Properties

Label 784.2.a
Level 784
Weight 2
Character orbit a
Rep. character \(\chi_{784}(1,\cdot)\)
Character field \(\Q\)
Dimension 18
Newform subspaces 14
Sturm bound 224
Trace bound 11

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Defining parameters

Level: \( N \) = \( 784 = 2^{4} \cdot 7^{2} \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 784.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 14 \)
Sturm bound: \(224\)
Trace bound: \(11\)
Distinguishing \(T_p\): \(3\), \(5\), \(11\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(784))\).

Total New Old
Modular forms 136 23 113
Cusp forms 89 18 71
Eisenstein series 47 5 42

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

\(2\)\(7\)FrickeDim.
\(+\)\(+\)\(+\)\(4\)
\(+\)\(-\)\(-\)\(6\)
\(-\)\(+\)\(-\)\(5\)
\(-\)\(-\)\(+\)\(3\)
Plus space\(+\)\(7\)
Minus space\(-\)\(11\)

Trace form

\( 18q + 2q^{5} + 16q^{9} + O(q^{10}) \) \( 18q + 2q^{5} + 16q^{9} - 6q^{11} + 2q^{13} - 2q^{15} + 2q^{17} + 8q^{19} + 6q^{23} + 4q^{25} - 4q^{29} + 8q^{31} + 10q^{37} + 20q^{39} - 6q^{41} + 16q^{43} + 10q^{45} - 24q^{47} + 14q^{51} + 2q^{53} - 8q^{55} - 6q^{57} - 6q^{61} - 12q^{65} - 30q^{67} - 16q^{69} + 8q^{71} + 2q^{73} + 32q^{75} + 6q^{79} + 2q^{81} + 8q^{83} - 10q^{85} + 16q^{87} + 2q^{89} - 42q^{93} + 66q^{95} + 18q^{97} + 20q^{99} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(784))\) into newform subspaces

Label Dim. \(A\) Field CM Traces A-L signs $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\) 2 7
784.2.a.a \(1\) \(6.260\) \(\Q\) None \(0\) \(-3\) \(-1\) \(0\) \(+\) \(+\) \(q-3q^{3}-q^{5}+6q^{9}+q^{11}+2q^{13}+\cdots\)
784.2.a.b \(1\) \(6.260\) \(\Q\) None \(0\) \(-2\) \(0\) \(0\) \(-\) \(-\) \(q-2q^{3}+q^{9}+4q^{13}-6q^{17}+2q^{19}+\cdots\)
784.2.a.c \(1\) \(6.260\) \(\Q\) None \(0\) \(-1\) \(1\) \(0\) \(+\) \(-\) \(q-q^{3}+q^{5}-2q^{9}-3q^{11}+6q^{13}+\cdots\)
784.2.a.d \(1\) \(6.260\) \(\Q\) None \(0\) \(-1\) \(3\) \(0\) \(-\) \(+\) \(q-q^{3}+3q^{5}-2q^{9}+3q^{11}+2q^{13}+\cdots\)
784.2.a.e \(1\) \(6.260\) \(\Q\) None \(0\) \(0\) \(-2\) \(0\) \(+\) \(-\) \(q-2q^{5}-3q^{9}+4q^{11}-2q^{13}+6q^{17}+\cdots\)
784.2.a.f \(1\) \(6.260\) \(\Q\) \(\Q(\sqrt{-7}) \) \(0\) \(0\) \(0\) \(0\) \(-\) \(-\) \(q-3q^{9}-4q^{11}-8q^{23}-5q^{25}+2q^{29}+\cdots\)
784.2.a.g \(1\) \(6.260\) \(\Q\) None \(0\) \(1\) \(-3\) \(0\) \(-\) \(-\) \(q+q^{3}-3q^{5}-2q^{9}+3q^{11}-2q^{13}+\cdots\)
784.2.a.h \(1\) \(6.260\) \(\Q\) None \(0\) \(1\) \(-1\) \(0\) \(+\) \(+\) \(q+q^{3}-q^{5}-2q^{9}-3q^{11}-6q^{13}+\cdots\)
784.2.a.i \(1\) \(6.260\) \(\Q\) None \(0\) \(2\) \(4\) \(0\) \(+\) \(-\) \(q+2q^{3}+4q^{5}+q^{9}+8q^{15}+2q^{17}+\cdots\)
784.2.a.j \(1\) \(6.260\) \(\Q\) None \(0\) \(3\) \(1\) \(0\) \(+\) \(-\) \(q+3q^{3}+q^{5}+6q^{9}+q^{11}-2q^{13}+\cdots\)
784.2.a.k \(2\) \(6.260\) \(\Q(\sqrt{2}) \) None \(0\) \(0\) \(0\) \(0\) \(+\) \(+\) \(q+\beta q^{3}-2\beta q^{5}-q^{9}-6q^{11}+4\beta q^{13}+\cdots\)
784.2.a.l \(2\) \(6.260\) \(\Q(\sqrt{2}) \) None \(0\) \(0\) \(0\) \(0\) \(-\) \(+\) \(q+\beta q^{3}+2\beta q^{5}-q^{9}+2q^{11}+4q^{15}+\cdots\)
784.2.a.m \(2\) \(6.260\) \(\Q(\sqrt{2}) \) None \(0\) \(0\) \(0\) \(0\) \(-\) \(+\) \(q+2\beta q^{3}+\beta q^{5}+5q^{9}-4q^{11}+3\beta q^{13}+\cdots\)
784.2.a.n \(2\) \(6.260\) \(\Q(\sqrt{2}) \) None \(0\) \(0\) \(0\) \(0\) \(+\) \(-\) \(q+\beta q^{3}-\beta q^{5}+5q^{9}+4q^{11}+\beta q^{13}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(784))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(784)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(14))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(49))\)\(^{\oplus 5}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(56))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(98))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(112))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(196))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(392))\)\(^{\oplus 2}\)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ 1
$3$ (\( 1 + 3 T + 3 T^{2} \))(\( 1 + 2 T + 3 T^{2} \))(\( 1 + T + 3 T^{2} \))(\( 1 + T + 3 T^{2} \))(\( 1 + 3 T^{2} \))(\( 1 + 3 T^{2} \))(\( 1 - T + 3 T^{2} \))(\( 1 - T + 3 T^{2} \))(\( 1 - 2 T + 3 T^{2} \))(\( 1 - 3 T + 3 T^{2} \))(\( 1 + 4 T^{2} + 9 T^{4} \))(\( 1 + 4 T^{2} + 9 T^{4} \))(\( 1 - 2 T^{2} + 9 T^{4} \))(\( 1 - 2 T^{2} + 9 T^{4} \))
$5$ (\( 1 + T + 5 T^{2} \))(\( 1 + 5 T^{2} \))(\( 1 - T + 5 T^{2} \))(\( 1 - 3 T + 5 T^{2} \))(\( 1 + 2 T + 5 T^{2} \))(\( 1 + 5 T^{2} \))(\( 1 + 3 T + 5 T^{2} \))(\( 1 + T + 5 T^{2} \))(\( 1 - 4 T + 5 T^{2} \))(\( 1 - T + 5 T^{2} \))(\( 1 + 2 T^{2} + 25 T^{4} \))(\( 1 + 2 T^{2} + 25 T^{4} \))(\( 1 + 8 T^{2} + 25 T^{4} \))(\( 1 + 2 T^{2} + 25 T^{4} \))
$7$ 1
$11$ (\( 1 - T + 11 T^{2} \))(\( 1 + 11 T^{2} \))(\( 1 + 3 T + 11 T^{2} \))(\( 1 - 3 T + 11 T^{2} \))(\( 1 - 4 T + 11 T^{2} \))(\( 1 + 4 T + 11 T^{2} \))(\( 1 - 3 T + 11 T^{2} \))(\( 1 + 3 T + 11 T^{2} \))(\( 1 + 11 T^{2} \))(\( 1 - T + 11 T^{2} \))(\( ( 1 + 6 T + 11 T^{2} )^{2} \))(\( ( 1 - 2 T + 11 T^{2} )^{2} \))(\( ( 1 + 4 T + 11 T^{2} )^{2} \))(\( ( 1 - 4 T + 11 T^{2} )^{2} \))
$13$ (\( 1 - 2 T + 13 T^{2} \))(\( 1 - 4 T + 13 T^{2} \))(\( 1 - 6 T + 13 T^{2} \))(\( 1 - 2 T + 13 T^{2} \))(\( 1 + 2 T + 13 T^{2} \))(\( 1 + 13 T^{2} \))(\( 1 + 2 T + 13 T^{2} \))(\( 1 + 6 T + 13 T^{2} \))(\( 1 + 13 T^{2} \))(\( 1 + 2 T + 13 T^{2} \))(\( 1 - 6 T^{2} + 169 T^{4} \))(\( ( 1 + 13 T^{2} )^{2} \))(\( 1 + 8 T^{2} + 169 T^{4} \))(\( 1 + 18 T^{2} + 169 T^{4} \))
$17$ (\( 1 - 3 T + 17 T^{2} \))(\( 1 + 6 T + 17 T^{2} \))(\( 1 - 5 T + 17 T^{2} \))(\( 1 - 3 T + 17 T^{2} \))(\( 1 - 6 T + 17 T^{2} \))(\( 1 + 17 T^{2} \))(\( 1 + 3 T + 17 T^{2} \))(\( 1 + 5 T + 17 T^{2} \))(\( 1 - 2 T + 17 T^{2} \))(\( 1 + 3 T + 17 T^{2} \))(\( 1 + 32 T^{2} + 289 T^{4} \))(\( 1 + 32 T^{2} + 289 T^{4} \))(\( 1 + 32 T^{2} + 289 T^{4} \))(\( 1 + 2 T^{2} + 289 T^{4} \))
$19$ (\( 1 + 5 T + 19 T^{2} \))(\( 1 - 2 T + 19 T^{2} \))(\( 1 - T + 19 T^{2} \))(\( 1 - T + 19 T^{2} \))(\( 1 - 8 T + 19 T^{2} \))(\( 1 + 19 T^{2} \))(\( 1 + T + 19 T^{2} \))(\( 1 + T + 19 T^{2} \))(\( 1 + 2 T + 19 T^{2} \))(\( 1 - 5 T + 19 T^{2} \))(\( 1 + 20 T^{2} + 361 T^{4} \))(\( 1 - 12 T^{2} + 361 T^{4} \))(\( 1 + 30 T^{2} + 361 T^{4} \))(\( 1 + 30 T^{2} + 361 T^{4} \))
$23$ (\( 1 - 3 T + 23 T^{2} \))(\( 1 + 23 T^{2} \))(\( 1 - 7 T + 23 T^{2} \))(\( 1 + 3 T + 23 T^{2} \))(\( 1 + 23 T^{2} \))(\( 1 + 8 T + 23 T^{2} \))(\( 1 + 3 T + 23 T^{2} \))(\( 1 - 7 T + 23 T^{2} \))(\( 1 + 8 T + 23 T^{2} \))(\( 1 - 3 T + 23 T^{2} \))(\( ( 1 + 4 T + 23 T^{2} )^{2} \))(\( ( 1 - 4 T + 23 T^{2} )^{2} \))(\( ( 1 - 4 T + 23 T^{2} )^{2} \))(\( ( 1 + 23 T^{2} )^{2} \))
$29$ (\( 1 + 6 T + 29 T^{2} \))(\( 1 + 6 T + 29 T^{2} \))(\( 1 - 2 T + 29 T^{2} \))(\( 1 + 6 T + 29 T^{2} \))(\( 1 - 6 T + 29 T^{2} \))(\( 1 - 2 T + 29 T^{2} \))(\( 1 + 6 T + 29 T^{2} \))(\( 1 - 2 T + 29 T^{2} \))(\( 1 - 2 T + 29 T^{2} \))(\( 1 + 6 T + 29 T^{2} \))(\( ( 1 + 6 T + 29 T^{2} )^{2} \))(\( ( 1 - 2 T + 29 T^{2} )^{2} \))(\( ( 1 - 8 T + 29 T^{2} )^{2} \))(\( ( 1 - 2 T + 29 T^{2} )^{2} \))
$31$ (\( 1 - T + 31 T^{2} \))(\( 1 + 4 T + 31 T^{2} \))(\( 1 + 5 T + 31 T^{2} \))(\( 1 - 7 T + 31 T^{2} \))(\( 1 - 8 T + 31 T^{2} \))(\( 1 + 31 T^{2} \))(\( 1 + 7 T + 31 T^{2} \))(\( 1 - 5 T + 31 T^{2} \))(\( 1 - 4 T + 31 T^{2} \))(\( 1 + T + 31 T^{2} \))(\( 1 + 54 T^{2} + 961 T^{4} \))(\( 1 - 10 T^{2} + 961 T^{4} \))(\( ( 1 + 31 T^{2} )^{2} \))(\( 1 + 30 T^{2} + 961 T^{4} \))
$37$ (\( 1 + 5 T + 37 T^{2} \))(\( 1 - 2 T + 37 T^{2} \))(\( 1 - 3 T + 37 T^{2} \))(\( 1 + T + 37 T^{2} \))(\( 1 + 2 T + 37 T^{2} \))(\( 1 + 6 T + 37 T^{2} \))(\( 1 + T + 37 T^{2} \))(\( 1 - 3 T + 37 T^{2} \))(\( 1 + 6 T + 37 T^{2} \))(\( 1 + 5 T + 37 T^{2} \))(\( ( 1 - 2 T + 37 T^{2} )^{2} \))(\( ( 1 - 10 T + 37 T^{2} )^{2} \))(\( ( 1 + 8 T + 37 T^{2} )^{2} \))(\( ( 1 - 10 T + 37 T^{2} )^{2} \))
$41$ (\( 1 + 10 T + 41 T^{2} \))(\( 1 + 6 T + 41 T^{2} \))(\( 1 - 2 T + 41 T^{2} \))(\( 1 - 6 T + 41 T^{2} \))(\( 1 + 2 T + 41 T^{2} \))(\( 1 + 41 T^{2} \))(\( 1 + 6 T + 41 T^{2} \))(\( 1 + 2 T + 41 T^{2} \))(\( 1 - 2 T + 41 T^{2} \))(\( 1 - 10 T + 41 T^{2} \))(\( 1 + 80 T^{2} + 1681 T^{4} \))(\( 1 - 16 T^{2} + 1681 T^{4} \))(\( 1 + 32 T^{2} + 1681 T^{4} \))(\( 1 + 50 T^{2} + 1681 T^{4} \))
$43$ (\( 1 - 4 T + 43 T^{2} \))(\( 1 + 8 T + 43 T^{2} \))(\( 1 - 4 T + 43 T^{2} \))(\( 1 - 4 T + 43 T^{2} \))(\( 1 - 4 T + 43 T^{2} \))(\( 1 - 12 T + 43 T^{2} \))(\( 1 - 4 T + 43 T^{2} \))(\( 1 - 4 T + 43 T^{2} \))(\( 1 + 8 T + 43 T^{2} \))(\( 1 - 4 T + 43 T^{2} \))(\( ( 1 + 10 T + 43 T^{2} )^{2} \))(\( ( 1 + 2 T + 43 T^{2} )^{2} \))(\( ( 1 - 4 T + 43 T^{2} )^{2} \))(\( ( 1 - 4 T + 43 T^{2} )^{2} \))
$47$ (\( 1 + T + 47 T^{2} \))(\( 1 + 12 T + 47 T^{2} \))(\( 1 - 5 T + 47 T^{2} \))(\( 1 - 9 T + 47 T^{2} \))(\( 1 + 8 T + 47 T^{2} \))(\( 1 + 47 T^{2} \))(\( 1 + 9 T + 47 T^{2} \))(\( 1 + 5 T + 47 T^{2} \))(\( 1 + 4 T + 47 T^{2} \))(\( 1 - T + 47 T^{2} \))(\( 1 + 86 T^{2} + 2209 T^{4} \))(\( 1 + 86 T^{2} + 2209 T^{4} \))(\( 1 + 62 T^{2} + 2209 T^{4} \))(\( 1 + 62 T^{2} + 2209 T^{4} \))
$53$ (\( 1 + 9 T + 53 T^{2} \))(\( 1 - 6 T + 53 T^{2} \))(\( 1 + T + 53 T^{2} \))(\( 1 - 3 T + 53 T^{2} \))(\( 1 - 6 T + 53 T^{2} \))(\( 1 + 10 T + 53 T^{2} \))(\( 1 - 3 T + 53 T^{2} \))(\( 1 + T + 53 T^{2} \))(\( 1 + 10 T + 53 T^{2} \))(\( 1 + 9 T + 53 T^{2} \))(\( ( 1 + 2 T + 53 T^{2} )^{2} \))(\( ( 1 + 2 T + 53 T^{2} )^{2} \))(\( ( 1 - 10 T + 53 T^{2} )^{2} \))(\( ( 1 - 6 T + 53 T^{2} )^{2} \))
$59$ (\( 1 + 3 T + 59 T^{2} \))(\( 1 + 6 T + 59 T^{2} \))(\( 1 - 15 T + 59 T^{2} \))(\( 1 + 9 T + 59 T^{2} \))(\( 1 + 59 T^{2} \))(\( 1 + 59 T^{2} \))(\( 1 - 9 T + 59 T^{2} \))(\( 1 + 15 T + 59 T^{2} \))(\( 1 - 6 T + 59 T^{2} \))(\( 1 - 3 T + 59 T^{2} \))(\( 1 + 116 T^{2} + 3481 T^{4} \))(\( 1 + 116 T^{2} + 3481 T^{4} \))(\( 1 - 82 T^{2} + 3481 T^{4} \))(\( 1 + 110 T^{2} + 3481 T^{4} \))
$61$ (\( 1 - 3 T + 61 T^{2} \))(\( 1 + 8 T + 61 T^{2} \))(\( 1 - 5 T + 61 T^{2} \))(\( 1 + T + 61 T^{2} \))(\( 1 - 6 T + 61 T^{2} \))(\( 1 + 61 T^{2} \))(\( 1 - T + 61 T^{2} \))(\( 1 + 5 T + 61 T^{2} \))(\( 1 + 4 T + 61 T^{2} \))(\( 1 + 3 T + 61 T^{2} \))(\( 1 + 50 T^{2} + 3721 T^{4} \))(\( 1 + 114 T^{2} + 3721 T^{4} \))(\( 1 + 72 T^{2} + 3721 T^{4} \))(\( 1 - 78 T^{2} + 3721 T^{4} \))
$67$ (\( 1 + 11 T + 67 T^{2} \))(\( 1 - 4 T + 67 T^{2} \))(\( 1 - 9 T + 67 T^{2} \))(\( 1 - 7 T + 67 T^{2} \))(\( 1 - 4 T + 67 T^{2} \))(\( 1 + 4 T + 67 T^{2} \))(\( 1 - 7 T + 67 T^{2} \))(\( 1 - 9 T + 67 T^{2} \))(\( 1 - 12 T + 67 T^{2} \))(\( 1 + 11 T + 67 T^{2} \))(\( ( 1 + 4 T + 67 T^{2} )^{2} \))(\( ( 1 + 12 T + 67 T^{2} )^{2} \))(\( ( 1 + 67 T^{2} )^{2} \))(\( ( 1 + 12 T + 67 T^{2} )^{2} \))
$71$ (\( 1 + 16 T + 71 T^{2} \))(\( 1 + 71 T^{2} \))(\( 1 + 71 T^{2} \))(\( 1 + 71 T^{2} \))(\( 1 - 8 T + 71 T^{2} \))(\( 1 + 16 T + 71 T^{2} \))(\( 1 + 71 T^{2} \))(\( 1 + 71 T^{2} \))(\( 1 + 71 T^{2} \))(\( 1 + 16 T + 71 T^{2} \))(\( ( 1 - 12 T + 71 T^{2} )^{2} \))(\( ( 1 - 12 T + 71 T^{2} )^{2} \))(\( ( 1 + 71 T^{2} )^{2} \))(\( ( 1 + 71 T^{2} )^{2} \))
$73$ (\( 1 - 7 T + 73 T^{2} \))(\( 1 + 2 T + 73 T^{2} \))(\( 1 + 7 T + 73 T^{2} \))(\( 1 + T + 73 T^{2} \))(\( 1 + 10 T + 73 T^{2} \))(\( 1 + 73 T^{2} \))(\( 1 - T + 73 T^{2} \))(\( 1 - 7 T + 73 T^{2} \))(\( 1 - 14 T + 73 T^{2} \))(\( 1 + 7 T + 73 T^{2} \))(\( 1 + 48 T^{2} + 5329 T^{4} \))(\( 1 + 144 T^{2} + 5329 T^{4} \))(\( 1 + 96 T^{2} + 5329 T^{4} \))(\( ( 1 + 73 T^{2} )^{2} \))
$79$ (\( 1 - 11 T + 79 T^{2} \))(\( 1 + 8 T + 79 T^{2} \))(\( 1 + T + 79 T^{2} \))(\( 1 - 13 T + 79 T^{2} \))(\( 1 + 16 T + 79 T^{2} \))(\( 1 + 8 T + 79 T^{2} \))(\( 1 - 13 T + 79 T^{2} \))(\( 1 + T + 79 T^{2} \))(\( 1 - 8 T + 79 T^{2} \))(\( 1 - 11 T + 79 T^{2} \))(\( ( 1 - 4 T + 79 T^{2} )^{2} \))(\( ( 1 - 4 T + 79 T^{2} )^{2} \))(\( ( 1 + 8 T + 79 T^{2} )^{2} \))(\( ( 1 + 8 T + 79 T^{2} )^{2} \))
$83$ (\( 1 - 4 T + 83 T^{2} \))(\( 1 + 6 T + 83 T^{2} \))(\( 1 - 12 T + 83 T^{2} \))(\( 1 + 12 T + 83 T^{2} \))(\( 1 - 8 T + 83 T^{2} \))(\( 1 + 83 T^{2} \))(\( 1 - 12 T + 83 T^{2} \))(\( 1 + 12 T + 83 T^{2} \))(\( 1 - 6 T + 83 T^{2} \))(\( 1 + 4 T + 83 T^{2} \))(\( 1 + 164 T^{2} + 6889 T^{4} \))(\( 1 + 68 T^{2} + 6889 T^{4} \))(\( 1 - 34 T^{2} + 6889 T^{4} \))(\( 1 - 34 T^{2} + 6889 T^{4} \))
$89$ (\( 1 + 9 T + 89 T^{2} \))(\( 1 - 6 T + 89 T^{2} \))(\( 1 + 7 T + 89 T^{2} \))(\( 1 - 15 T + 89 T^{2} \))(\( 1 - 6 T + 89 T^{2} \))(\( 1 + 89 T^{2} \))(\( 1 + 15 T + 89 T^{2} \))(\( 1 - 7 T + 89 T^{2} \))(\( 1 + 10 T + 89 T^{2} \))(\( 1 - 9 T + 89 T^{2} \))(\( 1 + 160 T^{2} + 7921 T^{4} \))(\( 1 + 128 T^{2} + 7921 T^{4} \))(\( 1 + 128 T^{2} + 7921 T^{4} \))(\( ( 1 + 89 T^{2} )^{2} \))
$97$ (\( 1 - 6 T + 97 T^{2} \))(\( 1 - 10 T + 97 T^{2} \))(\( 1 - 2 T + 97 T^{2} \))(\( 1 + 10 T + 97 T^{2} \))(\( 1 - 6 T + 97 T^{2} \))(\( 1 + 97 T^{2} \))(\( 1 - 10 T + 97 T^{2} \))(\( 1 + 2 T + 97 T^{2} \))(\( 1 - 2 T + 97 T^{2} \))(\( 1 + 6 T + 97 T^{2} \))(\( 1 + 32 T^{2} + 9409 T^{4} \))(\( 1 + 96 T^{2} + 9409 T^{4} \))(\( 1 + 192 T^{2} + 9409 T^{4} \))(\( 1 + 162 T^{2} + 9409 T^{4} \))
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