Properties

Label 1225.2.a
Level 1225
Weight 2
Character orbit a
Rep. character \(\chi_{1225}(1,\cdot)\)
Character field \(\Q\)
Dimension 58
Newform subspaces 29
Sturm bound 280
Trace bound 3

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Defining parameters

Level: \( N \) \(=\) \( 1225 = 5^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1225.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 29 \)
Sturm bound: \(280\)
Trace bound: \(3\)
Distinguishing \(T_p\): \(2\), \(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(1225))\).

Total New Old
Modular forms 164 73 91
Cusp forms 117 58 59
Eisenstein series 47 15 32

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

\(5\)\(7\)FrickeDim.
\(+\)\(+\)\(+\)\(13\)
\(+\)\(-\)\(-\)\(15\)
\(-\)\(+\)\(-\)\(17\)
\(-\)\(-\)\(+\)\(13\)
Plus space\(+\)\(26\)
Minus space\(-\)\(32\)

Trace form

\( 58q + 54q^{4} - 4q^{6} + 38q^{9} + O(q^{10}) \) \( 58q + 54q^{4} - 4q^{6} + 38q^{9} + 4q^{12} + 10q^{13} + 54q^{16} - 2q^{17} + 20q^{18} + 4q^{19} + 16q^{22} - 12q^{23} + 24q^{24} + 2q^{26} - 12q^{27} - 8q^{29} + 12q^{31} - 12q^{33} + 2q^{34} + 22q^{36} + 20q^{38} - 8q^{39} - 22q^{41} - 4q^{43} + 40q^{44} + 24q^{46} + 4q^{47} + 28q^{48} + 20q^{51} - 6q^{52} + 36q^{53} + 8q^{54} - 32q^{57} - 8q^{58} - 32q^{59} + 14q^{61} + 34q^{64} - 4q^{67} - 10q^{68} + 12q^{69} - 36q^{71} + 44q^{72} - 6q^{73} + 16q^{76} + 32q^{78} + 4q^{79} - 46q^{81} - 18q^{82} + 20q^{83} - 48q^{86} + 28q^{87} - 32q^{88} - 54q^{89} - 12q^{92} - 8q^{93} - 12q^{94} - 8q^{96} - 10q^{97} - 60q^{99} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(1225))\) into newform subspaces

Label Dim. \(A\) Field CM Traces A-L signs $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\) 5 7
1225.2.a.a \(1\) \(9.782\) \(\Q\) None \(-2\) \(1\) \(0\) \(0\) \(-\) \(-\) \(q-2q^{2}+q^{3}+2q^{4}-2q^{6}-2q^{9}+\cdots\)
1225.2.a.b \(1\) \(9.782\) \(\Q\) None \(-1\) \(-1\) \(0\) \(0\) \(-\) \(-\) \(q-q^{2}-q^{3}-q^{4}+q^{6}+3q^{8}-2q^{9}+\cdots\)
1225.2.a.c \(1\) \(9.782\) \(\Q\) \(\Q(\sqrt{-7}) \) \(-1\) \(0\) \(0\) \(0\) \(+\) \(-\) \(q-q^{2}-q^{4}+3q^{8}-3q^{9}+4q^{11}+\cdots\)
1225.2.a.d \(1\) \(9.782\) \(\Q\) None \(-1\) \(1\) \(0\) \(0\) \(-\) \(+\) \(q-q^{2}+q^{3}-q^{4}-q^{6}+3q^{8}-2q^{9}+\cdots\)
1225.2.a.e \(1\) \(9.782\) \(\Q\) None \(0\) \(1\) \(0\) \(0\) \(+\) \(-\) \(q+q^{3}-2q^{4}-2q^{9}-3q^{11}-2q^{12}+\cdots\)
1225.2.a.f \(1\) \(9.782\) \(\Q\) None \(1\) \(-1\) \(0\) \(0\) \(-\) \(+\) \(q+q^{2}-q^{3}-q^{4}-q^{6}-3q^{8}-2q^{9}+\cdots\)
1225.2.a.g \(1\) \(9.782\) \(\Q\) None \(1\) \(1\) \(0\) \(0\) \(-\) \(-\) \(q+q^{2}+q^{3}-q^{4}+q^{6}-3q^{8}-2q^{9}+\cdots\)
1225.2.a.h \(1\) \(9.782\) \(\Q\) None \(2\) \(-3\) \(0\) \(0\) \(+\) \(-\) \(q+2q^{2}-3q^{3}+2q^{4}-6q^{6}+6q^{9}+\cdots\)
1225.2.a.i \(1\) \(9.782\) \(\Q\) None \(2\) \(-1\) \(0\) \(0\) \(-\) \(-\) \(q+2q^{2}-q^{3}+2q^{4}-2q^{6}-2q^{9}+\cdots\)
1225.2.a.j \(1\) \(9.782\) \(\Q\) None \(2\) \(3\) \(0\) \(0\) \(+\) \(-\) \(q+2q^{2}+3q^{3}+2q^{4}+6q^{6}+6q^{9}+\cdots\)
1225.2.a.k \(2\) \(9.782\) \(\Q(\sqrt{2}) \) None \(-2\) \(-2\) \(0\) \(0\) \(+\) \(+\) \(q+(-1+\beta )q^{2}+(-1-\beta )q^{3}+(1-2\beta )q^{4}+\cdots\)
1225.2.a.l \(2\) \(9.782\) \(\Q(\sqrt{2}) \) None \(-2\) \(0\) \(0\) \(0\) \(-\) \(+\) \(q-q^{2}+2\beta q^{3}-q^{4}-2\beta q^{6}+3q^{8}+\cdots\)
1225.2.a.m \(2\) \(9.782\) \(\Q(\sqrt{2}) \) None \(-2\) \(2\) \(0\) \(0\) \(+\) \(-\) \(q+(-1+\beta )q^{2}+(1+\beta )q^{3}+(1-2\beta )q^{4}+\cdots\)
1225.2.a.n \(2\) \(9.782\) \(\Q(\sqrt{5}) \) None \(-1\) \(-2\) \(0\) \(0\) \(-\) \(-\) \(q-\beta q^{2}+(-2+2\beta )q^{3}+(-1+\beta )q^{4}+\cdots\)
1225.2.a.o \(2\) \(9.782\) \(\Q(\sqrt{21}) \) \(\Q(\sqrt{-7}) \) \(-1\) \(0\) \(0\) \(0\) \(-\) \(-\) \(q-\beta q^{2}+(3+\beta )q^{4}+(-5-2\beta )q^{8}+\cdots\)
1225.2.a.p \(2\) \(9.782\) \(\Q(\sqrt{2}) \) None \(0\) \(-2\) \(0\) \(0\) \(+\) \(+\) \(q+\beta q^{2}+(-1+\beta )q^{3}+(2-\beta )q^{6}-2\beta q^{8}+\cdots\)
1225.2.a.q \(2\) \(9.782\) \(\Q(\sqrt{5}) \) \(\Q(\sqrt{-35}) \) \(0\) \(0\) \(0\) \(0\) \(-\) \(-\) \(q-\beta q^{3}-2q^{4}+2q^{9}-3q^{11}+2\beta q^{12}+\cdots\)
1225.2.a.r \(2\) \(9.782\) \(\Q(\sqrt{2}) \) None \(0\) \(2\) \(0\) \(0\) \(+\) \(+\) \(q+\beta q^{2}+(1-\beta )q^{3}+(-2+\beta )q^{6}-2\beta q^{8}+\cdots\)
1225.2.a.s \(2\) \(9.782\) \(\Q(\sqrt{17}) \) None \(1\) \(-1\) \(0\) \(0\) \(+\) \(-\) \(q+\beta q^{2}+(-1+\beta )q^{3}+(2+\beta )q^{4}+4q^{6}+\cdots\)
1225.2.a.t \(2\) \(9.782\) \(\Q(\sqrt{21}) \) \(\Q(\sqrt{-7}) \) \(1\) \(0\) \(0\) \(0\) \(+\) \(-\) \(q+\beta q^{2}+(3+\beta )q^{4}+(5+2\beta )q^{8}-3q^{9}+\cdots\)
1225.2.a.u \(2\) \(9.782\) \(\Q(\sqrt{5}) \) None \(1\) \(2\) \(0\) \(0\) \(+\) \(-\) \(q+\beta q^{2}+(2-2\beta )q^{3}+(-1+\beta )q^{4}+\cdots\)
1225.2.a.v \(2\) \(9.782\) \(\Q(\sqrt{2}) \) None \(2\) \(0\) \(0\) \(0\) \(-\) \(+\) \(q+q^{2}-2\beta q^{3}-q^{4}-2\beta q^{6}-3q^{8}+\cdots\)
1225.2.a.w \(3\) \(9.782\) 3.3.257.1 None \(-1\) \(-3\) \(0\) \(0\) \(-\) \(-\) \(q-\beta _{1}q^{2}+(-1-\beta _{2})q^{3}+(1+\beta _{2})q^{4}+\cdots\)
1225.2.a.x \(3\) \(9.782\) 3.3.257.1 None \(-1\) \(3\) \(0\) \(0\) \(-\) \(+\) \(q-\beta _{1}q^{2}+(1+\beta _{2})q^{3}+(1+\beta _{2})q^{4}+\cdots\)
1225.2.a.y \(3\) \(9.782\) 3.3.257.1 None \(1\) \(-3\) \(0\) \(0\) \(+\) \(+\) \(q+\beta _{1}q^{2}+(-1-\beta _{2})q^{3}+(1+\beta _{2})q^{4}+\cdots\)
1225.2.a.z \(3\) \(9.782\) 3.3.257.1 None \(1\) \(3\) \(0\) \(0\) \(+\) \(-\) \(q+\beta _{1}q^{2}+(1+\beta _{2})q^{3}+(1+\beta _{2})q^{4}+\cdots\)
1225.2.a.ba \(4\) \(9.782\) \(\Q(\sqrt{2}, \sqrt{5})\) None \(-6\) \(0\) \(0\) \(0\) \(+\) \(+\) \(q+(-1-\beta _{2})q^{2}+\beta _{1}q^{3}+3\beta _{2}q^{4}+\cdots\)
1225.2.a.bb \(4\) \(9.782\) \(\Q(\zeta_{24})^+\) None \(0\) \(0\) \(0\) \(0\) \(-\) \(+\) \(q-\beta _{3}q^{2}-\beta _{1}q^{3}+4q^{4}+3\beta _{2}q^{6}+\cdots\)
1225.2.a.bc \(4\) \(9.782\) \(\Q(\sqrt{2}, \sqrt{5})\) None \(6\) \(0\) \(0\) \(0\) \(-\) \(+\) \(q+(1+\beta _{2})q^{2}+\beta _{1}q^{3}+3\beta _{2}q^{4}+(2\beta _{1}+\cdots)q^{6}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(1225))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(1225)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(35))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(49))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(175))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(245))\)\(^{\oplus 2}\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ (\( 1 + 2 T + 2 T^{2} \))(\( 1 + T + 2 T^{2} \))(\( 1 + T + 2 T^{2} \))(\( 1 + T + 2 T^{2} \))(\( 1 + 2 T^{2} \))(\( 1 - T + 2 T^{2} \))(\( 1 - T + 2 T^{2} \))(\( 1 - 2 T + 2 T^{2} \))(\( 1 - 2 T + 2 T^{2} \))(\( 1 - 2 T + 2 T^{2} \))(\( 1 + 2 T + 3 T^{2} + 4 T^{3} + 4 T^{4} \))(\( ( 1 + T + 2 T^{2} )^{2} \))(\( 1 + 2 T + 3 T^{2} + 4 T^{3} + 4 T^{4} \))(\( 1 + T + 3 T^{2} + 2 T^{3} + 4 T^{4} \))(\( 1 + T - T^{2} + 2 T^{3} + 4 T^{4} \))(\( 1 + 2 T^{2} + 4 T^{4} \))(\( ( 1 + 2 T^{2} )^{2} \))(\( 1 + 2 T^{2} + 4 T^{4} \))(\( 1 - T - 2 T^{3} + 4 T^{4} \))(\( 1 - T - T^{2} - 2 T^{3} + 4 T^{4} \))(\( 1 - T + 3 T^{2} - 2 T^{3} + 4 T^{4} \))(\( ( 1 - T + 2 T^{2} )^{2} \))(\( 1 + T + 2 T^{2} + T^{3} + 4 T^{4} + 4 T^{5} + 8 T^{6} \))(\( 1 + T + 2 T^{2} + T^{3} + 4 T^{4} + 4 T^{5} + 8 T^{6} \))(\( 1 - T + 2 T^{2} - T^{3} + 4 T^{4} - 4 T^{5} + 8 T^{6} \))(\( 1 - T + 2 T^{2} - T^{3} + 4 T^{4} - 4 T^{5} + 8 T^{6} \))(\( ( 1 + 3 T + 5 T^{2} + 6 T^{3} + 4 T^{4} )^{2} \))(\( ( 1 - 2 T^{2} + 4 T^{4} )^{2} \))(\( ( 1 - 3 T + 5 T^{2} - 6 T^{3} + 4 T^{4} )^{2} \))
$3$ (\( 1 - T + 3 T^{2} \))(\( 1 + T + 3 T^{2} \))(\( 1 + 3 T^{2} \))(\( 1 - T + 3 T^{2} \))(\( 1 - T + 3 T^{2} \))(\( 1 + T + 3 T^{2} \))(\( 1 - T + 3 T^{2} \))(\( 1 + 3 T + 3 T^{2} \))(\( 1 + T + 3 T^{2} \))(\( 1 - 3 T + 3 T^{2} \))(\( 1 + 2 T + 5 T^{2} + 6 T^{3} + 9 T^{4} \))(\( 1 - 2 T^{2} + 9 T^{4} \))(\( 1 - 2 T + 5 T^{2} - 6 T^{3} + 9 T^{4} \))(\( 1 + 2 T + 2 T^{2} + 6 T^{3} + 9 T^{4} \))(\( ( 1 + 3 T^{2} )^{2} \))(\( 1 + 2 T + 5 T^{2} + 6 T^{3} + 9 T^{4} \))(\( 1 + T^{2} + 9 T^{4} \))(\( 1 - 2 T + 5 T^{2} - 6 T^{3} + 9 T^{4} \))(\( 1 + T + 2 T^{2} + 3 T^{3} + 9 T^{4} \))(\( ( 1 + 3 T^{2} )^{2} \))(\( 1 - 2 T + 2 T^{2} - 6 T^{3} + 9 T^{4} \))(\( 1 - 2 T^{2} + 9 T^{4} \))(\( 1 + 3 T + 7 T^{2} + 11 T^{3} + 21 T^{4} + 27 T^{5} + 27 T^{6} \))(\( 1 - 3 T + 7 T^{2} - 11 T^{3} + 21 T^{4} - 27 T^{5} + 27 T^{6} \))(\( 1 + 3 T + 7 T^{2} + 11 T^{3} + 21 T^{4} + 27 T^{5} + 27 T^{6} \))(\( 1 - 3 T + 7 T^{2} - 11 T^{3} + 21 T^{4} - 27 T^{5} + 27 T^{6} \))(\( 1 + 6 T^{2} + 22 T^{4} + 54 T^{6} + 81 T^{8} \))(\( ( 1 + 3 T^{2} + 9 T^{4} )^{2} \))(\( 1 + 6 T^{2} + 22 T^{4} + 54 T^{6} + 81 T^{8} \))
$5$ 1
$7$ 1
$11$ (\( 1 + 3 T + 11 T^{2} \))(\( 1 + 11 T^{2} \))(\( 1 - 4 T + 11 T^{2} \))(\( 1 + 11 T^{2} \))(\( 1 + 3 T + 11 T^{2} \))(\( 1 + 11 T^{2} \))(\( 1 + 11 T^{2} \))(\( 1 - T + 11 T^{2} \))(\( 1 + 3 T + 11 T^{2} \))(\( 1 - T + 11 T^{2} \))(\( 1 - 4 T + 18 T^{2} - 44 T^{3} + 121 T^{4} \))(\( ( 1 + 11 T^{2} )^{2} \))(\( 1 - 4 T + 18 T^{2} - 44 T^{3} + 121 T^{4} \))(\( 1 - 4 T + 21 T^{2} - 44 T^{3} + 121 T^{4} \))(\( 1 + 4 T + 5 T^{2} + 44 T^{3} + 121 T^{4} \))(\( 1 + 6 T + 23 T^{2} + 66 T^{3} + 121 T^{4} \))(\( ( 1 + 3 T + 11 T^{2} )^{2} \))(\( 1 + 6 T + 23 T^{2} + 66 T^{3} + 121 T^{4} \))(\( 1 - T + 18 T^{2} - 11 T^{3} + 121 T^{4} \))(\( 1 + 4 T + 5 T^{2} + 44 T^{3} + 121 T^{4} \))(\( 1 - 4 T + 21 T^{2} - 44 T^{3} + 121 T^{4} \))(\( ( 1 + 11 T^{2} )^{2} \))(\( 1 + 2 T + 14 T^{2} - T^{3} + 154 T^{4} + 242 T^{5} + 1331 T^{6} \))(\( 1 + 2 T + 14 T^{2} - T^{3} + 154 T^{4} + 242 T^{5} + 1331 T^{6} \))(\( 1 + 2 T + 14 T^{2} - T^{3} + 154 T^{4} + 242 T^{5} + 1331 T^{6} \))(\( 1 + 2 T + 14 T^{2} - T^{3} + 154 T^{4} + 242 T^{5} + 1331 T^{6} \))(\( ( 1 + 17 T^{2} + 121 T^{4} )^{2} \))(\( ( 1 - 5 T + 11 T^{2} )^{4} \))(\( ( 1 + 17 T^{2} + 121 T^{4} )^{2} \))
$13$ (\( 1 - T + 13 T^{2} \))(\( 1 - 2 T + 13 T^{2} \))(\( 1 + 13 T^{2} \))(\( 1 + 2 T + 13 T^{2} \))(\( 1 - 5 T + 13 T^{2} \))(\( 1 - 2 T + 13 T^{2} \))(\( 1 + 2 T + 13 T^{2} \))(\( 1 + 3 T + 13 T^{2} \))(\( 1 + T + 13 T^{2} \))(\( 1 - 3 T + 13 T^{2} \))(\( 1 - 4 T + 22 T^{2} - 52 T^{3} + 169 T^{4} \))(\( 1 + 8 T^{2} + 169 T^{4} \))(\( 1 + 4 T + 22 T^{2} + 52 T^{3} + 169 T^{4} \))(\( 1 + 2 T + 22 T^{2} + 26 T^{3} + 169 T^{4} \))(\( ( 1 + 13 T^{2} )^{2} \))(\( 1 + 6 T + 33 T^{2} + 78 T^{3} + 169 T^{4} \))(\( 1 - 19 T^{2} + 169 T^{4} \))(\( 1 - 6 T + 33 T^{2} - 78 T^{3} + 169 T^{4} \))(\( 1 - 5 T + 28 T^{2} - 65 T^{3} + 169 T^{4} \))(\( ( 1 + 13 T^{2} )^{2} \))(\( 1 - 2 T + 22 T^{2} - 26 T^{3} + 169 T^{4} \))(\( 1 + 8 T^{2} + 169 T^{4} \))(\( 1 + 4 T + 24 T^{2} + 111 T^{3} + 312 T^{4} + 676 T^{5} + 2197 T^{6} \))(\( 1 - 4 T + 24 T^{2} - 111 T^{3} + 312 T^{4} - 676 T^{5} + 2197 T^{6} \))(\( 1 + 4 T + 24 T^{2} + 111 T^{3} + 312 T^{4} + 676 T^{5} + 2197 T^{6} \))(\( 1 - 4 T + 24 T^{2} - 111 T^{3} + 312 T^{4} - 676 T^{5} + 2197 T^{6} \))(\( 1 + 6 T^{2} + 302 T^{4} + 1014 T^{6} + 28561 T^{8} \))(\( ( 1 + 23 T^{2} + 169 T^{4} )^{2} \))(\( 1 + 6 T^{2} + 302 T^{4} + 1014 T^{6} + 28561 T^{8} \))
$17$ (\( 1 - 7 T + 17 T^{2} \))(\( 1 - 2 T + 17 T^{2} \))(\( 1 + 17 T^{2} \))(\( 1 + 2 T + 17 T^{2} \))(\( 1 - 3 T + 17 T^{2} \))(\( 1 - 2 T + 17 T^{2} \))(\( 1 + 2 T + 17 T^{2} \))(\( 1 - 3 T + 17 T^{2} \))(\( 1 + 7 T + 17 T^{2} \))(\( 1 + 3 T + 17 T^{2} \))(\( 1 + 4 T + 30 T^{2} + 68 T^{3} + 289 T^{4} \))(\( 1 + 16 T^{2} + 289 T^{4} \))(\( 1 - 4 T + 30 T^{2} - 68 T^{3} + 289 T^{4} \))(\( 1 + 4 T + 18 T^{2} + 68 T^{3} + 289 T^{4} \))(\( ( 1 + 17 T^{2} )^{2} \))(\( 1 - 2 T + 17 T^{2} - 34 T^{3} + 289 T^{4} \))(\( 1 + 29 T^{2} + 289 T^{4} \))(\( 1 + 2 T + 17 T^{2} + 34 T^{3} + 289 T^{4} \))(\( 1 + 5 T + 36 T^{2} + 85 T^{3} + 289 T^{4} \))(\( ( 1 + 17 T^{2} )^{2} \))(\( 1 - 4 T + 18 T^{2} - 68 T^{3} + 289 T^{4} \))(\( 1 + 16 T^{2} + 289 T^{4} \))(\( 1 + 10 T + 80 T^{2} + 361 T^{3} + 1360 T^{4} + 2890 T^{5} + 4913 T^{6} \))(\( 1 - 10 T + 80 T^{2} - 361 T^{3} + 1360 T^{4} - 2890 T^{5} + 4913 T^{6} \))(\( 1 + 10 T + 80 T^{2} + 361 T^{3} + 1360 T^{4} + 2890 T^{5} + 4913 T^{6} \))(\( 1 - 10 T + 80 T^{2} - 361 T^{3} + 1360 T^{4} - 2890 T^{5} + 4913 T^{6} \))(\( 1 + 12 T^{2} - 106 T^{4} + 3468 T^{6} + 83521 T^{8} \))(\( ( 1 + 31 T^{2} + 289 T^{4} )^{2} \))(\( 1 + 12 T^{2} - 106 T^{4} + 3468 T^{6} + 83521 T^{8} \))
$19$ (\( 1 + 19 T^{2} \))(\( 1 + 6 T + 19 T^{2} \))(\( 1 + 19 T^{2} \))(\( 1 - 6 T + 19 T^{2} \))(\( 1 + 2 T + 19 T^{2} \))(\( 1 - 6 T + 19 T^{2} \))(\( 1 + 6 T + 19 T^{2} \))(\( 1 - 6 T + 19 T^{2} \))(\( 1 + 19 T^{2} \))(\( 1 + 6 T + 19 T^{2} \))(\( 1 + 30 T^{2} + 361 T^{4} \))(\( 1 + 30 T^{2} + 361 T^{4} \))(\( 1 + 30 T^{2} + 361 T^{4} \))(\( 1 + 18 T^{2} + 361 T^{4} \))(\( ( 1 + 19 T^{2} )^{2} \))(\( ( 1 - 6 T + 19 T^{2} )^{2} \))(\( ( 1 + 19 T^{2} )^{2} \))(\( ( 1 + 6 T + 19 T^{2} )^{2} \))(\( 1 - 6 T + 30 T^{2} - 114 T^{3} + 361 T^{4} \))(\( ( 1 + 19 T^{2} )^{2} \))(\( 1 + 18 T^{2} + 361 T^{4} \))(\( 1 + 30 T^{2} + 361 T^{4} \))(\( 1 - 4 T + 40 T^{2} - 159 T^{3} + 760 T^{4} - 1444 T^{5} + 6859 T^{6} \))(\( 1 + 4 T + 40 T^{2} + 159 T^{3} + 760 T^{4} + 1444 T^{5} + 6859 T^{6} \))(\( 1 + 4 T + 40 T^{2} + 159 T^{3} + 760 T^{4} + 1444 T^{5} + 6859 T^{6} \))(\( 1 - 4 T + 40 T^{2} - 159 T^{3} + 760 T^{4} - 1444 T^{5} + 6859 T^{6} \))(\( ( 1 + 20 T^{2} + 361 T^{4} )^{2} \))(\( ( 1 + 30 T^{2} + 361 T^{4} )^{2} \))(\( ( 1 + 20 T^{2} + 361 T^{4} )^{2} \))
$23$ (\( 1 + 6 T + 23 T^{2} \))(\( 1 - 3 T + 23 T^{2} \))(\( 1 + 8 T + 23 T^{2} \))(\( 1 - 3 T + 23 T^{2} \))(\( 1 - 6 T + 23 T^{2} \))(\( 1 + 3 T + 23 T^{2} \))(\( 1 + 3 T + 23 T^{2} \))(\( 1 - 4 T + 23 T^{2} \))(\( 1 - 6 T + 23 T^{2} \))(\( 1 - 4 T + 23 T^{2} \))(\( 1 - 2 T + 45 T^{2} - 46 T^{3} + 529 T^{4} \))(\( ( 1 + 4 T + 23 T^{2} )^{2} \))(\( 1 - 2 T + 45 T^{2} - 46 T^{3} + 529 T^{4} \))(\( 1 + 8 T + 57 T^{2} + 184 T^{3} + 529 T^{4} \))(\( 1 + 8 T + 41 T^{2} + 184 T^{3} + 529 T^{4} \))(\( 1 + 12 T + 80 T^{2} + 276 T^{3} + 529 T^{4} \))(\( ( 1 + 23 T^{2} )^{2} \))(\( 1 + 12 T + 80 T^{2} + 276 T^{3} + 529 T^{4} \))(\( 1 - 2 T + 30 T^{2} - 46 T^{3} + 529 T^{4} \))(\( 1 - 8 T + 41 T^{2} - 184 T^{3} + 529 T^{4} \))(\( 1 - 8 T + 57 T^{2} - 184 T^{3} + 529 T^{4} \))(\( ( 1 - 4 T + 23 T^{2} )^{2} \))(\( 1 - T + 59 T^{2} - 55 T^{3} + 1357 T^{4} - 529 T^{5} + 12167 T^{6} \))(\( 1 - T + 59 T^{2} - 55 T^{3} + 1357 T^{4} - 529 T^{5} + 12167 T^{6} \))(\( 1 + T + 59 T^{2} + 55 T^{3} + 1357 T^{4} + 529 T^{5} + 12167 T^{6} \))(\( 1 + T + 59 T^{2} + 55 T^{3} + 1357 T^{4} + 529 T^{5} + 12167 T^{6} \))(\( ( 1 + 12 T + 77 T^{2} + 276 T^{3} + 529 T^{4} )^{2} \))(\( ( 1 + 40 T^{2} + 529 T^{4} )^{2} \))(\( ( 1 - 12 T + 77 T^{2} - 276 T^{3} + 529 T^{4} )^{2} \))
$29$ (\( 1 + 5 T + 29 T^{2} \))(\( 1 - 7 T + 29 T^{2} \))(\( 1 - 2 T + 29 T^{2} \))(\( 1 - 7 T + 29 T^{2} \))(\( 1 - 3 T + 29 T^{2} \))(\( 1 - 7 T + 29 T^{2} \))(\( 1 - 7 T + 29 T^{2} \))(\( 1 + T + 29 T^{2} \))(\( 1 + 5 T + 29 T^{2} \))(\( 1 + T + 29 T^{2} \))(\( ( 1 + T + 29 T^{2} )^{2} \))(\( ( 1 + 29 T^{2} )^{2} \))(\( ( 1 + T + 29 T^{2} )^{2} \))(\( ( 1 - 5 T + 29 T^{2} )^{2} \))(\( 1 + 2 T - 25 T^{2} + 58 T^{3} + 841 T^{4} \))(\( 1 + 6 T + 35 T^{2} + 174 T^{3} + 841 T^{4} \))(\( ( 1 + 9 T + 29 T^{2} )^{2} \))(\( 1 + 6 T + 35 T^{2} + 174 T^{3} + 841 T^{4} \))(\( 1 - T + 20 T^{2} - 29 T^{3} + 841 T^{4} \))(\( 1 + 2 T - 25 T^{2} + 58 T^{3} + 841 T^{4} \))(\( ( 1 - 5 T + 29 T^{2} )^{2} \))(\( ( 1 + 29 T^{2} )^{2} \))(\( 1 + 8 T + 82 T^{2} + 389 T^{3} + 2378 T^{4} + 6728 T^{5} + 24389 T^{6} \))(\( 1 + 8 T + 82 T^{2} + 389 T^{3} + 2378 T^{4} + 6728 T^{5} + 24389 T^{6} \))(\( 1 + 8 T + 82 T^{2} + 389 T^{3} + 2378 T^{4} + 6728 T^{5} + 24389 T^{6} \))(\( 1 + 8 T + 82 T^{2} + 389 T^{3} + 2378 T^{4} + 6728 T^{5} + 24389 T^{6} \))(\( ( 1 + 53 T^{2} + 841 T^{4} )^{2} \))(\( ( 1 - 5 T + 29 T^{2} )^{4} \))(\( ( 1 + 53 T^{2} + 841 T^{4} )^{2} \))
$31$ (\( 1 + 2 T + 31 T^{2} \))(\( 1 + 2 T + 31 T^{2} \))(\( 1 + 31 T^{2} \))(\( 1 - 2 T + 31 T^{2} \))(\( 1 - 4 T + 31 T^{2} \))(\( 1 - 2 T + 31 T^{2} \))(\( 1 + 2 T + 31 T^{2} \))(\( 1 - 6 T + 31 T^{2} \))(\( 1 + 2 T + 31 T^{2} \))(\( 1 + 6 T + 31 T^{2} \))(\( ( 1 + 6 T + 31 T^{2} )^{2} \))(\( 1 + 30 T^{2} + 961 T^{4} \))(\( ( 1 - 6 T + 31 T^{2} )^{2} \))(\( 1 - 6 T + 26 T^{2} - 186 T^{3} + 961 T^{4} \))(\( ( 1 + 31 T^{2} )^{2} \))(\( 1 - 12 T + 80 T^{2} - 372 T^{3} + 961 T^{4} \))(\( ( 1 + 31 T^{2} )^{2} \))(\( 1 + 12 T + 80 T^{2} + 372 T^{3} + 961 T^{4} \))(\( ( 1 + 31 T^{2} )^{2} \))(\( ( 1 + 31 T^{2} )^{2} \))(\( 1 - 6 T + 26 T^{2} - 186 T^{3} + 961 T^{4} \))(\( 1 + 30 T^{2} + 961 T^{4} \))(\( 1 + 5 T + 75 T^{2} + 247 T^{3} + 2325 T^{4} + 4805 T^{5} + 29791 T^{6} \))(\( 1 - 5 T + 75 T^{2} - 247 T^{3} + 2325 T^{4} - 4805 T^{5} + 29791 T^{6} \))(\( 1 - 5 T + 75 T^{2} - 247 T^{3} + 2325 T^{4} - 4805 T^{5} + 29791 T^{6} \))(\( 1 + 5 T + 75 T^{2} + 247 T^{3} + 2325 T^{4} + 4805 T^{5} + 29791 T^{6} \))(\( 1 + 70 T^{2} + 2742 T^{4} + 67270 T^{6} + 923521 T^{8} \))(\( ( 1 + 60 T^{2} + 961 T^{4} )^{2} \))(\( 1 + 70 T^{2} + 2742 T^{4} + 67270 T^{6} + 923521 T^{8} \))
$37$ (\( 1 + 2 T + 37 T^{2} \))(\( 1 - 8 T + 37 T^{2} \))(\( 1 - 6 T + 37 T^{2} \))(\( 1 - 8 T + 37 T^{2} \))(\( 1 + 2 T + 37 T^{2} \))(\( 1 + 8 T + 37 T^{2} \))(\( 1 + 8 T + 37 T^{2} \))(\( 1 + 37 T^{2} \))(\( 1 - 2 T + 37 T^{2} \))(\( 1 + 37 T^{2} \))(\( ( 1 + 37 T^{2} )^{2} \))(\( ( 1 + 6 T + 37 T^{2} )^{2} \))(\( ( 1 + 37 T^{2} )^{2} \))(\( ( 1 + 3 T + 37 T^{2} )^{2} \))(\( 1 - 6 T - T^{2} - 222 T^{3} + 1369 T^{4} \))(\( 1 - 4 T + 60 T^{2} - 148 T^{3} + 1369 T^{4} \))(\( ( 1 + 37 T^{2} )^{2} \))(\( 1 - 4 T + 60 T^{2} - 148 T^{3} + 1369 T^{4} \))(\( ( 1 + 6 T + 37 T^{2} )^{2} \))(\( 1 + 6 T - T^{2} + 222 T^{3} + 1369 T^{4} \))(\( ( 1 - 3 T + 37 T^{2} )^{2} \))(\( ( 1 - 6 T + 37 T^{2} )^{2} \))(\( 1 + 10 T + 120 T^{2} + 659 T^{3} + 4440 T^{4} + 13690 T^{5} + 50653 T^{6} \))(\( 1 + 10 T + 120 T^{2} + 659 T^{3} + 4440 T^{4} + 13690 T^{5} + 50653 T^{6} \))(\( 1 - 10 T + 120 T^{2} - 659 T^{3} + 4440 T^{4} - 13690 T^{5} + 50653 T^{6} \))(\( 1 - 10 T + 120 T^{2} - 659 T^{3} + 4440 T^{4} - 13690 T^{5} + 50653 T^{6} \))(\( ( 1 + 8 T + 45 T^{2} + 296 T^{3} + 1369 T^{4} )^{2} \))(\( ( 1 + 68 T^{2} + 1369 T^{4} )^{2} \))(\( ( 1 - 8 T + 45 T^{2} - 296 T^{3} + 1369 T^{4} )^{2} \))
$41$ (\( 1 + 2 T + 41 T^{2} \))(\( 1 + 5 T + 41 T^{2} \))(\( 1 + 41 T^{2} \))(\( 1 - 5 T + 41 T^{2} \))(\( 1 - 12 T + 41 T^{2} \))(\( 1 - 5 T + 41 T^{2} \))(\( 1 + 5 T + 41 T^{2} \))(\( 1 - 6 T + 41 T^{2} \))(\( 1 + 2 T + 41 T^{2} \))(\( 1 + 6 T + 41 T^{2} \))(\( 1 + 10 T + 99 T^{2} + 410 T^{3} + 1681 T^{4} \))(\( 1 + 64 T^{2} + 1681 T^{4} \))(\( 1 - 10 T + 99 T^{2} - 410 T^{3} + 1681 T^{4} \))(\( 1 + 14 T + 126 T^{2} + 574 T^{3} + 1681 T^{4} \))(\( ( 1 + 41 T^{2} )^{2} \))(\( 1 + 4 T + 68 T^{2} + 164 T^{3} + 1681 T^{4} \))(\( ( 1 + 41 T^{2} )^{2} \))(\( 1 - 4 T + 68 T^{2} - 164 T^{3} + 1681 T^{4} \))(\( 1 + 2 T + 66 T^{2} + 82 T^{3} + 1681 T^{4} \))(\( ( 1 + 41 T^{2} )^{2} \))(\( 1 + 14 T + 126 T^{2} + 574 T^{3} + 1681 T^{4} \))(\( 1 + 64 T^{2} + 1681 T^{4} \))(\( 1 - T + 77 T^{2} - 187 T^{3} + 3157 T^{4} - 1681 T^{5} + 68921 T^{6} \))(\( 1 + T + 77 T^{2} + 187 T^{3} + 3157 T^{4} + 1681 T^{5} + 68921 T^{6} \))(\( 1 + T + 77 T^{2} + 187 T^{3} + 3157 T^{4} + 1681 T^{5} + 68921 T^{6} \))(\( 1 - T + 77 T^{2} - 187 T^{3} + 3157 T^{4} - 1681 T^{5} + 68921 T^{6} \))(\( 1 + 78 T^{2} + 4478 T^{4} + 131118 T^{6} + 2825761 T^{8} \))(\( ( 1 - 16 T^{2} + 1681 T^{4} )^{2} \))(\( 1 + 78 T^{2} + 4478 T^{4} + 131118 T^{6} + 2825761 T^{8} \))
$43$ (\( 1 - 4 T + 43 T^{2} \))(\( 1 + 7 T + 43 T^{2} \))(\( 1 - 12 T + 43 T^{2} \))(\( 1 + 7 T + 43 T^{2} \))(\( 1 - 10 T + 43 T^{2} \))(\( 1 - 7 T + 43 T^{2} \))(\( 1 - 7 T + 43 T^{2} \))(\( 1 - 6 T + 43 T^{2} \))(\( 1 + 4 T + 43 T^{2} \))(\( 1 - 6 T + 43 T^{2} \))(\( 1 + 10 T + 109 T^{2} + 430 T^{3} + 1849 T^{4} \))(\( ( 1 + 43 T^{2} )^{2} \))(\( 1 + 10 T + 109 T^{2} + 430 T^{3} + 1849 T^{4} \))(\( 1 + 8 T + 97 T^{2} + 344 T^{3} + 1849 T^{4} \))(\( 1 - 12 T + 101 T^{2} - 516 T^{3} + 1849 T^{4} \))(\( ( 1 + 2 T + 43 T^{2} )^{2} \))(\( ( 1 + 43 T^{2} )^{2} \))(\( ( 1 + 2 T + 43 T^{2} )^{2} \))(\( 1 + 10 T + 94 T^{2} + 430 T^{3} + 1849 T^{4} \))(\( 1 + 12 T + 101 T^{2} + 516 T^{3} + 1849 T^{4} \))(\( 1 - 8 T + 97 T^{2} - 344 T^{3} + 1849 T^{4} \))(\( ( 1 + 43 T^{2} )^{2} \))(\( 1 - 9 T + 109 T^{2} - 721 T^{3} + 4687 T^{4} - 16641 T^{5} + 79507 T^{6} \))(\( 1 - 9 T + 109 T^{2} - 721 T^{3} + 4687 T^{4} - 16641 T^{5} + 79507 T^{6} \))(\( 1 + 9 T + 109 T^{2} + 721 T^{3} + 4687 T^{4} + 16641 T^{5} + 79507 T^{6} \))(\( 1 + 9 T + 109 T^{2} + 721 T^{3} + 4687 T^{4} + 16641 T^{5} + 79507 T^{6} \))(\( ( 1 + 5 T + 43 T^{2} )^{4} \))(\( ( 1 + 43 T^{2} )^{4} \))(\( ( 1 - 5 T + 43 T^{2} )^{4} \))
$47$ (\( 1 + 3 T + 47 T^{2} \))(\( 1 + 47 T^{2} \))(\( 1 + 47 T^{2} \))(\( 1 + 47 T^{2} \))(\( 1 - 9 T + 47 T^{2} \))(\( 1 + 47 T^{2} \))(\( 1 + 47 T^{2} \))(\( 1 - 9 T + 47 T^{2} \))(\( 1 - 3 T + 47 T^{2} \))(\( 1 + 9 T + 47 T^{2} \))(\( ( 1 + 2 T + 47 T^{2} )^{2} \))(\( ( 1 + 47 T^{2} )^{2} \))(\( ( 1 - 2 T + 47 T^{2} )^{2} \))(\( ( 1 + 2 T + 47 T^{2} )^{2} \))(\( ( 1 + 47 T^{2} )^{2} \))(\( 1 - 6 T + 85 T^{2} - 282 T^{3} + 2209 T^{4} \))(\( 1 - 31 T^{2} + 2209 T^{4} \))(\( 1 + 6 T + 85 T^{2} + 282 T^{3} + 2209 T^{4} \))(\( 1 + 5 T + 62 T^{2} + 235 T^{3} + 2209 T^{4} \))(\( ( 1 + 47 T^{2} )^{2} \))(\( ( 1 - 2 T + 47 T^{2} )^{2} \))(\( ( 1 + 47 T^{2} )^{2} \))(\( 1 + 28 T + 394 T^{2} + 3367 T^{3} + 18518 T^{4} + 61852 T^{5} + 103823 T^{6} \))(\( 1 - 28 T + 394 T^{2} - 3367 T^{3} + 18518 T^{4} - 61852 T^{5} + 103823 T^{6} \))(\( 1 + 28 T + 394 T^{2} + 3367 T^{3} + 18518 T^{4} + 61852 T^{5} + 103823 T^{6} \))(\( 1 - 28 T + 394 T^{2} - 3367 T^{3} + 18518 T^{4} - 61852 T^{5} + 103823 T^{6} \))(\( ( 1 + 4 T^{2} + 2209 T^{4} )^{2} \))(\( ( 1 + 19 T^{2} + 2209 T^{4} )^{2} \))(\( ( 1 + 4 T^{2} + 2209 T^{4} )^{2} \))
$53$ (\( 1 + 6 T + 53 T^{2} \))(\( 1 + 6 T + 53 T^{2} \))(\( 1 - 10 T + 53 T^{2} \))(\( 1 + 6 T + 53 T^{2} \))(\( 1 + 12 T + 53 T^{2} \))(\( 1 - 6 T + 53 T^{2} \))(\( 1 - 6 T + 53 T^{2} \))(\( 1 - 10 T + 53 T^{2} \))(\( 1 - 6 T + 53 T^{2} \))(\( 1 - 10 T + 53 T^{2} \))(\( 1 - 8 T + 114 T^{2} - 424 T^{3} + 2809 T^{4} \))(\( ( 1 - 8 T + 53 T^{2} )^{2} \))(\( 1 - 8 T + 114 T^{2} - 424 T^{3} + 2809 T^{4} \))(\( 1 + 8 T + 102 T^{2} + 424 T^{3} + 2809 T^{4} \))(\( ( 1 + 10 T + 53 T^{2} )^{2} \))(\( 1 + 88 T^{2} + 2809 T^{4} \))(\( ( 1 + 53 T^{2} )^{2} \))(\( 1 + 88 T^{2} + 2809 T^{4} \))(\( 1 - 2 T + 90 T^{2} - 106 T^{3} + 2809 T^{4} \))(\( ( 1 - 10 T + 53 T^{2} )^{2} \))(\( 1 - 8 T + 102 T^{2} - 424 T^{3} + 2809 T^{4} \))(\( ( 1 + 8 T + 53 T^{2} )^{2} \))(\( 1 - 10 T + 140 T^{2} - 937 T^{3} + 7420 T^{4} - 28090 T^{5} + 148877 T^{6} \))(\( 1 - 10 T + 140 T^{2} - 937 T^{3} + 7420 T^{4} - 28090 T^{5} + 148877 T^{6} \))(\( 1 + 10 T + 140 T^{2} + 937 T^{3} + 7420 T^{4} + 28090 T^{5} + 148877 T^{6} \))(\( 1 + 10 T + 140 T^{2} + 937 T^{3} + 7420 T^{4} + 28090 T^{5} + 148877 T^{6} \))(\( ( 1 + 6 T + 70 T^{2} + 318 T^{3} + 2809 T^{4} )^{2} \))(\( ( 1 + 52 T^{2} + 2809 T^{4} )^{2} \))(\( ( 1 - 6 T + 70 T^{2} - 318 T^{3} + 2809 T^{4} )^{2} \))
$59$ (\( 1 + 10 T + 59 T^{2} \))(\( 1 + 10 T + 59 T^{2} \))(\( 1 + 59 T^{2} \))(\( 1 - 10 T + 59 T^{2} \))(\( 1 + 59 T^{2} \))(\( 1 - 10 T + 59 T^{2} \))(\( 1 + 10 T + 59 T^{2} \))(\( 1 + 6 T + 59 T^{2} \))(\( 1 + 10 T + 59 T^{2} \))(\( 1 - 6 T + 59 T^{2} \))(\( 1 + 8 T + 62 T^{2} + 472 T^{3} + 3481 T^{4} \))(\( 1 + 46 T^{2} + 3481 T^{4} \))(\( 1 - 8 T + 62 T^{2} - 472 T^{3} + 3481 T^{4} \))(\( 1 + 10 T + 98 T^{2} + 590 T^{3} + 3481 T^{4} \))(\( ( 1 + 59 T^{2} )^{2} \))(\( 1 - 4 T + 104 T^{2} - 236 T^{3} + 3481 T^{4} \))(\( ( 1 + 59 T^{2} )^{2} \))(\( 1 + 4 T + 104 T^{2} + 236 T^{3} + 3481 T^{4} \))(\( ( 1 - 4 T + 59 T^{2} )^{2} \))(\( ( 1 + 59 T^{2} )^{2} \))(\( 1 + 10 T + 98 T^{2} + 590 T^{3} + 3481 T^{4} \))(\( 1 + 46 T^{2} + 3481 T^{4} \))(\( 1 - 5 T + 115 T^{2} - 695 T^{3} + 6785 T^{4} - 17405 T^{5} + 205379 T^{6} \))(\( 1 + 5 T + 115 T^{2} + 695 T^{3} + 6785 T^{4} + 17405 T^{5} + 205379 T^{6} \))(\( 1 + 5 T + 115 T^{2} + 695 T^{3} + 6785 T^{4} + 17405 T^{5} + 205379 T^{6} \))(\( 1 - 5 T + 115 T^{2} - 695 T^{3} + 6785 T^{4} - 17405 T^{5} + 205379 T^{6} \))(\( 1 - 18 T^{2} + 4838 T^{4} - 62658 T^{6} + 12117361 T^{8} \))(\( ( 1 + 116 T^{2} + 3481 T^{4} )^{2} \))(\( 1 - 18 T^{2} + 4838 T^{4} - 62658 T^{6} + 12117361 T^{8} \))
$61$ (\( 1 - 8 T + 61 T^{2} \))(\( 1 + 7 T + 61 T^{2} \))(\( 1 + 61 T^{2} \))(\( 1 - 7 T + 61 T^{2} \))(\( 1 + 8 T + 61 T^{2} \))(\( 1 - 7 T + 61 T^{2} \))(\( 1 + 7 T + 61 T^{2} \))(\( 1 + 61 T^{2} \))(\( 1 - 8 T + 61 T^{2} \))(\( 1 + 61 T^{2} \))(\( 1 + 6 T + 59 T^{2} + 366 T^{3} + 3721 T^{4} \))(\( 1 + 24 T^{2} + 3721 T^{4} \))(\( 1 - 6 T + 59 T^{2} - 366 T^{3} + 3721 T^{4} \))(\( 1 - 6 T + 86 T^{2} - 366 T^{3} + 3721 T^{4} \))(\( ( 1 + 61 T^{2} )^{2} \))(\( 1 + 114 T^{2} + 3721 T^{4} \))(\( ( 1 + 61 T^{2} )^{2} \))(\( 1 + 114 T^{2} + 3721 T^{4} \))(\( 1 + 6 T - 22 T^{2} + 366 T^{3} + 3721 T^{4} \))(\( ( 1 + 61 T^{2} )^{2} \))(\( 1 - 6 T + 86 T^{2} - 366 T^{3} + 3721 T^{4} \))(\( 1 + 24 T^{2} + 3721 T^{4} \))(\( 1 + 5 T + 165 T^{2} + 547 T^{3} + 10065 T^{4} + 18605 T^{5} + 226981 T^{6} \))(\( 1 - 5 T + 165 T^{2} - 547 T^{3} + 10065 T^{4} - 18605 T^{5} + 226981 T^{6} \))(\( 1 - 5 T + 165 T^{2} - 547 T^{3} + 10065 T^{4} - 18605 T^{5} + 226981 T^{6} \))(\( 1 + 5 T + 165 T^{2} + 547 T^{3} + 10065 T^{4} + 18605 T^{5} + 226981 T^{6} \))(\( 1 + 78 T^{2} + 3518 T^{4} + 290238 T^{6} + 13845841 T^{8} \))(\( ( 1 + 114 T^{2} + 3721 T^{4} )^{2} \))(\( 1 + 78 T^{2} + 3518 T^{4} + 290238 T^{6} + 13845841 T^{8} \))
$67$ (\( 1 + 2 T + 67 T^{2} \))(\( 1 - 5 T + 67 T^{2} \))(\( 1 + 4 T + 67 T^{2} \))(\( 1 - 5 T + 67 T^{2} \))(\( 1 - 4 T + 67 T^{2} \))(\( 1 + 5 T + 67 T^{2} \))(\( 1 + 5 T + 67 T^{2} \))(\( 1 - 14 T + 67 T^{2} \))(\( 1 - 2 T + 67 T^{2} \))(\( 1 - 14 T + 67 T^{2} \))(\( 1 + 22 T + 253 T^{2} + 1474 T^{3} + 4489 T^{4} \))(\( ( 1 - 12 T + 67 T^{2} )^{2} \))(\( 1 + 22 T + 253 T^{2} + 1474 T^{3} + 4489 T^{4} \))(\( 1 - 4 T + 133 T^{2} - 268 T^{3} + 4489 T^{4} \))(\( 1 + 4 T - 51 T^{2} + 268 T^{3} + 4489 T^{4} \))(\( 1 - 8 T + 132 T^{2} - 536 T^{3} + 4489 T^{4} \))(\( ( 1 + 67 T^{2} )^{2} \))(\( 1 - 8 T + 132 T^{2} - 536 T^{3} + 4489 T^{4} \))(\( 1 + 4 T + 70 T^{2} + 268 T^{3} + 4489 T^{4} \))(\( 1 - 4 T - 51 T^{2} - 268 T^{3} + 4489 T^{4} \))(\( 1 + 4 T + 133 T^{2} + 268 T^{3} + 4489 T^{4} \))(\( ( 1 + 12 T + 67 T^{2} )^{2} \))(\( 1 + 12 T + 229 T^{2} + 1568 T^{3} + 15343 T^{4} + 53868 T^{5} + 300763 T^{6} \))(\( 1 + 12 T + 229 T^{2} + 1568 T^{3} + 15343 T^{4} + 53868 T^{5} + 300763 T^{6} \))(\( 1 - 12 T + 229 T^{2} - 1568 T^{3} + 15343 T^{4} - 53868 T^{5} + 300763 T^{6} \))(\( 1 - 12 T + 229 T^{2} - 1568 T^{3} + 15343 T^{4} - 53868 T^{5} + 300763 T^{6} \))(\( ( 1 + 4 T + 93 T^{2} + 268 T^{3} + 4489 T^{4} )^{2} \))(\( ( 1 + 80 T^{2} + 4489 T^{4} )^{2} \))(\( ( 1 - 4 T + 93 T^{2} - 268 T^{3} + 4489 T^{4} )^{2} \))
$71$ (\( 1 + 8 T + 71 T^{2} \))(\( 1 + 2 T + 71 T^{2} \))(\( 1 - 16 T + 71 T^{2} \))(\( 1 + 2 T + 71 T^{2} \))(\( 1 + 71 T^{2} \))(\( 1 + 2 T + 71 T^{2} \))(\( 1 + 2 T + 71 T^{2} \))(\( 1 + 8 T + 71 T^{2} \))(\( 1 + 8 T + 71 T^{2} \))(\( 1 + 8 T + 71 T^{2} \))(\( 1 + 8 T + 86 T^{2} + 568 T^{3} + 5041 T^{4} \))(\( ( 1 - 12 T + 71 T^{2} )^{2} \))(\( 1 + 8 T + 86 T^{2} + 568 T^{3} + 5041 T^{4} \))(\( 1 - 4 T + 101 T^{2} - 284 T^{3} + 5041 T^{4} \))(\( 1 + 16 T + 185 T^{2} + 1136 T^{3} + 5041 T^{4} \))(\( 1 + 12 T + 170 T^{2} + 852 T^{3} + 5041 T^{4} \))(\( ( 1 + 12 T + 71 T^{2} )^{2} \))(\( 1 + 12 T + 170 T^{2} + 852 T^{3} + 5041 T^{4} \))(\( ( 1 - 8 T + 71 T^{2} )^{2} \))(\( 1 + 16 T + 185 T^{2} + 1136 T^{3} + 5041 T^{4} \))(\( 1 - 4 T + 101 T^{2} - 284 T^{3} + 5041 T^{4} \))(\( ( 1 - 12 T + 71 T^{2} )^{2} \))(\( 1 - 7 T + 124 T^{2} - 571 T^{3} + 8804 T^{4} - 35287 T^{5} + 357911 T^{6} \))(\( 1 - 7 T + 124 T^{2} - 571 T^{3} + 8804 T^{4} - 35287 T^{5} + 357911 T^{6} \))(\( 1 - 7 T + 124 T^{2} - 571 T^{3} + 8804 T^{4} - 35287 T^{5} + 357911 T^{6} \))(\( 1 - 7 T + 124 T^{2} - 571 T^{3} + 8804 T^{4} - 35287 T^{5} + 357911 T^{6} \))(\( ( 1 + 6 T + 131 T^{2} + 426 T^{3} + 5041 T^{4} )^{2} \))(\( ( 1 - 2 T + 71 T^{2} )^{4} \))(\( ( 1 + 6 T + 131 T^{2} + 426 T^{3} + 5041 T^{4} )^{2} \))
$73$ (\( 1 - 6 T + 73 T^{2} \))(\( 1 + 6 T + 73 T^{2} \))(\( 1 + 73 T^{2} \))(\( 1 - 6 T + 73 T^{2} \))(\( 1 - 2 T + 73 T^{2} \))(\( 1 + 6 T + 73 T^{2} \))(\( 1 - 6 T + 73 T^{2} \))(\( 1 + 6 T + 73 T^{2} \))(\( 1 + 6 T + 73 T^{2} \))(\( 1 - 6 T + 73 T^{2} \))(\( 1 + 4 T + 142 T^{2} + 292 T^{3} + 5329 T^{4} \))(\( 1 - 16 T^{2} + 5329 T^{4} \))(\( 1 - 4 T + 142 T^{2} - 292 T^{3} + 5329 T^{4} \))(\( 1 + 22 T + 262 T^{2} + 1606 T^{3} + 5329 T^{4} \))(\( ( 1 + 73 T^{2} )^{2} \))(\( 1 + 74 T^{2} + 5329 T^{4} \))(\( 1 - 34 T^{2} + 5329 T^{4} \))(\( 1 + 74 T^{2} + 5329 T^{4} \))(\( 1 + 8 T + 94 T^{2} + 584 T^{3} + 5329 T^{4} \))(\( ( 1 + 73 T^{2} )^{2} \))(\( 1 - 22 T + 262 T^{2} - 1606 T^{3} + 5329 T^{4} \))(\( 1 - 16 T^{2} + 5329 T^{4} \))(\( 1 - 12 T + 187 T^{2} - 1304 T^{3} + 13651 T^{4} - 63948 T^{5} + 389017 T^{6} \))(\( 1 + 12 T + 187 T^{2} + 1304 T^{3} + 13651 T^{4} + 63948 T^{5} + 389017 T^{6} \))(\( 1 - 12 T + 187 T^{2} - 1304 T^{3} + 13651 T^{4} - 63948 T^{5} + 389017 T^{6} \))(\( 1 + 12 T + 187 T^{2} + 1304 T^{3} + 13651 T^{4} + 63948 T^{5} + 389017 T^{6} \))(\( 1 + 238 T^{2} + 24414 T^{4} + 1268302 T^{6} + 28398241 T^{8} \))(\( ( 1 - 46 T^{2} + 5329 T^{4} )^{2} \))(\( 1 + 238 T^{2} + 24414 T^{4} + 1268302 T^{6} + 28398241 T^{8} \))
$79$ (\( 1 + 5 T + 79 T^{2} \))(\( 1 + 2 T + 79 T^{2} \))(\( 1 - 8 T + 79 T^{2} \))(\( 1 + 2 T + 79 T^{2} \))(\( 1 + T + 79 T^{2} \))(\( 1 + 2 T + 79 T^{2} \))(\( 1 + 2 T + 79 T^{2} \))(\( 1 + T + 79 T^{2} \))(\( 1 + 5 T + 79 T^{2} \))(\( 1 + T + 79 T^{2} \))(\( 1 - 24 T + 294 T^{2} - 1896 T^{3} + 6241 T^{4} \))(\( ( 1 - 12 T + 79 T^{2} )^{2} \))(\( 1 - 24 T + 294 T^{2} - 1896 T^{3} + 6241 T^{4} \))(\( 1 + 33 T^{2} + 6241 T^{4} \))(\( 1 + 8 T - 15 T^{2} + 632 T^{3} + 6241 T^{4} \))(\( 1 + 14 T + 135 T^{2} + 1106 T^{3} + 6241 T^{4} \))(\( ( 1 + T + 79 T^{2} )^{2} \))(\( 1 + 14 T + 135 T^{2} + 1106 T^{3} + 6241 T^{4} \))(\( 1 + 9 T + 174 T^{2} + 711 T^{3} + 6241 T^{4} \))(\( 1 + 8 T - 15 T^{2} + 632 T^{3} + 6241 T^{4} \))(\( 1 + 33 T^{2} + 6241 T^{4} \))(\( ( 1 - 12 T + 79 T^{2} )^{2} \))(\( 1 + 7 T + 145 T^{2} + 1257 T^{3} + 11455 T^{4} + 43687 T^{5} + 493039 T^{6} \))(\( 1 + 7 T + 145 T^{2} + 1257 T^{3} + 11455 T^{4} + 43687 T^{5} + 493039 T^{6} \))(\( 1 + 7 T + 145 T^{2} + 1257 T^{3} + 11455 T^{4} + 43687 T^{5} + 493039 T^{6} \))(\( 1 + 7 T + 145 T^{2} + 1257 T^{3} + 11455 T^{4} + 43687 T^{5} + 493039 T^{6} \))(\( ( 1 - 4 T + 117 T^{2} - 316 T^{3} + 6241 T^{4} )^{2} \))(\( ( 1 + 3 T + 79 T^{2} )^{4} \))(\( ( 1 - 4 T + 117 T^{2} - 316 T^{3} + 6241 T^{4} )^{2} \))
$83$ (\( 1 + 4 T + 83 T^{2} \))(\( 1 + 11 T + 83 T^{2} \))(\( 1 + 83 T^{2} \))(\( 1 - 11 T + 83 T^{2} \))(\( 1 - 12 T + 83 T^{2} \))(\( 1 + 11 T + 83 T^{2} \))(\( 1 - 11 T + 83 T^{2} \))(\( 1 + 12 T + 83 T^{2} \))(\( 1 - 4 T + 83 T^{2} \))(\( 1 - 12 T + 83 T^{2} \))(\( 1 + 2 T + 5 T^{2} + 166 T^{3} + 6889 T^{4} \))(\( 1 + 94 T^{2} + 6889 T^{4} \))(\( 1 - 2 T + 5 T^{2} - 166 T^{3} + 6889 T^{4} \))(\( 1 + 2 T + 122 T^{2} + 166 T^{3} + 6889 T^{4} \))(\( ( 1 + 83 T^{2} )^{2} \))(\( ( 1 + 83 T^{2} )^{2} \))(\( 1 + 86 T^{2} + 6889 T^{4} \))(\( ( 1 + 83 T^{2} )^{2} \))(\( ( 1 - 4 T + 83 T^{2} )^{2} \))(\( ( 1 + 83 T^{2} )^{2} \))(\( 1 - 2 T + 122 T^{2} - 166 T^{3} + 6889 T^{4} \))(\( 1 + 94 T^{2} + 6889 T^{4} \))(\( 1 + 26 T + 448 T^{2} + 4715 T^{3} + 37184 T^{4} + 179114 T^{5} + 571787 T^{6} \))(\( 1 - 26 T + 448 T^{2} - 4715 T^{3} + 37184 T^{4} - 179114 T^{5} + 571787 T^{6} \))(\( 1 + 26 T + 448 T^{2} + 4715 T^{3} + 37184 T^{4} + 179114 T^{5} + 571787 T^{6} \))(\( 1 - 26 T + 448 T^{2} - 4715 T^{3} + 37184 T^{4} - 179114 T^{5} + 571787 T^{6} \))(\( 1 + 278 T^{2} + 32694 T^{4} + 1915142 T^{6} + 47458321 T^{8} \))(\( ( 1 - 26 T^{2} + 6889 T^{4} )^{2} \))(\( 1 + 278 T^{2} + 32694 T^{4} + 1915142 T^{6} + 47458321 T^{8} \))
$89$ (\( 1 + 89 T^{2} \))(\( 1 + 9 T + 89 T^{2} \))(\( 1 + 89 T^{2} \))(\( 1 - 9 T + 89 T^{2} \))(\( 1 - 12 T + 89 T^{2} \))(\( 1 - 9 T + 89 T^{2} \))(\( 1 + 9 T + 89 T^{2} \))(\( 1 - 12 T + 89 T^{2} \))(\( 1 + 89 T^{2} \))(\( 1 + 12 T + 89 T^{2} \))(\( 1 + 6 T + 155 T^{2} + 534 T^{3} + 7921 T^{4} \))(\( 1 + 160 T^{2} + 7921 T^{4} \))(\( 1 - 6 T + 155 T^{2} - 534 T^{3} + 7921 T^{4} \))(\( 1 + 30 T + 398 T^{2} + 2670 T^{3} + 7921 T^{4} \))(\( ( 1 + 89 T^{2} )^{2} \))(\( ( 1 + 8 T + 89 T^{2} )^{2} \))(\( ( 1 + 89 T^{2} )^{2} \))(\( ( 1 - 8 T + 89 T^{2} )^{2} \))(\( 1 + 6 T + 170 T^{2} + 534 T^{3} + 7921 T^{4} \))(\( ( 1 + 89 T^{2} )^{2} \))(\( 1 + 30 T + 398 T^{2} + 2670 T^{3} + 7921 T^{4} \))(\( 1 + 160 T^{2} + 7921 T^{4} \))(\( 1 - 6 T + 130 T^{2} - 291 T^{3} + 11570 T^{4} - 47526 T^{5} + 704969 T^{6} \))(\( 1 + 6 T + 130 T^{2} + 291 T^{3} + 11570 T^{4} + 47526 T^{5} + 704969 T^{6} \))(\( 1 + 6 T + 130 T^{2} + 291 T^{3} + 11570 T^{4} + 47526 T^{5} + 704969 T^{6} \))(\( 1 - 6 T + 130 T^{2} - 291 T^{3} + 11570 T^{4} - 47526 T^{5} + 704969 T^{6} \))(\( 1 + 270 T^{2} + 33662 T^{4} + 2138670 T^{6} + 62742241 T^{8} \))(\( ( 1 - 110 T^{2} + 7921 T^{4} )^{2} \))(\( 1 + 270 T^{2} + 33662 T^{4} + 2138670 T^{6} + 62742241 T^{8} \))
$97$ (\( 1 - 7 T + 97 T^{2} \))(\( 1 + 16 T + 97 T^{2} \))(\( 1 + 97 T^{2} \))(\( 1 - 16 T + 97 T^{2} \))(\( 1 + T + 97 T^{2} \))(\( 1 + 16 T + 97 T^{2} \))(\( 1 - 16 T + 97 T^{2} \))(\( 1 - 15 T + 97 T^{2} \))(\( 1 + 7 T + 97 T^{2} \))(\( 1 + 15 T + 97 T^{2} \))(\( 1 + 12 T + 198 T^{2} + 1164 T^{3} + 9409 T^{4} \))(\( 1 + 176 T^{2} + 9409 T^{4} \))(\( 1 - 12 T + 198 T^{2} - 1164 T^{3} + 9409 T^{4} \))(\( 1 - 6 T + 198 T^{2} - 582 T^{3} + 9409 T^{4} \))(\( ( 1 + 97 T^{2} )^{2} \))(\( 1 + 18 T + 257 T^{2} + 1746 T^{3} + 9409 T^{4} \))(\( 1 + 149 T^{2} + 9409 T^{4} \))(\( 1 - 18 T + 257 T^{2} - 1746 T^{3} + 9409 T^{4} \))(\( 1 + 9 T + 108 T^{2} + 873 T^{3} + 9409 T^{4} \))(\( ( 1 + 97 T^{2} )^{2} \))(\( 1 + 6 T + 198 T^{2} + 582 T^{3} + 9409 T^{4} \))(\( 1 + 176 T^{2} + 9409 T^{4} \))(\( 1 - 7 T + 181 T^{2} - 1617 T^{3} + 17557 T^{4} - 65863 T^{5} + 912673 T^{6} \))(\( 1 + 7 T + 181 T^{2} + 1617 T^{3} + 17557 T^{4} + 65863 T^{5} + 912673 T^{6} \))(\( 1 - 7 T + 181 T^{2} - 1617 T^{3} + 17557 T^{4} - 65863 T^{5} + 912673 T^{6} \))(\( 1 + 7 T + 181 T^{2} + 1617 T^{3} + 17557 T^{4} + 65863 T^{5} + 912673 T^{6} \))(\( 1 - 98 T^{2} + 1374 T^{4} - 922082 T^{6} + 88529281 T^{8} \))(\( ( 1 + 191 T^{2} + 9409 T^{4} )^{2} \))(\( 1 - 98 T^{2} + 1374 T^{4} - 922082 T^{6} + 88529281 T^{8} \))
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