+\beta )q^{5}+\cdots\)
1694.2.a.q \(2\) \(13.527\) \(\Q(\sqrt{5}) \) None \(2\) \(-2\) \(-3\) \(2\) \(-\) \(-\) \(+\) \(q+q^{2}-q^{3}+q^{4}+(-1-\beta )q^{5}-q^{6}+\cdots\)
1694.2.a.r \(2\) \(13.527\) \(\Q(\sqrt{5}) \) None \(2\) \(-1\) \(4\) \(2\) \(-\) \(-\) \(-\) \(q+q^{2}-\beta q^{3}+q^{4}+2q^{5}-\beta q^{6}+q^{7}+\cdots\)
1694.2.a.s \(2\) \(13.527\) \(\Q(\sqrt{5}) \) None \(2\) \(0\) \(-3\) \(2\) \(-\) \(-\) \(-\) \(q+q^{2}+(1-2\beta )q^{3}+q^{4}+(-1-\beta )q^{5}+\cdots\)
1694.2.a.t \(2\) \(13.527\) \(\Q(\sqrt{5}) \) None \(2\) \(0\) \(-1\) \(-2\) \(-\) \(+\) \(+\) \(q+q^{2}+(1-2\beta )q^{3}+q^{4}+(-1+\beta )q^{5}+\cdots\)
1694.2.a.u \(2\) \(13.527\) \(\Q(\sqrt{3}) \) None \(2\) \(2\) \(2\) \(-2\) \(-\) \(+\) \(+\) \(q+q^{2}+(1+\beta )q^{3}+q^{4}+(1+\beta )q^{5}+\cdots\)
1694.2.a.v \(3\) \(13.527\) 3.3.2024.1 None \(-3\) \(2\) \(-1\) \(-3\) \(+\) \(+\) \(-\) \(q-q^{2}+(1-\beta _{1})q^{3}+q^{4}-\beta _{2}q^{5}+(-1+\cdots)q^{6}+\cdots\)
1694.2.a.w \(3\) \(13.527\) 3.3.2024.1 None \(3\) \(2\) \(-1\) \(3\) \(-\) \(-\) \(-\) \(q+q^{2}+(1-\beta _{1})q^{3}+q^{4}-\beta _{2}q^{5}+(1+\cdots)q^{6}+\cdots\)
1694.2.a.x \(4\) \(13.527\) 4.4.7625.1 None \(-4\) \(-3\) \(-5\) \(4\) \(+\) \(-\) \(+\) \(q-q^{2}+(-1+\beta _{1})q^{3}+q^{4}+(-1-\beta _{1}+\cdots)q^{5}+\cdots\)
1694.2.a.y \(4\) \(13.527\) 4.4.13824.1 None \(-4\) \(4\) \(4\) \(4\) \(+\) \(-\) \(+\) \(q-q^{2}+(1+\beta _{1})q^{3}+q^{4}+(1+\beta _{2}-\beta _{3})q^{5}+\cdots\)
1694.2.a.z \(4\) \(13.527\) 4.4.7625.1 None \(4\) \(-3\) \(-5\) \(-4\) \(-\) \(+\) \(-\) \(q+q^{2}+(-1+\beta _{1})q^{3}+q^{4}+(-1-\beta _{1}+\cdots)q^{5}+\cdots\)
1694.2.a.ba \(4\) \(13.527\) 4.4.13824.1 None \(4\) \(4\) \(4\) \(-4\) \(-\) \(+\) \(+\) \(q+q^{2}+(1+\beta _{1})q^{3}+q^{4}+(1+\beta _{2}-\beta _{3})q^{5}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(1694))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(1694)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(11))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(14))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(77))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(121))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(154))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(242))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(847))\)\(^{\oplus 2}\)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ (\( 1 + T \))(\( 1 + T \))(\( 1 + T \))(\( 1 + T \))(\( 1 - T \))(\( 1 - T \))(\( 1 - T \))(\( 1 - T \))(\( 1 - T \))(\( 1 - T \))(\( ( 1 + T )^{2} \))(\( ( 1 + T )^{2} \))(\( ( 1 + T )^{2} \))(\( ( 1 + T )^{2} \))(\( ( 1 + T )^{2} \))(\( ( 1 + T )^{2} \))(\( ( 1 - T )^{2} \))(\( ( 1 - T )^{2} \))(\( ( 1 - T )^{2} \))(\( ( 1 - T )^{2} \))(\( ( 1 - T )^{2} \))(\( ( 1 + T )^{3} \))(\( ( 1 - T )^{3} \))(\( ( 1 + T )^{4} \))(\( ( 1 + T )^{4} \))(\( ( 1 - T )^{4} \))(\( ( 1 - T )^{4} \))
$3$ (\( 1 + T + 3 T^{2} \))(\( 1 + 3 T^{2} \))(\( 1 + 3 T^{2} \))(\( 1 - 3 T + 3 T^{2} \))(\( 1 + 2 T + 3 T^{2} \))(\( 1 + T + 3 T^{2} \))(\( 1 + 3 T^{2} \))(\( 1 + 3 T^{2} \))(\( 1 - 2 T + 3 T^{2} \))(\( 1 - 3 T + 3 T^{2} \))(\( ( 1 + T + 3 T^{2} )^{2} \))(\( 1 + 2 T + 2 T^{2} + 6 T^{3} + 9 T^{4} \))(\( 1 + T + 5 T^{2} + 3 T^{3} + 9 T^{4} \))(\( 1 + T^{2} + 9 T^{4} \))(\( 1 + T^{2} + 9 T^{4} \))(\( 1 - 2 T + 4 T^{2} - 6 T^{3} + 9 T^{4} \))(\( ( 1 + T + 3 T^{2} )^{2} \))(\( 1 + T + 5 T^{2} + 3 T^{3} + 9 T^{4} \))(\( 1 + T^{2} + 9 T^{4} \))(\( 1 + T^{2} + 9 T^{4} \))(\( 1 - 2 T + 4 T^{2} - 6 T^{3} + 9 T^{4} \))(\( 1 - 2 T + 4 T^{3} - 18 T^{5} + 27 T^{6} \))(\( 1 - 2 T + 4 T^{3} - 18 T^{5} + 27 T^{6} \))(\( 1 + 3 T + 6 T^{2} + 14 T^{3} + 29 T^{4} + 42 T^{5} + 54 T^{6} + 81 T^{7} + 81 T^{8} \))(\( 1 - 4 T + 12 T^{2} - 28 T^{3} + 55 T^{4} - 84 T^{5} + 108 T^{6} - 108 T^{7} + 81 T^{8} \))(\( 1 + 3 T + 6 T^{2} + 14 T^{3} + 29 T^{4} + 42 T^{5} + 54 T^{6} + 81 T^{7} + 81 T^{8} \))(\( 1 - 4 T + 12 T^{2} - 28 T^{3} + 55 T^{4} - 84 T^{5} + 108 T^{6} - 108 T^{7} + 81 T^{8} \))
$5$ (\( 1 - 2 T + 5 T^{2} \))(\( 1 + T + 5 T^{2} \))(\( 1 - 2 T + 5 T^{2} \))(\( 1 - 2 T + 5 T^{2} \))(\( 1 + 5 T^{2} \))(\( 1 - 2 T + 5 T^{2} \))(\( 1 + 4 T + 5 T^{2} \))(\( 1 + T + 5 T^{2} \))(\( 1 - 2 T + 5 T^{2} \))(\( 1 - 2 T + 5 T^{2} \))(\( 1 + 3 T + 11 T^{2} + 15 T^{3} + 25 T^{4} \))(\( 1 - 2 T + 6 T^{2} - 10 T^{3} + 25 T^{4} \))(\( ( 1 - 2 T + 5 T^{2} )^{2} \))(\( 1 + 3 T + 11 T^{2} + 15 T^{3} + 25 T^{4} \))(\( 1 + T + 9 T^{2} + 5 T^{3} + 25 T^{4} \))(\( 1 - 2 T + 8 T^{2} - 10 T^{3} + 25 T^{4} \))(\( 1 + 3 T + 11 T^{2} + 15 T^{3} + 25 T^{4} \))(\( ( 1 - 2 T + 5 T^{2} )^{2} \))(\( 1 + 3 T + 11 T^{2} + 15 T^{3} + 25 T^{4} \))(\( 1 + T + 9 T^{2} + 5 T^{3} + 25 T^{4} \))(\( 1 - 2 T + 8 T^{2} - 10 T^{3} + 25 T^{4} \))(\( 1 + T - T^{2} - 14 T^{3} - 5 T^{4} + 25 T^{5} + 125 T^{6} \))(\( 1 + T - T^{2} - 14 T^{3} - 5 T^{4} + 25 T^{5} + 125 T^{6} \))(\( 1 + 5 T + 15 T^{2} + 35 T^{3} + 80 T^{4} + 175 T^{5} + 375 T^{6} + 625 T^{7} + 625 T^{8} \))(\( 1 - 4 T + 8 T^{2} - 4 T^{3} - 14 T^{4} - 20 T^{5} + 200 T^{6} - 500 T^{7} + 625 T^{8} \))(\( 1 + 5 T + 15 T^{2} + 35 T^{3} + 80 T^{4} + 175 T^{5} + 375 T^{6} + 625 T^{7} + 625 T^{8} \))(\( 1 - 4 T + 8 T^{2} - 4 T^{3} - 14 T^{4} - 20 T^{5} + 200 T^{6} - 500 T^{7} + 625 T^{8} \))
$7$ (\( 1 + T \))(\( 1 - T \))(\( 1 - T \))(\( 1 + T \))(\( 1 + T \))(\( 1 - T \))(\( 1 - T \))(\( 1 + T \))(\( 1 - T \))(\( 1 - T \))(\( ( 1 + T )^{2} \))(\( ( 1 + T )^{2} \))(\( ( 1 + T )^{2} \))(\( ( 1 + T )^{2} \))(\( ( 1 - T )^{2} \))(\( ( 1 - T )^{2} \))(\( ( 1 - T )^{2} \))(\( ( 1 - T )^{2} \))(\( ( 1 - T )^{2} \))(\( ( 1 + T )^{2} \))(\( ( 1 + T )^{2} \))(\( ( 1 + T )^{3} \))(\( ( 1 - T )^{3} \))(\( ( 1 - T )^{4} \))(\( ( 1 - T )^{4} \))(\( ( 1 + T )^{4} \))(\( ( 1 + T )^{4} \))
$11$ 1
$13$ (\( 1 + 7 T + 13 T^{2} \))(\( 1 - T + 13 T^{2} \))(\( 1 + 2 T + 13 T^{2} \))(\( 1 - 5 T + 13 T^{2} \))(\( 1 - 4 T + 13 T^{2} \))(\( 1 - 7 T + 13 T^{2} \))(\( 1 + 2 T + 13 T^{2} \))(\( 1 + T + 13 T^{2} \))(\( 1 - 4 T + 13 T^{2} \))(\( 1 + 5 T + 13 T^{2} \))(\( 1 - 2 T + 7 T^{2} - 26 T^{3} + 169 T^{4} \))(\( 1 - 2 T + 22 T^{2} - 26 T^{3} + 169 T^{4} \))(\( 1 - 2 T + 22 T^{2} - 26 T^{3} + 169 T^{4} \))(\( ( 1 + 3 T + 13 T^{2} )^{2} \))(\( 1 + 4 T + 25 T^{2} + 52 T^{3} + 169 T^{4} \))(\( 1 + 6 T + 32 T^{2} + 78 T^{3} + 169 T^{4} \))(\( 1 + 2 T + 7 T^{2} + 26 T^{3} + 169 T^{4} \))(\( 1 + 2 T + 22 T^{2} + 26 T^{3} + 169 T^{4} \))(\( ( 1 - 3 T + 13 T^{2} )^{2} \))(\( 1 - 4 T + 25 T^{2} - 52 T^{3} + 169 T^{4} \))(\( 1 - 6 T + 32 T^{2} - 78 T^{3} + 169 T^{4} \))(\( 1 - 3 T + 16 T^{2} - 5 T^{3} + 208 T^{4} - 507 T^{5} + 2197 T^{6} \))(\( 1 + 3 T + 16 T^{2} + 5 T^{3} + 208 T^{4} + 507 T^{5} + 2197 T^{6} \))(\( 1 - 14 T + 103 T^{2} - 520 T^{3} + 2056 T^{4} - 6760 T^{5} + 17407 T^{6} - 30758 T^{7} + 28561 T^{8} \))(\( 1 + 28 T^{2} + 60 T^{3} + 351 T^{4} + 780 T^{5} + 4732 T^{6} + 28561 T^{8} \))(\( 1 + 14 T + 103 T^{2} + 520 T^{3} + 2056 T^{4} + 6760 T^{5} + 17407 T^{6} + 30758 T^{7} + 28561 T^{8} \))(\( 1 + 28 T^{2} - 60 T^{3} + 351 T^{4} - 780 T^{5} + 4732 T^{6} + 28561 T^{8} \))
$17$ (\( 1 + 17 T^{2} \))(\( 1 - T + 17 T^{2} \))(\( 1 + 2 T + 17 T^{2} \))(\( 1 - 8 T + 17 T^{2} \))(\( 1 + 6 T + 17 T^{2} \))(\( 1 + 17 T^{2} \))(\( 1 - 4 T + 17 T^{2} \))(\( 1 + T + 17 T^{2} \))(\( 1 + 17 T^{2} \))(\( 1 + 8 T + 17 T^{2} \))(\( 1 + 6 T + 23 T^{2} + 102 T^{3} + 289 T^{4} \))(\( 1 - 4 T + 18 T^{2} - 68 T^{3} + 289 T^{4} \))(\( 1 + 5 T + 29 T^{2} + 85 T^{3} + 289 T^{4} \))(\( 1 - 2 T + 15 T^{2} - 34 T^{3} + 289 T^{4} \))(\( 1 + 4 T + 33 T^{2} + 68 T^{3} + 289 T^{4} \))(\( 1 - 14 T^{2} + 289 T^{4} \))(\( 1 - 6 T + 23 T^{2} - 102 T^{3} + 289 T^{4} \))(\( 1 - 5 T + 29 T^{2} - 85 T^{3} + 289 T^{4} \))(\( 1 + 2 T + 15 T^{2} + 34 T^{3} + 289 T^{4} \))(\( 1 - 4 T + 33 T^{2} - 68 T^{3} + 289 T^{4} \))(\( 1 - 14 T^{2} + 289 T^{4} \))(\( 1 - T + 35 T^{2} - 10 T^{3} + 595 T^{4} - 289 T^{5} + 4913 T^{6} \))(\( 1 + T + 35 T^{2} + 10 T^{3} + 595 T^{4} + 289 T^{5} + 4913 T^{6} \))(\( 1 + T + 34 T^{2} - 58 T^{3} + 499 T^{4} - 986 T^{5} + 9826 T^{6} + 4913 T^{7} + 83521 T^{8} \))(\( 1 + 44 T^{2} + 1014 T^{4} + 12716 T^{6} + 83521 T^{8} \))(\( 1 - T + 34 T^{2} + 58 T^{3} + 499 T^{4} + 986 T^{5} + 9826 T^{6} - 4913 T^{7} + 83521 T^{8} \))(\( 1 + 44 T^{2} + 1014 T^{4} + 12716 T^{6} + 83521 T^{8} \))
$19$ (\( 1 - T + 19 T^{2} \))(\( 1 + 19 T^{2} \))(\( 1 + 19 T^{2} \))(\( 1 + 3 T + 19 T^{2} \))(\( 1 + 2 T + 19 T^{2} \))(\( 1 + T + 19 T^{2} \))(\( 1 - 6 T + 19 T^{2} \))(\( 1 + 19 T^{2} \))(\( 1 + 4 T + 19 T^{2} \))(\( 1 - 3 T + 19 T^{2} \))(\( 1 - 7 T^{2} + 361 T^{4} \))(\( 1 - 10 T + 58 T^{2} - 190 T^{3} + 361 T^{4} \))(\( 1 + 7 T + 39 T^{2} + 133 T^{3} + 361 T^{4} \))(\( 1 + 2 T + 19 T^{2} + 38 T^{3} + 361 T^{4} \))(\( ( 1 + 5 T + 19 T^{2} )^{2} \))(\( 1 - 2 T + 36 T^{2} - 38 T^{3} + 361 T^{4} \))(\( 1 - 7 T^{2} + 361 T^{4} \))(\( 1 - 7 T + 39 T^{2} - 133 T^{3} + 361 T^{4} \))(\( 1 - 2 T + 19 T^{2} - 38 T^{3} + 361 T^{4} \))(\( ( 1 - 5 T + 19 T^{2} )^{2} \))(\( 1 + 2 T + 36 T^{2} + 38 T^{3} + 361 T^{4} \))(\( 1 - 2 T + 48 T^{2} - 60 T^{3} + 912 T^{4} - 722 T^{5} + 6859 T^{6} \))(\( 1 + 2 T + 48 T^{2} + 60 T^{3} + 912 T^{4} + 722 T^{5} + 6859 T^{6} \))(\( 1 - 13 T + 120 T^{2} - 788 T^{3} + 3849 T^{4} - 14972 T^{5} + 43320 T^{6} - 89167 T^{7} + 130321 T^{8} \))(\( 1 + 8 T + 28 T^{2} - 28 T^{3} - 569 T^{4} - 532 T^{5} + 10108 T^{6} + 54872 T^{7} + 130321 T^{8} \))(\( 1 + 13 T + 120 T^{2} + 788 T^{3} + 3849 T^{4} + 14972 T^{5} + 43320 T^{6} + 89167 T^{7} + 130321 T^{8} \))(\( 1 - 8 T + 28 T^{2} + 28 T^{3} - 569 T^{4} + 532 T^{5} + 10108 T^{6} - 54872 T^{7} + 130321 T^{8} \))
$23$ (\( 1 - T + 23 T^{2} \))(\( 1 - 4 T + 23 T^{2} \))(\( 1 + 8 T + 23 T^{2} \))(\( 1 - T + 23 T^{2} \))(\( 1 + 23 T^{2} \))(\( 1 - T + 23 T^{2} \))(\( 1 - 4 T + 23 T^{2} \))(\( 1 - 4 T + 23 T^{2} \))(\( 1 - 4 T + 23 T^{2} \))(\( 1 - T + 23 T^{2} \))(\( 1 + 12 T + 77 T^{2} + 276 T^{3} + 529 T^{4} \))(\( ( 1 - 4 T + 23 T^{2} )^{2} \))(\( 1 - 8 T + 42 T^{2} - 184 T^{3} + 529 T^{4} \))(\( 1 + 2 T + 27 T^{2} + 46 T^{3} + 529 T^{4} \))(\( 1 - 4 T + 45 T^{2} - 92 T^{3} + 529 T^{4} \))(\( 1 - 2 T^{2} + 529 T^{4} \))(\( 1 + 12 T + 77 T^{2} + 276 T^{3} + 529 T^{4} \))(\( 1 - 8 T + 42 T^{2} - 184 T^{3} + 529 T^{4} \))(\( 1 + 2 T + 27 T^{2} + 46 T^{3} + 529 T^{4} \))(\( 1 - 4 T + 45 T^{2} - 92 T^{3} + 529 T^{4} \))(\( 1 - 2 T^{2} + 529 T^{4} \))(\( 1 + 10 T + 46 T^{2} + 184 T^{3} + 1058 T^{4} + 5290 T^{5} + 12167 T^{6} \))(\( 1 + 10 T + 46 T^{2} + 184 T^{3} + 1058 T^{4} + 5290 T^{5} + 12167 T^{6} \))(\( 1 + 10 T + 117 T^{2} + 690 T^{3} + 4304 T^{4} + 15870 T^{5} + 61893 T^{6} + 121670 T^{7} + 279841 T^{8} \))(\( 1 - 12 T + 98 T^{2} - 648 T^{3} + 3483 T^{4} - 14904 T^{5} + 51842 T^{6} - 146004 T^{7} + 279841 T^{8} \))(\( 1 + 10 T + 117 T^{2} + 690 T^{3} + 4304 T^{4} + 15870 T^{5} + 61893 T^{6} + 121670 T^{7} + 279841 T^{8} \))(\( 1 - 12 T + 98 T^{2} - 648 T^{3} + 3483 T^{4} - 14904 T^{5} + 51842 T^{6} - 146004 T^{7} + 279841 T^{8} \))
$29$ (\( 1 - 8 T + 29 T^{2} \))(\( 1 - 5 T + 29 T^{2} \))(\( 1 - 2 T + 29 T^{2} \))(\( 1 + 8 T + 29 T^{2} \))(\( 1 - 6 T + 29 T^{2} \))(\( 1 + 8 T + 29 T^{2} \))(\( 1 - 2 T + 29 T^{2} \))(\( 1 + 5 T + 29 T^{2} \))(\( 1 + 2 T + 29 T^{2} \))(\( 1 - 8 T + 29 T^{2} \))(\( 1 - 10 T + 63 T^{2} - 290 T^{3} + 841 T^{4} \))(\( 1 + 38 T^{2} + 841 T^{4} \))(\( ( 1 + 8 T + 29 T^{2} )^{2} \))(\( ( 1 - 3 T + 29 T^{2} )^{2} \))(\( 1 + 6 T + 47 T^{2} + 174 T^{3} + 841 T^{4} \))(\( 1 - 8 T + 62 T^{2} - 232 T^{3} + 841 T^{4} \))(\( 1 + 10 T + 63 T^{2} + 290 T^{3} + 841 T^{4} \))(\( ( 1 - 8 T + 29 T^{2} )^{2} \))(\( ( 1 + 3 T + 29 T^{2} )^{2} \))(\( 1 - 6 T + 47 T^{2} - 174 T^{3} + 841 T^{4} \))(\( 1 + 8 T + 62 T^{2} + 232 T^{3} + 841 T^{4} \))(\( 1 + 3 T + 31 T^{2} + 222 T^{3} + 899 T^{4} + 2523 T^{5} + 24389 T^{6} \))(\( 1 - 3 T + 31 T^{2} - 222 T^{3} + 899 T^{4} - 2523 T^{5} + 24389 T^{6} \))(\( 1 - 6 T + 77 T^{2} - 298 T^{3} + 2720 T^{4} - 8642 T^{5} + 64757 T^{6} - 146334 T^{7} + 707281 T^{8} \))(\( 1 + 8 T + 68 T^{2} + 248 T^{3} + 1558 T^{4} + 7192 T^{5} + 57188 T^{6} + 195112 T^{7} + 707281 T^{8} \))(\( 1 + 6 T + 77 T^{2} + 298 T^{3} + 2720 T^{4} + 8642 T^{5} + 64757 T^{6} + 146334 T^{7} + 707281 T^{8} \))(\( 1 - 8 T + 68 T^{2} - 248 T^{3} + 1558 T^{4} - 7192 T^{5} + 57188 T^{6} - 195112 T^{7} + 707281 T^{8} \))
$31$ (\( 1 - 8 T + 31 T^{2} \))(\( 1 + 8 T + 31 T^{2} \))(\( 1 + 8 T + 31 T^{2} \))(\( 1 + 8 T + 31 T^{2} \))(\( 1 + 4 T + 31 T^{2} \))(\( 1 - 8 T + 31 T^{2} \))(\( 1 + 2 T + 31 T^{2} \))(\( 1 + 8 T + 31 T^{2} \))(\( 1 + 10 T + 31 T^{2} \))(\( 1 + 8 T + 31 T^{2} \))(\( ( 1 + 3 T + 31 T^{2} )^{2} \))(\( ( 1 - 2 T + 31 T^{2} )^{2} \))(\( 1 + 42 T^{2} + 961 T^{4} \))(\( 1 + 4 T + 21 T^{2} + 124 T^{3} + 961 T^{4} \))(\( 1 - 4 T + 21 T^{2} - 124 T^{3} + 961 T^{4} \))(\( 1 - 8 T + 66 T^{2} - 248 T^{3} + 961 T^{4} \))(\( ( 1 + 3 T + 31 T^{2} )^{2} \))(\( 1 + 42 T^{2} + 961 T^{4} \))(\( 1 + 4 T + 21 T^{2} + 124 T^{3} + 961 T^{4} \))(\( 1 - 4 T + 21 T^{2} - 124 T^{3} + 961 T^{4} \))(\( 1 - 8 T + 66 T^{2} - 248 T^{3} + 961 T^{4} \))(\( ( 1 - 8 T + 31 T^{2} )^{3} \))(\( ( 1 - 8 T + 31 T^{2} )^{3} \))(\( 1 + 6 T + 125 T^{2} + 534 T^{3} + 5824 T^{4} + 16554 T^{5} + 120125 T^{6} + 178746 T^{7} + 923521 T^{8} \))(\( 1 - 16 T + 172 T^{2} - 1360 T^{3} + 8614 T^{4} - 42160 T^{5} + 165292 T^{6} - 476656 T^{7} + 923521 T^{8} \))(\( 1 + 6 T + 125 T^{2} + 534 T^{3} + 5824 T^{4} + 16554 T^{5} + 120125 T^{6} + 178746 T^{7} + 923521 T^{8} \))(\( 1 - 16 T + 172 T^{2} - 1360 T^{3} + 8614 T^{4} - 42160 T^{5} + 165292 T^{6} - 476656 T^{7} + 923521 T^{8} \))
$37$ (\( 1 - 6 T + 37 T^{2} \))(\( 1 + 5 T + 37 T^{2} \))(\( 1 + 2 T + 37 T^{2} \))(\( 1 + 2 T + 37 T^{2} \))(\( 1 - 2 T + 37 T^{2} \))(\( 1 - 6 T + 37 T^{2} \))(\( 1 - 10 T + 37 T^{2} \))(\( 1 + 5 T + 37 T^{2} \))(\( 1 + 6 T + 37 T^{2} \))(\( 1 + 2 T + 37 T^{2} \))(\( 1 + 9 T + 83 T^{2} + 333 T^{3} + 1369 T^{4} \))(\( 1 + 4 T - 2 T^{2} + 148 T^{3} + 1369 T^{4} \))(\( 1 + 2 T + 30 T^{2} + 74 T^{3} + 1369 T^{4} \))(\( 1 - 5 T + 19 T^{2} - 185 T^{3} + 1369 T^{4} \))(\( 1 + 9 T + 83 T^{2} + 333 T^{3} + 1369 T^{4} \))(\( ( 1 - 6 T + 37 T^{2} )^{2} \))(\( 1 + 9 T + 83 T^{2} + 333 T^{3} + 1369 T^{4} \))(\( 1 + 2 T + 30 T^{2} + 74 T^{3} + 1369 T^{4} \))(\( 1 - 5 T + 19 T^{2} - 185 T^{3} + 1369 T^{4} \))(\( 1 + 9 T + 83 T^{2} + 333 T^{3} + 1369 T^{4} \))(\( ( 1 - 6 T + 37 T^{2} )^{2} \))(\( 1 - 7 T + 111 T^{2} - 474 T^{3} + 4107 T^{4} - 9583 T^{5} + 50653 T^{6} \))(\( 1 - 7 T + 111 T^{2} - 474 T^{3} + 4107 T^{4} - 9583 T^{5} + 50653 T^{6} \))(\( 1 - 3 T + 47 T^{2} - 255 T^{3} + 2776 T^{4} - 9435 T^{5} + 64343 T^{6} - 151959 T^{7} + 1874161 T^{8} \))(\( 1 + 52 T^{2} + 288 T^{3} + 1254 T^{4} + 10656 T^{5} + 71188 T^{6} + 1874161 T^{8} \))(\( 1 - 3 T + 47 T^{2} - 255 T^{3} + 2776 T^{4} - 9435 T^{5} + 64343 T^{6} - 151959 T^{7} + 1874161 T^{8} \))(\( 1 + 52 T^{2} + 288 T^{3} + 1254 T^{4} + 10656 T^{5} + 71188 T^{6} + 1874161 T^{8} \))
$41$ (\( 1 - 6 T + 41 T^{2} \))(\( 1 - 5 T + 41 T^{2} \))(\( 1 + 10 T + 41 T^{2} \))(\( 1 + 2 T + 41 T^{2} \))(\( 1 + 6 T + 41 T^{2} \))(\( 1 + 6 T + 41 T^{2} \))(\( 1 + 4 T + 41 T^{2} \))(\( 1 + 5 T + 41 T^{2} \))(\( 1 + 41 T^{2} \))(\( 1 - 2 T + 41 T^{2} \))(\( 1 + 4 T + 66 T^{2} + 164 T^{3} + 1681 T^{4} \))(\( 1 + 4 T + 66 T^{2} + 164 T^{3} + 1681 T^{4} \))(\( 1 + 5 T + 77 T^{2} + 205 T^{3} + 1681 T^{4} \))(\( 1 + 12 T + 98 T^{2} + 492 T^{3} + 1681 T^{4} \))(\( 1 + 2 T^{2} + 1681 T^{4} \))(\( 1 - 8 T + 50 T^{2} - 328 T^{3} + 1681 T^{4} \))(\( 1 - 4 T + 66 T^{2} - 164 T^{3} + 1681 T^{4} \))(\( 1 - 5 T + 77 T^{2} - 205 T^{3} + 1681 T^{4} \))(\( 1 - 12 T + 98 T^{2} - 492 T^{3} + 1681 T^{4} \))(\( 1 + 2 T^{2} + 1681 T^{4} \))(\( 1 + 8 T + 50 T^{2} + 328 T^{3} + 1681 T^{4} \))(\( 1 - 17 T + 203 T^{2} - 1502 T^{3} + 8323 T^{4} - 28577 T^{5} + 68921 T^{6} \))(\( 1 + 17 T + 203 T^{2} + 1502 T^{3} + 8323 T^{4} + 28577 T^{5} + 68921 T^{6} \))(\( 1 + T + 135 T^{2} + 139 T^{3} + 7724 T^{4} + 5699 T^{5} + 226935 T^{6} + 68921 T^{7} + 2825761 T^{8} \))(\( 1 + 8 T + 68 T^{2} + 344 T^{3} + 2614 T^{4} + 14104 T^{5} + 114308 T^{6} + 551368 T^{7} + 2825761 T^{8} \))(\( 1 - T + 135 T^{2} - 139 T^{3} + 7724 T^{4} - 5699 T^{5} + 226935 T^{6} - 68921 T^{7} + 2825761 T^{8} \))(\( 1 - 8 T + 68 T^{2} - 344 T^{3} + 2614 T^{4} - 14104 T^{5} + 114308 T^{6} - 551368 T^{7} + 2825761 T^{8} \))
$43$ (\( 1 + 10 T + 43 T^{2} \))(\( 1 + 4 T + 43 T^{2} \))(\( 1 + 4 T + 43 T^{2} \))(\( 1 + 2 T + 43 T^{2} \))(\( 1 + 8 T + 43 T^{2} \))(\( 1 - 10 T + 43 T^{2} \))(\( 1 - 8 T + 43 T^{2} \))(\( 1 - 4 T + 43 T^{2} \))(\( 1 - 4 T + 43 T^{2} \))(\( 1 - 2 T + 43 T^{2} \))(\( 1 - 17 T + 157 T^{2} - 731 T^{3} + 1849 T^{4} \))(\( 1 - 12 T + 102 T^{2} - 516 T^{3} + 1849 T^{4} \))(\( 1 + 3 T + 77 T^{2} + 129 T^{3} + 1849 T^{4} \))(\( 1 + T + 75 T^{2} + 43 T^{3} + 1849 T^{4} \))(\( 1 + 3 T + 77 T^{2} + 129 T^{3} + 1849 T^{4} \))(\( 1 + 4 T + 78 T^{2} + 172 T^{3} + 1849 T^{4} \))(\( 1 + 17 T + 157 T^{2} + 731 T^{3} + 1849 T^{4} \))(\( 1 - 3 T + 77 T^{2} - 129 T^{3} + 1849 T^{4} \))(\( 1 - T + 75 T^{2} - 43 T^{3} + 1849 T^{4} \))(\( 1 - 3 T + 77 T^{2} - 129 T^{3} + 1849 T^{4} \))(\( 1 - 4 T + 78 T^{2} - 172 T^{3} + 1849 T^{4} \))(\( 1 + 8 T + 109 T^{2} + 672 T^{3} + 4687 T^{4} + 14792 T^{5} + 79507 T^{6} \))(\( 1 - 8 T + 109 T^{2} - 672 T^{3} + 4687 T^{4} - 14792 T^{5} + 79507 T^{6} \))(\( 1 + 8 T + 151 T^{2} + 874 T^{3} + 9259 T^{4} + 37582 T^{5} + 279199 T^{6} + 636056 T^{7} + 3418801 T^{8} \))(\( 1 - 16 T + 220 T^{2} - 1840 T^{3} + 14278 T^{4} - 79120 T^{5} + 406780 T^{6} - 1272112 T^{7} + 3418801 T^{8} \))(\( 1 - 8 T + 151 T^{2} - 874 T^{3} + 9259 T^{4} - 37582 T^{5} + 279199 T^{6} - 636056 T^{7} + 3418801 T^{8} \))(\( 1 + 16 T + 220 T^{2} + 1840 T^{3} + 14278 T^{4} + 79120 T^{5} + 406780 T^{6} + 1272112 T^{7} + 3418801 T^{8} \))
$47$ (\( 1 - 4 T + 47 T^{2} \))(\( 1 + 4 T + 47 T^{2} \))(\( 1 - 8 T + 47 T^{2} \))(\( 1 + 4 T + 47 T^{2} \))(\( 1 + 12 T + 47 T^{2} \))(\( 1 - 4 T + 47 T^{2} \))(\( 1 - 2 T + 47 T^{2} \))(\( 1 + 4 T + 47 T^{2} \))(\( 1 - 10 T + 47 T^{2} \))(\( 1 + 4 T + 47 T^{2} \))(\( 1 - T + 93 T^{2} - 47 T^{3} + 2209 T^{4} \))(\( ( 1 + 2 T + 47 T^{2} )^{2} \))(\( 1 + 10 T + 74 T^{2} + 470 T^{3} + 2209 T^{4} \))(\( 1 - 9 T + 103 T^{2} - 423 T^{3} + 2209 T^{4} \))(\( 1 - 11 T + 123 T^{2} - 517 T^{3} + 2209 T^{4} \))(\( 1 + 16 T + 146 T^{2} + 752 T^{3} + 2209 T^{4} \))(\( 1 - T + 93 T^{2} - 47 T^{3} + 2209 T^{4} \))(\( 1 + 10 T + 74 T^{2} + 470 T^{3} + 2209 T^{4} \))(\( 1 - 9 T + 103 T^{2} - 423 T^{3} + 2209 T^{4} \))(\( 1 - 11 T + 123 T^{2} - 517 T^{3} + 2209 T^{4} \))(\( 1 + 16 T + 146 T^{2} + 752 T^{3} + 2209 T^{4} \))(\( 1 + 4 T + 37 T^{2} + 88 T^{3} + 1739 T^{4} + 8836 T^{5} + 103823 T^{6} \))(\( 1 + 4 T + 37 T^{2} + 88 T^{3} + 1739 T^{4} + 8836 T^{5} + 103823 T^{6} \))(\( 1 + 3 T + 57 T^{2} - 175 T^{3} + 1856 T^{4} - 8225 T^{5} + 125913 T^{6} + 311469 T^{7} + 4879681 T^{8} \))(\( 1 - 16 T + 260 T^{2} - 2320 T^{3} + 19990 T^{4} - 109040 T^{5} + 574340 T^{6} - 1661168 T^{7} + 4879681 T^{8} \))(\( 1 + 3 T + 57 T^{2} - 175 T^{3} + 1856 T^{4} - 8225 T^{5} + 125913 T^{6} + 311469 T^{7} + 4879681 T^{8} \))(\( 1 - 16 T + 260 T^{2} - 2320 T^{3} + 19990 T^{4} - 109040 T^{5} + 574340 T^{6} - 1661168 T^{7} + 4879681 T^{8} \))
$53$ (\( 1 - 4 T + 53 T^{2} \))(\( 1 + 9 T + 53 T^{2} \))(\( 1 - 6 T + 53 T^{2} \))(\( 1 - 12 T + 53 T^{2} \))(\( 1 - 6 T + 53 T^{2} \))(\( 1 - 4 T + 53 T^{2} \))(\( 1 - 6 T + 53 T^{2} \))(\( 1 + 9 T + 53 T^{2} \))(\( 1 + 14 T + 53 T^{2} \))(\( 1 - 12 T + 53 T^{2} \))(\( 1 + 12 T + 97 T^{2} + 636 T^{3} + 2809 T^{4} \))(\( 1 - 8 T + 102 T^{2} - 424 T^{3} + 2809 T^{4} \))(\( 1 + 8 T + 42 T^{2} + 424 T^{3} + 2809 T^{4} \))(\( 1 - 12 T + 137 T^{2} - 636 T^{3} + 2809 T^{4} \))(\( 1 - 12 T + 137 T^{2} - 636 T^{3} + 2809 T^{4} \))(\( 1 + 94 T^{2} + 2809 T^{4} \))(\( 1 + 12 T + 97 T^{2} + 636 T^{3} + 2809 T^{4} \))(\( 1 + 8 T + 42 T^{2} + 424 T^{3} + 2809 T^{4} \))(\( 1 - 12 T + 137 T^{2} - 636 T^{3} + 2809 T^{4} \))(\( 1 - 12 T + 137 T^{2} - 636 T^{3} + 2809 T^{4} \))(\( 1 + 94 T^{2} + 2809 T^{4} \))(\( 1 + T + 143 T^{2} + 82 T^{3} + 7579 T^{4} + 2809 T^{5} + 148877 T^{6} \))(\( 1 + T + 143 T^{2} + 82 T^{3} + 7579 T^{4} + 2809 T^{5} + 148877 T^{6} \))(\( 1 - 4 T + 163 T^{2} - 740 T^{3} + 11596 T^{4} - 39220 T^{5} + 457867 T^{6} - 595508 T^{7} + 7890481 T^{8} \))(\( 1 + 24 T + 332 T^{2} + 3048 T^{3} + 23910 T^{4} + 161544 T^{5} + 932588 T^{6} + 3573048 T^{7} + 7890481 T^{8} \))(\( 1 - 4 T + 163 T^{2} - 740 T^{3} + 11596 T^{4} - 39220 T^{5} + 457867 T^{6} - 595508 T^{7} + 7890481 T^{8} \))(\( 1 + 24 T + 332 T^{2} + 3048 T^{3} + 23910 T^{4} + 161544 T^{5} + 932588 T^{6} + 3573048 T^{7} + 7890481 T^{8} \))
$59$ (\( 1 + 11 T + 59 T^{2} \))(\( 1 + 12 T + 59 T^{2} \))(\( 1 + 59 T^{2} \))(\( 1 + 15 T + 59 T^{2} \))(\( 1 + 6 T + 59 T^{2} \))(\( 1 + 11 T + 59 T^{2} \))(\( 1 + 12 T + 59 T^{2} \))(\( 1 + 12 T + 59 T^{2} \))(\( 1 - 10 T + 59 T^{2} \))(\( 1 + 15 T + 59 T^{2} \))(\( 1 - 5 T + 63 T^{2} - 295 T^{3} + 3481 T^{4} \))(\( 1 - 10 T + 138 T^{2} - 590 T^{3} + 3481 T^{4} \))(\( 1 - 15 T + 163 T^{2} - 885 T^{3} + 3481 T^{4} \))(\( 1 + 9 T + 37 T^{2} + 531 T^{3} + 3481 T^{4} \))(\( 1 - 15 T + 173 T^{2} - 885 T^{3} + 3481 T^{4} \))(\( 1 - 18 T + 196 T^{2} - 1062 T^{3} + 3481 T^{4} \))(\( 1 - 5 T + 63 T^{2} - 295 T^{3} + 3481 T^{4} \))(\( 1 - 15 T + 163 T^{2} - 885 T^{3} + 3481 T^{4} \))(\( 1 + 9 T + 37 T^{2} + 531 T^{3} + 3481 T^{4} \))(\( 1 - 15 T + 173 T^{2} - 885 T^{3} + 3481 T^{4} \))(\( 1 - 18 T + 196 T^{2} - 1062 T^{3} + 3481 T^{4} \))(\( 1 - 10 T + 200 T^{2} - 1192 T^{3} + 11800 T^{4} - 34810 T^{5} + 205379 T^{6} \))(\( 1 - 10 T + 200 T^{2} - 1192 T^{3} + 11800 T^{4} - 34810 T^{5} + 205379 T^{6} \))(\( 1 + 4 T + 57 T^{2} + 512 T^{3} + 6405 T^{4} + 30208 T^{5} + 198417 T^{6} + 821516 T^{7} + 12117361 T^{8} \))(\( 1 + 12 T + 284 T^{2} + 2196 T^{3} + 26583 T^{4} + 129564 T^{5} + 988604 T^{6} + 2464548 T^{7} + 12117361 T^{8} \))(\( 1 + 4 T + 57 T^{2} + 512 T^{3} + 6405 T^{4} + 30208 T^{5} + 198417 T^{6} + 821516 T^{7} + 12117361 T^{8} \))(\( 1 + 12 T + 284 T^{2} + 2196 T^{3} + 26583 T^{4} + 129564 T^{5} + 988604 T^{6} + 2464548 T^{7} + 12117361 T^{8} \))
$61$ (\( 1 - 10 T + 61 T^{2} \))(\( 1 - 2 T + 61 T^{2} \))(\( 1 + 10 T + 61 T^{2} \))(\( 1 - 10 T + 61 T^{2} \))(\( 1 + 8 T + 61 T^{2} \))(\( 1 + 10 T + 61 T^{2} \))(\( 1 - 14 T + 61 T^{2} \))(\( 1 + 2 T + 61 T^{2} \))(\( 1 - 8 T + 61 T^{2} \))(\( 1 + 10 T + 61 T^{2} \))(\( 1 - 21 T + 221 T^{2} - 1281 T^{3} + 3721 T^{4} \))(\( 1 - 6 T + 126 T^{2} - 366 T^{3} + 3721 T^{4} \))(\( 1 - 4 T + 106 T^{2} - 244 T^{3} + 3721 T^{4} \))(\( 1 + 13 T + 153 T^{2} + 793 T^{3} + 3721 T^{4} \))(\( 1 - 5 T + 67 T^{2} - 305 T^{3} + 3721 T^{4} \))(\( 1 + 2 T + 120 T^{2} + 122 T^{3} + 3721 T^{4} \))(\( 1 + 21 T + 221 T^{2} + 1281 T^{3} + 3721 T^{4} \))(\( 1 + 4 T + 106 T^{2} + 244 T^{3} + 3721 T^{4} \))(\( 1 - 13 T + 153 T^{2} - 793 T^{3} + 3721 T^{4} \))(\( 1 + 5 T + 67 T^{2} + 305 T^{3} + 3721 T^{4} \))(\( 1 - 2 T + 120 T^{2} - 122 T^{3} + 3721 T^{4} \))(\( 1 - 10 T + 175 T^{2} - 1044 T^{3} + 10675 T^{4} - 37210 T^{5} + 226981 T^{6} \))(\( 1 + 10 T + 175 T^{2} + 1044 T^{3} + 10675 T^{4} + 37210 T^{5} + 226981 T^{6} \))(\( 1 - 27 T + 473 T^{2} - 5569 T^{3} + 50760 T^{4} - 339709 T^{5} + 1760033 T^{6} - 6128487 T^{7} + 13845841 T^{8} \))(\( 1 + 4 T + 232 T^{2} + 676 T^{3} + 20818 T^{4} + 41236 T^{5} + 863272 T^{6} + 907924 T^{7} + 13845841 T^{8} \))(\( 1 + 27 T + 473 T^{2} + 5569 T^{3} + 50760 T^{4} + 339709 T^{5} + 1760033 T^{6} + 6128487 T^{7} + 13845841 T^{8} \))(\( 1 - 4 T + 232 T^{2} - 676 T^{3} + 20818 T^{4} - 41236 T^{5} + 863272 T^{6} - 907924 T^{7} + 13845841 T^{8} \))
$67$ (\( 1 - 8 T + 67 T^{2} \))(\( 1 + 67 T^{2} \))(\( 1 + 12 T + 67 T^{2} \))(\( 1 + 67 T^{2} \))(\( 1 + 4 T + 67 T^{2} \))(\( 1 - 8 T + 67 T^{2} \))(\( 1 + 12 T + 67 T^{2} \))(\( 1 + 67 T^{2} \))(\( 1 - 8 T + 67 T^{2} \))(\( 1 + 67 T^{2} \))(\( 1 + 4 T + 93 T^{2} + 268 T^{3} + 4489 T^{4} \))(\( 1 + 4 T - 42 T^{2} + 268 T^{3} + 4489 T^{4} \))(\( 1 + 9 T + 123 T^{2} + 603 T^{3} + 4489 T^{4} \))(\( ( 1 - 13 T + 67 T^{2} )^{2} \))(\( 1 + 4 T + 13 T^{2} + 268 T^{3} + 4489 T^{4} \))(\( 1 - 12 T + 158 T^{2} - 804 T^{3} + 4489 T^{4} \))(\( 1 + 4 T + 93 T^{2} + 268 T^{3} + 4489 T^{4} \))(\( 1 + 9 T + 123 T^{2} + 603 T^{3} + 4489 T^{4} \))(\( ( 1 - 13 T + 67 T^{2} )^{2} \))(\( 1 + 4 T + 13 T^{2} + 268 T^{3} + 4489 T^{4} \))(\( 1 - 12 T + 158 T^{2} - 804 T^{3} + 4489 T^{4} \))(\( 1 + 8 T + 113 T^{2} + 944 T^{3} + 7571 T^{4} + 35912 T^{5} + 300763 T^{6} \))(\( 1 + 8 T + 113 T^{2} + 944 T^{3} + 7571 T^{4} + 35912 T^{5} + 300763 T^{6} \))(\( 1 - 9 T + 224 T^{2} - 1698 T^{3} + 21379 T^{4} - 113766 T^{5} + 1005536 T^{6} - 2706867 T^{7} + 20151121 T^{8} \))(\( 1 + 100 T^{2} - 960 T^{3} + 3366 T^{4} - 64320 T^{5} + 448900 T^{6} + 20151121 T^{8} \))(\( 1 - 9 T + 224 T^{2} - 1698 T^{3} + 21379 T^{4} - 113766 T^{5} + 1005536 T^{6} - 2706867 T^{7} + 20151121 T^{8} \))(\( 1 + 100 T^{2} - 960 T^{3} + 3366 T^{4} - 64320 T^{5} + 448900 T^{6} + 20151121 T^{8} \))
$71$ (\( 1 - 8 T + 71 T^{2} \))(\( 1 - 4 T + 71 T^{2} \))(\( 1 - 16 T + 71 T^{2} \))(\( 1 + 8 T + 71 T^{2} \))(\( 1 + 71 T^{2} \))(\( 1 - 8 T + 71 T^{2} \))(\( 1 + 8 T + 71 T^{2} \))(\( 1 - 4 T + 71 T^{2} \))(\( 1 + 4 T + 71 T^{2} \))(\( 1 + 8 T + 71 T^{2} \))(\( ( 1 + 3 T + 71 T^{2} )^{2} \))(\( 1 - 4 T + 126 T^{2} - 284 T^{3} + 5041 T^{4} \))(\( 1 - 12 T + 158 T^{2} - 852 T^{3} + 5041 T^{4} \))(\( 1 - 4 T + 141 T^{2} - 284 T^{3} + 5041 T^{4} \))(\( 1 + 18 T + 203 T^{2} + 1278 T^{3} + 5041 T^{4} \))(\( 1 - 8 T + 146 T^{2} - 568 T^{3} + 5041 T^{4} \))(\( ( 1 + 3 T + 71 T^{2} )^{2} \))(\( 1 - 12 T + 158 T^{2} - 852 T^{3} + 5041 T^{4} \))(\( 1 - 4 T + 141 T^{2} - 284 T^{3} + 5041 T^{4} \))(\( 1 + 18 T + 203 T^{2} + 1278 T^{3} + 5041 T^{4} \))(\( 1 - 8 T + 146 T^{2} - 568 T^{3} + 5041 T^{4} \))(\( 1 - 4 T + 53 T^{2} - 952 T^{3} + 3763 T^{4} - 20164 T^{5} + 357911 T^{6} \))(\( 1 - 4 T + 53 T^{2} - 952 T^{3} + 3763 T^{4} - 20164 T^{5} + 357911 T^{6} \))(\( 1 + 20 T + 279 T^{2} + 2720 T^{3} + 25616 T^{4} + 193120 T^{5} + 1406439 T^{6} + 7158220 T^{7} + 25411681 T^{8} \))(\( 1 - 16 T + 164 T^{2} - 1168 T^{3} + 10918 T^{4} - 82928 T^{5} + 826724 T^{6} - 5726576 T^{7} + 25411681 T^{8} \))(\( 1 + 20 T + 279 T^{2} + 2720 T^{3} + 25616 T^{4} + 193120 T^{5} + 1406439 T^{6} + 7158220 T^{7} + 25411681 T^{8} \))(\( 1 - 16 T + 164 T^{2} - 1168 T^{3} + 10918 T^{4} - 82928 T^{5} + 826724 T^{6} - 5726576 T^{7} + 25411681 T^{8} \))
$73$ (\( 1 - 4 T + 73 T^{2} \))(\( 1 + 10 T + 73 T^{2} \))(\( 1 - 14 T + 73 T^{2} \))(\( 1 - 4 T + 73 T^{2} \))(\( 1 + 2 T + 73 T^{2} \))(\( 1 + 4 T + 73 T^{2} \))(\( 1 + 4 T + 73 T^{2} \))(\( 1 - 10 T + 73 T^{2} \))(\( 1 + 4 T + 73 T^{2} \))(\( 1 + 4 T + 73 T^{2} \))(\( 1 - 7 T + 147 T^{2} - 511 T^{3} + 5329 T^{4} \))(\( 1 + 8 T + 82 T^{2} + 584 T^{3} + 5329 T^{4} \))(\( 1 + 7 T + 157 T^{2} + 511 T^{3} + 5329 T^{4} \))(\( 1 - 3 T + 47 T^{2} - 219 T^{3} + 5329 T^{4} \))(\( 1 + 17 T + 217 T^{2} + 1241 T^{3} + 5329 T^{4} \))(\( 1 + 4 T + 42 T^{2} + 292 T^{3} + 5329 T^{4} \))(\( 1 + 7 T + 147 T^{2} + 511 T^{3} + 5329 T^{4} \))(\( 1 - 7 T + 157 T^{2} - 511 T^{3} + 5329 T^{4} \))(\( 1 + 3 T + 47 T^{2} + 219 T^{3} + 5329 T^{4} \))(\( 1 - 17 T + 217 T^{2} - 1241 T^{3} + 5329 T^{4} \))(\( 1 - 4 T + 42 T^{2} - 292 T^{3} + 5329 T^{4} \))(\( 1 - 14 T + 219 T^{2} - 1692 T^{3} + 15987 T^{4} - 74606 T^{5} + 389017 T^{6} \))(\( 1 + 14 T + 219 T^{2} + 1692 T^{3} + 15987 T^{4} + 74606 T^{5} + 389017 T^{6} \))(\( 1 - 28 T + 441 T^{2} - 4964 T^{3} + 46509 T^{4} - 362372 T^{5} + 2350089 T^{6} - 10892476 T^{7} + 28398241 T^{8} \))(\( 1 - 16 T + 148 T^{2} - 688 T^{3} + 1606 T^{4} - 50224 T^{5} + 788692 T^{6} - 6224272 T^{7} + 28398241 T^{8} \))(\( 1 + 28 T + 441 T^{2} + 4964 T^{3} + 46509 T^{4} + 362372 T^{5} + 2350089 T^{6} + 10892476 T^{7} + 28398241 T^{8} \))(\( 1 + 16 T + 148 T^{2} + 688 T^{3} + 1606 T^{4} + 50224 T^{5} + 788692 T^{6} + 6224272 T^{7} + 28398241 T^{8} \))
$79$ (\( 1 + 11 T + 79 T^{2} \))(\( 1 + 12 T + 79 T^{2} \))(\( 1 + 79 T^{2} \))(\( 1 + 3 T + 79 T^{2} \))(\( 1 + 8 T + 79 T^{2} \))(\( 1 - 11 T + 79 T^{2} \))(\( 1 + 79 T^{2} \))(\( 1 - 12 T + 79 T^{2} \))(\( 1 + 16 T + 79 T^{2} \))(\( 1 - 3 T + 79 T^{2} \))(\( 1 - 10 T + 103 T^{2} - 790 T^{3} + 6241 T^{4} \))(\( ( 1 + 79 T^{2} )^{2} \))(\( 1 - 2 T + 154 T^{2} - 158 T^{3} + 6241 T^{4} \))(\( 1 + 20 T + 253 T^{2} + 1580 T^{3} + 6241 T^{4} \))(\( ( 1 + 15 T + 79 T^{2} )^{2} \))(\( 1 - 34 T^{2} + 6241 T^{4} \))(\( 1 + 10 T + 103 T^{2} + 790 T^{3} + 6241 T^{4} \))(\( 1 + 2 T + 154 T^{2} + 158 T^{3} + 6241 T^{4} \))(\( 1 - 20 T + 253 T^{2} - 1580 T^{3} + 6241 T^{4} \))(\( ( 1 - 15 T + 79 T^{2} )^{2} \))(\( 1 - 34 T^{2} + 6241 T^{4} \))(\( 1 + 2 T + 54 T^{2} + 1144 T^{3} + 4266 T^{4} + 12482 T^{5} + 493039 T^{6} \))(\( 1 - 2 T + 54 T^{2} - 1144 T^{3} + 4266 T^{4} - 12482 T^{5} + 493039 T^{6} \))(\( 1 - 22 T + 375 T^{2} - 4892 T^{3} + 46964 T^{4} - 386468 T^{5} + 2340375 T^{6} - 10846858 T^{7} + 38950081 T^{8} \))(\( 1 - 12 T + 322 T^{2} - 2568 T^{3} + 37707 T^{4} - 202872 T^{5} + 2009602 T^{6} - 5916468 T^{7} + 38950081 T^{8} \))(\( 1 + 22 T + 375 T^{2} + 4892 T^{3} + 46964 T^{4} + 386468 T^{5} + 2340375 T^{6} + 10846858 T^{7} + 38950081 T^{8} \))(\( 1 + 12 T + 322 T^{2} + 2568 T^{3} + 37707 T^{4} + 202872 T^{5} + 2009602 T^{6} + 5916468 T^{7} + 38950081 T^{8} \))
$83$ (\( 1 - 7 T + 83 T^{2} \))(\( 1 - 12 T + 83 T^{2} \))(\( 1 + 83 T^{2} \))(\( 1 - 3 T + 83 T^{2} \))(\( 1 - 6 T + 83 T^{2} \))(\( 1 + 7 T + 83 T^{2} \))(\( 1 - 6 T + 83 T^{2} \))(\( 1 + 12 T + 83 T^{2} \))(\( 1 + 4 T + 83 T^{2} \))(\( 1 + 3 T + 83 T^{2} \))(\( 1 + 3 T + 107 T^{2} + 249 T^{3} + 6889 T^{4} \))(\( 1 - 2 T + 42 T^{2} - 166 T^{3} + 6889 T^{4} \))(\( 1 - 3 T - 43 T^{2} - 249 T^{3} + 6889 T^{4} \))(\( 1 + 7 T + 177 T^{2} + 581 T^{3} + 6889 T^{4} \))(\( 1 - 15 T + 221 T^{2} - 1245 T^{3} + 6889 T^{4} \))(\( 1 + 26 T + 332 T^{2} + 2158 T^{3} + 6889 T^{4} \))(\( 1 - 3 T + 107 T^{2} - 249 T^{3} + 6889 T^{4} \))(\( 1 + 3 T - 43 T^{2} + 249 T^{3} + 6889 T^{4} \))(\( 1 - 7 T + 177 T^{2} - 581 T^{3} + 6889 T^{4} \))(\( 1 + 15 T + 221 T^{2} + 1245 T^{3} + 6889 T^{4} \))(\( 1 - 26 T + 332 T^{2} - 2158 T^{3} + 6889 T^{4} \))(\( 1 + 10 T + 128 T^{2} + 472 T^{3} + 10624 T^{4} + 68890 T^{5} + 571787 T^{6} \))(\( 1 - 10 T + 128 T^{2} - 472 T^{3} + 10624 T^{4} - 68890 T^{5} + 571787 T^{6} \))(\( 1 - 6 T + 193 T^{2} - 1890 T^{3} + 18151 T^{4} - 156870 T^{5} + 1329577 T^{6} - 3430722 T^{7} + 47458321 T^{8} \))(\( 1 + 4 T + 236 T^{2} + 844 T^{3} + 27367 T^{4} + 70052 T^{5} + 1625804 T^{6} + 2287148 T^{7} + 47458321 T^{8} \))(\( 1 + 6 T + 193 T^{2} + 1890 T^{3} + 18151 T^{4} + 156870 T^{5} + 1329577 T^{6} + 3430722 T^{7} + 47458321 T^{8} \))(\( 1 - 4 T + 236 T^{2} - 844 T^{3} + 27367 T^{4} - 70052 T^{5} + 1625804 T^{6} - 2287148 T^{7} + 47458321 T^{8} \))
$89$ (\( 1 + 2 T + 89 T^{2} \))(\( 1 - 15 T + 89 T^{2} \))(\( 1 + 6 T + 89 T^{2} \))(\( 1 - 6 T + 89 T^{2} \))(\( 1 + 6 T + 89 T^{2} \))(\( 1 + 2 T + 89 T^{2} \))(\( 1 + 6 T + 89 T^{2} \))(\( 1 - 15 T + 89 T^{2} \))(\( 1 - 10 T + 89 T^{2} \))(\( 1 - 6 T + 89 T^{2} \))(\( 1 - 5 T + 153 T^{2} - 445 T^{3} + 7921 T^{4} \))(\( ( 1 - 10 T + 89 T^{2} )^{2} \))(\( 1 + 15 T + 173 T^{2} + 1335 T^{3} + 7921 T^{4} \))(\( 1 - 3 T + 169 T^{2} - 267 T^{3} + 7921 T^{4} \))(\( 1 + 17 T + 219 T^{2} + 1513 T^{3} + 7921 T^{4} \))(\( 1 + 4 T + 134 T^{2} + 356 T^{3} + 7921 T^{4} \))(\( 1 - 5 T + 153 T^{2} - 445 T^{3} + 7921 T^{4} \))(\( 1 + 15 T + 173 T^{2} + 1335 T^{3} + 7921 T^{4} \))(\( 1 - 3 T + 169 T^{2} - 267 T^{3} + 7921 T^{4} \))(\( 1 + 17 T + 219 T^{2} + 1513 T^{3} + 7921 T^{4} \))(\( 1 + 4 T + 134 T^{2} + 356 T^{3} + 7921 T^{4} \))(\( 1 + 13 T + 139 T^{2} + 646 T^{3} + 12371 T^{4} + 102973 T^{5} + 704969 T^{6} \))(\( 1 + 13 T + 139 T^{2} + 646 T^{3} + 12371 T^{4} + 102973 T^{5} + 704969 T^{6} \))(\( 1 - 6 T + 297 T^{2} - 1158 T^{3} + 36375 T^{4} - 103062 T^{5} + 2352537 T^{6} - 4229814 T^{7} + 62742241 T^{8} \))(\( 1 - 16 T + 164 T^{2} - 784 T^{3} + 7222 T^{4} - 69776 T^{5} + 1299044 T^{6} - 11279504 T^{7} + 62742241 T^{8} \))(\( 1 - 6 T + 297 T^{2} - 1158 T^{3} + 36375 T^{4} - 103062 T^{5} + 2352537 T^{6} - 4229814 T^{7} + 62742241 T^{8} \))(\( 1 - 16 T + 164 T^{2} - 784 T^{3} + 7222 T^{4} - 69776 T^{5} + 1299044 T^{6} - 11279504 T^{7} + 62742241 T^{8} \))
$97$ (\( 1 + 97 T^{2} \))(\( 1 + 17 T + 97 T^{2} \))(\( 1 - 10 T + 97 T^{2} \))(\( 1 - 16 T + 97 T^{2} \))(\( 1 + 10 T + 97 T^{2} \))(\( 1 + 97 T^{2} \))(\( 1 + 14 T + 97 T^{2} \))(\( 1 + 17 T + 97 T^{2} \))(\( 1 - 6 T + 97 T^{2} \))(\( 1 - 16 T + 97 T^{2} \))(\( 1 - 16 T + 238 T^{2} - 1552 T^{3} + 9409 T^{4} \))(\( 1 - 16 T + 238 T^{2} - 1552 T^{3} + 9409 T^{4} \))(\( 1 + 13 T + 225 T^{2} + 1261 T^{3} + 9409 T^{4} \))(\( 1 + 8 T + 190 T^{2} + 776 T^{3} + 9409 T^{4} \))(\( 1 + 14 T^{2} + 9409 T^{4} \))(\( 1 - 16 T + 246 T^{2} - 1552 T^{3} + 9409 T^{4} \))(\( 1 - 16 T + 238 T^{2} - 1552 T^{3} + 9409 T^{4} \))(\( 1 + 13 T + 225 T^{2} + 1261 T^{3} + 9409 T^{4} \))(\( 1 + 8 T + 190 T^{2} + 776 T^{3} + 9409 T^{4} \))(\( 1 + 14 T^{2} + 9409 T^{4} \))(\( 1 - 16 T + 246 T^{2} - 1552 T^{3} + 9409 T^{4} \))(\( 1 + 9 T + 259 T^{2} + 1490 T^{3} + 25123 T^{4} + 84681 T^{5} + 912673 T^{6} \))(\( 1 + 9 T + 259 T^{2} + 1490 T^{3} + 25123 T^{4} + 84681 T^{5} + 912673 T^{6} \))(\( 1 - 15 T + 343 T^{2} - 3265 T^{3} + 46304 T^{4} - 316705 T^{5} + 3227287 T^{6} - 13690095 T^{7} + 88529281 T^{8} \))(\( 1 + 16 T + 52 T^{2} + 1456 T^{3} + 31078 T^{4} + 141232 T^{5} + 489268 T^{6} + 14602768 T^{7} + 88529281 T^{8} \))(\( 1 - 15 T + 343 T^{2} - 3265 T^{3} + 46304 T^{4} - 316705 T^{5} + 3227287 T^{6} - 13690095 T^{7} + 88529281 T^{8} \))(\( 1 + 16 T + 52 T^{2} + 1456 T^{3} + 31078 T^{4} + 141232 T^{5} + 489268 T^{6} + 14602768 T^{7} + 88529281 T^{8} \))
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