Properties

Label 1176.2.a
Level 1176
Weight 2
Character orbit a
Rep. character \(\chi_{1176}(1,\cdot)\)
Character field \(\Q\)
Dimension 21
Newform subspaces 15
Sturm bound 448
Trace bound 11

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Defining parameters

Level: \( N \) \(=\) \( 1176 = 2^{3} \cdot 3 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1176.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 15 \)
Sturm bound: \(448\)
Trace bound: \(11\)
Distinguishing \(T_p\): \(5\), \(11\), \(13\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(1176))\).

Total New Old
Modular forms 256 21 235
Cusp forms 193 21 172
Eisenstein series 63 0 63

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.
\(+\)\(+\)\(+\)\(+\)\(3\)
\(+\)\(+\)\(-\)\(-\)\(2\)
\(+\)\(-\)\(+\)\(-\)\(3\)
\(+\)\(-\)\(-\)\(+\)\(2\)
\(-\)\(+\)\(+\)\(-\)\(4\)
\(-\)\(+\)\(-\)\(+\)\(1\)
\(-\)\(-\)\(+\)\(+\)\(2\)
\(-\)\(-\)\(-\)\(-\)\(4\)
Plus space\(+\)\(8\)
Minus space\(-\)\(13\)

Trace form

\( 21q + q^{3} - 2q^{5} + 21q^{9} + O(q^{10}) \) \( 21q + q^{3} - 2q^{5} + 21q^{9} - 4q^{11} - 2q^{13} - 2q^{15} - 6q^{17} + 4q^{19} + 16q^{23} + 23q^{25} + q^{27} - 2q^{29} + 8q^{31} + 4q^{33} - 6q^{37} + 6q^{39} + 18q^{41} - 16q^{43} - 2q^{45} - 10q^{51} + 46q^{53} + 8q^{55} - 20q^{59} + 14q^{61} + 28q^{65} - 24q^{67} - 8q^{69} - 8q^{71} + 2q^{73} - q^{75} + 21q^{81} - 12q^{83} + 44q^{85} - 10q^{87} + 2q^{89} + 20q^{93} + 8q^{95} - 22q^{97} - 4q^{99} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(1176))\) 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
1176.2.a.a \(1\) \(9.390\) \(\Q\) None \(0\) \(-1\) \(-2\) \(0\) \(+\) \(+\) \(-\) \(q-q^{3}-2q^{5}+q^{9}+2q^{13}+2q^{15}+\cdots\)
1176.2.a.b \(1\) \(9.390\) \(\Q\) None \(0\) \(-1\) \(0\) \(0\) \(-\) \(+\) \(-\) \(q-q^{3}+q^{9}-4q^{13}-4q^{17}+4q^{19}+\cdots\)
1176.2.a.c \(1\) \(9.390\) \(\Q\) None \(0\) \(-1\) \(1\) \(0\) \(+\) \(+\) \(-\) \(q-q^{3}+q^{5}+q^{9}+3q^{11}-4q^{13}+\cdots\)
1176.2.a.d \(1\) \(9.390\) \(\Q\) None \(0\) \(-1\) \(2\) \(0\) \(+\) \(+\) \(+\) \(q-q^{3}+2q^{5}+q^{9}-6q^{11}-3q^{13}+\cdots\)
1176.2.a.e \(1\) \(9.390\) \(\Q\) None \(0\) \(1\) \(-2\) \(0\) \(+\) \(-\) \(-\) \(q+q^{3}-2q^{5}+q^{9}-6q^{11}+3q^{13}+\cdots\)
1176.2.a.f \(1\) \(9.390\) \(\Q\) None \(0\) \(1\) \(-2\) \(0\) \(+\) \(-\) \(-\) \(q+q^{3}-2q^{5}+q^{9}-6q^{13}-2q^{15}+\cdots\)
1176.2.a.g \(1\) \(9.390\) \(\Q\) None \(0\) \(1\) \(-1\) \(0\) \(+\) \(-\) \(+\) \(q+q^{3}-q^{5}+q^{9}+3q^{11}+4q^{13}+\cdots\)
1176.2.a.h \(1\) \(9.390\) \(\Q\) None \(0\) \(1\) \(0\) \(0\) \(-\) \(-\) \(-\) \(q+q^{3}+q^{9}+4q^{13}+4q^{17}-4q^{19}+\cdots\)
1176.2.a.i \(1\) \(9.390\) \(\Q\) None \(0\) \(1\) \(2\) \(0\) \(-\) \(-\) \(-\) \(q+q^{3}+2q^{5}+q^{9}+4q^{11}+2q^{13}+\cdots\)
1176.2.a.j \(2\) \(9.390\) \(\Q(\sqrt{2}) \) None \(0\) \(-2\) \(-4\) \(0\) \(+\) \(+\) \(+\) \(q-q^{3}+(-2+\beta )q^{5}+q^{9}+(2-2\beta )q^{11}+\cdots\)
1176.2.a.k \(2\) \(9.390\) \(\Q(\sqrt{57}) \) None \(0\) \(-2\) \(-1\) \(0\) \(-\) \(+\) \(+\) \(q-q^{3}-\beta q^{5}+q^{9}-\beta q^{11}+(3-\beta )q^{13}+\cdots\)
1176.2.a.l \(2\) \(9.390\) \(\Q(\sqrt{2}) \) None \(0\) \(-2\) \(4\) \(0\) \(-\) \(+\) \(+\) \(q-q^{3}+(2+\beta )q^{5}+q^{9}+(-2+2\beta )q^{11}+\cdots\)
1176.2.a.m \(2\) \(9.390\) \(\Q(\sqrt{2}) \) None \(0\) \(2\) \(-4\) \(0\) \(-\) \(-\) \(+\) \(q+q^{3}+(-2+\beta )q^{5}+q^{9}+(-2-2\beta )q^{11}+\cdots\)
1176.2.a.n \(2\) \(9.390\) \(\Q(\sqrt{57}) \) None \(0\) \(2\) \(1\) \(0\) \(-\) \(-\) \(-\) \(q+q^{3}+\beta q^{5}+q^{9}-\beta q^{11}+(-3+\beta )q^{13}+\cdots\)
1176.2.a.o \(2\) \(9.390\) \(\Q(\sqrt{2}) \) None \(0\) \(2\) \(4\) \(0\) \(+\) \(-\) \(+\) \(q+q^{3}+(2+\beta )q^{5}+q^{9}+(2+2\beta )q^{11}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(1176))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(1176)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(14))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(21))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(24))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(42))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(49))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(56))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(84))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(98))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(147))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(168))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(196))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(294))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(392))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(588))\)\(^{\oplus 2}\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ (\( 1 + T \))(\( 1 + T \))(\( 1 + T \))(\( 1 + T \))(\( 1 - T \))(\( 1 - T \))(\( 1 - T \))(\( 1 - T \))(\( 1 - T \))(\( ( 1 + T )^{2} \))(\( ( 1 + T )^{2} \))(\( ( 1 + T )^{2} \))(\( ( 1 - T )^{2} \))(\( ( 1 - T )^{2} \))(\( ( 1 - T )^{2} \))
$5$ (\( 1 + 2 T + 5 T^{2} \))(\( 1 + 5 T^{2} \))(\( 1 - T + 5 T^{2} \))(\( 1 - 2 T + 5 T^{2} \))(\( 1 + 2 T + 5 T^{2} \))(\( 1 + 2 T + 5 T^{2} \))(\( 1 + T + 5 T^{2} \))(\( 1 + 5 T^{2} \))(\( 1 - 2 T + 5 T^{2} \))(\( 1 + 4 T + 12 T^{2} + 20 T^{3} + 25 T^{4} \))(\( 1 + T - 4 T^{2} + 5 T^{3} + 25 T^{4} \))(\( 1 - 4 T + 12 T^{2} - 20 T^{3} + 25 T^{4} \))(\( 1 + 4 T + 12 T^{2} + 20 T^{3} + 25 T^{4} \))(\( 1 - T - 4 T^{2} - 5 T^{3} + 25 T^{4} \))(\( 1 - 4 T + 12 T^{2} - 20 T^{3} + 25 T^{4} \))
$7$ 1
$11$ (\( 1 + 11 T^{2} \))(\( 1 + 11 T^{2} \))(\( 1 - 3 T + 11 T^{2} \))(\( 1 + 6 T + 11 T^{2} \))(\( 1 + 6 T + 11 T^{2} \))(\( 1 + 11 T^{2} \))(\( 1 - 3 T + 11 T^{2} \))(\( 1 + 11 T^{2} \))(\( 1 - 4 T + 11 T^{2} \))(\( 1 - 4 T + 18 T^{2} - 44 T^{3} + 121 T^{4} \))(\( 1 + T + 8 T^{2} + 11 T^{3} + 121 T^{4} \))(\( 1 + 4 T + 18 T^{2} + 44 T^{3} + 121 T^{4} \))(\( 1 + 4 T + 18 T^{2} + 44 T^{3} + 121 T^{4} \))(\( 1 + T + 8 T^{2} + 11 T^{3} + 121 T^{4} \))(\( 1 - 4 T + 18 T^{2} - 44 T^{3} + 121 T^{4} \))
$13$ (\( 1 - 2 T + 13 T^{2} \))(\( 1 + 4 T + 13 T^{2} \))(\( 1 + 4 T + 13 T^{2} \))(\( 1 + 3 T + 13 T^{2} \))(\( 1 - 3 T + 13 T^{2} \))(\( 1 + 6 T + 13 T^{2} \))(\( 1 - 4 T + 13 T^{2} \))(\( 1 - 4 T + 13 T^{2} \))(\( 1 - 2 T + 13 T^{2} \))(\( 1 + 24 T^{2} + 169 T^{4} \))(\( 1 - 5 T + 18 T^{2} - 65 T^{3} + 169 T^{4} \))(\( 1 + 8 T^{2} + 169 T^{4} \))(\( 1 + 8 T^{2} + 169 T^{4} \))(\( 1 + 5 T + 18 T^{2} + 65 T^{3} + 169 T^{4} \))(\( 1 + 24 T^{2} + 169 T^{4} \))
$17$ (\( 1 + 6 T + 17 T^{2} \))(\( 1 + 4 T + 17 T^{2} \))(\( 1 + 17 T^{2} \))(\( 1 - 4 T + 17 T^{2} \))(\( 1 + 4 T + 17 T^{2} \))(\( 1 - 2 T + 17 T^{2} \))(\( 1 + 17 T^{2} \))(\( 1 - 4 T + 17 T^{2} \))(\( 1 + 2 T + 17 T^{2} \))(\( 1 - 4 T + 20 T^{2} - 68 T^{3} + 289 T^{4} \))(\( ( 1 + 4 T + 17 T^{2} )^{2} \))(\( 1 - 12 T + 68 T^{2} - 204 T^{3} + 289 T^{4} \))(\( 1 + 12 T + 68 T^{2} + 204 T^{3} + 289 T^{4} \))(\( ( 1 - 4 T + 17 T^{2} )^{2} \))(\( 1 + 4 T + 20 T^{2} + 68 T^{3} + 289 T^{4} \))
$19$ (\( 1 - 4 T + 19 T^{2} \))(\( 1 - 4 T + 19 T^{2} \))(\( 1 - 4 T + 19 T^{2} \))(\( 1 + 5 T + 19 T^{2} \))(\( 1 - 5 T + 19 T^{2} \))(\( 1 + 4 T + 19 T^{2} \))(\( 1 + 4 T + 19 T^{2} \))(\( 1 + 4 T + 19 T^{2} \))(\( 1 - 4 T + 19 T^{2} \))(\( 1 + 8 T + 46 T^{2} + 152 T^{3} + 361 T^{4} \))(\( 1 + 5 T + 30 T^{2} + 95 T^{3} + 361 T^{4} \))(\( 1 - 8 T + 46 T^{2} - 152 T^{3} + 361 T^{4} \))(\( 1 + 8 T + 46 T^{2} + 152 T^{3} + 361 T^{4} \))(\( 1 - 5 T + 30 T^{2} - 95 T^{3} + 361 T^{4} \))(\( 1 - 8 T + 46 T^{2} - 152 T^{3} + 361 T^{4} \))
$23$ (\( 1 + 4 T + 23 T^{2} \))(\( 1 - 4 T + 23 T^{2} \))(\( 1 - 8 T + 23 T^{2} \))(\( 1 + 4 T + 23 T^{2} \))(\( 1 + 4 T + 23 T^{2} \))(\( 1 + 4 T + 23 T^{2} \))(\( 1 - 8 T + 23 T^{2} \))(\( 1 - 4 T + 23 T^{2} \))(\( 1 + 8 T + 23 T^{2} \))(\( 1 - 4 T + 42 T^{2} - 92 T^{3} + 529 T^{4} \))(\( ( 1 - 4 T + 23 T^{2} )^{2} \))(\( 1 + 4 T + 42 T^{2} + 92 T^{3} + 529 T^{4} \))(\( 1 + 4 T + 42 T^{2} + 92 T^{3} + 529 T^{4} \))(\( ( 1 - 4 T + 23 T^{2} )^{2} \))(\( 1 - 4 T + 42 T^{2} - 92 T^{3} + 529 T^{4} \))
$29$ (\( 1 - 6 T + 29 T^{2} \))(\( 1 - 2 T + 29 T^{2} \))(\( 1 + 3 T + 29 T^{2} \))(\( 1 + 4 T + 29 T^{2} \))(\( 1 + 4 T + 29 T^{2} \))(\( 1 + 10 T + 29 T^{2} \))(\( 1 + 3 T + 29 T^{2} \))(\( 1 - 2 T + 29 T^{2} \))(\( 1 - 6 T + 29 T^{2} \))(\( 1 - 14 T^{2} + 841 T^{4} \))(\( 1 - 3 T + 46 T^{2} - 87 T^{3} + 841 T^{4} \))(\( 1 + 50 T^{2} + 841 T^{4} \))(\( 1 + 50 T^{2} + 841 T^{4} \))(\( 1 - 3 T + 46 T^{2} - 87 T^{3} + 841 T^{4} \))(\( 1 - 14 T^{2} + 841 T^{4} \))
$31$ (\( 1 - 8 T + 31 T^{2} \))(\( 1 + 8 T + 31 T^{2} \))(\( 1 - 5 T + 31 T^{2} \))(\( 1 - 7 T + 31 T^{2} \))(\( 1 + 7 T + 31 T^{2} \))(\( 1 - 8 T + 31 T^{2} \))(\( 1 + 5 T + 31 T^{2} \))(\( 1 - 8 T + 31 T^{2} \))(\( 1 + 8 T + 31 T^{2} \))(\( 1 + 16 T + 118 T^{2} + 496 T^{3} + 961 T^{4} \))(\( ( 1 + T + 31 T^{2} )^{2} \))(\( 1 + 54 T^{2} + 961 T^{4} \))(\( 1 + 54 T^{2} + 961 T^{4} \))(\( ( 1 - T + 31 T^{2} )^{2} \))(\( 1 - 16 T + 118 T^{2} - 496 T^{3} + 961 T^{4} \))
$37$ (\( 1 + 10 T + 37 T^{2} \))(\( 1 + 6 T + 37 T^{2} \))(\( 1 - 8 T + 37 T^{2} \))(\( 1 + 9 T + 37 T^{2} \))(\( 1 + 9 T + 37 T^{2} \))(\( 1 - 6 T + 37 T^{2} \))(\( 1 - 8 T + 37 T^{2} \))(\( 1 + 6 T + 37 T^{2} \))(\( 1 - 6 T + 37 T^{2} \))(\( 1 + 8 T + 58 T^{2} + 296 T^{3} + 1369 T^{4} \))(\( 1 - 3 T + 62 T^{2} - 111 T^{3} + 1369 T^{4} \))(\( 1 - 8 T + 58 T^{2} - 296 T^{3} + 1369 T^{4} \))(\( 1 - 8 T + 58 T^{2} - 296 T^{3} + 1369 T^{4} \))(\( 1 - 3 T + 62 T^{2} - 111 T^{3} + 1369 T^{4} \))(\( 1 + 8 T + 58 T^{2} + 296 T^{3} + 1369 T^{4} \))
$41$ (\( 1 - 10 T + 41 T^{2} \))(\( 1 + 12 T + 41 T^{2} \))(\( 1 + 8 T + 41 T^{2} \))(\( 1 + 2 T + 41 T^{2} \))(\( 1 - 2 T + 41 T^{2} \))(\( 1 - 2 T + 41 T^{2} \))(\( 1 - 8 T + 41 T^{2} \))(\( 1 - 12 T + 41 T^{2} \))(\( 1 - 6 T + 41 T^{2} \))(\( 1 - 4 T + 84 T^{2} - 164 T^{3} + 1681 T^{4} \))(\( 1 - 6 T + 34 T^{2} - 246 T^{3} + 1681 T^{4} \))(\( 1 - 12 T + 100 T^{2} - 492 T^{3} + 1681 T^{4} \))(\( 1 + 12 T + 100 T^{2} + 492 T^{3} + 1681 T^{4} \))(\( 1 + 6 T + 34 T^{2} + 246 T^{3} + 1681 T^{4} \))(\( 1 + 4 T + 84 T^{2} + 164 T^{3} + 1681 T^{4} \))
$43$ (\( 1 - 12 T + 43 T^{2} \))(\( 1 - 4 T + 43 T^{2} \))(\( 1 - 6 T + 43 T^{2} \))(\( 1 + T + 43 T^{2} \))(\( 1 + T + 43 T^{2} \))(\( 1 + 4 T + 43 T^{2} \))(\( 1 - 6 T + 43 T^{2} \))(\( 1 - 4 T + 43 T^{2} \))(\( 1 - 4 T + 43 T^{2} \))(\( ( 1 + 8 T + 43 T^{2} )^{2} \))(\( 1 + 7 T + 84 T^{2} + 301 T^{3} + 1849 T^{4} \))(\( 1 - 42 T^{2} + 1849 T^{4} \))(\( 1 - 42 T^{2} + 1849 T^{4} \))(\( 1 + 7 T + 84 T^{2} + 301 T^{3} + 1849 T^{4} \))(\( ( 1 + 8 T + 43 T^{2} )^{2} \))
$47$ (\( 1 - 8 T + 47 T^{2} \))(\( 1 - 8 T + 47 T^{2} \))(\( 1 + 10 T + 47 T^{2} \))(\( 1 - 2 T + 47 T^{2} \))(\( 1 + 2 T + 47 T^{2} \))(\( 1 + 8 T + 47 T^{2} \))(\( 1 - 10 T + 47 T^{2} \))(\( 1 + 8 T + 47 T^{2} \))(\( 1 + 47 T^{2} \))(\( 1 + 8 T + 102 T^{2} + 376 T^{3} + 2209 T^{4} \))(\( ( 1 + 6 T + 47 T^{2} )^{2} \))(\( 1 - 8 T + 38 T^{2} - 376 T^{3} + 2209 T^{4} \))(\( 1 + 8 T + 38 T^{2} + 376 T^{3} + 2209 T^{4} \))(\( ( 1 - 6 T + 47 T^{2} )^{2} \))(\( 1 - 8 T + 102 T^{2} - 376 T^{3} + 2209 T^{4} \))
$53$ (\( 1 - 6 T + 53 T^{2} \))(\( 1 - 6 T + 53 T^{2} \))(\( 1 - 9 T + 53 T^{2} \))(\( 1 - 8 T + 53 T^{2} \))(\( 1 - 8 T + 53 T^{2} \))(\( 1 + 10 T + 53 T^{2} \))(\( 1 - 9 T + 53 T^{2} \))(\( 1 - 6 T + 53 T^{2} \))(\( 1 + 2 T + 53 T^{2} \))(\( 1 + 4 T - 18 T^{2} + 212 T^{3} + 2809 T^{4} \))(\( 1 - 11 T + 122 T^{2} - 583 T^{3} + 2809 T^{4} \))(\( ( 1 + 2 T + 53 T^{2} )^{2} \))(\( ( 1 + 2 T + 53 T^{2} )^{2} \))(\( 1 - 11 T + 122 T^{2} - 583 T^{3} + 2809 T^{4} \))(\( 1 + 4 T - 18 T^{2} + 212 T^{3} + 2809 T^{4} \))
$59$ (\( 1 + 4 T + 59 T^{2} \))(\( 1 + 12 T + 59 T^{2} \))(\( 1 - 5 T + 59 T^{2} \))(\( 1 + 59 T^{2} \))(\( 1 + 59 T^{2} \))(\( 1 + 12 T + 59 T^{2} \))(\( 1 + 5 T + 59 T^{2} \))(\( 1 - 12 T + 59 T^{2} \))(\( 1 + 4 T + 59 T^{2} \))(\( 1 + 16 T + 174 T^{2} + 944 T^{3} + 3481 T^{4} \))(\( 1 - 5 T + 110 T^{2} - 295 T^{3} + 3481 T^{4} \))(\( 1 + 46 T^{2} + 3481 T^{4} \))(\( 1 + 46 T^{2} + 3481 T^{4} \))(\( 1 + 5 T + 110 T^{2} + 295 T^{3} + 3481 T^{4} \))(\( 1 - 16 T + 174 T^{2} - 944 T^{3} + 3481 T^{4} \))
$61$ (\( 1 - 10 T + 61 T^{2} \))(\( 1 + 4 T + 61 T^{2} \))(\( 1 - 10 T + 61 T^{2} \))(\( 1 - 10 T + 61 T^{2} \))(\( 1 + 10 T + 61 T^{2} \))(\( 1 - 2 T + 61 T^{2} \))(\( 1 + 10 T + 61 T^{2} \))(\( 1 - 4 T + 61 T^{2} \))(\( 1 - 2 T + 61 T^{2} \))(\( 1 + 8 T + 40 T^{2} + 488 T^{3} + 3721 T^{4} \))(\( ( 1 - 10 T + 61 T^{2} )^{2} \))(\( 1 - 8 T + 88 T^{2} - 488 T^{3} + 3721 T^{4} \))(\( 1 + 8 T + 88 T^{2} + 488 T^{3} + 3721 T^{4} \))(\( ( 1 + 10 T + 61 T^{2} )^{2} \))(\( 1 - 8 T + 40 T^{2} - 488 T^{3} + 3721 T^{4} \))
$67$ (\( 1 - 12 T + 67 T^{2} \))(\( 1 + 4 T + 67 T^{2} \))(\( 1 - 6 T + 67 T^{2} \))(\( 1 + 15 T + 67 T^{2} \))(\( 1 + 15 T + 67 T^{2} \))(\( 1 - 12 T + 67 T^{2} \))(\( 1 - 6 T + 67 T^{2} \))(\( 1 + 4 T + 67 T^{2} \))(\( 1 + 4 T + 67 T^{2} \))(\( ( 1 + 8 T + 67 T^{2} )^{2} \))(\( 1 - 7 T + 132 T^{2} - 469 T^{3} + 4489 T^{4} \))(\( 1 + 6 T^{2} + 4489 T^{4} \))(\( 1 + 6 T^{2} + 4489 T^{4} \))(\( 1 - 7 T + 132 T^{2} - 469 T^{3} + 4489 T^{4} \))(\( ( 1 + 8 T + 67 T^{2} )^{2} \))
$71$ (\( 1 - 4 T + 71 T^{2} \))(\( 1 + 12 T + 71 T^{2} \))(\( 1 - 10 T + 71 T^{2} \))(\( 1 + 6 T + 71 T^{2} \))(\( 1 + 6 T + 71 T^{2} \))(\( 1 + 12 T + 71 T^{2} \))(\( 1 - 10 T + 71 T^{2} \))(\( 1 + 12 T + 71 T^{2} \))(\( 1 - 8 T + 71 T^{2} \))(\( 1 - 4 T + 138 T^{2} - 284 T^{3} + 5041 T^{4} \))(\( ( 1 - 2 T + 71 T^{2} )^{2} \))(\( 1 + 4 T + 74 T^{2} + 284 T^{3} + 5041 T^{4} \))(\( 1 + 4 T + 74 T^{2} + 284 T^{3} + 5041 T^{4} \))(\( ( 1 - 2 T + 71 T^{2} )^{2} \))(\( 1 - 4 T + 138 T^{2} - 284 T^{3} + 5041 T^{4} \))
$73$ (\( 1 + 2 T + 73 T^{2} \))(\( 1 + 8 T + 73 T^{2} \))(\( 1 + 2 T + 73 T^{2} \))(\( 1 + 11 T + 73 T^{2} \))(\( 1 - 11 T + 73 T^{2} \))(\( 1 - 14 T + 73 T^{2} \))(\( 1 - 2 T + 73 T^{2} \))(\( 1 - 8 T + 73 T^{2} \))(\( 1 + 10 T + 73 T^{2} \))(\( 1 - 8 T + 112 T^{2} - 584 T^{3} + 5329 T^{4} \))(\( 1 + T + 132 T^{2} + 73 T^{3} + 5329 T^{4} \))(\( 1 - 24 T + 272 T^{2} - 1752 T^{3} + 5329 T^{4} \))(\( 1 + 24 T + 272 T^{2} + 1752 T^{3} + 5329 T^{4} \))(\( 1 - T + 132 T^{2} - 73 T^{3} + 5329 T^{4} \))(\( 1 + 8 T + 112 T^{2} + 584 T^{3} + 5329 T^{4} \))
$79$ (\( 1 - 8 T + 79 T^{2} \))(\( 1 + 16 T + 79 T^{2} \))(\( 1 - 11 T + 79 T^{2} \))(\( 1 - T + 79 T^{2} \))(\( 1 - T + 79 T^{2} \))(\( 1 + 8 T + 79 T^{2} \))(\( 1 - 11 T + 79 T^{2} \))(\( 1 + 16 T + 79 T^{2} \))(\( 1 + 8 T + 79 T^{2} \))(\( 1 + 16 T + 190 T^{2} + 1264 T^{3} + 6241 T^{4} \))(\( 1 - 8 T + 117 T^{2} - 632 T^{3} + 6241 T^{4} \))(\( 1 - 16 T + 190 T^{2} - 1264 T^{3} + 6241 T^{4} \))(\( 1 - 16 T + 190 T^{2} - 1264 T^{3} + 6241 T^{4} \))(\( 1 - 8 T + 117 T^{2} - 632 T^{3} + 6241 T^{4} \))(\( 1 + 16 T + 190 T^{2} + 1264 T^{3} + 6241 T^{4} \))
$83$ (\( 1 + 4 T + 83 T^{2} \))(\( 1 - 4 T + 83 T^{2} \))(\( 1 + 7 T + 83 T^{2} \))(\( 1 - 6 T + 83 T^{2} \))(\( 1 + 6 T + 83 T^{2} \))(\( 1 + 12 T + 83 T^{2} \))(\( 1 - 7 T + 83 T^{2} \))(\( 1 + 4 T + 83 T^{2} \))(\( 1 - 4 T + 83 T^{2} \))(\( 1 + 8 T + 54 T^{2} + 664 T^{3} + 6889 T^{4} \))(\( 1 - 7 T + 164 T^{2} - 581 T^{3} + 6889 T^{4} \))(\( ( 1 + 4 T + 83 T^{2} )^{2} \))(\( ( 1 - 4 T + 83 T^{2} )^{2} \))(\( 1 + 7 T + 164 T^{2} + 581 T^{3} + 6889 T^{4} \))(\( 1 - 8 T + 54 T^{2} - 664 T^{3} + 6889 T^{4} \))
$89$ (\( 1 + 6 T + 89 T^{2} \))(\( 1 + 4 T + 89 T^{2} \))(\( 1 - 18 T + 89 T^{2} \))(\( 1 + 8 T + 89 T^{2} \))(\( 1 - 8 T + 89 T^{2} \))(\( 1 - 2 T + 89 T^{2} \))(\( 1 + 18 T + 89 T^{2} \))(\( 1 - 4 T + 89 T^{2} \))(\( 1 - 6 T + 89 T^{2} \))(\( 1 + 4 T + 20 T^{2} + 356 T^{3} + 7921 T^{4} \))(\( 1 + 6 T + 130 T^{2} + 534 T^{3} + 7921 T^{4} \))(\( 1 - 20 T + 260 T^{2} - 1780 T^{3} + 7921 T^{4} \))(\( 1 + 20 T + 260 T^{2} + 1780 T^{3} + 7921 T^{4} \))(\( 1 - 6 T + 130 T^{2} - 534 T^{3} + 7921 T^{4} \))(\( 1 - 4 T + 20 T^{2} - 356 T^{3} + 7921 T^{4} \))
$97$ (\( 1 + 10 T + 97 T^{2} \))(\( 1 + 16 T + 97 T^{2} \))(\( 1 - 17 T + 97 T^{2} \))(\( 1 + 14 T + 97 T^{2} \))(\( 1 - 14 T + 97 T^{2} \))(\( 1 + 10 T + 97 T^{2} \))(\( 1 + 17 T + 97 T^{2} \))(\( 1 - 16 T + 97 T^{2} \))(\( 1 + 2 T + 97 T^{2} \))(\( 1 - 24 T + 320 T^{2} - 2328 T^{3} + 9409 T^{4} \))(\( 1 - 25 T + 336 T^{2} - 2425 T^{3} + 9409 T^{4} \))(\( 1 - 8 T + 192 T^{2} - 776 T^{3} + 9409 T^{4} \))(\( 1 + 8 T + 192 T^{2} + 776 T^{3} + 9409 T^{4} \))(\( 1 + 25 T + 336 T^{2} + 2425 T^{3} + 9409 T^{4} \))(\( 1 + 24 T + 320 T^{2} + 2328 T^{3} + 9409 T^{4} \))
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