Properties

Label 6300.2.a
Level 6300
Weight 2
Character orbit a
Rep. character \(\chi_{6300}(1,\cdot)\)
Character field \(\Q\)
Dimension 48
Newform subspaces 38
Sturm bound 2880
Trace bound 17

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Defining parameters

Level: \( N \) \(=\) \( 6300 = 2^{2} \cdot 3^{2} \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6300.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 38 \)
Sturm bound: \(2880\)
Trace bound: \(17\)
Distinguishing \(T_p\): \(11\), \(13\), \(17\), \(37\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(6300))\).

Total New Old
Modular forms 1512 48 1464
Cusp forms 1369 48 1321
Eisenstein series 143 0 143

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

\(2\)\(3\)\(5\)\(7\)FrickeDim.
\(-\)\(+\)\(+\)\(+\)\(-\)\(4\)
\(-\)\(+\)\(+\)\(-\)\(+\)\(4\)
\(-\)\(+\)\(-\)\(+\)\(+\)\(6\)
\(-\)\(+\)\(-\)\(-\)\(-\)\(6\)
\(-\)\(-\)\(+\)\(+\)\(+\)\(7\)
\(-\)\(-\)\(+\)\(-\)\(-\)\(7\)
\(-\)\(-\)\(-\)\(+\)\(-\)\(7\)
\(-\)\(-\)\(-\)\(-\)\(+\)\(7\)
Plus space\(+\)\(24\)
Minus space\(-\)\(24\)

Trace form

\( 48q + O(q^{10}) \) \( 48q + 4q^{11} + 8q^{13} + 8q^{17} - 4q^{23} - 24q^{29} - 20q^{31} - 12q^{37} - 4q^{41} - 4q^{43} + 4q^{47} + 48q^{49} - 8q^{53} + 20q^{59} - 4q^{61} - 20q^{67} - 40q^{71} - 4q^{73} - 8q^{79} - 48q^{83} - 44q^{89} + 8q^{97} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(6300))\) into newform subspaces

Label Dim. \(A\) Field CM Traces A-L signs $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\) 2 3 5 7
6300.2.a.a \(1\) \(50.306\) \(\Q\) None \(0\) \(0\) \(0\) \(-1\) \(-\) \(-\) \(+\) \(+\) \(q-q^{7}-6q^{11}+4q^{13}+6q^{17}+2q^{19}+\cdots\)
6300.2.a.b \(1\) \(50.306\) \(\Q\) None \(0\) \(0\) \(0\) \(-1\) \(-\) \(-\) \(-\) \(+\) \(q-q^{7}-4q^{11}-2q^{13}+2q^{17}-2q^{19}+\cdots\)
6300.2.a.c \(1\) \(50.306\) \(\Q\) None \(0\) \(0\) \(0\) \(-1\) \(-\) \(-\) \(-\) \(+\) \(q-q^{7}-3q^{11}-q^{13}-5q^{17}-8q^{19}+\cdots\)
6300.2.a.d \(1\) \(50.306\) \(\Q\) None \(0\) \(0\) \(0\) \(-1\) \(-\) \(-\) \(+\) \(+\) \(q-q^{7}-3q^{11}+q^{13}-3q^{17}+2q^{19}+\cdots\)
6300.2.a.e \(1\) \(50.306\) \(\Q\) None \(0\) \(0\) \(0\) \(-1\) \(-\) \(-\) \(+\) \(+\) \(q-q^{7}-3q^{11}+4q^{13}+2q^{19}-3q^{23}+\cdots\)
6300.2.a.f \(1\) \(50.306\) \(\Q\) None \(0\) \(0\) \(0\) \(-1\) \(-\) \(-\) \(+\) \(+\) \(q-q^{7}-q^{11}-4q^{13}+2q^{17}-4q^{19}+\cdots\)
6300.2.a.g \(1\) \(50.306\) \(\Q\) None \(0\) \(0\) \(0\) \(-1\) \(-\) \(-\) \(-\) \(+\) \(q-q^{7}-4q^{13}+4q^{17}+4q^{19}+8q^{23}+\cdots\)
6300.2.a.h \(1\) \(50.306\) \(\Q\) None \(0\) \(0\) \(0\) \(-1\) \(-\) \(+\) \(+\) \(+\) \(q-q^{7}+4q^{13}-6q^{17}+2q^{19}-6q^{23}+\cdots\)
6300.2.a.i \(1\) \(50.306\) \(\Q\) None \(0\) \(0\) \(0\) \(-1\) \(-\) \(+\) \(+\) \(+\) \(q-q^{7}+4q^{13}+6q^{17}+2q^{19}+6q^{23}+\cdots\)
6300.2.a.j \(1\) \(50.306\) \(\Q\) None \(0\) \(0\) \(0\) \(-1\) \(-\) \(-\) \(-\) \(+\) \(q-q^{7}+q^{11}-2q^{13}-8q^{17}-2q^{19}+\cdots\)
6300.2.a.k \(1\) \(50.306\) \(\Q\) None \(0\) \(0\) \(0\) \(-1\) \(-\) \(-\) \(-\) \(+\) \(q-q^{7}+q^{11}-2q^{13}+6q^{19}-q^{23}+\cdots\)
6300.2.a.l \(1\) \(50.306\) \(\Q\) None \(0\) \(0\) \(0\) \(-1\) \(-\) \(-\) \(+\) \(+\) \(q-q^{7}+2q^{11}-4q^{13}+2q^{17}+2q^{19}+\cdots\)
6300.2.a.m \(1\) \(50.306\) \(\Q\) None \(0\) \(0\) \(0\) \(-1\) \(-\) \(-\) \(+\) \(+\) \(q-q^{7}+3q^{11}+4q^{13}-6q^{17}-4q^{19}+\cdots\)
6300.2.a.n \(1\) \(50.306\) \(\Q\) None \(0\) \(0\) \(0\) \(-1\) \(-\) \(-\) \(-\) \(+\) \(q-q^{7}+4q^{11}+6q^{13}+2q^{17}+6q^{19}+\cdots\)
6300.2.a.o \(1\) \(50.306\) \(\Q\) None \(0\) \(0\) \(0\) \(-1\) \(-\) \(-\) \(-\) \(+\) \(q-q^{7}+5q^{11}+6q^{13}+4q^{17}-6q^{19}+\cdots\)
6300.2.a.p \(1\) \(50.306\) \(\Q\) None \(0\) \(0\) \(0\) \(-1\) \(-\) \(-\) \(+\) \(+\) \(q-q^{7}+6q^{11}-2q^{13}-4q^{19}-6q^{23}+\cdots\)
6300.2.a.q \(1\) \(50.306\) \(\Q\) None \(0\) \(0\) \(0\) \(1\) \(-\) \(+\) \(+\) \(-\) \(q+q^{7}-4q^{11}+2q^{17}-6q^{19}+6q^{23}+\cdots\)
6300.2.a.r \(1\) \(50.306\) \(\Q\) None \(0\) \(0\) \(0\) \(1\) \(-\) \(-\) \(-\) \(-\) \(q+q^{7}-4q^{11}+2q^{13}-2q^{17}-2q^{19}+\cdots\)
6300.2.a.s \(1\) \(50.306\) \(\Q\) None \(0\) \(0\) \(0\) \(1\) \(-\) \(-\) \(-\) \(-\) \(q+q^{7}-3q^{11}-4q^{13}+2q^{19}+3q^{23}+\cdots\)
6300.2.a.t \(1\) \(50.306\) \(\Q\) None \(0\) \(0\) \(0\) \(1\) \(-\) \(-\) \(-\) \(-\) \(q+q^{7}-3q^{11}+q^{13}+5q^{17}-8q^{19}+\cdots\)
6300.2.a.u \(1\) \(50.306\) \(\Q\) None \(0\) \(0\) \(0\) \(1\) \(-\) \(-\) \(+\) \(-\) \(q+q^{7}-2q^{11}-4q^{13}+2q^{17}-2q^{19}+\cdots\)
6300.2.a.v \(1\) \(50.306\) \(\Q\) None \(0\) \(0\) \(0\) \(1\) \(-\) \(-\) \(+\) \(-\) \(q+q^{7}-2q^{11}-4q^{13}+6q^{17}+6q^{19}+\cdots\)
6300.2.a.w \(1\) \(50.306\) \(\Q\) None \(0\) \(0\) \(0\) \(1\) \(-\) \(-\) \(+\) \(-\) \(q+q^{7}-2q^{11}+6q^{13}-4q^{17}-4q^{19}+\cdots\)
6300.2.a.x \(1\) \(50.306\) \(\Q\) None \(0\) \(0\) \(0\) \(1\) \(-\) \(-\) \(-\) \(-\) \(q+q^{7}-q^{11}+4q^{13}-2q^{17}-4q^{19}+\cdots\)
6300.2.a.y \(1\) \(50.306\) \(\Q\) None \(0\) \(0\) \(0\) \(1\) \(-\) \(-\) \(-\) \(-\) \(q+q^{7}+4q^{13}-4q^{17}+4q^{19}-8q^{23}+\cdots\)
6300.2.a.z \(1\) \(50.306\) \(\Q\) None \(0\) \(0\) \(0\) \(1\) \(-\) \(-\) \(+\) \(-\) \(q+q^{7}+q^{11}+2q^{13}+6q^{19}+q^{23}+\cdots\)
6300.2.a.ba \(1\) \(50.306\) \(\Q\) None \(0\) \(0\) \(0\) \(1\) \(-\) \(-\) \(+\) \(-\) \(q+q^{7}+q^{11}+2q^{13}+8q^{17}-2q^{19}+\cdots\)
6300.2.a.bb \(1\) \(50.306\) \(\Q\) None \(0\) \(0\) \(0\) \(1\) \(-\) \(-\) \(-\) \(-\) \(q+q^{7}+3q^{11}-4q^{13}+6q^{17}-4q^{19}+\cdots\)
6300.2.a.bc \(1\) \(50.306\) \(\Q\) None \(0\) \(0\) \(0\) \(1\) \(-\) \(-\) \(-\) \(-\) \(q+q^{7}+4q^{11}-6q^{13}-2q^{17}+6q^{19}+\cdots\)
6300.2.a.bd \(1\) \(50.306\) \(\Q\) None \(0\) \(0\) \(0\) \(1\) \(-\) \(+\) \(+\) \(-\) \(q+q^{7}+4q^{11}-2q^{17}-6q^{19}-6q^{23}+\cdots\)
6300.2.a.be \(1\) \(50.306\) \(\Q\) None \(0\) \(0\) \(0\) \(1\) \(-\) \(-\) \(+\) \(-\) \(q+q^{7}+5q^{11}-6q^{13}-4q^{17}-6q^{19}+\cdots\)
6300.2.a.bf \(1\) \(50.306\) \(\Q\) None \(0\) \(0\) \(0\) \(1\) \(-\) \(-\) \(+\) \(-\) \(q+q^{7}+5q^{11}+3q^{13}-q^{17}+6q^{19}+\cdots\)
6300.2.a.bg \(2\) \(50.306\) \(\Q(\sqrt{3}) \) None \(0\) \(0\) \(0\) \(-2\) \(-\) \(+\) \(+\) \(+\) \(q-q^{7}+\beta q^{11}-2q^{13}+2q^{19}-\beta q^{23}+\cdots\)
6300.2.a.bh \(2\) \(50.306\) \(\Q(\sqrt{7}) \) None \(0\) \(0\) \(0\) \(-2\) \(-\) \(+\) \(-\) \(+\) \(q-q^{7}+\beta q^{11}+2\beta q^{17}-3\beta q^{23}+\cdots\)
6300.2.a.bi \(2\) \(50.306\) \(\Q(\sqrt{7}) \) None \(0\) \(0\) \(0\) \(2\) \(-\) \(+\) \(+\) \(-\) \(q+q^{7}+\beta q^{11}-2\beta q^{17}+3\beta q^{23}+\cdots\)
6300.2.a.bj \(2\) \(50.306\) \(\Q(\sqrt{3}) \) None \(0\) \(0\) \(0\) \(2\) \(-\) \(+\) \(-\) \(-\) \(q+q^{7}+\beta q^{11}+2q^{13}+2q^{19}+\beta q^{23}+\cdots\)
6300.2.a.bk \(4\) \(50.306\) \(\Q(\sqrt{3}, \sqrt{7})\) None \(0\) \(0\) \(0\) \(-4\) \(-\) \(+\) \(-\) \(+\) \(q-q^{7}+(-\beta _{1}-\beta _{3})q^{11}+(-1+\beta _{2}+\cdots)q^{13}+\cdots\)
6300.2.a.bl \(4\) \(50.306\) \(\Q(\sqrt{3}, \sqrt{7})\) None \(0\) \(0\) \(0\) \(4\) \(-\) \(+\) \(-\) \(-\) \(q+q^{7}+(-\beta _{1}-\beta _{3})q^{11}+(1-\beta _{2})q^{13}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(6300))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(6300)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(14))\)\(^{\oplus 18}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(15))\)\(^{\oplus 24}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(20))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(21))\)\(^{\oplus 18}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(30))\)\(^{\oplus 16}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(35))\)\(^{\oplus 18}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(36))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(42))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(45))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(50))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(63))\)\(^{\oplus 9}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(70))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(75))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(84))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(90))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(100))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(105))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(126))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(140))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(150))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(175))\)\(^{\oplus 9}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(180))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(210))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(225))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(252))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(300))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(315))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(350))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(420))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(450))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(525))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(630))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(700))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(900))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(1050))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(1260))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(1575))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(2100))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(3150))\)\(^{\oplus 2}\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ 1
$5$ 1
$7$ (\( 1 + T \))(\( 1 + T \))(\( 1 + T \))(\( 1 + T \))(\( 1 + T \))(\( 1 + T \))(\( 1 + T \))(\( 1 + T \))(\( 1 + T \))(\( 1 + T \))(\( 1 + T \))(\( 1 + T \))(\( 1 + T \))(\( 1 + T \))(\( 1 + T \))(\( 1 + T \))(\( 1 - T \))(\( 1 - T \))(\( 1 - T \))(\( 1 - T \))(\( 1 - T \))(\( 1 - T \))(\( 1 - T \))(\( 1 - T \))(\( 1 - T \))(\( 1 - T \))(\( 1 - T \))(\( 1 - T \))(\( 1 - T \))(\( 1 - T \))(\( 1 - T \))(\( 1 - T \))(\( ( 1 + T )^{2} \))(\( ( 1 + T )^{2} \))(\( ( 1 - T )^{2} \))(\( ( 1 - T )^{2} \))(\( ( 1 + T )^{4} \))(\( ( 1 - T )^{4} \))
$11$ (\( 1 + 6 T + 11 T^{2} \))(\( 1 + 4 T + 11 T^{2} \))(\( 1 + 3 T + 11 T^{2} \))(\( 1 + 3 T + 11 T^{2} \))(\( 1 + 3 T + 11 T^{2} \))(\( 1 + T + 11 T^{2} \))(\( 1 + 11 T^{2} \))(\( 1 + 11 T^{2} \))(\( 1 + 11 T^{2} \))(\( 1 - T + 11 T^{2} \))(\( 1 - T + 11 T^{2} \))(\( 1 - 2 T + 11 T^{2} \))(\( 1 - 3 T + 11 T^{2} \))(\( 1 - 4 T + 11 T^{2} \))(\( 1 - 5 T + 11 T^{2} \))(\( 1 - 6 T + 11 T^{2} \))(\( 1 + 4 T + 11 T^{2} \))(\( 1 + 4 T + 11 T^{2} \))(\( 1 + 3 T + 11 T^{2} \))(\( 1 + 3 T + 11 T^{2} \))(\( 1 + 2 T + 11 T^{2} \))(\( 1 + 2 T + 11 T^{2} \))(\( 1 + 2 T + 11 T^{2} \))(\( 1 + T + 11 T^{2} \))(\( 1 + 11 T^{2} \))(\( 1 - T + 11 T^{2} \))(\( 1 - T + 11 T^{2} \))(\( 1 - 3 T + 11 T^{2} \))(\( 1 - 4 T + 11 T^{2} \))(\( 1 - 4 T + 11 T^{2} \))(\( 1 - 5 T + 11 T^{2} \))(\( 1 - 5 T + 11 T^{2} \))(\( 1 - 5 T^{2} + 121 T^{4} \))(\( 1 + 15 T^{2} + 121 T^{4} \))(\( 1 + 15 T^{2} + 121 T^{4} \))(\( 1 - 5 T^{2} + 121 T^{4} \))(\( ( 1 + 10 T^{2} + 121 T^{4} )^{2} \))(\( ( 1 + 10 T^{2} + 121 T^{4} )^{2} \))
$13$ (\( 1 - 4 T + 13 T^{2} \))(\( 1 + 2 T + 13 T^{2} \))(\( 1 + T + 13 T^{2} \))(\( 1 - T + 13 T^{2} \))(\( 1 - 4 T + 13 T^{2} \))(\( 1 + 4 T + 13 T^{2} \))(\( 1 + 4 T + 13 T^{2} \))(\( 1 - 4 T + 13 T^{2} \))(\( 1 - 4 T + 13 T^{2} \))(\( 1 + 2 T + 13 T^{2} \))(\( 1 + 2 T + 13 T^{2} \))(\( 1 + 4 T + 13 T^{2} \))(\( 1 - 4 T + 13 T^{2} \))(\( 1 - 6 T + 13 T^{2} \))(\( 1 - 6 T + 13 T^{2} \))(\( 1 + 2 T + 13 T^{2} \))(\( 1 + 13 T^{2} \))(\( 1 - 2 T + 13 T^{2} \))(\( 1 + 4 T + 13 T^{2} \))(\( 1 - T + 13 T^{2} \))(\( 1 + 4 T + 13 T^{2} \))(\( 1 + 4 T + 13 T^{2} \))(\( 1 - 6 T + 13 T^{2} \))(\( 1 - 4 T + 13 T^{2} \))(\( 1 - 4 T + 13 T^{2} \))(\( 1 - 2 T + 13 T^{2} \))(\( 1 - 2 T + 13 T^{2} \))(\( 1 + 4 T + 13 T^{2} \))(\( 1 + 6 T + 13 T^{2} \))(\( 1 + 13 T^{2} \))(\( 1 + 6 T + 13 T^{2} \))(\( 1 - 3 T + 13 T^{2} \))(\( ( 1 + 2 T + 13 T^{2} )^{2} \))(\( ( 1 + 13 T^{2} )^{2} \))(\( ( 1 + 13 T^{2} )^{2} \))(\( ( 1 - 2 T + 13 T^{2} )^{2} \))(\( ( 1 + 2 T + 6 T^{2} + 26 T^{3} + 169 T^{4} )^{2} \))(\( ( 1 - 2 T + 6 T^{2} - 26 T^{3} + 169 T^{4} )^{2} \))
$17$ (\( 1 - 6 T + 17 T^{2} \))(\( 1 - 2 T + 17 T^{2} \))(\( 1 + 5 T + 17 T^{2} \))(\( 1 + 3 T + 17 T^{2} \))(\( 1 + 17 T^{2} \))(\( 1 - 2 T + 17 T^{2} \))(\( 1 - 4 T + 17 T^{2} \))(\( 1 + 6 T + 17 T^{2} \))(\( 1 - 6 T + 17 T^{2} \))(\( 1 + 8 T + 17 T^{2} \))(\( 1 + 17 T^{2} \))(\( 1 - 2 T + 17 T^{2} \))(\( 1 + 6 T + 17 T^{2} \))(\( 1 - 2 T + 17 T^{2} \))(\( 1 - 4 T + 17 T^{2} \))(\( 1 + 17 T^{2} \))(\( 1 - 2 T + 17 T^{2} \))(\( 1 + 2 T + 17 T^{2} \))(\( 1 + 17 T^{2} \))(\( 1 - 5 T + 17 T^{2} \))(\( 1 - 2 T + 17 T^{2} \))(\( 1 - 6 T + 17 T^{2} \))(\( 1 + 4 T + 17 T^{2} \))(\( 1 + 2 T + 17 T^{2} \))(\( 1 + 4 T + 17 T^{2} \))(\( 1 + 17 T^{2} \))(\( 1 - 8 T + 17 T^{2} \))(\( 1 - 6 T + 17 T^{2} \))(\( 1 + 2 T + 17 T^{2} \))(\( 1 + 2 T + 17 T^{2} \))(\( 1 + 4 T + 17 T^{2} \))(\( 1 + T + 17 T^{2} \))(\( ( 1 + 17 T^{2} )^{2} \))(\( 1 + 6 T^{2} + 289 T^{4} \))(\( 1 + 6 T^{2} + 289 T^{4} \))(\( ( 1 + 17 T^{2} )^{2} \))(\( 1 - 178 T^{4} + 83521 T^{8} \))(\( 1 - 178 T^{4} + 83521 T^{8} \))
$19$ (\( 1 - 2 T + 19 T^{2} \))(\( 1 + 2 T + 19 T^{2} \))(\( 1 + 8 T + 19 T^{2} \))(\( 1 - 2 T + 19 T^{2} \))(\( 1 - 2 T + 19 T^{2} \))(\( 1 + 4 T + 19 T^{2} \))(\( 1 - 4 T + 19 T^{2} \))(\( 1 - 2 T + 19 T^{2} \))(\( 1 - 2 T + 19 T^{2} \))(\( 1 + 2 T + 19 T^{2} \))(\( 1 - 6 T + 19 T^{2} \))(\( 1 - 2 T + 19 T^{2} \))(\( 1 + 4 T + 19 T^{2} \))(\( 1 - 6 T + 19 T^{2} \))(\( 1 + 6 T + 19 T^{2} \))(\( 1 + 4 T + 19 T^{2} \))(\( 1 + 6 T + 19 T^{2} \))(\( 1 + 2 T + 19 T^{2} \))(\( 1 - 2 T + 19 T^{2} \))(\( 1 + 8 T + 19 T^{2} \))(\( 1 + 2 T + 19 T^{2} \))(\( 1 - 6 T + 19 T^{2} \))(\( 1 + 4 T + 19 T^{2} \))(\( 1 + 4 T + 19 T^{2} \))(\( 1 - 4 T + 19 T^{2} \))(\( 1 - 6 T + 19 T^{2} \))(\( 1 + 2 T + 19 T^{2} \))(\( 1 + 4 T + 19 T^{2} \))(\( 1 - 6 T + 19 T^{2} \))(\( 1 + 6 T + 19 T^{2} \))(\( 1 + 6 T + 19 T^{2} \))(\( 1 - 6 T + 19 T^{2} \))(\( ( 1 - 2 T + 19 T^{2} )^{2} \))(\( ( 1 + 19 T^{2} )^{2} \))(\( ( 1 + 19 T^{2} )^{2} \))(\( ( 1 - 2 T + 19 T^{2} )^{2} \))(\( ( 1 - 2 T + 18 T^{2} - 38 T^{3} + 361 T^{4} )^{2} \))(\( ( 1 - 2 T + 18 T^{2} - 38 T^{3} + 361 T^{4} )^{2} \))
$23$ (\( 1 + 23 T^{2} \))(\( 1 + 6 T + 23 T^{2} \))(\( 1 - 2 T + 23 T^{2} \))(\( 1 + 6 T + 23 T^{2} \))(\( 1 + 3 T + 23 T^{2} \))(\( 1 - 7 T + 23 T^{2} \))(\( 1 - 8 T + 23 T^{2} \))(\( 1 + 6 T + 23 T^{2} \))(\( 1 - 6 T + 23 T^{2} \))(\( 1 + T + 23 T^{2} \))(\( 1 + T + 23 T^{2} \))(\( 1 - 4 T + 23 T^{2} \))(\( 1 - 3 T + 23 T^{2} \))(\( 1 - 2 T + 23 T^{2} \))(\( 1 - 3 T + 23 T^{2} \))(\( 1 + 6 T + 23 T^{2} \))(\( 1 - 6 T + 23 T^{2} \))(\( 1 - 6 T + 23 T^{2} \))(\( 1 - 3 T + 23 T^{2} \))(\( 1 + 2 T + 23 T^{2} \))(\( 1 - 4 T + 23 T^{2} \))(\( 1 + 8 T + 23 T^{2} \))(\( 1 - 2 T + 23 T^{2} \))(\( 1 + 7 T + 23 T^{2} \))(\( 1 + 8 T + 23 T^{2} \))(\( 1 - T + 23 T^{2} \))(\( 1 - T + 23 T^{2} \))(\( 1 + 3 T + 23 T^{2} \))(\( 1 + 2 T + 23 T^{2} \))(\( 1 + 6 T + 23 T^{2} \))(\( 1 + 3 T + 23 T^{2} \))(\( 1 - 6 T + 23 T^{2} \))(\( 1 + 19 T^{2} + 529 T^{4} \))(\( 1 - 17 T^{2} + 529 T^{4} \))(\( 1 - 17 T^{2} + 529 T^{4} \))(\( 1 + 19 T^{2} + 529 T^{4} \))(\( 1 + 72 T^{2} + 2270 T^{4} + 38088 T^{6} + 279841 T^{8} \))(\( 1 + 72 T^{2} + 2270 T^{4} + 38088 T^{6} + 279841 T^{8} \))
$29$ (\( 1 + 6 T + 29 T^{2} \))(\( 1 + 6 T + 29 T^{2} \))(\( 1 - T + 29 T^{2} \))(\( 1 - 9 T + 29 T^{2} \))(\( 1 + 9 T + 29 T^{2} \))(\( 1 - 9 T + 29 T^{2} \))(\( 1 + 2 T + 29 T^{2} \))(\( 1 + 29 T^{2} \))(\( 1 + 29 T^{2} \))(\( 1 + T + 29 T^{2} \))(\( 1 + T + 29 T^{2} \))(\( 1 + 6 T + 29 T^{2} \))(\( 1 + 3 T + 29 T^{2} \))(\( 1 + 6 T + 29 T^{2} \))(\( 1 - 3 T + 29 T^{2} \))(\( 1 + 6 T + 29 T^{2} \))(\( 1 + 29 T^{2} \))(\( 1 + 6 T + 29 T^{2} \))(\( 1 + 9 T + 29 T^{2} \))(\( 1 - T + 29 T^{2} \))(\( 1 - 2 T + 29 T^{2} \))(\( 1 - 2 T + 29 T^{2} \))(\( 1 - 2 T + 29 T^{2} \))(\( 1 - 9 T + 29 T^{2} \))(\( 1 + 2 T + 29 T^{2} \))(\( 1 + T + 29 T^{2} \))(\( 1 + T + 29 T^{2} \))(\( 1 + 3 T + 29 T^{2} \))(\( 1 + 6 T + 29 T^{2} \))(\( 1 + 29 T^{2} \))(\( 1 - 3 T + 29 T^{2} \))(\( 1 - 9 T + 29 T^{2} \))(\( 1 + 31 T^{2} + 841 T^{4} \))(\( 1 - 5 T^{2} + 841 T^{4} \))(\( 1 - 5 T^{2} + 841 T^{4} \))(\( 1 + 31 T^{2} + 841 T^{4} \))(\( 1 + 36 T^{2} + 662 T^{4} + 30276 T^{6} + 707281 T^{8} \))(\( 1 + 36 T^{2} + 662 T^{4} + 30276 T^{6} + 707281 T^{8} \))
$31$ (\( 1 + 10 T + 31 T^{2} \))(\( 1 - 6 T + 31 T^{2} \))(\( 1 + 2 T + 31 T^{2} \))(\( 1 - 8 T + 31 T^{2} \))(\( 1 - 8 T + 31 T^{2} \))(\( 1 + 2 T + 31 T^{2} \))(\( 1 + 8 T + 31 T^{2} \))(\( 1 - 2 T + 31 T^{2} \))(\( 1 - 2 T + 31 T^{2} \))(\( 1 - 6 T + 31 T^{2} \))(\( 1 + 2 T + 31 T^{2} \))(\( 1 + 2 T + 31 T^{2} \))(\( 1 + 10 T + 31 T^{2} \))(\( 1 + 2 T + 31 T^{2} \))(\( 1 - 2 T + 31 T^{2} \))(\( 1 - 8 T + 31 T^{2} \))(\( 1 - 2 T + 31 T^{2} \))(\( 1 - 6 T + 31 T^{2} \))(\( 1 - 8 T + 31 T^{2} \))(\( 1 + 2 T + 31 T^{2} \))(\( 1 + 6 T + 31 T^{2} \))(\( 1 - 10 T + 31 T^{2} \))(\( 1 + 31 T^{2} \))(\( 1 + 2 T + 31 T^{2} \))(\( 1 + 8 T + 31 T^{2} \))(\( 1 + 2 T + 31 T^{2} \))(\( 1 - 6 T + 31 T^{2} \))(\( 1 + 10 T + 31 T^{2} \))(\( 1 + 2 T + 31 T^{2} \))(\( 1 - 2 T + 31 T^{2} \))(\( 1 - 2 T + 31 T^{2} \))(\( 1 + 4 T + 31 T^{2} \))(\( ( 1 - 2 T + 31 T^{2} )^{2} \))(\( ( 1 + 10 T + 31 T^{2} )^{2} \))(\( ( 1 + 10 T + 31 T^{2} )^{2} \))(\( ( 1 - 2 T + 31 T^{2} )^{2} \))(\( ( 1 - 2 T + 42 T^{2} - 62 T^{3} + 961 T^{4} )^{2} \))(\( ( 1 - 2 T + 42 T^{2} - 62 T^{3} + 961 T^{4} )^{2} \))
$37$ (\( 1 + 2 T + 37 T^{2} \))(\( 1 - 4 T + 37 T^{2} \))(\( 1 + 10 T + 37 T^{2} \))(\( 1 - 10 T + 37 T^{2} \))(\( 1 + 5 T + 37 T^{2} \))(\( 1 + T + 37 T^{2} \))(\( 1 - 8 T + 37 T^{2} \))(\( 1 + 2 T + 37 T^{2} \))(\( 1 + 2 T + 37 T^{2} \))(\( 1 - 9 T + 37 T^{2} \))(\( 1 + 7 T + 37 T^{2} \))(\( 1 + 10 T + 37 T^{2} \))(\( 1 - 7 T + 37 T^{2} \))(\( 1 - 4 T + 37 T^{2} \))(\( 1 + 7 T + 37 T^{2} \))(\( 1 + 2 T + 37 T^{2} \))(\( 1 + 2 T + 37 T^{2} \))(\( 1 + 4 T + 37 T^{2} \))(\( 1 - 5 T + 37 T^{2} \))(\( 1 - 10 T + 37 T^{2} \))(\( 1 - 6 T + 37 T^{2} \))(\( 1 + 2 T + 37 T^{2} \))(\( 1 + 2 T + 37 T^{2} \))(\( 1 - T + 37 T^{2} \))(\( 1 + 8 T + 37 T^{2} \))(\( 1 - 7 T + 37 T^{2} \))(\( 1 + 9 T + 37 T^{2} \))(\( 1 + 7 T + 37 T^{2} \))(\( 1 + 4 T + 37 T^{2} \))(\( 1 + 2 T + 37 T^{2} \))(\( 1 - 7 T + 37 T^{2} \))(\( 1 + 2 T + 37 T^{2} \))(\( ( 1 - T + 37 T^{2} )^{2} \))(\( ( 1 - 11 T + 37 T^{2} )^{2} \))(\( ( 1 + 11 T + 37 T^{2} )^{2} \))(\( ( 1 + T + 37 T^{2} )^{2} \))(\( ( 1 + 4 T + 37 T^{2} )^{4} \))(\( ( 1 - 4 T + 37 T^{2} )^{4} \))
$41$ (\( 1 - 6 T + 41 T^{2} \))(\( 1 + 41 T^{2} \))(\( 1 - 6 T + 41 T^{2} \))(\( 1 + 41 T^{2} \))(\( 1 - 6 T + 41 T^{2} \))(\( 1 + 8 T + 41 T^{2} \))(\( 1 + 6 T + 41 T^{2} \))(\( 1 - 6 T + 41 T^{2} \))(\( 1 + 6 T + 41 T^{2} \))(\( 1 + 41 T^{2} \))(\( 1 - 8 T + 41 T^{2} \))(\( 1 - 10 T + 41 T^{2} \))(\( 1 + 41 T^{2} \))(\( 1 + 8 T + 41 T^{2} \))(\( 1 - 4 T + 41 T^{2} \))(\( 1 + 12 T + 41 T^{2} \))(\( 1 + 2 T + 41 T^{2} \))(\( 1 + 41 T^{2} \))(\( 1 - 6 T + 41 T^{2} \))(\( 1 - 6 T + 41 T^{2} \))(\( 1 + 6 T + 41 T^{2} \))(\( 1 + 10 T + 41 T^{2} \))(\( 1 + 41 T^{2} \))(\( 1 + 8 T + 41 T^{2} \))(\( 1 + 6 T + 41 T^{2} \))(\( 1 - 8 T + 41 T^{2} \))(\( 1 + 41 T^{2} \))(\( 1 + 41 T^{2} \))(\( 1 + 8 T + 41 T^{2} \))(\( 1 - 2 T + 41 T^{2} \))(\( 1 - 4 T + 41 T^{2} \))(\( 1 - 4 T + 41 T^{2} \))(\( ( 1 + 41 T^{2} )^{2} \))(\( 1 - 30 T^{2} + 1681 T^{4} \))(\( 1 - 30 T^{2} + 1681 T^{4} \))(\( ( 1 + 41 T^{2} )^{2} \))(\( 1 + 32 T^{2} + 2862 T^{4} + 53792 T^{6} + 2825761 T^{8} \))(\( 1 + 32 T^{2} + 2862 T^{4} + 53792 T^{6} + 2825761 T^{8} \))
$43$ (\( 1 - 4 T + 43 T^{2} \))(\( 1 - 4 T + 43 T^{2} \))(\( 1 - 4 T + 43 T^{2} \))(\( 1 + 2 T + 43 T^{2} \))(\( 1 + 11 T + 43 T^{2} \))(\( 1 - 9 T + 43 T^{2} \))(\( 1 + 8 T + 43 T^{2} \))(\( 1 - 4 T + 43 T^{2} \))(\( 1 - 4 T + 43 T^{2} \))(\( 1 + T + 43 T^{2} \))(\( 1 + T + 43 T^{2} \))(\( 1 + 12 T + 43 T^{2} \))(\( 1 - T + 43 T^{2} \))(\( 1 - 4 T + 43 T^{2} \))(\( 1 - 7 T + 43 T^{2} \))(\( 1 - 4 T + 43 T^{2} \))(\( 1 + 4 T + 43 T^{2} \))(\( 1 + 4 T + 43 T^{2} \))(\( 1 - 11 T + 43 T^{2} \))(\( 1 + 4 T + 43 T^{2} \))(\( 1 - 4 T + 43 T^{2} \))(\( 1 - 4 T + 43 T^{2} \))(\( 1 - 4 T + 43 T^{2} \))(\( 1 + 9 T + 43 T^{2} \))(\( 1 - 8 T + 43 T^{2} \))(\( 1 - T + 43 T^{2} \))(\( 1 - T + 43 T^{2} \))(\( 1 + T + 43 T^{2} \))(\( 1 + 4 T + 43 T^{2} \))(\( 1 + 4 T + 43 T^{2} \))(\( 1 + 7 T + 43 T^{2} \))(\( 1 + 10 T + 43 T^{2} \))(\( ( 1 + 5 T + 43 T^{2} )^{2} \))(\( ( 1 - T + 43 T^{2} )^{2} \))(\( ( 1 + T + 43 T^{2} )^{2} \))(\( ( 1 - 5 T + 43 T^{2} )^{2} \))(\( ( 1 + 4 T + 6 T^{2} + 172 T^{3} + 1849 T^{4} )^{2} \))(\( ( 1 - 4 T + 6 T^{2} - 172 T^{3} + 1849 T^{4} )^{2} \))
$47$ (\( 1 + 47 T^{2} \))(\( 1 - 4 T + 47 T^{2} \))(\( 1 - 11 T + 47 T^{2} \))(\( 1 + 3 T + 47 T^{2} \))(\( 1 - 6 T + 47 T^{2} \))(\( 1 - 4 T + 47 T^{2} \))(\( 1 - 8 T + 47 T^{2} \))(\( 1 + 47 T^{2} \))(\( 1 + 47 T^{2} \))(\( 1 + 6 T + 47 T^{2} \))(\( 1 - 2 T + 47 T^{2} \))(\( 1 + 8 T + 47 T^{2} \))(\( 1 + 12 T + 47 T^{2} \))(\( 1 - 4 T + 47 T^{2} \))(\( 1 + 2 T + 47 T^{2} \))(\( 1 - 12 T + 47 T^{2} \))(\( 1 - 8 T + 47 T^{2} \))(\( 1 + 4 T + 47 T^{2} \))(\( 1 + 6 T + 47 T^{2} \))(\( 1 + 11 T + 47 T^{2} \))(\( 1 + 47 T^{2} \))(\( 1 + 8 T + 47 T^{2} \))(\( 1 - 12 T + 47 T^{2} \))(\( 1 + 4 T + 47 T^{2} \))(\( 1 + 8 T + 47 T^{2} \))(\( 1 + 2 T + 47 T^{2} \))(\( 1 - 6 T + 47 T^{2} \))(\( 1 - 12 T + 47 T^{2} \))(\( 1 + 4 T + 47 T^{2} \))(\( 1 + 8 T + 47 T^{2} \))(\( 1 - 2 T + 47 T^{2} \))(\( 1 + T + 47 T^{2} \))(\( 1 - 14 T^{2} + 2209 T^{4} \))(\( 1 - 18 T^{2} + 2209 T^{4} \))(\( 1 - 18 T^{2} + 2209 T^{4} \))(\( 1 - 14 T^{2} + 2209 T^{4} \))(\( 1 + 108 T^{2} + 5990 T^{4} + 238572 T^{6} + 4879681 T^{8} \))(\( 1 + 108 T^{2} + 5990 T^{4} + 238572 T^{6} + 4879681 T^{8} \))
$53$ (\( 1 + 12 T + 53 T^{2} \))(\( 1 + 2 T + 53 T^{2} \))(\( 1 - 6 T + 53 T^{2} \))(\( 1 + 53 T^{2} \))(\( 1 - 6 T + 53 T^{2} \))(\( 1 - 6 T + 53 T^{2} \))(\( 1 + 53 T^{2} \))(\( 1 - 6 T + 53 T^{2} \))(\( 1 + 6 T + 53 T^{2} \))(\( 1 + 2 T + 53 T^{2} \))(\( 1 - 14 T + 53 T^{2} \))(\( 1 + 53 T^{2} \))(\( 1 - 6 T + 53 T^{2} \))(\( 1 - 6 T + 53 T^{2} \))(\( 1 + 10 T + 53 T^{2} \))(\( 1 + 6 T + 53 T^{2} \))(\( 1 - 10 T + 53 T^{2} \))(\( 1 - 2 T + 53 T^{2} \))(\( 1 + 6 T + 53 T^{2} \))(\( 1 + 6 T + 53 T^{2} \))(\( 1 - 8 T + 53 T^{2} \))(\( 1 - 4 T + 53 T^{2} \))(\( 1 + 6 T + 53 T^{2} \))(\( 1 + 6 T + 53 T^{2} \))(\( 1 + 53 T^{2} \))(\( 1 + 14 T + 53 T^{2} \))(\( 1 - 2 T + 53 T^{2} \))(\( 1 + 6 T + 53 T^{2} \))(\( 1 + 6 T + 53 T^{2} \))(\( 1 + 10 T + 53 T^{2} \))(\( 1 - 10 T + 53 T^{2} \))(\( 1 - 4 T + 53 T^{2} \))(\( ( 1 + 53 T^{2} )^{2} \))(\( 1 - 6 T^{2} + 2809 T^{4} \))(\( 1 - 6 T^{2} + 2809 T^{4} \))(\( ( 1 + 53 T^{2} )^{2} \))(\( 1 + 80 T^{2} + 6462 T^{4} + 224720 T^{6} + 7890481 T^{8} \))(\( 1 + 80 T^{2} + 6462 T^{4} + 224720 T^{6} + 7890481 T^{8} \))
$59$ (\( 1 + 59 T^{2} \))(\( 1 + 4 T + 59 T^{2} \))(\( 1 - 10 T + 59 T^{2} \))(\( 1 + 12 T + 59 T^{2} \))(\( 1 + 59 T^{2} \))(\( 1 + 4 T + 59 T^{2} \))(\( 1 - 4 T + 59 T^{2} \))(\( 1 - 12 T + 59 T^{2} \))(\( 1 + 12 T + 59 T^{2} \))(\( 1 - 6 T + 59 T^{2} \))(\( 1 + 10 T + 59 T^{2} \))(\( 1 - 8 T + 59 T^{2} \))(\( 1 + 12 T + 59 T^{2} \))(\( 1 + 4 T + 59 T^{2} \))(\( 1 - 14 T + 59 T^{2} \))(\( 1 + 59 T^{2} \))(\( 1 + 4 T + 59 T^{2} \))(\( 1 + 4 T + 59 T^{2} \))(\( 1 + 59 T^{2} \))(\( 1 - 10 T + 59 T^{2} \))(\( 1 + 59 T^{2} \))(\( 1 - 8 T + 59 T^{2} \))(\( 1 - 8 T + 59 T^{2} \))(\( 1 + 4 T + 59 T^{2} \))(\( 1 - 4 T + 59 T^{2} \))(\( 1 + 10 T + 59 T^{2} \))(\( 1 - 6 T + 59 T^{2} \))(\( 1 + 12 T + 59 T^{2} \))(\( 1 + 4 T + 59 T^{2} \))(\( 1 - 4 T + 59 T^{2} \))(\( 1 - 14 T + 59 T^{2} \))(\( 1 - 8 T + 59 T^{2} \))(\( 1 + 10 T^{2} + 3481 T^{4} \))(\( 1 + 6 T^{2} + 3481 T^{4} \))(\( 1 + 6 T^{2} + 3481 T^{4} \))(\( 1 + 10 T^{2} + 3481 T^{4} \))(\( 1 + 156 T^{2} + 11702 T^{4} + 543036 T^{6} + 12117361 T^{8} \))(\( 1 + 156 T^{2} + 11702 T^{4} + 543036 T^{6} + 12117361 T^{8} \))
$61$ (\( 1 - 14 T + 61 T^{2} \))(\( 1 + 2 T + 61 T^{2} \))(\( 1 + 61 T^{2} \))(\( 1 - 8 T + 61 T^{2} \))(\( 1 + 10 T + 61 T^{2} \))(\( 1 - 4 T + 61 T^{2} \))(\( 1 + 6 T + 61 T^{2} \))(\( 1 + 10 T + 61 T^{2} \))(\( 1 + 10 T + 61 T^{2} \))(\( 1 - 8 T + 61 T^{2} \))(\( 1 + 61 T^{2} \))(\( 1 + 2 T + 61 T^{2} \))(\( 1 + 4 T + 61 T^{2} \))(\( 1 - 14 T + 61 T^{2} \))(\( 1 - 4 T + 61 T^{2} \))(\( 1 + 10 T + 61 T^{2} \))(\( 1 + 2 T + 61 T^{2} \))(\( 1 + 2 T + 61 T^{2} \))(\( 1 + 10 T + 61 T^{2} \))(\( 1 + 61 T^{2} \))(\( 1 + 10 T + 61 T^{2} \))(\( 1 - 6 T + 61 T^{2} \))(\( 1 - 6 T + 61 T^{2} \))(\( 1 - 4 T + 61 T^{2} \))(\( 1 + 6 T + 61 T^{2} \))(\( 1 + 61 T^{2} \))(\( 1 - 8 T + 61 T^{2} \))(\( 1 + 4 T + 61 T^{2} \))(\( 1 - 14 T + 61 T^{2} \))(\( 1 + 2 T + 61 T^{2} \))(\( 1 - 4 T + 61 T^{2} \))(\( 1 + 8 T + 61 T^{2} \))(\( ( 1 - 8 T + 61 T^{2} )^{2} \))(\( ( 1 + 8 T + 61 T^{2} )^{2} \))(\( ( 1 + 8 T + 61 T^{2} )^{2} \))(\( ( 1 - 8 T + 61 T^{2} )^{2} \))(\( ( 1 + 38 T^{2} + 3721 T^{4} )^{2} \))(\( ( 1 + 38 T^{2} + 3721 T^{4} )^{2} \))
$67$ (\( 1 - 4 T + 67 T^{2} \))(\( 1 - 12 T + 67 T^{2} \))(\( 1 - 10 T + 67 T^{2} \))(\( 1 + 8 T + 67 T^{2} \))(\( 1 + 5 T + 67 T^{2} \))(\( 1 + 9 T + 67 T^{2} \))(\( 1 + 8 T + 67 T^{2} \))(\( 1 - 4 T + 67 T^{2} \))(\( 1 - 4 T + 67 T^{2} \))(\( 1 + 3 T + 67 T^{2} \))(\( 1 + 3 T + 67 T^{2} \))(\( 1 - 12 T + 67 T^{2} \))(\( 1 - 7 T + 67 T^{2} \))(\( 1 + 4 T + 67 T^{2} \))(\( 1 + 3 T + 67 T^{2} \))(\( 1 + 8 T + 67 T^{2} \))(\( 1 + 12 T + 67 T^{2} \))(\( 1 + 12 T + 67 T^{2} \))(\( 1 - 5 T + 67 T^{2} \))(\( 1 + 10 T + 67 T^{2} \))(\( 1 - 12 T + 67 T^{2} \))(\( 1 + 12 T + 67 T^{2} \))(\( 1 - 8 T + 67 T^{2} \))(\( 1 - 9 T + 67 T^{2} \))(\( 1 - 8 T + 67 T^{2} \))(\( 1 - 3 T + 67 T^{2} \))(\( 1 - 3 T + 67 T^{2} \))(\( 1 + 7 T + 67 T^{2} \))(\( 1 - 4 T + 67 T^{2} \))(\( 1 + 12 T + 67 T^{2} \))(\( 1 - 3 T + 67 T^{2} \))(\( 1 + 12 T + 67 T^{2} \))(\( ( 1 - 7 T + 67 T^{2} )^{2} \))(\( ( 1 + 3 T + 67 T^{2} )^{2} \))(\( ( 1 - 3 T + 67 T^{2} )^{2} \))(\( ( 1 + 7 T + 67 T^{2} )^{2} \))(\( ( 1 + 8 T + 67 T^{2} )^{4} \))(\( ( 1 - 8 T + 67 T^{2} )^{4} \))
$71$ (\( 1 + 6 T + 71 T^{2} \))(\( 1 - 8 T + 71 T^{2} \))(\( 1 + 71 T^{2} \))(\( 1 + 71 T^{2} \))(\( 1 + 15 T + 71 T^{2} \))(\( 1 + 5 T + 71 T^{2} \))(\( 1 + 12 T + 71 T^{2} \))(\( 1 - 12 T + 71 T^{2} \))(\( 1 + 12 T + 71 T^{2} \))(\( 1 + 7 T + 71 T^{2} \))(\( 1 - 9 T + 71 T^{2} \))(\( 1 - 10 T + 71 T^{2} \))(\( 1 + 9 T + 71 T^{2} \))(\( 1 + 71 T^{2} \))(\( 1 - 13 T + 71 T^{2} \))(\( 1 + 6 T + 71 T^{2} \))(\( 1 + 8 T + 71 T^{2} \))(\( 1 - 8 T + 71 T^{2} \))(\( 1 + 15 T + 71 T^{2} \))(\( 1 + 71 T^{2} \))(\( 1 - 14 T + 71 T^{2} \))(\( 1 - 6 T + 71 T^{2} \))(\( 1 + 14 T + 71 T^{2} \))(\( 1 + 5 T + 71 T^{2} \))(\( 1 + 12 T + 71 T^{2} \))(\( 1 - 9 T + 71 T^{2} \))(\( 1 + 7 T + 71 T^{2} \))(\( 1 + 9 T + 71 T^{2} \))(\( 1 + 71 T^{2} \))(\( 1 - 8 T + 71 T^{2} \))(\( 1 - 13 T + 71 T^{2} \))(\( 1 + 8 T + 71 T^{2} \))(\( 1 + 115 T^{2} + 5041 T^{4} \))(\( 1 + 135 T^{2} + 5041 T^{4} \))(\( 1 + 135 T^{2} + 5041 T^{4} \))(\( 1 + 115 T^{2} + 5041 T^{4} \))(\( ( 1 + 130 T^{2} + 5041 T^{4} )^{2} \))(\( ( 1 + 130 T^{2} + 5041 T^{4} )^{2} \))
$73$ (\( 1 - 4 T + 73 T^{2} \))(\( 1 + 14 T + 73 T^{2} \))(\( 1 - 10 T + 73 T^{2} \))(\( 1 + 14 T + 73 T^{2} \))(\( 1 - 10 T + 73 T^{2} \))(\( 1 + 10 T + 73 T^{2} \))(\( 1 - 4 T + 73 T^{2} \))(\( 1 - 4 T + 73 T^{2} \))(\( 1 - 4 T + 73 T^{2} \))(\( 1 - 16 T + 73 T^{2} \))(\( 1 + 73 T^{2} \))(\( 1 + 4 T + 73 T^{2} \))(\( 1 + 2 T + 73 T^{2} \))(\( 1 - 10 T + 73 T^{2} \))(\( 1 + 16 T + 73 T^{2} \))(\( 1 - 10 T + 73 T^{2} \))(\( 1 + 8 T + 73 T^{2} \))(\( 1 - 14 T + 73 T^{2} \))(\( 1 + 10 T + 73 T^{2} \))(\( 1 + 10 T + 73 T^{2} \))(\( 1 + 4 T + 73 T^{2} \))(\( 1 - 12 T + 73 T^{2} \))(\( 1 - 2 T + 73 T^{2} \))(\( 1 - 10 T + 73 T^{2} \))(\( 1 + 4 T + 73 T^{2} \))(\( 1 + 73 T^{2} \))(\( 1 + 16 T + 73 T^{2} \))(\( 1 - 2 T + 73 T^{2} \))(\( 1 + 10 T + 73 T^{2} \))(\( 1 + 8 T + 73 T^{2} \))(\( 1 - 16 T + 73 T^{2} \))(\( 1 + 2 T + 73 T^{2} \))(\( ( 1 + 8 T + 73 T^{2} )^{2} \))(\( ( 1 + 10 T + 73 T^{2} )^{2} \))(\( ( 1 - 10 T + 73 T^{2} )^{2} \))(\( ( 1 - 8 T + 73 T^{2} )^{2} \))(\( ( 1 - 2 T + 126 T^{2} - 146 T^{3} + 5329 T^{4} )^{2} \))(\( ( 1 + 2 T + 126 T^{2} + 146 T^{3} + 5329 T^{4} )^{2} \))
$79$ (\( 1 + 16 T + 79 T^{2} \))(\( 1 - 16 T + 79 T^{2} \))(\( 1 + 7 T + 79 T^{2} \))(\( 1 - 5 T + 79 T^{2} \))(\( 1 + 7 T + 79 T^{2} \))(\( 1 + 15 T + 79 T^{2} \))(\( 1 + 4 T + 79 T^{2} \))(\( 1 - 8 T + 79 T^{2} \))(\( 1 - 8 T + 79 T^{2} \))(\( 1 - T + 79 T^{2} \))(\( 1 - T + 79 T^{2} \))(\( 1 + 79 T^{2} \))(\( 1 - 17 T + 79 T^{2} \))(\( 1 + 79 T^{2} \))(\( 1 - T + 79 T^{2} \))(\( 1 + 4 T + 79 T^{2} \))(\( 1 + 8 T + 79 T^{2} \))(\( 1 - 16 T + 79 T^{2} \))(\( 1 + 7 T + 79 T^{2} \))(\( 1 + 7 T + 79 T^{2} \))(\( 1 + 8 T + 79 T^{2} \))(\( 1 + 8 T + 79 T^{2} \))(\( 1 - 12 T + 79 T^{2} \))(\( 1 + 15 T + 79 T^{2} \))(\( 1 + 4 T + 79 T^{2} \))(\( 1 - T + 79 T^{2} \))(\( 1 - T + 79 T^{2} \))(\( 1 - 17 T + 79 T^{2} \))(\( 1 + 79 T^{2} \))(\( 1 + 8 T + 79 T^{2} \))(\( 1 - T + 79 T^{2} \))(\( 1 - 13 T + 79 T^{2} \))(\( ( 1 - 5 T + 79 T^{2} )^{2} \))(\( ( 1 + 11 T + 79 T^{2} )^{2} \))(\( ( 1 + 11 T + 79 T^{2} )^{2} \))(\( ( 1 - 5 T + 79 T^{2} )^{2} \))(\( ( 1 - 4 T + 78 T^{2} - 316 T^{3} + 6241 T^{4} )^{2} \))(\( ( 1 - 4 T + 78 T^{2} - 316 T^{3} + 6241 T^{4} )^{2} \))
$83$ (\( 1 + 12 T + 83 T^{2} \))(\( 1 - 16 T + 83 T^{2} \))(\( 1 - 12 T + 83 T^{2} \))(\( 1 + 12 T + 83 T^{2} \))(\( 1 - 12 T + 83 T^{2} \))(\( 1 + 6 T + 83 T^{2} \))(\( 1 + 83 T^{2} \))(\( 1 + 12 T + 83 T^{2} \))(\( 1 - 12 T + 83 T^{2} \))(\( 1 - 6 T + 83 T^{2} \))(\( 1 + 2 T + 83 T^{2} \))(\( 1 + 12 T + 83 T^{2} \))(\( 1 + 6 T + 83 T^{2} \))(\( 1 + 16 T + 83 T^{2} \))(\( 1 - 10 T + 83 T^{2} \))(\( 1 + 12 T + 83 T^{2} \))(\( 1 + 4 T + 83 T^{2} \))(\( 1 + 16 T + 83 T^{2} \))(\( 1 + 12 T + 83 T^{2} \))(\( 1 + 12 T + 83 T^{2} \))(\( 1 - 12 T + 83 T^{2} \))(\( 1 + 4 T + 83 T^{2} \))(\( 1 + 4 T + 83 T^{2} \))(\( 1 - 6 T + 83 T^{2} \))(\( 1 + 83 T^{2} \))(\( 1 - 2 T + 83 T^{2} \))(\( 1 + 6 T + 83 T^{2} \))(\( 1 - 6 T + 83 T^{2} \))(\( 1 - 16 T + 83 T^{2} \))(\( 1 - 4 T + 83 T^{2} \))(\( 1 + 10 T + 83 T^{2} \))(\( 1 + 4 T + 83 T^{2} \))(\( 1 + 58 T^{2} + 6889 T^{4} \))(\( 1 + 138 T^{2} + 6889 T^{4} \))(\( 1 + 138 T^{2} + 6889 T^{4} \))(\( 1 + 58 T^{2} + 6889 T^{4} \))(\( ( 1 + 118 T^{2} + 6889 T^{4} )^{2} \))(\( ( 1 + 118 T^{2} + 6889 T^{4} )^{2} \))
$89$ (\( 1 + 6 T + 89 T^{2} \))(\( 1 + 16 T + 89 T^{2} \))(\( 1 + 8 T + 89 T^{2} \))(\( 1 + 12 T + 89 T^{2} \))(\( 1 - 12 T + 89 T^{2} \))(\( 1 + 8 T + 89 T^{2} \))(\( 1 - 10 T + 89 T^{2} \))(\( 1 - 6 T + 89 T^{2} \))(\( 1 + 6 T + 89 T^{2} \))(\( 1 - 14 T + 89 T^{2} \))(\( 1 + 2 T + 89 T^{2} \))(\( 1 + 2 T + 89 T^{2} \))(\( 1 + 89 T^{2} \))(\( 1 + 8 T + 89 T^{2} \))(\( 1 + 10 T + 89 T^{2} \))(\( 1 + 12 T + 89 T^{2} \))(\( 1 + 10 T + 89 T^{2} \))(\( 1 + 16 T + 89 T^{2} \))(\( 1 - 12 T + 89 T^{2} \))(\( 1 + 8 T + 89 T^{2} \))(\( 1 - 14 T + 89 T^{2} \))(\( 1 - 10 T + 89 T^{2} \))(\( 1 + 89 T^{2} \))(\( 1 + 8 T + 89 T^{2} \))(\( 1 - 10 T + 89 T^{2} \))(\( 1 + 2 T + 89 T^{2} \))(\( 1 - 14 T + 89 T^{2} \))(\( 1 + 89 T^{2} \))(\( 1 + 8 T + 89 T^{2} \))(\( 1 - 10 T + 89 T^{2} \))(\( 1 + 10 T + 89 T^{2} \))(\( 1 + 4 T + 89 T^{2} \))(\( 1 + 70 T^{2} + 7921 T^{4} \))(\( 1 + 66 T^{2} + 7921 T^{4} \))(\( 1 + 66 T^{2} + 7921 T^{4} \))(\( 1 + 70 T^{2} + 7921 T^{4} \))(\( 1 - 144 T^{2} + 14222 T^{4} - 1140624 T^{6} + 62742241 T^{8} \))(\( 1 - 144 T^{2} + 14222 T^{4} - 1140624 T^{6} + 62742241 T^{8} \))
$97$ (\( 1 - 16 T + 97 T^{2} \))(\( 1 - 14 T + 97 T^{2} \))(\( 1 + 3 T + 97 T^{2} \))(\( 1 + 17 T + 97 T^{2} \))(\( 1 + 8 T + 97 T^{2} \))(\( 1 + 10 T + 97 T^{2} \))(\( 1 - 12 T + 97 T^{2} \))(\( 1 + 8 T + 97 T^{2} \))(\( 1 + 8 T + 97 T^{2} \))(\( 1 - 14 T + 97 T^{2} \))(\( 1 + 10 T + 97 T^{2} \))(\( 1 - 8 T + 97 T^{2} \))(\( 1 + 2 T + 97 T^{2} \))(\( 1 + 10 T + 97 T^{2} \))(\( 1 - 2 T + 97 T^{2} \))(\( 1 - 10 T + 97 T^{2} \))(\( 1 - 4 T + 97 T^{2} \))(\( 1 + 14 T + 97 T^{2} \))(\( 1 - 8 T + 97 T^{2} \))(\( 1 - 3 T + 97 T^{2} \))(\( 1 + 8 T + 97 T^{2} \))(\( 1 + 8 T + 97 T^{2} \))(\( 1 - 2 T + 97 T^{2} \))(\( 1 - 10 T + 97 T^{2} \))(\( 1 + 12 T + 97 T^{2} \))(\( 1 - 10 T + 97 T^{2} \))(\( 1 + 14 T + 97 T^{2} \))(\( 1 - 2 T + 97 T^{2} \))(\( 1 - 10 T + 97 T^{2} \))(\( 1 - 4 T + 97 T^{2} \))(\( 1 + 2 T + 97 T^{2} \))(\( 1 - 13 T + 97 T^{2} \))(\( ( 1 - 10 T + 97 T^{2} )^{2} \))(\( ( 1 - 2 T + 97 T^{2} )^{2} \))(\( ( 1 + 2 T + 97 T^{2} )^{2} \))(\( ( 1 + 10 T + 97 T^{2} )^{2} \))(\( ( 1 + 18 T + 254 T^{2} + 1746 T^{3} + 9409 T^{4} )^{2} \))(\( ( 1 - 18 T + 254 T^{2} - 1746 T^{3} + 9409 T^{4} )^{2} \))
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