Properties

Label 1008.2.a
Level 1008
Weight 2
Character orbit a
Rep. character \(\chi_{1008}(1,\cdot)\)
Character field \(\Q\)
Dimension 15
Newform subspaces 14
Sturm bound 384
Trace bound 11

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Defining parameters

Level: \( N \) \(=\) \( 1008 = 2^{4} \cdot 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1008.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 14 \)
Sturm bound: \(384\)
Trace bound: \(11\)
Distinguishing \(T_p\): \(5\), \(11\), \(13\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(1008))\).

Total New Old
Modular forms 216 15 201
Cusp forms 169 15 154
Eisenstein series 47 0 47

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

\(2\)\(3\)\(7\)FrickeDim.
\(+\)\(+\)\(+\)\(+\)\(2\)
\(+\)\(+\)\(-\)\(-\)\(2\)
\(+\)\(-\)\(+\)\(-\)\(2\)
\(+\)\(-\)\(-\)\(+\)\(2\)
\(-\)\(+\)\(+\)\(-\)\(2\)
\(-\)\(-\)\(+\)\(+\)\(2\)
\(-\)\(-\)\(-\)\(-\)\(3\)
Plus space\(+\)\(6\)
Minus space\(-\)\(9\)

Trace form

\( 15q - 2q^{5} - q^{7} + O(q^{10}) \) \( 15q - 2q^{5} - q^{7} - 8q^{11} - 2q^{13} + 6q^{17} + 8q^{19} + 4q^{23} + 17q^{25} + 2q^{29} + 8q^{31} - 6q^{35} + 2q^{37} - 2q^{41} + 20q^{43} + 15q^{49} + 2q^{53} - 8q^{55} + 24q^{59} - 18q^{61} + 20q^{65} - 20q^{67} + 12q^{71} - 2q^{73} + 4q^{77} + 8q^{79} - 8q^{83} - 12q^{85} + 6q^{89} + 6q^{91} + 8q^{95} - 18q^{97} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(1008))\) into newform subspaces

Label Dim. \(A\) Field CM Traces A-L signs $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\) 2 3 7
1008.2.a.a \(1\) \(8.049\) \(\Q\) None \(0\) \(0\) \(-4\) \(1\) \(-\) \(-\) \(-\) \(q-4q^{5}+q^{7}+2q^{11}-6q^{13}+4q^{17}+\cdots\)
1008.2.a.b \(1\) \(8.049\) \(\Q\) None \(0\) \(0\) \(-2\) \(-1\) \(+\) \(-\) \(+\) \(q-2q^{5}-q^{7}+6q^{13}+2q^{17}-4q^{19}+\cdots\)
1008.2.a.c \(1\) \(8.049\) \(\Q\) None \(0\) \(0\) \(-2\) \(-1\) \(+\) \(+\) \(+\) \(q-2q^{5}-q^{7}+6q^{11}-6q^{13}+2q^{17}+\cdots\)
1008.2.a.d \(1\) \(8.049\) \(\Q\) None \(0\) \(0\) \(-2\) \(1\) \(+\) \(-\) \(-\) \(q-2q^{5}+q^{7}-4q^{11}+2q^{13}+6q^{17}+\cdots\)
1008.2.a.e \(1\) \(8.049\) \(\Q\) None \(0\) \(0\) \(-2\) \(1\) \(+\) \(-\) \(-\) \(q-2q^{5}+q^{7}-2q^{13}-6q^{17}+4q^{19}+\cdots\)
1008.2.a.f \(1\) \(8.049\) \(\Q\) None \(0\) \(0\) \(-2\) \(1\) \(+\) \(+\) \(-\) \(q-2q^{5}+q^{7}+2q^{11}+2q^{13}-6q^{17}+\cdots\)
1008.2.a.g \(1\) \(8.049\) \(\Q\) None \(0\) \(0\) \(0\) \(-1\) \(-\) \(-\) \(+\) \(q-q^{7}-6q^{11}+2q^{13}+4q^{19}-6q^{23}+\cdots\)
1008.2.a.h \(1\) \(8.049\) \(\Q\) None \(0\) \(0\) \(0\) \(-1\) \(-\) \(-\) \(+\) \(q-q^{7}-4q^{13}-6q^{17}-2q^{19}-5q^{25}+\cdots\)
1008.2.a.i \(1\) \(8.049\) \(\Q\) None \(0\) \(0\) \(2\) \(-1\) \(+\) \(+\) \(+\) \(q+2q^{5}-q^{7}-6q^{11}-6q^{13}-2q^{17}+\cdots\)
1008.2.a.j \(1\) \(8.049\) \(\Q\) None \(0\) \(0\) \(2\) \(1\) \(-\) \(-\) \(-\) \(q+2q^{5}+q^{7}-4q^{11}+6q^{13}-2q^{17}+\cdots\)
1008.2.a.k \(1\) \(8.049\) \(\Q\) None \(0\) \(0\) \(2\) \(1\) \(+\) \(+\) \(-\) \(q+2q^{5}+q^{7}-2q^{11}+2q^{13}+6q^{17}+\cdots\)
1008.2.a.l \(1\) \(8.049\) \(\Q\) None \(0\) \(0\) \(2\) \(1\) \(-\) \(-\) \(-\) \(q+2q^{5}+q^{7}+4q^{11}-2q^{13}+6q^{17}+\cdots\)
1008.2.a.m \(1\) \(8.049\) \(\Q\) None \(0\) \(0\) \(4\) \(-1\) \(+\) \(-\) \(+\) \(q+4q^{5}-q^{7}+2q^{17}+2q^{19}+8q^{23}+\cdots\)
1008.2.a.n \(2\) \(8.049\) \(\Q(\sqrt{3}) \) None \(0\) \(0\) \(0\) \(-2\) \(-\) \(+\) \(+\) \(q+\beta q^{5}-q^{7}+\beta q^{11}+2q^{13}-\beta q^{17}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(1008))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(1008)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(14))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(21))\)\(^{\oplus 10}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(24))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(36))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(42))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(48))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(56))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(63))\)\(^{\oplus 5}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(72))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(84))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(112))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(126))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(144))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(168))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(252))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(336))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(504))\)\(^{\oplus 2}\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ 1
$5$ (\( 1 + 4 T + 5 T^{2} \))(\( 1 + 2 T + 5 T^{2} \))(\( 1 + 2 T + 5 T^{2} \))(\( 1 + 2 T + 5 T^{2} \))(\( 1 + 2 T + 5 T^{2} \))(\( 1 + 2 T + 5 T^{2} \))(\( 1 + 5 T^{2} \))(\( 1 + 5 T^{2} \))(\( 1 - 2 T + 5 T^{2} \))(\( 1 - 2 T + 5 T^{2} \))(\( 1 - 2 T + 5 T^{2} \))(\( 1 - 2 T + 5 T^{2} \))(\( 1 - 4 T + 5 T^{2} \))(\( 1 - 2 T^{2} + 25 T^{4} \))
$7$ (\( 1 - T \))(\( 1 + T \))(\( 1 + T \))(\( 1 - T \))(\( 1 - T \))(\( 1 - T \))(\( 1 + T \))(\( 1 + T \))(\( 1 + T \))(\( 1 - T \))(\( 1 - T \))(\( 1 - T \))(\( 1 + T \))(\( ( 1 + T )^{2} \))
$11$ (\( 1 - 2 T + 11 T^{2} \))(\( 1 + 11 T^{2} \))(\( 1 - 6 T + 11 T^{2} \))(\( 1 + 4 T + 11 T^{2} \))(\( 1 + 11 T^{2} \))(\( 1 - 2 T + 11 T^{2} \))(\( 1 + 6 T + 11 T^{2} \))(\( 1 + 11 T^{2} \))(\( 1 + 6 T + 11 T^{2} \))(\( 1 + 4 T + 11 T^{2} \))(\( 1 + 2 T + 11 T^{2} \))(\( 1 - 4 T + 11 T^{2} \))(\( 1 + 11 T^{2} \))(\( 1 + 10 T^{2} + 121 T^{4} \))
$13$ (\( 1 + 6 T + 13 T^{2} \))(\( 1 - 6 T + 13 T^{2} \))(\( 1 + 6 T + 13 T^{2} \))(\( 1 - 2 T + 13 T^{2} \))(\( 1 + 2 T + 13 T^{2} \))(\( 1 - 2 T + 13 T^{2} \))(\( 1 - 2 T + 13 T^{2} \))(\( 1 + 4 T + 13 T^{2} \))(\( 1 + 6 T + 13 T^{2} \))(\( 1 - 6 T + 13 T^{2} \))(\( 1 - 2 T + 13 T^{2} \))(\( 1 + 2 T + 13 T^{2} \))(\( 1 + 13 T^{2} \))(\( ( 1 - 2 T + 13 T^{2} )^{2} \))
$17$ (\( 1 - 4 T + 17 T^{2} \))(\( 1 - 2 T + 17 T^{2} \))(\( 1 - 2 T + 17 T^{2} \))(\( 1 - 6 T + 17 T^{2} \))(\( 1 + 6 T + 17 T^{2} \))(\( 1 + 6 T + 17 T^{2} \))(\( 1 + 17 T^{2} \))(\( 1 + 6 T + 17 T^{2} \))(\( 1 + 2 T + 17 T^{2} \))(\( 1 + 2 T + 17 T^{2} \))(\( 1 - 6 T + 17 T^{2} \))(\( 1 - 6 T + 17 T^{2} \))(\( 1 - 2 T + 17 T^{2} \))(\( 1 + 22 T^{2} + 289 T^{4} \))
$19$ (\( 1 - 4 T + 19 T^{2} \))(\( 1 + 4 T + 19 T^{2} \))(\( 1 + 4 T + 19 T^{2} \))(\( 1 + 8 T + 19 T^{2} \))(\( 1 - 4 T + 19 T^{2} \))(\( 1 - 4 T + 19 T^{2} \))(\( 1 - 4 T + 19 T^{2} \))(\( 1 + 2 T + 19 T^{2} \))(\( 1 + 4 T + 19 T^{2} \))(\( 1 - 4 T + 19 T^{2} \))(\( 1 - 4 T + 19 T^{2} \))(\( 1 + 4 T + 19 T^{2} \))(\( 1 - 2 T + 19 T^{2} \))(\( ( 1 - 4 T + 19 T^{2} )^{2} \))
$23$ (\( 1 - 2 T + 23 T^{2} \))(\( 1 + 4 T + 23 T^{2} \))(\( 1 - 2 T + 23 T^{2} \))(\( 1 + 23 T^{2} \))(\( 1 + 4 T + 23 T^{2} \))(\( 1 - 6 T + 23 T^{2} \))(\( 1 + 6 T + 23 T^{2} \))(\( 1 + 23 T^{2} \))(\( 1 + 2 T + 23 T^{2} \))(\( 1 - 8 T + 23 T^{2} \))(\( 1 + 6 T + 23 T^{2} \))(\( 1 + 23 T^{2} \))(\( 1 - 8 T + 23 T^{2} \))(\( 1 + 34 T^{2} + 529 T^{4} \))
$29$ (\( 1 - 2 T + 29 T^{2} \))(\( 1 - 10 T + 29 T^{2} \))(\( 1 + 8 T + 29 T^{2} \))(\( 1 + 6 T + 29 T^{2} \))(\( 1 + 6 T + 29 T^{2} \))(\( 1 + 29 T^{2} \))(\( 1 + 6 T + 29 T^{2} \))(\( 1 - 6 T + 29 T^{2} \))(\( 1 - 8 T + 29 T^{2} \))(\( 1 - 2 T + 29 T^{2} \))(\( 1 + 29 T^{2} \))(\( 1 - 2 T + 29 T^{2} \))(\( 1 + 2 T + 29 T^{2} \))(\( ( 1 + 29 T^{2} )^{2} \))
$31$ (\( 1 + 31 T^{2} \))(\( 1 - 8 T + 31 T^{2} \))(\( 1 + 4 T + 31 T^{2} \))(\( 1 + 8 T + 31 T^{2} \))(\( 1 - 8 T + 31 T^{2} \))(\( 1 - 4 T + 31 T^{2} \))(\( 1 + 8 T + 31 T^{2} \))(\( 1 - 4 T + 31 T^{2} \))(\( 1 + 4 T + 31 T^{2} \))(\( 1 + 31 T^{2} \))(\( 1 - 4 T + 31 T^{2} \))(\( 1 + 31 T^{2} \))(\( 1 + 4 T + 31 T^{2} \))(\( ( 1 - 4 T + 31 T^{2} )^{2} \))
$37$ (\( 1 - 2 T + 37 T^{2} \))(\( 1 - 6 T + 37 T^{2} \))(\( 1 + 6 T + 37 T^{2} \))(\( 1 + 2 T + 37 T^{2} \))(\( 1 + 10 T + 37 T^{2} \))(\( 1 - 10 T + 37 T^{2} \))(\( 1 - 2 T + 37 T^{2} \))(\( 1 - 2 T + 37 T^{2} \))(\( 1 + 6 T + 37 T^{2} \))(\( 1 + 10 T + 37 T^{2} \))(\( 1 - 10 T + 37 T^{2} \))(\( 1 - 6 T + 37 T^{2} \))(\( 1 + 6 T + 37 T^{2} \))(\( ( 1 - 2 T + 37 T^{2} )^{2} \))
$41$ (\( 1 + 41 T^{2} \))(\( 1 - 2 T + 41 T^{2} \))(\( 1 + 10 T + 41 T^{2} \))(\( 1 + 2 T + 41 T^{2} \))(\( 1 - 10 T + 41 T^{2} \))(\( 1 + 2 T + 41 T^{2} \))(\( 1 + 12 T + 41 T^{2} \))(\( 1 + 6 T + 41 T^{2} \))(\( 1 - 10 T + 41 T^{2} \))(\( 1 - 6 T + 41 T^{2} \))(\( 1 - 2 T + 41 T^{2} \))(\( 1 + 2 T + 41 T^{2} \))(\( 1 - 2 T + 41 T^{2} \))(\( 1 - 26 T^{2} + 1681 T^{4} \))
$43$ (\( 1 - 4 T + 43 T^{2} \))(\( 1 - 4 T + 43 T^{2} \))(\( 1 - 4 T + 43 T^{2} \))(\( 1 - 4 T + 43 T^{2} \))(\( 1 + 12 T + 43 T^{2} \))(\( 1 - 4 T + 43 T^{2} \))(\( 1 - 4 T + 43 T^{2} \))(\( 1 + 8 T + 43 T^{2} \))(\( 1 - 4 T + 43 T^{2} \))(\( 1 - 4 T + 43 T^{2} \))(\( 1 - 4 T + 43 T^{2} \))(\( 1 - 4 T + 43 T^{2} \))(\( 1 + 8 T + 43 T^{2} \))(\( ( 1 - 4 T + 43 T^{2} )^{2} \))
$47$ (\( 1 - 12 T + 47 T^{2} \))(\( 1 - 8 T + 47 T^{2} \))(\( 1 + 4 T + 47 T^{2} \))(\( 1 + 8 T + 47 T^{2} \))(\( 1 + 8 T + 47 T^{2} \))(\( 1 - 4 T + 47 T^{2} \))(\( 1 - 12 T + 47 T^{2} \))(\( 1 + 12 T + 47 T^{2} \))(\( 1 - 4 T + 47 T^{2} \))(\( 1 + 47 T^{2} \))(\( 1 + 4 T + 47 T^{2} \))(\( 1 + 47 T^{2} \))(\( 1 + 4 T + 47 T^{2} \))(\( 1 + 46 T^{2} + 2209 T^{4} \))
$53$ (\( 1 - 6 T + 53 T^{2} \))(\( 1 - 10 T + 53 T^{2} \))(\( 1 - 4 T + 53 T^{2} \))(\( 1 + 6 T + 53 T^{2} \))(\( 1 + 6 T + 53 T^{2} \))(\( 1 - 12 T + 53 T^{2} \))(\( 1 - 6 T + 53 T^{2} \))(\( 1 + 6 T + 53 T^{2} \))(\( 1 + 4 T + 53 T^{2} \))(\( 1 + 6 T + 53 T^{2} \))(\( 1 + 12 T + 53 T^{2} \))(\( 1 + 6 T + 53 T^{2} \))(\( 1 - 10 T + 53 T^{2} \))(\( 1 + 58 T^{2} + 2809 T^{4} \))
$59$ (\( 1 + 8 T + 59 T^{2} \))(\( 1 - 12 T + 59 T^{2} \))(\( 1 + 12 T + 59 T^{2} \))(\( 1 + 59 T^{2} \))(\( 1 - 4 T + 59 T^{2} \))(\( 1 - 12 T + 59 T^{2} \))(\( 1 + 59 T^{2} \))(\( 1 + 6 T + 59 T^{2} \))(\( 1 - 12 T + 59 T^{2} \))(\( 1 - 4 T + 59 T^{2} \))(\( 1 + 12 T + 59 T^{2} \))(\( 1 - 12 T + 59 T^{2} \))(\( 1 - 6 T + 59 T^{2} \))(\( 1 + 70 T^{2} + 3481 T^{4} \))
$61$ (\( 1 - 6 T + 61 T^{2} \))(\( 1 + 2 T + 61 T^{2} \))(\( 1 + 2 T + 61 T^{2} \))(\( 1 + 6 T + 61 T^{2} \))(\( 1 + 10 T + 61 T^{2} \))(\( 1 - 6 T + 61 T^{2} \))(\( 1 + 10 T + 61 T^{2} \))(\( 1 - 8 T + 61 T^{2} \))(\( 1 + 2 T + 61 T^{2} \))(\( 1 - 6 T + 61 T^{2} \))(\( 1 - 6 T + 61 T^{2} \))(\( 1 + 2 T + 61 T^{2} \))(\( 1 - 4 T + 61 T^{2} \))(\( ( 1 + 10 T + 61 T^{2} )^{2} \))
$67$ (\( 1 - 8 T + 67 T^{2} \))(\( 1 + 12 T + 67 T^{2} \))(\( 1 + 12 T + 67 T^{2} \))(\( 1 - 4 T + 67 T^{2} \))(\( 1 + 12 T + 67 T^{2} \))(\( 1 - 4 T + 67 T^{2} \))(\( 1 + 8 T + 67 T^{2} \))(\( 1 - 4 T + 67 T^{2} \))(\( 1 + 12 T + 67 T^{2} \))(\( 1 + 4 T + 67 T^{2} \))(\( 1 - 4 T + 67 T^{2} \))(\( 1 + 4 T + 67 T^{2} \))(\( 1 - 12 T + 67 T^{2} \))(\( ( 1 - 4 T + 67 T^{2} )^{2} \))
$71$ (\( 1 - 14 T + 71 T^{2} \))(\( 1 + 12 T + 71 T^{2} \))(\( 1 - 6 T + 71 T^{2} \))(\( 1 + 8 T + 71 T^{2} \))(\( 1 - 4 T + 71 T^{2} \))(\( 1 + 14 T + 71 T^{2} \))(\( 1 - 6 T + 71 T^{2} \))(\( 1 + 71 T^{2} \))(\( 1 + 6 T + 71 T^{2} \))(\( 1 - 8 T + 71 T^{2} \))(\( 1 - 14 T + 71 T^{2} \))(\( 1 + 71 T^{2} \))(\( 1 + 71 T^{2} \))(\( 1 + 34 T^{2} + 5041 T^{4} \))
$73$ (\( 1 + 2 T + 73 T^{2} \))(\( 1 + 14 T + 73 T^{2} \))(\( 1 + 2 T + 73 T^{2} \))(\( 1 - 10 T + 73 T^{2} \))(\( 1 - 2 T + 73 T^{2} \))(\( 1 + 2 T + 73 T^{2} \))(\( 1 + 10 T + 73 T^{2} \))(\( 1 - 2 T + 73 T^{2} \))(\( 1 + 2 T + 73 T^{2} \))(\( 1 - 10 T + 73 T^{2} \))(\( 1 + 2 T + 73 T^{2} \))(\( 1 + 6 T + 73 T^{2} \))(\( 1 + 14 T + 73 T^{2} \))(\( ( 1 - 14 T + 73 T^{2} )^{2} \))
$79$ (\( 1 + 12 T + 79 T^{2} \))(\( 1 - 8 T + 79 T^{2} \))(\( 1 - 8 T + 79 T^{2} \))(\( 1 + 16 T + 79 T^{2} \))(\( 1 + 8 T + 79 T^{2} \))(\( 1 - 8 T + 79 T^{2} \))(\( 1 - 4 T + 79 T^{2} \))(\( 1 + 8 T + 79 T^{2} \))(\( 1 - 8 T + 79 T^{2} \))(\( 1 + 79 T^{2} \))(\( 1 - 8 T + 79 T^{2} \))(\( 1 - 16 T + 79 T^{2} \))(\( 1 - 8 T + 79 T^{2} \))(\( ( 1 + 8 T + 79 T^{2} )^{2} \))
$83$ (\( 1 + 4 T + 83 T^{2} \))(\( 1 - 12 T + 83 T^{2} \))(\( 1 + 83 T^{2} \))(\( 1 - 8 T + 83 T^{2} \))(\( 1 - 4 T + 83 T^{2} \))(\( 1 + 16 T + 83 T^{2} \))(\( 1 + 12 T + 83 T^{2} \))(\( 1 + 6 T + 83 T^{2} \))(\( 1 + 83 T^{2} \))(\( 1 + 4 T + 83 T^{2} \))(\( 1 - 16 T + 83 T^{2} \))(\( 1 + 12 T + 83 T^{2} \))(\( 1 - 6 T + 83 T^{2} \))(\( ( 1 + 83 T^{2} )^{2} \))
$89$ (\( 1 + 89 T^{2} \))(\( 1 - 2 T + 89 T^{2} \))(\( 1 - 14 T + 89 T^{2} \))(\( 1 - 6 T + 89 T^{2} \))(\( 1 + 6 T + 89 T^{2} \))(\( 1 - 6 T + 89 T^{2} \))(\( 1 + 12 T + 89 T^{2} \))(\( 1 - 6 T + 89 T^{2} \))(\( 1 + 14 T + 89 T^{2} \))(\( 1 - 6 T + 89 T^{2} \))(\( 1 + 6 T + 89 T^{2} \))(\( 1 - 14 T + 89 T^{2} \))(\( 1 + 10 T + 89 T^{2} \))(\( 1 + 166 T^{2} + 7921 T^{4} \))
$97$ (\( 1 + 2 T + 97 T^{2} \))(\( 1 - 10 T + 97 T^{2} \))(\( 1 + 2 T + 97 T^{2} \))(\( 1 + 6 T + 97 T^{2} \))(\( 1 - 10 T + 97 T^{2} \))(\( 1 + 18 T + 97 T^{2} \))(\( 1 + 10 T + 97 T^{2} \))(\( 1 + 10 T + 97 T^{2} \))(\( 1 + 2 T + 97 T^{2} \))(\( 1 + 14 T + 97 T^{2} \))(\( 1 + 18 T + 97 T^{2} \))(\( 1 - 18 T + 97 T^{2} \))(\( 1 + 2 T + 97 T^{2} \))(\( ( 1 - 14 T + 97 T^{2} )^{2} \))
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