Properties

Label 864.2.a
Level 864
Weight 2
Character orbit a
Rep. character \(\chi_{864}(1,\cdot)\)
Character field \(\Q\)
Dimension 16
Newform subspaces 14
Sturm bound 288
Trace bound 11

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Defining parameters

Level: \( N \) = \( 864 = 2^{5} \cdot 3^{3} \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 864.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 14 \)
Sturm bound: \(288\)
Trace bound: \(11\)
Distinguishing \(T_p\): \(5\), \(7\), \(11\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(864))\).

Total New Old
Modular forms 168 16 152
Cusp forms 121 16 105
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.

\(2\)\(3\)FrickeDim.
\(+\)\(+\)\(+\)\(3\)
\(+\)\(-\)\(-\)\(5\)
\(-\)\(+\)\(-\)\(5\)
\(-\)\(-\)\(+\)\(3\)
Plus space\(+\)\(6\)
Minus space\(-\)\(10\)

Trace form

\( 16q + O(q^{10}) \) \( 16q + 8q^{13} + 8q^{25} + 40q^{37} + 16q^{49} + 40q^{61} + 16q^{85} + 32q^{97} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(864))\) into newform subspaces

Label Dim. \(A\) Field CM Traces A-L signs $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\) 2 3
864.2.a.a \(1\) \(6.899\) \(\Q\) None \(0\) \(0\) \(-2\) \(-3\) \(-\) \(+\) \(q-2q^{5}-3q^{7}+6q^{11}-3q^{13}+2q^{17}+\cdots\)
864.2.a.b \(1\) \(6.899\) \(\Q\) None \(0\) \(0\) \(-2\) \(-1\) \(-\) \(-\) \(q-2q^{5}-q^{7}+2q^{11}+q^{13}-6q^{17}+\cdots\)
864.2.a.c \(1\) \(6.899\) \(\Q\) None \(0\) \(0\) \(-2\) \(1\) \(+\) \(+\) \(q-2q^{5}+q^{7}-2q^{11}+q^{13}-6q^{17}+\cdots\)
864.2.a.d \(1\) \(6.899\) \(\Q\) None \(0\) \(0\) \(-2\) \(3\) \(-\) \(-\) \(q-2q^{5}+3q^{7}-6q^{11}-3q^{13}+2q^{17}+\cdots\)
864.2.a.e \(1\) \(6.899\) \(\Q\) None \(0\) \(0\) \(-1\) \(-3\) \(+\) \(+\) \(q-q^{5}-3q^{7}+3q^{11}+4q^{17}-6q^{19}+\cdots\)
864.2.a.f \(1\) \(6.899\) \(\Q\) None \(0\) \(0\) \(-1\) \(3\) \(+\) \(-\) \(q-q^{5}+3q^{7}-3q^{11}+4q^{17}+6q^{19}+\cdots\)
864.2.a.g \(1\) \(6.899\) \(\Q\) None \(0\) \(0\) \(1\) \(-3\) \(-\) \(-\) \(q+q^{5}-3q^{7}-3q^{11}-4q^{17}-6q^{19}+\cdots\)
864.2.a.h \(1\) \(6.899\) \(\Q\) None \(0\) \(0\) \(1\) \(3\) \(-\) \(+\) \(q+q^{5}+3q^{7}+3q^{11}-4q^{17}+6q^{19}+\cdots\)
864.2.a.i \(1\) \(6.899\) \(\Q\) None \(0\) \(0\) \(2\) \(-3\) \(+\) \(+\) \(q+2q^{5}-3q^{7}-6q^{11}-3q^{13}-2q^{17}+\cdots\)
864.2.a.j \(1\) \(6.899\) \(\Q\) None \(0\) \(0\) \(2\) \(-1\) \(+\) \(-\) \(q+2q^{5}-q^{7}-2q^{11}+q^{13}+6q^{17}+\cdots\)
864.2.a.k \(1\) \(6.899\) \(\Q\) None \(0\) \(0\) \(2\) \(1\) \(-\) \(+\) \(q+2q^{5}+q^{7}+2q^{11}+q^{13}+6q^{17}+\cdots\)
864.2.a.l \(1\) \(6.899\) \(\Q\) None \(0\) \(0\) \(2\) \(3\) \(+\) \(-\) \(q+2q^{5}+3q^{7}+6q^{11}-3q^{13}-2q^{17}+\cdots\)
864.2.a.m \(2\) \(6.899\) \(\Q(\sqrt{13}) \) None \(0\) \(0\) \(0\) \(0\) \(-\) \(+\) \(q-\beta q^{5}+\beta q^{7}-q^{11}+4q^{13}-2\beta q^{19}+\cdots\)
864.2.a.n \(2\) \(6.899\) \(\Q(\sqrt{13}) \) None \(0\) \(0\) \(0\) \(0\) \(+\) \(-\) \(q-\beta q^{5}-\beta q^{7}+q^{11}+4q^{13}+2\beta q^{19}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(864))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(864)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(24))\)\(^{\oplus 9}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(27))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(32))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(36))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(48))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(54))\)\(^{\oplus 5}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(72))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(96))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(108))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(144))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(216))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(288))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(432))\)\(^{\oplus 2}\)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ (\( \))(\( \))(\( \))(\( \))(\( \))(\( \))(\( \))(\( \))(\( \))(\( \))(\( \))(\( \))(\( \))(\( \))
$3$ (\( \))(\( \))(\( \))(\( \))(\( \))(\( \))(\( \))(\( \))(\( \))(\( \))(\( \))(\( \))(\( \))(\( \))
$5$ (\( 1 + 2 T + 5 T^{2} \))(\( 1 + 2 T + 5 T^{2} \))(\( 1 + 2 T + 5 T^{2} \))(\( 1 + 2 T + 5 T^{2} \))(\( 1 + T + 5 T^{2} \))(\( 1 + T + 5 T^{2} \))(\( 1 - T + 5 T^{2} \))(\( 1 - T + 5 T^{2} \))(\( 1 - 2 T + 5 T^{2} \))(\( 1 - 2 T + 5 T^{2} \))(\( 1 - 2 T + 5 T^{2} \))(\( 1 - 2 T + 5 T^{2} \))(\( 1 - 3 T^{2} + 25 T^{4} \))(\( 1 - 3 T^{2} + 25 T^{4} \))
$7$ (\( 1 + 3 T + 7 T^{2} \))(\( 1 + T + 7 T^{2} \))(\( 1 - T + 7 T^{2} \))(\( 1 - 3 T + 7 T^{2} \))(\( 1 + 3 T + 7 T^{2} \))(\( 1 - 3 T + 7 T^{2} \))(\( 1 + 3 T + 7 T^{2} \))(\( 1 - 3 T + 7 T^{2} \))(\( 1 + 3 T + 7 T^{2} \))(\( 1 + T + 7 T^{2} \))(\( 1 - T + 7 T^{2} \))(\( 1 - 3 T + 7 T^{2} \))(\( 1 + T^{2} + 49 T^{4} \))(\( 1 + T^{2} + 49 T^{4} \))
$11$ (\( 1 - 6 T + 11 T^{2} \))(\( 1 - 2 T + 11 T^{2} \))(\( 1 + 2 T + 11 T^{2} \))(\( 1 + 6 T + 11 T^{2} \))(\( 1 - 3 T + 11 T^{2} \))(\( 1 + 3 T + 11 T^{2} \))(\( 1 + 3 T + 11 T^{2} \))(\( 1 - 3 T + 11 T^{2} \))(\( 1 + 6 T + 11 T^{2} \))(\( 1 + 2 T + 11 T^{2} \))(\( 1 - 2 T + 11 T^{2} \))(\( 1 - 6 T + 11 T^{2} \))(\( ( 1 + T + 11 T^{2} )^{2} \))(\( ( 1 - T + 11 T^{2} )^{2} \))
$13$ (\( 1 + 3 T + 13 T^{2} \))(\( 1 - T + 13 T^{2} \))(\( 1 - T + 13 T^{2} \))(\( 1 + 3 T + 13 T^{2} \))(\( 1 + 13 T^{2} \))(\( 1 + 13 T^{2} \))(\( 1 + 13 T^{2} \))(\( 1 + 13 T^{2} \))(\( 1 + 3 T + 13 T^{2} \))(\( 1 - T + 13 T^{2} \))(\( 1 - T + 13 T^{2} \))(\( 1 + 3 T + 13 T^{2} \))(\( ( 1 - 4 T + 13 T^{2} )^{2} \))(\( ( 1 - 4 T + 13 T^{2} )^{2} \))
$17$ (\( 1 - 2 T + 17 T^{2} \))(\( 1 + 6 T + 17 T^{2} \))(\( 1 + 6 T + 17 T^{2} \))(\( 1 - 2 T + 17 T^{2} \))(\( 1 - 4 T + 17 T^{2} \))(\( 1 - 4 T + 17 T^{2} \))(\( 1 + 4 T + 17 T^{2} \))(\( 1 + 4 T + 17 T^{2} \))(\( 1 + 2 T + 17 T^{2} \))(\( 1 - 6 T + 17 T^{2} \))(\( 1 - 6 T + 17 T^{2} \))(\( 1 + 2 T + 17 T^{2} \))(\( ( 1 + 17 T^{2} )^{2} \))(\( ( 1 + 17 T^{2} )^{2} \))
$19$ (\( 1 - 3 T + 19 T^{2} \))(\( 1 - 5 T + 19 T^{2} \))(\( 1 + 5 T + 19 T^{2} \))(\( 1 + 3 T + 19 T^{2} \))(\( 1 + 6 T + 19 T^{2} \))(\( 1 - 6 T + 19 T^{2} \))(\( 1 + 6 T + 19 T^{2} \))(\( 1 - 6 T + 19 T^{2} \))(\( 1 - 3 T + 19 T^{2} \))(\( 1 - 5 T + 19 T^{2} \))(\( 1 + 5 T + 19 T^{2} \))(\( 1 + 3 T + 19 T^{2} \))(\( 1 - 14 T^{2} + 361 T^{4} \))(\( 1 - 14 T^{2} + 361 T^{4} \))
$23$ (\( 1 - 6 T + 23 T^{2} \))(\( 1 + 6 T + 23 T^{2} \))(\( 1 - 6 T + 23 T^{2} \))(\( 1 + 6 T + 23 T^{2} \))(\( 1 + 6 T + 23 T^{2} \))(\( 1 - 6 T + 23 T^{2} \))(\( 1 - 6 T + 23 T^{2} \))(\( 1 + 6 T + 23 T^{2} \))(\( 1 + 6 T + 23 T^{2} \))(\( 1 - 6 T + 23 T^{2} \))(\( 1 + 6 T + 23 T^{2} \))(\( 1 - 6 T + 23 T^{2} \))(\( ( 1 - 6 T + 23 T^{2} )^{2} \))(\( ( 1 + 6 T + 23 T^{2} )^{2} \))
$29$ (\( 1 - 8 T + 29 T^{2} \))(\( 1 + 8 T + 29 T^{2} \))(\( 1 + 8 T + 29 T^{2} \))(\( 1 - 8 T + 29 T^{2} \))(\( 1 + 2 T + 29 T^{2} \))(\( 1 + 2 T + 29 T^{2} \))(\( 1 - 2 T + 29 T^{2} \))(\( 1 - 2 T + 29 T^{2} \))(\( 1 + 8 T + 29 T^{2} \))(\( 1 - 8 T + 29 T^{2} \))(\( 1 - 8 T + 29 T^{2} \))(\( 1 + 8 T + 29 T^{2} \))(\( 1 + 6 T^{2} + 841 T^{4} \))(\( 1 + 6 T^{2} + 841 T^{4} \))
$31$ (\( 1 + 31 T^{2} \))(\( 1 + 8 T + 31 T^{2} \))(\( 1 - 8 T + 31 T^{2} \))(\( 1 + 31 T^{2} \))(\( 1 + 9 T + 31 T^{2} \))(\( 1 - 9 T + 31 T^{2} \))(\( 1 + 9 T + 31 T^{2} \))(\( 1 - 9 T + 31 T^{2} \))(\( 1 + 31 T^{2} \))(\( 1 + 8 T + 31 T^{2} \))(\( 1 - 8 T + 31 T^{2} \))(\( 1 + 31 T^{2} \))(\( 1 + 49 T^{2} + 961 T^{4} \))(\( 1 + 49 T^{2} + 961 T^{4} \))
$37$ (\( 1 - 7 T + 37 T^{2} \))(\( 1 + 5 T + 37 T^{2} \))(\( 1 + 5 T + 37 T^{2} \))(\( 1 - 7 T + 37 T^{2} \))(\( 1 + 2 T + 37 T^{2} \))(\( 1 + 2 T + 37 T^{2} \))(\( 1 + 2 T + 37 T^{2} \))(\( 1 + 2 T + 37 T^{2} \))(\( 1 - 7 T + 37 T^{2} \))(\( 1 + 5 T + 37 T^{2} \))(\( 1 + 5 T + 37 T^{2} \))(\( 1 - 7 T + 37 T^{2} \))(\( ( 1 - 10 T + 37 T^{2} )^{2} \))(\( ( 1 - 10 T + 37 T^{2} )^{2} \))
$41$ (\( 1 + 8 T + 41 T^{2} \))(\( 1 + 8 T + 41 T^{2} \))(\( 1 + 8 T + 41 T^{2} \))(\( 1 + 8 T + 41 T^{2} \))(\( 1 + 10 T + 41 T^{2} \))(\( 1 + 10 T + 41 T^{2} \))(\( 1 - 10 T + 41 T^{2} \))(\( 1 - 10 T + 41 T^{2} \))(\( 1 - 8 T + 41 T^{2} \))(\( 1 - 8 T + 41 T^{2} \))(\( 1 - 8 T + 41 T^{2} \))(\( 1 - 8 T + 41 T^{2} \))(\( 1 + 30 T^{2} + 1681 T^{4} \))(\( 1 + 30 T^{2} + 1681 T^{4} \))
$43$ (\( 1 - 12 T + 43 T^{2} \))(\( 1 - 4 T + 43 T^{2} \))(\( 1 + 4 T + 43 T^{2} \))(\( 1 + 12 T + 43 T^{2} \))(\( 1 + 6 T + 43 T^{2} \))(\( 1 - 6 T + 43 T^{2} \))(\( 1 + 6 T + 43 T^{2} \))(\( 1 - 6 T + 43 T^{2} \))(\( 1 - 12 T + 43 T^{2} \))(\( 1 - 4 T + 43 T^{2} \))(\( 1 + 4 T + 43 T^{2} \))(\( 1 + 12 T + 43 T^{2} \))(\( 1 + 34 T^{2} + 1849 T^{4} \))(\( 1 + 34 T^{2} + 1849 T^{4} \))
$47$ (\( 1 - 6 T + 47 T^{2} \))(\( 1 - 10 T + 47 T^{2} \))(\( 1 + 10 T + 47 T^{2} \))(\( 1 + 6 T + 47 T^{2} \))(\( 1 + 6 T + 47 T^{2} \))(\( 1 - 6 T + 47 T^{2} \))(\( 1 - 6 T + 47 T^{2} \))(\( 1 + 6 T + 47 T^{2} \))(\( 1 + 6 T + 47 T^{2} \))(\( 1 + 10 T + 47 T^{2} \))(\( 1 - 10 T + 47 T^{2} \))(\( 1 - 6 T + 47 T^{2} \))(\( ( 1 - 10 T + 47 T^{2} )^{2} \))(\( ( 1 + 10 T + 47 T^{2} )^{2} \))
$53$ (\( 1 + 4 T + 53 T^{2} \))(\( 1 + 4 T + 53 T^{2} \))(\( 1 + 4 T + 53 T^{2} \))(\( 1 + 4 T + 53 T^{2} \))(\( 1 - 13 T + 53 T^{2} \))(\( 1 - 13 T + 53 T^{2} \))(\( 1 + 13 T + 53 T^{2} \))(\( 1 + 13 T + 53 T^{2} \))(\( 1 - 4 T + 53 T^{2} \))(\( 1 - 4 T + 53 T^{2} \))(\( 1 - 4 T + 53 T^{2} \))(\( 1 - 4 T + 53 T^{2} \))(\( 1 + 93 T^{2} + 2809 T^{4} \))(\( 1 + 93 T^{2} + 2809 T^{4} \))
$59$ (\( 1 - 6 T + 59 T^{2} \))(\( 1 + 14 T + 59 T^{2} \))(\( 1 - 14 T + 59 T^{2} \))(\( 1 + 6 T + 59 T^{2} \))(\( 1 - 12 T + 59 T^{2} \))(\( 1 + 12 T + 59 T^{2} \))(\( 1 + 12 T + 59 T^{2} \))(\( 1 - 12 T + 59 T^{2} \))(\( 1 + 6 T + 59 T^{2} \))(\( 1 - 14 T + 59 T^{2} \))(\( 1 + 14 T + 59 T^{2} \))(\( 1 - 6 T + 59 T^{2} \))(\( ( 1 - 4 T + 59 T^{2} )^{2} \))(\( ( 1 + 4 T + 59 T^{2} )^{2} \))
$61$ (\( 1 + T + 61 T^{2} \))(\( 1 - 3 T + 61 T^{2} \))(\( 1 - 3 T + 61 T^{2} \))(\( 1 + T + 61 T^{2} \))(\( 1 - 8 T + 61 T^{2} \))(\( 1 - 8 T + 61 T^{2} \))(\( 1 - 8 T + 61 T^{2} \))(\( 1 - 8 T + 61 T^{2} \))(\( 1 + T + 61 T^{2} \))(\( 1 - 3 T + 61 T^{2} \))(\( 1 - 3 T + 61 T^{2} \))(\( 1 + T + 61 T^{2} \))(\( ( 1 + 61 T^{2} )^{2} \))(\( ( 1 + 61 T^{2} )^{2} \))
$67$ (\( 1 - 3 T + 67 T^{2} \))(\( 1 - 13 T + 67 T^{2} \))(\( 1 + 13 T + 67 T^{2} \))(\( 1 + 3 T + 67 T^{2} \))(\( 1 + 6 T + 67 T^{2} \))(\( 1 - 6 T + 67 T^{2} \))(\( 1 + 6 T + 67 T^{2} \))(\( 1 - 6 T + 67 T^{2} \))(\( 1 - 3 T + 67 T^{2} \))(\( 1 - 13 T + 67 T^{2} \))(\( 1 + 13 T + 67 T^{2} \))(\( 1 + 3 T + 67 T^{2} \))(\( 1 + 82 T^{2} + 4489 T^{4} \))(\( 1 + 82 T^{2} + 4489 T^{4} \))
$71$ (\( 1 - 12 T + 71 T^{2} \))(\( 1 - 4 T + 71 T^{2} \))(\( 1 + 4 T + 71 T^{2} \))(\( 1 + 12 T + 71 T^{2} \))(\( 1 + 12 T + 71 T^{2} \))(\( 1 - 12 T + 71 T^{2} \))(\( 1 - 12 T + 71 T^{2} \))(\( 1 + 12 T + 71 T^{2} \))(\( 1 + 12 T + 71 T^{2} \))(\( 1 + 4 T + 71 T^{2} \))(\( 1 - 4 T + 71 T^{2} \))(\( 1 - 12 T + 71 T^{2} \))(\( ( 1 + 8 T + 71 T^{2} )^{2} \))(\( ( 1 - 8 T + 71 T^{2} )^{2} \))
$73$ (\( 1 + 15 T + 73 T^{2} \))(\( 1 - 9 T + 73 T^{2} \))(\( 1 - 9 T + 73 T^{2} \))(\( 1 + 15 T + 73 T^{2} \))(\( 1 - 9 T + 73 T^{2} \))(\( 1 - 9 T + 73 T^{2} \))(\( 1 - 9 T + 73 T^{2} \))(\( 1 - 9 T + 73 T^{2} \))(\( 1 + 15 T + 73 T^{2} \))(\( 1 - 9 T + 73 T^{2} \))(\( 1 - 9 T + 73 T^{2} \))(\( 1 + 15 T + 73 T^{2} \))(\( ( 1 + 3 T + 73 T^{2} )^{2} \))(\( ( 1 + 3 T + 73 T^{2} )^{2} \))
$79$ (\( 1 + 9 T + 79 T^{2} \))(\( 1 + 11 T + 79 T^{2} \))(\( 1 - 11 T + 79 T^{2} \))(\( 1 - 9 T + 79 T^{2} \))(\( 1 + 79 T^{2} \))(\( 1 + 79 T^{2} \))(\( 1 + 79 T^{2} \))(\( 1 + 79 T^{2} \))(\( 1 + 9 T + 79 T^{2} \))(\( 1 + 11 T + 79 T^{2} \))(\( 1 - 11 T + 79 T^{2} \))(\( 1 - 9 T + 79 T^{2} \))(\( 1 - 50 T^{2} + 6241 T^{4} \))(\( 1 - 50 T^{2} + 6241 T^{4} \))
$83$ (\( 1 + 12 T + 83 T^{2} \))(\( 1 - 12 T + 83 T^{2} \))(\( 1 + 12 T + 83 T^{2} \))(\( 1 - 12 T + 83 T^{2} \))(\( 1 - 3 T + 83 T^{2} \))(\( 1 + 3 T + 83 T^{2} \))(\( 1 + 3 T + 83 T^{2} \))(\( 1 - 3 T + 83 T^{2} \))(\( 1 - 12 T + 83 T^{2} \))(\( 1 + 12 T + 83 T^{2} \))(\( 1 - 12 T + 83 T^{2} \))(\( 1 + 12 T + 83 T^{2} \))(\( ( 1 + 9 T + 83 T^{2} )^{2} \))(\( ( 1 - 9 T + 83 T^{2} )^{2} \))
$89$ (\( 1 - 10 T + 89 T^{2} \))(\( 1 - 2 T + 89 T^{2} \))(\( 1 - 2 T + 89 T^{2} \))(\( 1 - 10 T + 89 T^{2} \))(\( 1 - 14 T + 89 T^{2} \))(\( 1 - 14 T + 89 T^{2} \))(\( 1 + 14 T + 89 T^{2} \))(\( 1 + 14 T + 89 T^{2} \))(\( 1 + 10 T + 89 T^{2} \))(\( 1 + 2 T + 89 T^{2} \))(\( 1 + 2 T + 89 T^{2} \))(\( 1 + 10 T + 89 T^{2} \))(\( 1 + 126 T^{2} + 7921 T^{4} \))(\( 1 + 126 T^{2} + 7921 T^{4} \))
$97$ (\( 1 - 9 T + 97 T^{2} \))(\( 1 - T + 97 T^{2} \))(\( 1 - T + 97 T^{2} \))(\( 1 - 9 T + 97 T^{2} \))(\( 1 + 9 T + 97 T^{2} \))(\( 1 + 9 T + 97 T^{2} \))(\( 1 + 9 T + 97 T^{2} \))(\( 1 + 9 T + 97 T^{2} \))(\( 1 - 9 T + 97 T^{2} \))(\( 1 - T + 97 T^{2} \))(\( 1 - T + 97 T^{2} \))(\( 1 - 9 T + 97 T^{2} \))(\( ( 1 - 7 T + 97 T^{2} )^{2} \))(\( ( 1 - 7 T + 97 T^{2} )^{2} \))
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