Properties

Label 1575.2.a
Level $1575$
Weight $2$
Character orbit 1575.a
Rep. character $\chi_{1575}(1,\cdot)$
Character field $\Q$
Dimension $47$
Newform subspaces $26$
Sturm bound $480$
Trace bound $13$

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Defining parameters

Level: \( N \) \(=\) \( 1575 = 3^{2} \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1575.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 26 \)
Sturm bound: \(480\)
Trace bound: \(13\)
Distinguishing \(T_p\): \(2\), \(11\), \(13\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(1575))\).

Total New Old
Modular forms 264 47 217
Cusp forms 217 47 170
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.

\(3\)\(5\)\(7\)FrickeDim.
\(+\)\(+\)\(+\)\(+\)\(2\)
\(+\)\(+\)\(-\)\(-\)\(8\)
\(+\)\(-\)\(+\)\(-\)\(6\)
\(+\)\(-\)\(-\)\(+\)\(2\)
\(-\)\(+\)\(+\)\(-\)\(8\)
\(-\)\(+\)\(-\)\(+\)\(5\)
\(-\)\(-\)\(+\)\(+\)\(7\)
\(-\)\(-\)\(-\)\(-\)\(9\)
Plus space\(+\)\(16\)
Minus space\(-\)\(31\)

Trace form

\( 47q - q^{2} + 51q^{4} + q^{7} - 9q^{8} + O(q^{10}) \) \( 47q - q^{2} + 51q^{4} + q^{7} - 9q^{8} + 8q^{11} + 2q^{13} + 3q^{14} + 67q^{16} - 10q^{17} + 20q^{19} + 12q^{22} + 8q^{23} + 18q^{26} + 3q^{28} + 18q^{29} + 12q^{31} - 9q^{32} + 34q^{34} - 2q^{37} + 28q^{38} + 10q^{41} + 24q^{43} + 50q^{44} + 2q^{46} + 20q^{47} + 47q^{49} + 34q^{52} + 10q^{53} + 21q^{56} - 6q^{58} - 24q^{59} + 22q^{61} + 24q^{62} + 133q^{64} + 24q^{67} - 18q^{68} - 16q^{71} - 6q^{73} + 60q^{74} - 8q^{76} - 4q^{77} - 4q^{79} - 34q^{82} - 12q^{83} - 66q^{86} + 8q^{88} - 62q^{89} - 14q^{91} + 40q^{92} - 92q^{94} - 2q^{97} - q^{98} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(1575))\) into newform subspaces

Label Dim. \(A\) Field CM Traces A-L signs $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\) 3 5 7
1575.2.a.a \(1\) \(12.576\) \(\Q\) None \(-2\) \(0\) \(0\) \(-1\) \(-\) \(-\) \(+\) \(q-2q^{2}+2q^{4}-q^{7}+3q^{11}+q^{13}+\cdots\)
1575.2.a.b \(1\) \(12.576\) \(\Q\) None \(-1\) \(0\) \(0\) \(-1\) \(+\) \(-\) \(+\) \(q-q^{2}-q^{4}-q^{7}+3q^{8}+4q^{13}+q^{14}+\cdots\)
1575.2.a.c \(1\) \(12.576\) \(\Q\) None \(-1\) \(0\) \(0\) \(1\) \(-\) \(+\) \(-\) \(q-q^{2}-q^{4}+q^{7}+3q^{8}-4q^{11}+2q^{13}+\cdots\)
1575.2.a.d \(1\) \(12.576\) \(\Q\) None \(-1\) \(0\) \(0\) \(1\) \(+\) \(-\) \(-\) \(q-q^{2}-q^{4}+q^{7}+3q^{8}-4q^{13}-q^{14}+\cdots\)
1575.2.a.e \(1\) \(12.576\) \(\Q\) None \(-1\) \(0\) \(0\) \(1\) \(-\) \(-\) \(-\) \(q-q^{2}-q^{4}+q^{7}+3q^{8}+6q^{11}+2q^{13}+\cdots\)
1575.2.a.f \(1\) \(12.576\) \(\Q\) None \(0\) \(0\) \(0\) \(-1\) \(-\) \(+\) \(+\) \(q-2q^{4}-q^{7}+3q^{11}-5q^{13}+4q^{16}+\cdots\)
1575.2.a.g \(1\) \(12.576\) \(\Q\) None \(1\) \(0\) \(0\) \(-1\) \(+\) \(-\) \(+\) \(q+q^{2}-q^{4}-q^{7}-3q^{8}+4q^{13}-q^{14}+\cdots\)
1575.2.a.h \(1\) \(12.576\) \(\Q\) None \(1\) \(0\) \(0\) \(-1\) \(-\) \(+\) \(+\) \(q+q^{2}-q^{4}-q^{7}-3q^{8}+6q^{13}-q^{14}+\cdots\)
1575.2.a.i \(1\) \(12.576\) \(\Q\) None \(1\) \(0\) \(0\) \(-1\) \(-\) \(-\) \(+\) \(q+q^{2}-q^{4}-q^{7}-3q^{8}+6q^{11}-2q^{13}+\cdots\)
1575.2.a.j \(1\) \(12.576\) \(\Q\) None \(1\) \(0\) \(0\) \(1\) \(+\) \(-\) \(-\) \(q+q^{2}-q^{4}+q^{7}-3q^{8}-4q^{13}+q^{14}+\cdots\)
1575.2.a.k \(1\) \(12.576\) \(\Q\) None \(2\) \(0\) \(0\) \(1\) \(-\) \(-\) \(-\) \(q+2q^{2}+2q^{4}+q^{7}+3q^{11}-q^{13}+\cdots\)
1575.2.a.l \(2\) \(12.576\) \(\Q(\sqrt{5}) \) None \(-3\) \(0\) \(0\) \(-2\) \(-\) \(+\) \(+\) \(q+(-1-\beta )q^{2}+3\beta q^{4}-q^{7}+(-1+\cdots)q^{8}+\cdots\)
1575.2.a.m \(2\) \(12.576\) \(\Q(\sqrt{2}) \) None \(-2\) \(0\) \(0\) \(2\) \(+\) \(+\) \(-\) \(q+(-1+\beta )q^{2}+(1-2\beta )q^{4}+q^{7}+(-3+\cdots)q^{8}+\cdots\)
1575.2.a.n \(2\) \(12.576\) \(\Q(\sqrt{5}) \) None \(-1\) \(0\) \(0\) \(2\) \(-\) \(+\) \(-\) \(q-\beta q^{2}+(-1+\beta )q^{4}+q^{7}+(-1+2\beta )q^{8}+\cdots\)
1575.2.a.o \(2\) \(12.576\) \(\Q(\sqrt{13}) \) None \(-1\) \(0\) \(0\) \(2\) \(-\) \(-\) \(-\) \(q-\beta q^{2}+(1+\beta )q^{4}+q^{7}-3q^{8}+3q^{11}+\cdots\)
1575.2.a.p \(2\) \(12.576\) \(\Q(\sqrt{17}) \) None \(-1\) \(0\) \(0\) \(2\) \(-\) \(+\) \(-\) \(q-\beta q^{2}+(2+\beta )q^{4}+q^{7}+(-4-\beta )q^{8}+\cdots\)
1575.2.a.q \(2\) \(12.576\) \(\Q(\sqrt{3}) \) None \(0\) \(0\) \(0\) \(-2\) \(+\) \(+\) \(+\) \(q+\beta q^{2}+q^{4}-q^{7}-\beta q^{8}-2\beta q^{11}+\cdots\)
1575.2.a.r \(2\) \(12.576\) \(\Q(\sqrt{5}) \) None \(0\) \(0\) \(0\) \(-2\) \(-\) \(+\) \(+\) \(q-\beta q^{2}+3q^{4}-q^{7}-\beta q^{8}+(-2-2\beta )q^{11}+\cdots\)
1575.2.a.s \(2\) \(12.576\) \(\Q(\sqrt{5}) \) None \(1\) \(0\) \(0\) \(-2\) \(-\) \(-\) \(+\) \(q+\beta q^{2}+(-1+\beta )q^{4}-q^{7}+(1-2\beta )q^{8}+\cdots\)
1575.2.a.t \(2\) \(12.576\) \(\Q(\sqrt{13}) \) None \(1\) \(0\) \(0\) \(-2\) \(-\) \(+\) \(+\) \(q+\beta q^{2}+(1+\beta )q^{4}-q^{7}+3q^{8}+3q^{11}+\cdots\)
1575.2.a.u \(2\) \(12.576\) \(\Q(\sqrt{2}) \) None \(2\) \(0\) \(0\) \(2\) \(+\) \(+\) \(-\) \(q+(1+\beta )q^{2}+(1+2\beta )q^{4}+q^{7}+(3+\beta )q^{8}+\cdots\)
1575.2.a.v \(2\) \(12.576\) \(\Q(\sqrt{5}) \) None \(3\) \(0\) \(0\) \(2\) \(-\) \(-\) \(-\) \(q+(1+\beta )q^{2}+3\beta q^{4}+q^{7}+(1+4\beta )q^{8}+\cdots\)
1575.2.a.w \(3\) \(12.576\) 3.3.148.1 None \(-1\) \(0\) \(0\) \(-3\) \(-\) \(-\) \(+\) \(q-\beta _{1}q^{2}+(1+\beta _{1}+\beta _{2})q^{4}-q^{7}+(-2+\cdots)q^{8}+\cdots\)
1575.2.a.x \(3\) \(12.576\) 3.3.148.1 None \(1\) \(0\) \(0\) \(3\) \(-\) \(-\) \(-\) \(q+\beta _{1}q^{2}+(1+\beta _{1}+\beta _{2})q^{4}+q^{7}+(2+\cdots)q^{8}+\cdots\)
1575.2.a.y \(4\) \(12.576\) 4.4.174928.1 None \(0\) \(0\) \(0\) \(-4\) \(+\) \(-\) \(+\) \(q+\beta _{1}q^{2}+(3+\beta _{2})q^{4}-q^{7}+(2\beta _{1}+\beta _{3})q^{8}+\cdots\)
1575.2.a.z \(4\) \(12.576\) 4.4.174928.1 None \(0\) \(0\) \(0\) \(4\) \(+\) \(+\) \(-\) \(q+\beta _{1}q^{2}+(3+\beta _{2})q^{4}+q^{7}+(2\beta _{1}+\beta _{3})q^{8}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(1575))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(1575)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(15))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(21))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(35))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(45))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(63))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(75))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(105))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(175))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(225))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(315))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(525))\)\(^{\oplus 2}\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ (\( 1 + 2 T + 2 T^{2} \))(\( 1 + T + 2 T^{2} \))(\( 1 + T + 2 T^{2} \))(\( 1 + T + 2 T^{2} \))(\( 1 + T + 2 T^{2} \))(\( 1 + 2 T^{2} \))(\( 1 - T + 2 T^{2} \))(\( 1 - T + 2 T^{2} \))(\( 1 - T + 2 T^{2} \))(\( 1 - T + 2 T^{2} \))(\( 1 - 2 T + 2 T^{2} \))(\( 1 + 3 T + 5 T^{2} + 6 T^{3} + 4 T^{4} \))(\( 1 + 2 T + 3 T^{2} + 4 T^{3} + 4 T^{4} \))(\( 1 + T + 3 T^{2} + 2 T^{3} + 4 T^{4} \))(\( 1 + T + T^{2} + 2 T^{3} + 4 T^{4} \))(\( 1 + T + 2 T^{3} + 4 T^{4} \))(\( 1 + T^{2} + 4 T^{4} \))(\( 1 - T^{2} + 4 T^{4} \))(\( 1 - T + 3 T^{2} - 2 T^{3} + 4 T^{4} \))(\( 1 - T + T^{2} - 2 T^{3} + 4 T^{4} \))(\( 1 - 2 T + 3 T^{2} - 4 T^{3} + 4 T^{4} \))(\( 1 - 3 T + 5 T^{2} - 6 T^{3} + 4 T^{4} \))(\( 1 + T + T^{2} + 3 T^{3} + 2 T^{4} + 4 T^{5} + 8 T^{6} \))(\( 1 - T + T^{2} - 3 T^{3} + 2 T^{4} - 4 T^{5} + 8 T^{6} \))(\( 1 - T^{2} + T^{4} - 4 T^{6} + 16 T^{8} \))(\( 1 - T^{2} + T^{4} - 4 T^{6} + 16 T^{8} \))
$3$ 1
$5$ 1
$7$ (\( 1 + T \))(\( 1 + T \))(\( 1 - T \))(\( 1 - T \))(\( 1 - T \))(\( 1 + T \))(\( 1 + T \))(\( 1 + T \))(\( 1 + T \))(\( 1 - T \))(\( 1 - T \))(\( ( 1 + T )^{2} \))(\( ( 1 - T )^{2} \))(\( ( 1 - T )^{2} \))(\( ( 1 - T )^{2} \))(\( ( 1 - T )^{2} \))(\( ( 1 + T )^{2} \))(\( ( 1 + T )^{2} \))(\( ( 1 + T )^{2} \))(\( ( 1 + T )^{2} \))(\( ( 1 - T )^{2} \))(\( ( 1 - T )^{2} \))(\( ( 1 + T )^{3} \))(\( ( 1 - T )^{3} \))(\( ( 1 + T )^{4} \))(\( ( 1 - T )^{4} \))
$11$ (\( 1 - 3 T + 11 T^{2} \))(\( 1 + 11 T^{2} \))(\( 1 + 4 T + 11 T^{2} \))(\( 1 + 11 T^{2} \))(\( 1 - 6 T + 11 T^{2} \))(\( 1 - 3 T + 11 T^{2} \))(\( 1 + 11 T^{2} \))(\( 1 + 11 T^{2} \))(\( 1 - 6 T + 11 T^{2} \))(\( 1 + 11 T^{2} \))(\( 1 - 3 T + 11 T^{2} \))(\( 1 - 2 T + 3 T^{2} - 22 T^{3} + 121 T^{4} \))(\( 1 - 4 T + 18 T^{2} - 44 T^{3} + 121 T^{4} \))(\( 1 + 4 T + 21 T^{2} + 44 T^{3} + 121 T^{4} \))(\( ( 1 - 3 T + 11 T^{2} )^{2} \))(\( 1 + T + 18 T^{2} + 11 T^{3} + 121 T^{4} \))(\( 1 + 10 T^{2} + 121 T^{4} \))(\( 1 + 4 T + 6 T^{2} + 44 T^{3} + 121 T^{4} \))(\( 1 + 4 T + 21 T^{2} + 44 T^{3} + 121 T^{4} \))(\( ( 1 - 3 T + 11 T^{2} )^{2} \))(\( 1 + 4 T + 18 T^{2} + 44 T^{3} + 121 T^{4} \))(\( 1 - 2 T + 3 T^{2} - 22 T^{3} + 121 T^{4} \))(\( ( 1 + 2 T + 11 T^{2} )^{3} \))(\( ( 1 + 2 T + 11 T^{2} )^{3} \))(\( 1 + 2 T^{2} + 127 T^{4} + 242 T^{6} + 14641 T^{8} \))(\( 1 + 2 T^{2} + 127 T^{4} + 242 T^{6} + 14641 T^{8} \))
$13$ (\( 1 - T + 13 T^{2} \))(\( 1 - 4 T + 13 T^{2} \))(\( 1 - 2 T + 13 T^{2} \))(\( 1 + 4 T + 13 T^{2} \))(\( 1 - 2 T + 13 T^{2} \))(\( 1 + 5 T + 13 T^{2} \))(\( 1 - 4 T + 13 T^{2} \))(\( 1 - 6 T + 13 T^{2} \))(\( 1 + 2 T + 13 T^{2} \))(\( 1 + 4 T + 13 T^{2} \))(\( 1 + T + 13 T^{2} \))(\( 1 - 6 T + 30 T^{2} - 78 T^{3} + 169 T^{4} \))(\( 1 - 4 T + 22 T^{2} - 52 T^{3} + 169 T^{4} \))(\( 1 + 2 T + 22 T^{2} + 26 T^{3} + 169 T^{4} \))(\( 1 - 2 T + 14 T^{2} - 26 T^{3} + 169 T^{4} \))(\( 1 + 5 T + 28 T^{2} + 65 T^{3} + 169 T^{4} \))(\( ( 1 + 2 T + 13 T^{2} )^{2} \))(\( 1 + 6 T^{2} + 169 T^{4} \))(\( 1 - 2 T + 22 T^{2} - 26 T^{3} + 169 T^{4} \))(\( 1 + 2 T + 14 T^{2} + 26 T^{3} + 169 T^{4} \))(\( 1 - 4 T + 22 T^{2} - 52 T^{3} + 169 T^{4} \))(\( 1 + 6 T + 30 T^{2} + 78 T^{3} + 169 T^{4} \))(\( 1 + 6 T + 35 T^{2} + 148 T^{3} + 455 T^{4} + 1014 T^{5} + 2197 T^{6} \))(\( 1 - 6 T + 35 T^{2} - 148 T^{3} + 455 T^{4} - 1014 T^{5} + 2197 T^{6} \))(\( ( 1 - 2 T - 2 T^{2} - 26 T^{3} + 169 T^{4} )^{2} \))(\( ( 1 + 2 T - 2 T^{2} + 26 T^{3} + 169 T^{4} )^{2} \))
$17$ (\( 1 + 7 T + 17 T^{2} \))(\( 1 + 2 T + 17 T^{2} \))(\( 1 + 6 T + 17 T^{2} \))(\( 1 + 2 T + 17 T^{2} \))(\( 1 - 4 T + 17 T^{2} \))(\( 1 - 3 T + 17 T^{2} \))(\( 1 - 2 T + 17 T^{2} \))(\( 1 - 2 T + 17 T^{2} \))(\( 1 + 4 T + 17 T^{2} \))(\( 1 - 2 T + 17 T^{2} \))(\( 1 - 7 T + 17 T^{2} \))(\( 1 - 2 T - 10 T^{2} - 34 T^{3} + 289 T^{4} \))(\( 1 + 4 T + 6 T^{2} + 68 T^{3} + 289 T^{4} \))(\( 1 - 4 T + 18 T^{2} - 68 T^{3} + 289 T^{4} \))(\( 1 + 2 T + 22 T^{2} + 34 T^{3} + 289 T^{4} \))(\( 1 + 5 T + 36 T^{2} + 85 T^{3} + 289 T^{4} \))(\( 1 + 22 T^{2} + 289 T^{4} \))(\( ( 1 + 2 T + 17 T^{2} )^{2} \))(\( 1 + 4 T + 18 T^{2} + 68 T^{3} + 289 T^{4} \))(\( 1 - 2 T + 22 T^{2} - 34 T^{3} + 289 T^{4} \))(\( 1 - 4 T + 6 T^{2} - 68 T^{3} + 289 T^{4} \))(\( 1 + 2 T - 10 T^{2} + 34 T^{3} + 289 T^{4} \))(\( 1 + 35 T^{2} + 16 T^{3} + 595 T^{4} + 4913 T^{6} \))(\( 1 + 35 T^{2} - 16 T^{3} + 595 T^{4} + 4913 T^{6} \))(\( 1 + 32 T^{2} + 718 T^{4} + 9248 T^{6} + 83521 T^{8} \))(\( 1 + 32 T^{2} + 718 T^{4} + 9248 T^{6} + 83521 T^{8} \))
$19$ (\( 1 + 19 T^{2} \))(\( 1 + 19 T^{2} \))(\( 1 - 4 T + 19 T^{2} \))(\( 1 + 19 T^{2} \))(\( 1 + 6 T + 19 T^{2} \))(\( 1 - 2 T + 19 T^{2} \))(\( 1 + 19 T^{2} \))(\( 1 + 8 T + 19 T^{2} \))(\( 1 + 6 T + 19 T^{2} \))(\( 1 + 19 T^{2} \))(\( 1 + 19 T^{2} \))(\( 1 + 2 T + 34 T^{2} + 38 T^{3} + 361 T^{4} \))(\( 1 + 30 T^{2} + 361 T^{4} \))(\( 1 + 18 T^{2} + 361 T^{4} \))(\( 1 - 6 T + 34 T^{2} - 114 T^{3} + 361 T^{4} \))(\( 1 + 6 T + 30 T^{2} + 114 T^{3} + 361 T^{4} \))(\( ( 1 + 4 T + 19 T^{2} )^{2} \))(\( 1 - 4 T + 22 T^{2} - 76 T^{3} + 361 T^{4} \))(\( 1 + 18 T^{2} + 361 T^{4} \))(\( 1 - 6 T + 34 T^{2} - 114 T^{3} + 361 T^{4} \))(\( 1 + 30 T^{2} + 361 T^{4} \))(\( 1 + 2 T + 34 T^{2} + 38 T^{3} + 361 T^{4} \))(\( 1 - 6 T + 53 T^{2} - 188 T^{3} + 1007 T^{4} - 2166 T^{5} + 6859 T^{6} \))(\( 1 - 6 T + 53 T^{2} - 188 T^{3} + 1007 T^{4} - 2166 T^{5} + 6859 T^{6} \))(\( ( 1 - 6 T + 18 T^{2} - 114 T^{3} + 361 T^{4} )^{2} \))(\( ( 1 - 6 T + 18 T^{2} - 114 T^{3} + 361 T^{4} )^{2} \))
$23$ (\( 1 + 6 T + 23 T^{2} \))(\( 1 + 23 T^{2} \))(\( 1 + 23 T^{2} \))(\( 1 + 23 T^{2} \))(\( 1 + 23 T^{2} \))(\( 1 + 6 T + 23 T^{2} \))(\( 1 + 23 T^{2} \))(\( 1 - 8 T + 23 T^{2} \))(\( 1 + 23 T^{2} \))(\( 1 + 23 T^{2} \))(\( 1 - 6 T + 23 T^{2} \))(\( ( 1 + 5 T + 23 T^{2} )^{2} \))(\( 1 + 4 T + 18 T^{2} + 92 T^{3} + 529 T^{4} \))(\( 1 + 8 T + 57 T^{2} + 184 T^{3} + 529 T^{4} \))(\( 1 + 2 T - 5 T^{2} + 46 T^{3} + 529 T^{4} \))(\( 1 + 2 T + 30 T^{2} + 46 T^{3} + 529 T^{4} \))(\( 1 + 34 T^{2} + 529 T^{4} \))(\( ( 1 - 4 T + 23 T^{2} )^{2} \))(\( 1 - 8 T + 57 T^{2} - 184 T^{3} + 529 T^{4} \))(\( 1 - 2 T - 5 T^{2} - 46 T^{3} + 529 T^{4} \))(\( 1 - 4 T + 18 T^{2} - 92 T^{3} + 529 T^{4} \))(\( ( 1 - 5 T + 23 T^{2} )^{2} \))(\( 1 + 4 T + 61 T^{2} + 168 T^{3} + 1403 T^{4} + 2116 T^{5} + 12167 T^{6} \))(\( 1 - 4 T + 61 T^{2} - 168 T^{3} + 1403 T^{4} - 2116 T^{5} + 12167 T^{6} \))(\( 1 + 50 T^{2} + 1567 T^{4} + 26450 T^{6} + 279841 T^{8} \))(\( 1 + 50 T^{2} + 1567 T^{4} + 26450 T^{6} + 279841 T^{8} \))
$29$ (\( 1 - 5 T + 29 T^{2} \))(\( 1 + 8 T + 29 T^{2} \))(\( 1 - 2 T + 29 T^{2} \))(\( 1 - 8 T + 29 T^{2} \))(\( 1 - 2 T + 29 T^{2} \))(\( 1 + 3 T + 29 T^{2} \))(\( 1 - 8 T + 29 T^{2} \))(\( 1 - 2 T + 29 T^{2} \))(\( 1 - 2 T + 29 T^{2} \))(\( 1 + 8 T + 29 T^{2} \))(\( 1 - 5 T + 29 T^{2} \))(\( 1 - 4 T + 17 T^{2} - 116 T^{3} + 841 T^{4} \))(\( ( 1 - 8 T + 29 T^{2} )^{2} \))(\( ( 1 + 5 T + 29 T^{2} )^{2} \))(\( 1 - 8 T + 61 T^{2} - 232 T^{3} + 841 T^{4} \))(\( 1 + T + 20 T^{2} + 29 T^{3} + 841 T^{4} \))(\( ( 1 + 29 T^{2} )^{2} \))(\( ( 1 - 2 T + 29 T^{2} )^{2} \))(\( ( 1 + 5 T + 29 T^{2} )^{2} \))(\( 1 - 8 T + 61 T^{2} - 232 T^{3} + 841 T^{4} \))(\( ( 1 + 8 T + 29 T^{2} )^{2} \))(\( 1 - 4 T + 17 T^{2} - 116 T^{3} + 841 T^{4} \))(\( 1 + 2 T + 35 T^{2} + 76 T^{3} + 1015 T^{4} + 1682 T^{5} + 24389 T^{6} \))(\( 1 + 2 T + 35 T^{2} + 76 T^{3} + 1015 T^{4} + 1682 T^{5} + 24389 T^{6} \))(\( 1 + 42 T^{2} + 1079 T^{4} + 35322 T^{6} + 707281 T^{8} \))(\( 1 + 42 T^{2} + 1079 T^{4} + 35322 T^{6} + 707281 T^{8} \))
$31$ (\( 1 - 2 T + 31 T^{2} \))(\( 1 + 4 T + 31 T^{2} \))(\( 1 + 31 T^{2} \))(\( 1 + 4 T + 31 T^{2} \))(\( 1 + 10 T + 31 T^{2} \))(\( 1 + 4 T + 31 T^{2} \))(\( 1 + 4 T + 31 T^{2} \))(\( 1 - 4 T + 31 T^{2} \))(\( 1 + 10 T + 31 T^{2} \))(\( 1 + 4 T + 31 T^{2} \))(\( 1 - 2 T + 31 T^{2} \))(\( 1 + 42 T^{2} + 961 T^{4} \))(\( 1 - 10 T^{2} + 961 T^{4} \))(\( 1 + 6 T + 26 T^{2} + 186 T^{3} + 961 T^{4} \))(\( 1 + 10 T^{2} + 961 T^{4} \))(\( ( 1 + 31 T^{2} )^{2} \))(\( ( 1 + 4 T + 31 T^{2} )^{2} \))(\( 1 - 12 T + 78 T^{2} - 372 T^{3} + 961 T^{4} \))(\( 1 + 6 T + 26 T^{2} + 186 T^{3} + 961 T^{4} \))(\( 1 + 10 T^{2} + 961 T^{4} \))(\( 1 - 10 T^{2} + 961 T^{4} \))(\( 1 + 42 T^{2} + 961 T^{4} \))(\( 1 - 2 T + 41 T^{2} + 60 T^{3} + 1271 T^{4} - 1922 T^{5} + 29791 T^{6} \))(\( 1 - 2 T + 41 T^{2} + 60 T^{3} + 1271 T^{4} - 1922 T^{5} + 29791 T^{6} \))(\( ( 1 - 6 T + 31 T^{2} )^{4} \))(\( ( 1 - 6 T + 31 T^{2} )^{4} \))
$37$ (\( 1 - 2 T + 37 T^{2} \))(\( 1 - 8 T + 37 T^{2} \))(\( 1 + 6 T + 37 T^{2} \))(\( 1 + 8 T + 37 T^{2} \))(\( 1 - 4 T + 37 T^{2} \))(\( 1 + 2 T + 37 T^{2} \))(\( 1 - 8 T + 37 T^{2} \))(\( 1 - 2 T + 37 T^{2} \))(\( 1 + 4 T + 37 T^{2} \))(\( 1 + 8 T + 37 T^{2} \))(\( 1 + 2 T + 37 T^{2} \))(\( 1 - 2 T + 55 T^{2} - 74 T^{3} + 1369 T^{4} \))(\( ( 1 - 6 T + 37 T^{2} )^{2} \))(\( ( 1 - 3 T + 37 T^{2} )^{2} \))(\( 1 - 6 T + 31 T^{2} - 222 T^{3} + 1369 T^{4} \))(\( ( 1 + 6 T + 37 T^{2} )^{2} \))(\( ( 1 + 2 T + 37 T^{2} )^{2} \))(\( 1 + 4 T - 2 T^{2} + 148 T^{3} + 1369 T^{4} \))(\( ( 1 + 3 T + 37 T^{2} )^{2} \))(\( 1 + 6 T + 31 T^{2} + 222 T^{3} + 1369 T^{4} \))(\( ( 1 - 6 T + 37 T^{2} )^{2} \))(\( 1 + 2 T + 55 T^{2} + 74 T^{3} + 1369 T^{4} \))(\( 1 + 4 T + 31 T^{2} + 232 T^{3} + 1147 T^{4} + 5476 T^{5} + 50653 T^{6} \))(\( 1 - 4 T + 31 T^{2} - 232 T^{3} + 1147 T^{4} - 5476 T^{5} + 50653 T^{6} \))(\( ( 1 - 3 T + 37 T^{2} )^{4} \))(\( ( 1 + 3 T + 37 T^{2} )^{4} \))
$41$ (\( 1 + 2 T + 41 T^{2} \))(\( 1 + 4 T + 41 T^{2} \))(\( 1 + 2 T + 41 T^{2} \))(\( 1 - 4 T + 41 T^{2} \))(\( 1 + 2 T + 41 T^{2} \))(\( 1 - 12 T + 41 T^{2} \))(\( 1 - 4 T + 41 T^{2} \))(\( 1 - 6 T + 41 T^{2} \))(\( 1 + 2 T + 41 T^{2} \))(\( 1 + 4 T + 41 T^{2} \))(\( 1 + 2 T + 41 T^{2} \))(\( ( 1 - 8 T + 41 T^{2} )^{2} \))(\( 1 + 4 T + 54 T^{2} + 164 T^{3} + 1681 T^{4} \))(\( 1 + 14 T + 126 T^{2} + 574 T^{3} + 1681 T^{4} \))(\( ( 1 + 41 T^{2} )^{2} \))(\( 1 + 2 T + 66 T^{2} + 82 T^{3} + 1681 T^{4} \))(\( 1 - 26 T^{2} + 1681 T^{4} \))(\( ( 1 - 2 T + 41 T^{2} )^{2} \))(\( 1 + 14 T + 126 T^{2} + 574 T^{3} + 1681 T^{4} \))(\( ( 1 + 41 T^{2} )^{2} \))(\( 1 - 4 T + 54 T^{2} - 164 T^{3} + 1681 T^{4} \))(\( ( 1 - 8 T + 41 T^{2} )^{2} \))(\( 1 + 2 T + 63 T^{2} - 36 T^{3} + 2583 T^{4} + 3362 T^{5} + 68921 T^{6} \))(\( 1 + 2 T + 63 T^{2} - 36 T^{3} + 2583 T^{4} + 3362 T^{5} + 68921 T^{6} \))(\( 1 + 20 T^{2} + 1606 T^{4} + 33620 T^{6} + 2825761 T^{8} \))(\( 1 + 20 T^{2} + 1606 T^{4} + 33620 T^{6} + 2825761 T^{8} \))
$43$ (\( 1 + 4 T + 43 T^{2} \))(\( 1 - 8 T + 43 T^{2} \))(\( 1 - 4 T + 43 T^{2} \))(\( 1 + 8 T + 43 T^{2} \))(\( 1 - 4 T + 43 T^{2} \))(\( 1 - 10 T + 43 T^{2} \))(\( 1 - 8 T + 43 T^{2} \))(\( 1 + 4 T + 43 T^{2} \))(\( 1 + 4 T + 43 T^{2} \))(\( 1 + 8 T + 43 T^{2} \))(\( 1 - 4 T + 43 T^{2} \))(\( 1 - 12 T + 117 T^{2} - 516 T^{3} + 1849 T^{4} \))(\( 1 - 8 T + 70 T^{2} - 344 T^{3} + 1849 T^{4} \))(\( 1 - 8 T + 97 T^{2} - 344 T^{3} + 1849 T^{4} \))(\( 1 - 12 T + 109 T^{2} - 516 T^{3} + 1849 T^{4} \))(\( 1 + 10 T + 94 T^{2} + 430 T^{3} + 1849 T^{4} \))(\( ( 1 - 4 T + 43 T^{2} )^{2} \))(\( 1 + 6 T^{2} + 1849 T^{4} \))(\( 1 + 8 T + 97 T^{2} + 344 T^{3} + 1849 T^{4} \))(\( 1 + 12 T + 109 T^{2} + 516 T^{3} + 1849 T^{4} \))(\( 1 - 8 T + 70 T^{2} - 344 T^{3} + 1849 T^{4} \))(\( 1 + 12 T + 117 T^{2} + 516 T^{3} + 1849 T^{4} \))(\( 1 - 4 T - 15 T^{2} + 488 T^{3} - 645 T^{4} - 7396 T^{5} + 79507 T^{6} \))(\( 1 + 4 T - 15 T^{2} - 488 T^{3} - 645 T^{4} + 7396 T^{5} + 79507 T^{6} \))(\( ( 1 + 8 T + 73 T^{2} + 344 T^{3} + 1849 T^{4} )^{2} \))(\( ( 1 - 8 T + 73 T^{2} - 344 T^{3} + 1849 T^{4} )^{2} \))
$47$ (\( 1 - 3 T + 47 T^{2} \))(\( 1 - 12 T + 47 T^{2} \))(\( 1 + 47 T^{2} \))(\( 1 - 12 T + 47 T^{2} \))(\( 1 + 47 T^{2} \))(\( 1 - 9 T + 47 T^{2} \))(\( 1 + 12 T + 47 T^{2} \))(\( 1 - 8 T + 47 T^{2} \))(\( 1 + 47 T^{2} \))(\( 1 + 12 T + 47 T^{2} \))(\( 1 + 3 T + 47 T^{2} \))(\( 1 + 10 T + 114 T^{2} + 470 T^{3} + 2209 T^{4} \))(\( ( 1 - 4 T + 47 T^{2} )^{2} \))(\( ( 1 - 2 T + 47 T^{2} )^{2} \))(\( 1 + 14 T + 130 T^{2} + 658 T^{3} + 2209 T^{4} \))(\( 1 + 5 T + 62 T^{2} + 235 T^{3} + 2209 T^{4} \))(\( 1 + 46 T^{2} + 2209 T^{4} \))(\( 1 - 8 T + 30 T^{2} - 376 T^{3} + 2209 T^{4} \))(\( ( 1 + 2 T + 47 T^{2} )^{2} \))(\( 1 - 14 T + 130 T^{2} - 658 T^{3} + 2209 T^{4} \))(\( ( 1 + 4 T + 47 T^{2} )^{2} \))(\( 1 - 10 T + 114 T^{2} - 470 T^{3} + 2209 T^{4} \))(\( 1 + 8 T + 109 T^{2} + 624 T^{3} + 5123 T^{4} + 17672 T^{5} + 103823 T^{6} \))(\( 1 - 8 T + 109 T^{2} - 624 T^{3} + 5123 T^{4} - 17672 T^{5} + 103823 T^{6} \))(\( 1 - 24 T^{2} + 3518 T^{4} - 53016 T^{6} + 4879681 T^{8} \))(\( 1 - 24 T^{2} + 3518 T^{4} - 53016 T^{6} + 4879681 T^{8} \))
$53$ (\( 1 + 6 T + 53 T^{2} \))(\( 1 + 6 T + 53 T^{2} \))(\( 1 - 6 T + 53 T^{2} \))(\( 1 + 6 T + 53 T^{2} \))(\( 1 - 6 T + 53 T^{2} \))(\( 1 - 12 T + 53 T^{2} \))(\( 1 - 6 T + 53 T^{2} \))(\( 1 - 10 T + 53 T^{2} \))(\( 1 + 6 T + 53 T^{2} \))(\( 1 - 6 T + 53 T^{2} \))(\( 1 - 6 T + 53 T^{2} \))(\( 1 - 8 T + 102 T^{2} - 424 T^{3} + 2809 T^{4} \))(\( 1 + 16 T + 162 T^{2} + 848 T^{3} + 2809 T^{4} \))(\( 1 + 8 T + 102 T^{2} + 424 T^{3} + 2809 T^{4} \))(\( 1 - 8 T + 70 T^{2} - 424 T^{3} + 2809 T^{4} \))(\( 1 + 2 T + 90 T^{2} + 106 T^{3} + 2809 T^{4} \))(\( 1 + 58 T^{2} + 2809 T^{4} \))(\( 1 + 16 T + 150 T^{2} + 848 T^{3} + 2809 T^{4} \))(\( 1 - 8 T + 102 T^{2} - 424 T^{3} + 2809 T^{4} \))(\( 1 + 8 T + 70 T^{2} + 424 T^{3} + 2809 T^{4} \))(\( 1 - 16 T + 162 T^{2} - 848 T^{3} + 2809 T^{4} \))(\( 1 + 8 T + 102 T^{2} + 424 T^{3} + 2809 T^{4} \))(\( 1 + 14 T + 171 T^{2} + 1188 T^{3} + 9063 T^{4} + 39326 T^{5} + 148877 T^{6} \))(\( 1 - 14 T + 171 T^{2} - 1188 T^{3} + 9063 T^{4} - 39326 T^{5} + 148877 T^{6} \))(\( 1 + 68 T^{2} + 4918 T^{4} + 191012 T^{6} + 7890481 T^{8} \))(\( 1 + 68 T^{2} + 4918 T^{4} + 191012 T^{6} + 7890481 T^{8} \))
$59$ (\( 1 + 10 T + 59 T^{2} \))(\( 1 + 8 T + 59 T^{2} \))(\( 1 + 12 T + 59 T^{2} \))(\( 1 - 8 T + 59 T^{2} \))(\( 1 - 8 T + 59 T^{2} \))(\( 1 + 59 T^{2} \))(\( 1 - 8 T + 59 T^{2} \))(\( 1 + 4 T + 59 T^{2} \))(\( 1 - 8 T + 59 T^{2} \))(\( 1 + 8 T + 59 T^{2} \))(\( 1 + 10 T + 59 T^{2} \))(\( 1 - 6 T + 122 T^{2} - 354 T^{3} + 3481 T^{4} \))(\( ( 1 + 4 T + 59 T^{2} )^{2} \))(\( 1 + 10 T + 98 T^{2} + 590 T^{3} + 3481 T^{4} \))(\( 1 - 14 T + 154 T^{2} - 826 T^{3} + 3481 T^{4} \))(\( ( 1 - 4 T + 59 T^{2} )^{2} \))(\( 1 + 70 T^{2} + 3481 T^{4} \))(\( 1 + 38 T^{2} + 3481 T^{4} \))(\( 1 + 10 T + 98 T^{2} + 590 T^{3} + 3481 T^{4} \))(\( 1 - 14 T + 154 T^{2} - 826 T^{3} + 3481 T^{4} \))(\( ( 1 - 4 T + 59 T^{2} )^{2} \))(\( 1 - 6 T + 122 T^{2} - 354 T^{3} + 3481 T^{4} \))(\( 1 + 16 T + 113 T^{2} + 608 T^{3} + 6667 T^{4} + 55696 T^{5} + 205379 T^{6} \))(\( 1 + 16 T + 113 T^{2} + 608 T^{3} + 6667 T^{4} + 55696 T^{5} + 205379 T^{6} \))(\( 1 + 24 T^{2} + 6062 T^{4} + 83544 T^{6} + 12117361 T^{8} \))(\( 1 + 24 T^{2} + 6062 T^{4} + 83544 T^{6} + 12117361 T^{8} \))
$61$ (\( 1 + 8 T + 61 T^{2} \))(\( 1 - 10 T + 61 T^{2} \))(\( 1 + 2 T + 61 T^{2} \))(\( 1 - 10 T + 61 T^{2} \))(\( 1 + 2 T + 61 T^{2} \))(\( 1 - 8 T + 61 T^{2} \))(\( 1 - 10 T + 61 T^{2} \))(\( 1 + 2 T + 61 T^{2} \))(\( 1 + 2 T + 61 T^{2} \))(\( 1 - 10 T + 61 T^{2} \))(\( 1 + 8 T + 61 T^{2} \))(\( 1 + 4 T - 54 T^{2} + 244 T^{3} + 3721 T^{4} \))(\( ( 1 - 6 T + 61 T^{2} )^{2} \))(\( 1 + 6 T + 86 T^{2} + 366 T^{3} + 3721 T^{4} \))(\( 1 - 12 T + 106 T^{2} - 732 T^{3} + 3721 T^{4} \))(\( 1 - 6 T - 22 T^{2} - 366 T^{3} + 3721 T^{4} \))(\( ( 1 + 10 T + 61 T^{2} )^{2} \))(\( ( 1 + 2 T + 61 T^{2} )^{2} \))(\( 1 + 6 T + 86 T^{2} + 366 T^{3} + 3721 T^{4} \))(\( 1 - 12 T + 106 T^{2} - 732 T^{3} + 3721 T^{4} \))(\( ( 1 - 6 T + 61 T^{2} )^{2} \))(\( 1 + 4 T - 54 T^{2} + 244 T^{3} + 3721 T^{4} \))(\( 1 + 6 T + 131 T^{2} + 484 T^{3} + 7991 T^{4} + 22326 T^{5} + 226981 T^{6} \))(\( 1 + 6 T + 131 T^{2} + 484 T^{3} + 7991 T^{4} + 22326 T^{5} + 226981 T^{6} \))(\( ( 1 + 61 T^{2} )^{4} \))(\( ( 1 + 61 T^{2} )^{4} \))
$67$ (\( 1 - 2 T + 67 T^{2} \))(\( 1 - 8 T + 67 T^{2} \))(\( 1 + 4 T + 67 T^{2} \))(\( 1 + 8 T + 67 T^{2} \))(\( 1 - 16 T + 67 T^{2} \))(\( 1 - 4 T + 67 T^{2} \))(\( 1 - 8 T + 67 T^{2} \))(\( 1 + 4 T + 67 T^{2} \))(\( 1 + 16 T + 67 T^{2} \))(\( 1 + 8 T + 67 T^{2} \))(\( 1 + 2 T + 67 T^{2} \))(\( 1 - 8 T + 105 T^{2} - 536 T^{3} + 4489 T^{4} \))(\( 1 - 8 T + 22 T^{2} - 536 T^{3} + 4489 T^{4} \))(\( 1 + 4 T + 133 T^{2} + 268 T^{3} + 4489 T^{4} \))(\( 1 - 24 T + 265 T^{2} - 1608 T^{3} + 4489 T^{4} \))(\( 1 + 4 T + 70 T^{2} + 268 T^{3} + 4489 T^{4} \))(\( ( 1 - 4 T + 67 T^{2} )^{2} \))(\( ( 1 - 4 T + 67 T^{2} )^{2} \))(\( 1 - 4 T + 133 T^{2} - 268 T^{3} + 4489 T^{4} \))(\( 1 + 24 T + 265 T^{2} + 1608 T^{3} + 4489 T^{4} \))(\( 1 - 8 T + 22 T^{2} - 536 T^{3} + 4489 T^{4} \))(\( 1 + 8 T + 105 T^{2} + 536 T^{3} + 4489 T^{4} \))(\( 1 - 8 T + 169 T^{2} - 944 T^{3} + 11323 T^{4} - 35912 T^{5} + 300763 T^{6} \))(\( 1 + 8 T + 169 T^{2} + 944 T^{3} + 11323 T^{4} + 35912 T^{5} + 300763 T^{6} \))(\( ( 1 + 8 T + 121 T^{2} + 536 T^{3} + 4489 T^{4} )^{2} \))(\( ( 1 - 8 T + 121 T^{2} - 536 T^{3} + 4489 T^{4} )^{2} \))
$71$ (\( 1 - 8 T + 71 T^{2} \))(\( 1 - 16 T + 71 T^{2} \))(\( 1 + 71 T^{2} \))(\( 1 + 16 T + 71 T^{2} \))(\( 1 + 10 T + 71 T^{2} \))(\( 1 + 71 T^{2} \))(\( 1 + 16 T + 71 T^{2} \))(\( 1 - 12 T + 71 T^{2} \))(\( 1 + 10 T + 71 T^{2} \))(\( 1 - 16 T + 71 T^{2} \))(\( 1 - 8 T + 71 T^{2} \))(\( 1 - 10 T + 147 T^{2} - 710 T^{3} + 5041 T^{4} \))(\( 1 - 4 T + 74 T^{2} - 284 T^{3} + 5041 T^{4} \))(\( 1 + 4 T + 101 T^{2} + 284 T^{3} + 5041 T^{4} \))(\( ( 1 - 3 T + 71 T^{2} )^{2} \))(\( ( 1 + 8 T + 71 T^{2} )^{2} \))(\( 1 + 34 T^{2} + 5041 T^{4} \))(\( 1 + 20 T + 222 T^{2} + 1420 T^{3} + 5041 T^{4} \))(\( 1 + 4 T + 101 T^{2} + 284 T^{3} + 5041 T^{4} \))(\( ( 1 - 3 T + 71 T^{2} )^{2} \))(\( 1 + 4 T + 74 T^{2} + 284 T^{3} + 5041 T^{4} \))(\( 1 - 10 T + 147 T^{2} - 710 T^{3} + 5041 T^{4} \))(\( ( 1 + 2 T + 71 T^{2} )^{3} \))(\( ( 1 + 2 T + 71 T^{2} )^{3} \))(\( 1 + 90 T^{2} + 2711 T^{4} + 453690 T^{6} + 25411681 T^{8} \))(\( 1 + 90 T^{2} + 2711 T^{4} + 453690 T^{6} + 25411681 T^{8} \))
$73$ (\( 1 - 6 T + 73 T^{2} \))(\( 1 - 12 T + 73 T^{2} \))(\( 1 - 6 T + 73 T^{2} \))(\( 1 + 12 T + 73 T^{2} \))(\( 1 - 6 T + 73 T^{2} \))(\( 1 + 2 T + 73 T^{2} \))(\( 1 - 12 T + 73 T^{2} \))(\( 1 - 2 T + 73 T^{2} \))(\( 1 + 6 T + 73 T^{2} \))(\( 1 + 12 T + 73 T^{2} \))(\( 1 + 6 T + 73 T^{2} \))(\( 1 + 2 T + 142 T^{2} + 146 T^{3} + 5329 T^{4} \))(\( 1 + 4 T - 50 T^{2} + 292 T^{3} + 5329 T^{4} \))(\( 1 + 22 T + 262 T^{2} + 1606 T^{3} + 5329 T^{4} \))(\( 1 + 6 T + 142 T^{2} + 438 T^{3} + 5329 T^{4} \))(\( 1 - 8 T + 94 T^{2} - 584 T^{3} + 5329 T^{4} \))(\( ( 1 + 14 T + 73 T^{2} )^{2} \))(\( 1 - 16 T + 190 T^{2} - 1168 T^{3} + 5329 T^{4} \))(\( 1 - 22 T + 262 T^{2} - 1606 T^{3} + 5329 T^{4} \))(\( 1 - 6 T + 142 T^{2} - 438 T^{3} + 5329 T^{4} \))(\( 1 + 4 T - 50 T^{2} + 292 T^{3} + 5329 T^{4} \))(\( 1 - 2 T + 142 T^{2} - 146 T^{3} + 5329 T^{4} \))(\( 1 + 18 T + 311 T^{2} + 2732 T^{3} + 22703 T^{4} + 95922 T^{5} + 389017 T^{6} \))(\( 1 - 18 T + 311 T^{2} - 2732 T^{3} + 22703 T^{4} - 95922 T^{5} + 389017 T^{6} \))(\( ( 1 + 2 T + 118 T^{2} + 146 T^{3} + 5329 T^{4} )^{2} \))(\( ( 1 - 2 T + 118 T^{2} - 146 T^{3} + 5329 T^{4} )^{2} \))
$79$ (\( 1 + 5 T + 79 T^{2} \))(\( 1 + 8 T + 79 T^{2} \))(\( 1 + 16 T + 79 T^{2} \))(\( 1 + 8 T + 79 T^{2} \))(\( 1 - 4 T + 79 T^{2} \))(\( 1 + T + 79 T^{2} \))(\( 1 + 8 T + 79 T^{2} \))(\( 1 - 8 T + 79 T^{2} \))(\( 1 - 4 T + 79 T^{2} \))(\( 1 + 8 T + 79 T^{2} \))(\( 1 + 5 T + 79 T^{2} \))(\( 1 - 8 T + 169 T^{2} - 632 T^{3} + 6241 T^{4} \))(\( 1 + 126 T^{2} + 6241 T^{4} \))(\( 1 + 33 T^{2} + 6241 T^{4} \))(\( 1 + 8 T + 57 T^{2} + 632 T^{3} + 6241 T^{4} \))(\( 1 + 9 T + 174 T^{2} + 711 T^{3} + 6241 T^{4} \))(\( ( 1 - 8 T + 79 T^{2} )^{2} \))(\( 1 - 8 T + 94 T^{2} - 632 T^{3} + 6241 T^{4} \))(\( 1 + 33 T^{2} + 6241 T^{4} \))(\( 1 + 8 T + 57 T^{2} + 632 T^{3} + 6241 T^{4} \))(\( 1 + 126 T^{2} + 6241 T^{4} \))(\( 1 - 8 T + 169 T^{2} - 632 T^{3} + 6241 T^{4} \))(\( 1 - 12 T + 221 T^{2} - 1576 T^{3} + 17459 T^{4} - 74892 T^{5} + 493039 T^{6} \))(\( 1 - 12 T + 221 T^{2} - 1576 T^{3} + 17459 T^{4} - 74892 T^{5} + 493039 T^{6} \))(\( ( 1 + 129 T^{2} + 6241 T^{4} )^{2} \))(\( ( 1 + 129 T^{2} + 6241 T^{4} )^{2} \))
$83$ (\( 1 - 4 T + 83 T^{2} \))(\( 1 + 16 T + 83 T^{2} \))(\( 1 + 12 T + 83 T^{2} \))(\( 1 + 16 T + 83 T^{2} \))(\( 1 - 8 T + 83 T^{2} \))(\( 1 - 12 T + 83 T^{2} \))(\( 1 - 16 T + 83 T^{2} \))(\( 1 + 4 T + 83 T^{2} \))(\( 1 + 8 T + 83 T^{2} \))(\( 1 - 16 T + 83 T^{2} \))(\( 1 + 4 T + 83 T^{2} \))(\( 1 + 16 T + 210 T^{2} + 1328 T^{3} + 6889 T^{4} \))(\( ( 1 - 8 T + 83 T^{2} )^{2} \))(\( 1 - 2 T + 122 T^{2} - 166 T^{3} + 6889 T^{4} \))(\( 1 + 8 T + 130 T^{2} + 664 T^{3} + 6889 T^{4} \))(\( ( 1 - 4 T + 83 T^{2} )^{2} \))(\( ( 1 + 83 T^{2} )^{2} \))(\( 1 + 16 T + 150 T^{2} + 1328 T^{3} + 6889 T^{4} \))(\( 1 + 2 T + 122 T^{2} + 166 T^{3} + 6889 T^{4} \))(\( 1 - 8 T + 130 T^{2} - 664 T^{3} + 6889 T^{4} \))(\( ( 1 + 8 T + 83 T^{2} )^{2} \))(\( 1 - 16 T + 210 T^{2} - 1328 T^{3} + 6889 T^{4} \))(\( 1 + 8 T + 185 T^{2} + 1072 T^{3} + 15355 T^{4} + 55112 T^{5} + 571787 T^{6} \))(\( 1 - 8 T + 185 T^{2} - 1072 T^{3} + 15355 T^{4} - 55112 T^{5} + 571787 T^{6} \))(\( 1 + 36 T^{2} - 2602 T^{4} + 248004 T^{6} + 47458321 T^{8} \))(\( 1 + 36 T^{2} - 2602 T^{4} + 248004 T^{6} + 47458321 T^{8} \))
$89$ (\( 1 + 89 T^{2} \))(\( 1 - 12 T + 89 T^{2} \))(\( 1 - 14 T + 89 T^{2} \))(\( 1 + 12 T + 89 T^{2} \))(\( 1 + 6 T + 89 T^{2} \))(\( 1 - 12 T + 89 T^{2} \))(\( 1 + 12 T + 89 T^{2} \))(\( 1 - 6 T + 89 T^{2} \))(\( 1 + 6 T + 89 T^{2} \))(\( 1 - 12 T + 89 T^{2} \))(\( 1 + 89 T^{2} \))(\( 1 + 2 T + 134 T^{2} + 178 T^{3} + 7921 T^{4} \))(\( 1 - 12 T + 86 T^{2} - 1068 T^{3} + 7921 T^{4} \))(\( 1 + 30 T + 398 T^{2} + 2670 T^{3} + 7921 T^{4} \))(\( 1 - 6 T + 70 T^{2} - 534 T^{3} + 7921 T^{4} \))(\( 1 + 6 T + 170 T^{2} + 534 T^{3} + 7921 T^{4} \))(\( 1 + 166 T^{2} + 7921 T^{4} \))(\( ( 1 - 2 T + 89 T^{2} )^{2} \))(\( 1 + 30 T + 398 T^{2} + 2670 T^{3} + 7921 T^{4} \))(\( 1 - 6 T + 70 T^{2} - 534 T^{3} + 7921 T^{4} \))(\( 1 + 12 T + 86 T^{2} + 1068 T^{3} + 7921 T^{4} \))(\( 1 + 2 T + 134 T^{2} + 178 T^{3} + 7921 T^{4} \))(\( 1 + 14 T + 319 T^{2} + 2532 T^{3} + 28391 T^{4} + 110894 T^{5} + 704969 T^{6} \))(\( 1 + 14 T + 319 T^{2} + 2532 T^{3} + 28391 T^{4} + 110894 T^{5} + 704969 T^{6} \))(\( 1 + 160 T^{2} + 12846 T^{4} + 1267360 T^{6} + 62742241 T^{8} \))(\( 1 + 160 T^{2} + 12846 T^{4} + 1267360 T^{6} + 62742241 T^{8} \))
$97$ (\( 1 - 7 T + 97 T^{2} \))(\( 1 - 4 T + 97 T^{2} \))(\( 1 + 18 T + 97 T^{2} \))(\( 1 + 4 T + 97 T^{2} \))(\( 1 - 2 T + 97 T^{2} \))(\( 1 - T + 97 T^{2} \))(\( 1 - 4 T + 97 T^{2} \))(\( 1 - 18 T + 97 T^{2} \))(\( 1 + 2 T + 97 T^{2} \))(\( 1 + 4 T + 97 T^{2} \))(\( 1 + 7 T + 97 T^{2} \))(\( 1 - 16 T + 238 T^{2} - 1552 T^{3} + 9409 T^{4} \))(\( 1 - 12 T + 222 T^{2} - 1164 T^{3} + 9409 T^{4} \))(\( 1 - 6 T + 198 T^{2} - 582 T^{3} + 9409 T^{4} \))(\( 1 + 16 T + 206 T^{2} + 1552 T^{3} + 9409 T^{4} \))(\( 1 - 9 T + 108 T^{2} - 873 T^{3} + 9409 T^{4} \))(\( ( 1 + 14 T + 97 T^{2} )^{2} \))(\( 1 + 8 T + 190 T^{2} + 776 T^{3} + 9409 T^{4} \))(\( 1 + 6 T + 198 T^{2} + 582 T^{3} + 9409 T^{4} \))(\( 1 - 16 T + 206 T^{2} - 1552 T^{3} + 9409 T^{4} \))(\( 1 - 12 T + 222 T^{2} - 1164 T^{3} + 9409 T^{4} \))(\( 1 + 16 T + 238 T^{2} + 1552 T^{3} + 9409 T^{4} \))(\( 1 + 22 T + 255 T^{2} + 2404 T^{3} + 24735 T^{4} + 206998 T^{5} + 912673 T^{6} \))(\( 1 - 22 T + 255 T^{2} - 2404 T^{3} + 24735 T^{4} - 206998 T^{5} + 912673 T^{6} \))(\( ( 1 + 78 T^{2} + 9409 T^{4} )^{2} \))(\( ( 1 + 78 T^{2} + 9409 T^{4} )^{2} \))
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