Properties

Label 1344.2.a
Level 1344
Weight 2
Character orbit a
Rep. character \(\chi_{1344}(1,\cdot)\)
Character field \(\Q\)
Dimension 24
Newform subspaces 22
Sturm bound 512
Trace bound 13

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Defining parameters

Level: \( N \) \(=\) \( 1344 = 2^{6} \cdot 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1344.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 22 \)
Sturm bound: \(512\)
Trace bound: \(13\)
Distinguishing \(T_p\): \(5\), \(11\), \(13\), \(19\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(1344))\).

Total New Old
Modular forms 280 24 256
Cusp forms 233 24 209
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.
\(+\)\(+\)\(+\)\(+\)\(2\)
\(+\)\(+\)\(-\)\(-\)\(3\)
\(+\)\(-\)\(+\)\(-\)\(4\)
\(+\)\(-\)\(-\)\(+\)\(1\)
\(-\)\(+\)\(+\)\(-\)\(4\)
\(-\)\(+\)\(-\)\(+\)\(3\)
\(-\)\(-\)\(+\)\(+\)\(2\)
\(-\)\(-\)\(-\)\(-\)\(5\)
Plus space\(+\)\(8\)
Minus space\(-\)\(16\)

Trace form

\( 24q + 24q^{9} + O(q^{10}) \) \( 24q + 24q^{9} + 16q^{17} + 40q^{25} + 16q^{29} + 16q^{37} + 16q^{41} + 24q^{49} + 16q^{53} + 32q^{61} + 32q^{69} - 16q^{73} + 16q^{77} + 24q^{81} + 32q^{85} - 16q^{89} + 16q^{97} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(1344))\) 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
1344.2.a.a \(1\) \(10.732\) \(\Q\) None \(0\) \(-1\) \(-4\) \(1\) \(-\) \(+\) \(-\) \(q-q^{3}-4q^{5}+q^{7}+q^{9}+2q^{11}+6q^{13}+\cdots\)
1344.2.a.b \(1\) \(10.732\) \(\Q\) None \(0\) \(-1\) \(-2\) \(-1\) \(-\) \(+\) \(+\) \(q-q^{3}-2q^{5}-q^{7}+q^{9}-6q^{13}+2q^{15}+\cdots\)
1344.2.a.c \(1\) \(10.732\) \(\Q\) None \(0\) \(-1\) \(-2\) \(-1\) \(+\) \(+\) \(+\) \(q-q^{3}-2q^{5}-q^{7}+q^{9}+2q^{13}+2q^{15}+\cdots\)
1344.2.a.d \(1\) \(10.732\) \(\Q\) None \(0\) \(-1\) \(-2\) \(1\) \(-\) \(+\) \(-\) \(q-q^{3}-2q^{5}+q^{7}+q^{9}-2q^{13}+2q^{15}+\cdots\)
1344.2.a.e \(1\) \(10.732\) \(\Q\) None \(0\) \(-1\) \(0\) \(1\) \(+\) \(+\) \(-\) \(q-q^{3}+q^{7}+q^{9}-2q^{11}+2q^{13}+\cdots\)
1344.2.a.f \(1\) \(10.732\) \(\Q\) None \(0\) \(-1\) \(0\) \(1\) \(+\) \(+\) \(-\) \(q-q^{3}+q^{7}+q^{9}+6q^{11}-2q^{13}+\cdots\)
1344.2.a.g \(1\) \(10.732\) \(\Q\) None \(0\) \(-1\) \(2\) \(-1\) \(+\) \(+\) \(+\) \(q-q^{3}+2q^{5}-q^{7}+q^{9}-4q^{11}+2q^{13}+\cdots\)
1344.2.a.h \(1\) \(10.732\) \(\Q\) None \(0\) \(-1\) \(2\) \(-1\) \(-\) \(+\) \(+\) \(q-q^{3}+2q^{5}-q^{7}+q^{9}+4q^{11}+6q^{13}+\cdots\)
1344.2.a.i \(1\) \(10.732\) \(\Q\) None \(0\) \(-1\) \(2\) \(1\) \(-\) \(+\) \(-\) \(q-q^{3}+2q^{5}+q^{7}+q^{9}-4q^{11}-6q^{13}+\cdots\)
1344.2.a.j \(1\) \(10.732\) \(\Q\) None \(0\) \(-1\) \(4\) \(1\) \(+\) \(+\) \(-\) \(q-q^{3}+4q^{5}+q^{7}+q^{9}+2q^{11}+2q^{13}+\cdots\)
1344.2.a.k \(1\) \(10.732\) \(\Q\) None \(0\) \(1\) \(-4\) \(-1\) \(+\) \(-\) \(+\) \(q+q^{3}-4q^{5}-q^{7}+q^{9}-2q^{11}+6q^{13}+\cdots\)
1344.2.a.l \(1\) \(10.732\) \(\Q\) None \(0\) \(1\) \(-2\) \(-1\) \(-\) \(-\) \(+\) \(q+q^{3}-2q^{5}-q^{7}+q^{9}-2q^{13}-2q^{15}+\cdots\)
1344.2.a.m \(1\) \(10.732\) \(\Q\) None \(0\) \(1\) \(-2\) \(1\) \(+\) \(-\) \(-\) \(q+q^{3}-2q^{5}+q^{7}+q^{9}-6q^{13}-2q^{15}+\cdots\)
1344.2.a.n \(1\) \(10.732\) \(\Q\) None \(0\) \(1\) \(-2\) \(1\) \(-\) \(-\) \(-\) \(q+q^{3}-2q^{5}+q^{7}+q^{9}+2q^{13}-2q^{15}+\cdots\)
1344.2.a.o \(1\) \(10.732\) \(\Q\) None \(0\) \(1\) \(0\) \(-1\) \(-\) \(-\) \(+\) \(q+q^{3}-q^{7}+q^{9}-6q^{11}-2q^{13}+\cdots\)
1344.2.a.p \(1\) \(10.732\) \(\Q\) None \(0\) \(1\) \(0\) \(-1\) \(+\) \(-\) \(+\) \(q+q^{3}-q^{7}+q^{9}+2q^{11}+2q^{13}+\cdots\)
1344.2.a.q \(1\) \(10.732\) \(\Q\) None \(0\) \(1\) \(2\) \(-1\) \(+\) \(-\) \(+\) \(q+q^{3}+2q^{5}-q^{7}+q^{9}+4q^{11}-6q^{13}+\cdots\)
1344.2.a.r \(1\) \(10.732\) \(\Q\) None \(0\) \(1\) \(2\) \(1\) \(-\) \(-\) \(-\) \(q+q^{3}+2q^{5}+q^{7}+q^{9}-4q^{11}+6q^{13}+\cdots\)
1344.2.a.s \(1\) \(10.732\) \(\Q\) None \(0\) \(1\) \(2\) \(1\) \(-\) \(-\) \(-\) \(q+q^{3}+2q^{5}+q^{7}+q^{9}+4q^{11}+2q^{13}+\cdots\)
1344.2.a.t \(1\) \(10.732\) \(\Q\) None \(0\) \(1\) \(4\) \(-1\) \(+\) \(-\) \(+\) \(q+q^{3}+4q^{5}-q^{7}+q^{9}-2q^{11}+2q^{13}+\cdots\)
1344.2.a.u \(2\) \(10.732\) \(\Q(\sqrt{3}) \) None \(0\) \(-2\) \(0\) \(-2\) \(-\) \(+\) \(+\) \(q-q^{3}+\beta q^{5}-q^{7}+q^{9}+(-2-\beta )q^{11}+\cdots\)
1344.2.a.v \(2\) \(10.732\) \(\Q(\sqrt{3}) \) None \(0\) \(2\) \(0\) \(2\) \(-\) \(-\) \(-\) \(q+q^{3}+\beta q^{5}+q^{7}+q^{9}+(2+\beta )q^{11}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(1344))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(1344)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(14))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(21))\)\(^{\oplus 7}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(24))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(32))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(42))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(48))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(56))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(64))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(84))\)\(^{\oplus 5}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(96))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(112))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(168))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(192))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(224))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(336))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(448))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(672))\)\(^{\oplus 2}\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ (\( 1 + T \))(\( 1 + T \))(\( 1 + T \))(\( 1 + T \))(\( 1 + T \))(\( 1 + T \))(\( 1 + T \))(\( 1 + T \))(\( 1 + T \))(\( 1 + T \))(\( 1 - T \))(\( 1 - T \))(\( 1 - T \))(\( 1 - T \))(\( 1 - T \))(\( 1 - T \))(\( 1 - T \))(\( 1 - T \))(\( 1 - T \))(\( 1 - T \))(\( ( 1 + T )^{2} \))(\( ( 1 - T )^{2} \))
$5$ (\( 1 + 4 T + 5 T^{2} \))(\( 1 + 2 T + 5 T^{2} \))(\( 1 + 2 T + 5 T^{2} \))(\( 1 + 2 T + 5 T^{2} \))(\( 1 + 5 T^{2} \))(\( 1 + 5 T^{2} \))(\( 1 - 2 T + 5 T^{2} \))(\( 1 - 2 T + 5 T^{2} \))(\( 1 - 2 T + 5 T^{2} \))(\( 1 - 4 T + 5 T^{2} \))(\( 1 + 4 T + 5 T^{2} \))(\( 1 + 2 T + 5 T^{2} \))(\( 1 + 2 T + 5 T^{2} \))(\( 1 + 2 T + 5 T^{2} \))(\( 1 + 5 T^{2} \))(\( 1 + 5 T^{2} \))(\( 1 - 2 T + 5 T^{2} \))(\( 1 - 2 T + 5 T^{2} \))(\( 1 - 2 T + 5 T^{2} \))(\( 1 - 4 T + 5 T^{2} \))(\( 1 - 2 T^{2} + 25 T^{4} \))(\( 1 - 2 T^{2} + 25 T^{4} \))
$7$ (\( 1 - T \))(\( 1 + T \))(\( 1 + T \))(\( 1 - T \))(\( 1 - T \))(\( 1 - T \))(\( 1 + T \))(\( 1 + T \))(\( 1 - T \))(\( 1 - T \))(\( 1 + T \))(\( 1 + T \))(\( 1 - T \))(\( 1 - T \))(\( 1 + T \))(\( 1 + T \))(\( 1 + T \))(\( 1 - T \))(\( 1 - T \))(\( 1 + T \))(\( ( 1 + T )^{2} \))(\( ( 1 - T )^{2} \))
$11$ (\( 1 - 2 T + 11 T^{2} \))(\( 1 + 11 T^{2} \))(\( 1 + 11 T^{2} \))(\( 1 + 11 T^{2} \))(\( 1 + 2 T + 11 T^{2} \))(\( 1 - 6 T + 11 T^{2} \))(\( 1 + 4 T + 11 T^{2} \))(\( 1 - 4 T + 11 T^{2} \))(\( 1 + 4 T + 11 T^{2} \))(\( 1 - 2 T + 11 T^{2} \))(\( 1 + 2 T + 11 T^{2} \))(\( 1 + 11 T^{2} \))(\( 1 + 11 T^{2} \))(\( 1 + 11 T^{2} \))(\( 1 + 6 T + 11 T^{2} \))(\( 1 - 2 T + 11 T^{2} \))(\( 1 - 4 T + 11 T^{2} \))(\( 1 + 4 T + 11 T^{2} \))(\( 1 - 4 T + 11 T^{2} \))(\( 1 + 2 T + 11 T^{2} \))(\( 1 + 4 T + 14 T^{2} + 44 T^{3} + 121 T^{4} \))(\( 1 - 4 T + 14 T^{2} - 44 T^{3} + 121 T^{4} \))
$13$ (\( 1 - 6 T + 13 T^{2} \))(\( 1 + 6 T + 13 T^{2} \))(\( 1 - 2 T + 13 T^{2} \))(\( 1 + 2 T + 13 T^{2} \))(\( 1 - 2 T + 13 T^{2} \))(\( 1 + 2 T + 13 T^{2} \))(\( 1 - 2 T + 13 T^{2} \))(\( 1 - 6 T + 13 T^{2} \))(\( 1 + 6 T + 13 T^{2} \))(\( 1 - 2 T + 13 T^{2} \))(\( 1 - 6 T + 13 T^{2} \))(\( 1 + 2 T + 13 T^{2} \))(\( 1 + 6 T + 13 T^{2} \))(\( 1 - 2 T + 13 T^{2} \))(\( 1 + 2 T + 13 T^{2} \))(\( 1 - 2 T + 13 T^{2} \))(\( 1 + 6 T + 13 T^{2} \))(\( 1 - 6 T + 13 T^{2} \))(\( 1 - 2 T + 13 T^{2} \))(\( 1 - 2 T + 13 T^{2} \))(\( ( 1 + 2 T + 13 T^{2} )^{2} \))(\( ( 1 + 2 T + 13 T^{2} )^{2} \))
$17$ (\( 1 + 4 T + 17 T^{2} \))(\( 1 + 2 T + 17 T^{2} \))(\( 1 - 6 T + 17 T^{2} \))(\( 1 - 2 T + 17 T^{2} \))(\( 1 - 4 T + 17 T^{2} \))(\( 1 + 17 T^{2} \))(\( 1 + 6 T + 17 T^{2} \))(\( 1 + 2 T + 17 T^{2} \))(\( 1 - 2 T + 17 T^{2} \))(\( 1 + 17 T^{2} \))(\( 1 + 4 T + 17 T^{2} \))(\( 1 - 2 T + 17 T^{2} \))(\( 1 + 2 T + 17 T^{2} \))(\( 1 - 6 T + 17 T^{2} \))(\( 1 + 17 T^{2} \))(\( 1 - 4 T + 17 T^{2} \))(\( 1 - 2 T + 17 T^{2} \))(\( 1 + 2 T + 17 T^{2} \))(\( 1 + 6 T + 17 T^{2} \))(\( 1 + 17 T^{2} \))(\( 1 - 8 T + 38 T^{2} - 136 T^{3} + 289 T^{4} \))(\( 1 - 8 T + 38 T^{2} - 136 T^{3} + 289 T^{4} \))
$19$ (\( 1 + 4 T + 19 T^{2} \))(\( 1 - 4 T + 19 T^{2} \))(\( 1 - 4 T + 19 T^{2} \))(\( 1 - 4 T + 19 T^{2} \))(\( 1 + 4 T + 19 T^{2} \))(\( 1 - 4 T + 19 T^{2} \))(\( 1 + 4 T + 19 T^{2} \))(\( 1 + 4 T + 19 T^{2} \))(\( 1 + 4 T + 19 T^{2} \))(\( 1 - 4 T + 19 T^{2} \))(\( 1 - 4 T + 19 T^{2} \))(\( 1 + 4 T + 19 T^{2} \))(\( 1 + 4 T + 19 T^{2} \))(\( 1 + 4 T + 19 T^{2} \))(\( 1 + 4 T + 19 T^{2} \))(\( 1 - 4 T + 19 T^{2} \))(\( 1 - 4 T + 19 T^{2} \))(\( 1 - 4 T + 19 T^{2} \))(\( 1 - 4 T + 19 T^{2} \))(\( 1 + 4 T + 19 T^{2} \))(\( 1 - 10 T^{2} + 361 T^{4} \))(\( 1 - 10 T^{2} + 361 T^{4} \))
$23$ (\( 1 + 2 T + 23 T^{2} \))(\( 1 - 4 T + 23 T^{2} \))(\( 1 + 4 T + 23 T^{2} \))(\( 1 + 23 T^{2} \))(\( 1 - 6 T + 23 T^{2} \))(\( 1 + 6 T + 23 T^{2} \))(\( 1 + 23 T^{2} \))(\( 1 + 4 T + 23 T^{2} \))(\( 1 + 8 T + 23 T^{2} \))(\( 1 + 6 T + 23 T^{2} \))(\( 1 - 2 T + 23 T^{2} \))(\( 1 + 23 T^{2} \))(\( 1 + 4 T + 23 T^{2} \))(\( 1 - 4 T + 23 T^{2} \))(\( 1 - 6 T + 23 T^{2} \))(\( 1 + 6 T + 23 T^{2} \))(\( 1 - 8 T + 23 T^{2} \))(\( 1 - 4 T + 23 T^{2} \))(\( 1 + 23 T^{2} \))(\( 1 - 6 T + 23 T^{2} \))(\( 1 - 4 T + 38 T^{2} - 92 T^{3} + 529 T^{4} \))(\( 1 + 4 T + 38 T^{2} + 92 T^{3} + 529 T^{4} \))
$29$ (\( 1 - 2 T + 29 T^{2} \))(\( 1 - 10 T + 29 T^{2} \))(\( 1 + 6 T + 29 T^{2} \))(\( 1 + 6 T + 29 T^{2} \))(\( 1 - 2 T + 29 T^{2} \))(\( 1 + 6 T + 29 T^{2} \))(\( 1 - 2 T + 29 T^{2} \))(\( 1 - 2 T + 29 T^{2} \))(\( 1 - 2 T + 29 T^{2} \))(\( 1 - 10 T + 29 T^{2} \))(\( 1 - 2 T + 29 T^{2} \))(\( 1 + 6 T + 29 T^{2} \))(\( 1 - 10 T + 29 T^{2} \))(\( 1 + 6 T + 29 T^{2} \))(\( 1 + 6 T + 29 T^{2} \))(\( 1 - 2 T + 29 T^{2} \))(\( 1 - 2 T + 29 T^{2} \))(\( 1 - 2 T + 29 T^{2} \))(\( 1 - 2 T + 29 T^{2} \))(\( 1 - 10 T + 29 T^{2} \))(\( 1 + 4 T + 14 T^{2} + 116 T^{3} + 841 T^{4} \))(\( 1 + 4 T + 14 T^{2} + 116 T^{3} + 841 T^{4} \))
$31$ (\( 1 + 31 T^{2} \))(\( 1 - 8 T + 31 T^{2} \))(\( 1 + 8 T + 31 T^{2} \))(\( 1 + 31 T^{2} \))(\( 1 + 31 T^{2} \))(\( 1 - 8 T + 31 T^{2} \))(\( 1 + 31 T^{2} \))(\( 1 - 8 T + 31 T^{2} \))(\( 1 + 31 T^{2} \))(\( 1 + 8 T + 31 T^{2} \))(\( 1 + 31 T^{2} \))(\( 1 + 31 T^{2} \))(\( 1 + 8 T + 31 T^{2} \))(\( 1 - 8 T + 31 T^{2} \))(\( 1 + 8 T + 31 T^{2} \))(\( 1 + 31 T^{2} \))(\( 1 + 31 T^{2} \))(\( 1 + 8 T + 31 T^{2} \))(\( 1 + 31 T^{2} \))(\( 1 - 8 T + 31 T^{2} \))(\( 1 + 8 T + 30 T^{2} + 248 T^{3} + 961 T^{4} \))(\( 1 - 8 T + 30 T^{2} - 248 T^{3} + 961 T^{4} \))
$37$ (\( 1 + 2 T + 37 T^{2} \))(\( 1 + 6 T + 37 T^{2} \))(\( 1 - 10 T + 37 T^{2} \))(\( 1 + 6 T + 37 T^{2} \))(\( 1 - 6 T + 37 T^{2} \))(\( 1 + 2 T + 37 T^{2} \))(\( 1 + 6 T + 37 T^{2} \))(\( 1 - 10 T + 37 T^{2} \))(\( 1 - 10 T + 37 T^{2} \))(\( 1 + 10 T + 37 T^{2} \))(\( 1 + 2 T + 37 T^{2} \))(\( 1 + 6 T + 37 T^{2} \))(\( 1 + 6 T + 37 T^{2} \))(\( 1 - 10 T + 37 T^{2} \))(\( 1 + 2 T + 37 T^{2} \))(\( 1 - 6 T + 37 T^{2} \))(\( 1 - 10 T + 37 T^{2} \))(\( 1 - 10 T + 37 T^{2} \))(\( 1 + 6 T + 37 T^{2} \))(\( 1 + 10 T + 37 T^{2} \))(\( ( 1 - 2 T + 37 T^{2} )^{2} \))(\( ( 1 - 2 T + 37 T^{2} )^{2} \))
$41$ (\( 1 + 41 T^{2} \))(\( 1 + 2 T + 41 T^{2} \))(\( 1 + 10 T + 41 T^{2} \))(\( 1 + 6 T + 41 T^{2} \))(\( 1 - 8 T + 41 T^{2} \))(\( 1 - 12 T + 41 T^{2} \))(\( 1 - 2 T + 41 T^{2} \))(\( 1 + 2 T + 41 T^{2} \))(\( 1 + 6 T + 41 T^{2} \))(\( 1 + 4 T + 41 T^{2} \))(\( 1 + 41 T^{2} \))(\( 1 + 6 T + 41 T^{2} \))(\( 1 + 2 T + 41 T^{2} \))(\( 1 + 10 T + 41 T^{2} \))(\( 1 - 12 T + 41 T^{2} \))(\( 1 - 8 T + 41 T^{2} \))(\( 1 + 6 T + 41 T^{2} \))(\( 1 + 2 T + 41 T^{2} \))(\( 1 - 2 T + 41 T^{2} \))(\( 1 + 4 T + 41 T^{2} \))(\( 1 - 16 T + 134 T^{2} - 656 T^{3} + 1681 T^{4} \))(\( 1 - 16 T + 134 T^{2} - 656 T^{3} + 1681 T^{4} \))
$43$ (\( 1 + 4 T + 43 T^{2} \))(\( 1 + 4 T + 43 T^{2} \))(\( 1 + 12 T + 43 T^{2} \))(\( 1 - 8 T + 43 T^{2} \))(\( 1 + 8 T + 43 T^{2} \))(\( 1 - 4 T + 43 T^{2} \))(\( 1 - 4 T + 43 T^{2} \))(\( 1 + 8 T + 43 T^{2} \))(\( 1 + 4 T + 43 T^{2} \))(\( 1 - 8 T + 43 T^{2} \))(\( 1 - 4 T + 43 T^{2} \))(\( 1 + 8 T + 43 T^{2} \))(\( 1 - 4 T + 43 T^{2} \))(\( 1 - 12 T + 43 T^{2} \))(\( 1 + 4 T + 43 T^{2} \))(\( 1 - 8 T + 43 T^{2} \))(\( 1 - 4 T + 43 T^{2} \))(\( 1 - 8 T + 43 T^{2} \))(\( 1 + 4 T + 43 T^{2} \))(\( 1 + 8 T + 43 T^{2} \))(\( ( 1 - 8 T + 43 T^{2} )^{2} \))(\( ( 1 + 8 T + 43 T^{2} )^{2} \))
$47$ (\( 1 + 12 T + 47 T^{2} \))(\( 1 + 8 T + 47 T^{2} \))(\( 1 + 8 T + 47 T^{2} \))(\( 1 + 8 T + 47 T^{2} \))(\( 1 - 4 T + 47 T^{2} \))(\( 1 - 12 T + 47 T^{2} \))(\( 1 + 47 T^{2} \))(\( 1 + 47 T^{2} \))(\( 1 + 47 T^{2} \))(\( 1 + 4 T + 47 T^{2} \))(\( 1 - 12 T + 47 T^{2} \))(\( 1 - 8 T + 47 T^{2} \))(\( 1 - 8 T + 47 T^{2} \))(\( 1 - 8 T + 47 T^{2} \))(\( 1 + 12 T + 47 T^{2} \))(\( 1 + 4 T + 47 T^{2} \))(\( 1 + 47 T^{2} \))(\( 1 + 47 T^{2} \))(\( 1 + 47 T^{2} \))(\( 1 - 4 T + 47 T^{2} \))(\( 1 - 8 T + 62 T^{2} - 376 T^{3} + 2209 T^{4} \))(\( 1 + 8 T + 62 T^{2} + 376 T^{3} + 2209 T^{4} \))
$53$ (\( 1 - 6 T + 53 T^{2} \))(\( 1 - 10 T + 53 T^{2} \))(\( 1 + 6 T + 53 T^{2} \))(\( 1 + 6 T + 53 T^{2} \))(\( 1 - 6 T + 53 T^{2} \))(\( 1 - 6 T + 53 T^{2} \))(\( 1 + 6 T + 53 T^{2} \))(\( 1 - 10 T + 53 T^{2} \))(\( 1 + 6 T + 53 T^{2} \))(\( 1 + 10 T + 53 T^{2} \))(\( 1 - 6 T + 53 T^{2} \))(\( 1 + 6 T + 53 T^{2} \))(\( 1 - 10 T + 53 T^{2} \))(\( 1 + 6 T + 53 T^{2} \))(\( 1 - 6 T + 53 T^{2} \))(\( 1 - 6 T + 53 T^{2} \))(\( 1 + 6 T + 53 T^{2} \))(\( 1 - 10 T + 53 T^{2} \))(\( 1 + 6 T + 53 T^{2} \))(\( 1 + 10 T + 53 T^{2} \))(\( ( 1 - 2 T + 53 T^{2} )^{2} \))(\( ( 1 - 2 T + 53 T^{2} )^{2} \))
$59$ (\( 1 + 8 T + 59 T^{2} \))(\( 1 - 12 T + 59 T^{2} \))(\( 1 + 4 T + 59 T^{2} \))(\( 1 + 12 T + 59 T^{2} \))(\( 1 + 59 T^{2} \))(\( 1 + 59 T^{2} \))(\( 1 + 12 T + 59 T^{2} \))(\( 1 - 12 T + 59 T^{2} \))(\( 1 - 4 T + 59 T^{2} \))(\( 1 + 8 T + 59 T^{2} \))(\( 1 - 8 T + 59 T^{2} \))(\( 1 - 12 T + 59 T^{2} \))(\( 1 + 12 T + 59 T^{2} \))(\( 1 - 4 T + 59 T^{2} \))(\( 1 + 59 T^{2} \))(\( 1 + 59 T^{2} \))(\( 1 + 4 T + 59 T^{2} \))(\( 1 + 12 T + 59 T^{2} \))(\( 1 - 12 T + 59 T^{2} \))(\( 1 - 8 T + 59 T^{2} \))(\( 1 - 16 T + 134 T^{2} - 944 T^{3} + 3481 T^{4} \))(\( 1 + 16 T + 134 T^{2} + 944 T^{3} + 3481 T^{4} \))
$61$ (\( 1 + 6 T + 61 T^{2} \))(\( 1 - 2 T + 61 T^{2} \))(\( 1 - 10 T + 61 T^{2} \))(\( 1 + 10 T + 61 T^{2} \))(\( 1 - 14 T + 61 T^{2} \))(\( 1 - 10 T + 61 T^{2} \))(\( 1 - 2 T + 61 T^{2} \))(\( 1 + 10 T + 61 T^{2} \))(\( 1 + 6 T + 61 T^{2} \))(\( 1 - 6 T + 61 T^{2} \))(\( 1 + 6 T + 61 T^{2} \))(\( 1 + 10 T + 61 T^{2} \))(\( 1 - 2 T + 61 T^{2} \))(\( 1 - 10 T + 61 T^{2} \))(\( 1 - 10 T + 61 T^{2} \))(\( 1 - 14 T + 61 T^{2} \))(\( 1 + 6 T + 61 T^{2} \))(\( 1 + 10 T + 61 T^{2} \))(\( 1 - 2 T + 61 T^{2} \))(\( 1 - 6 T + 61 T^{2} \))(\( 1 - 4 T + 78 T^{2} - 244 T^{3} + 3721 T^{4} \))(\( 1 - 4 T + 78 T^{2} - 244 T^{3} + 3721 T^{4} \))
$67$ (\( 1 + 8 T + 67 T^{2} \))(\( 1 - 12 T + 67 T^{2} \))(\( 1 + 12 T + 67 T^{2} \))(\( 1 - 16 T + 67 T^{2} \))(\( 1 - 4 T + 67 T^{2} \))(\( 1 + 8 T + 67 T^{2} \))(\( 1 + 4 T + 67 T^{2} \))(\( 1 - 8 T + 67 T^{2} \))(\( 1 - 4 T + 67 T^{2} \))(\( 1 + 4 T + 67 T^{2} \))(\( 1 - 8 T + 67 T^{2} \))(\( 1 + 16 T + 67 T^{2} \))(\( 1 + 12 T + 67 T^{2} \))(\( 1 - 12 T + 67 T^{2} \))(\( 1 - 8 T + 67 T^{2} \))(\( 1 + 4 T + 67 T^{2} \))(\( 1 + 4 T + 67 T^{2} \))(\( 1 + 8 T + 67 T^{2} \))(\( 1 - 4 T + 67 T^{2} \))(\( 1 - 4 T + 67 T^{2} \))(\( 1 + 8 T + 102 T^{2} + 536 T^{3} + 4489 T^{4} \))(\( 1 - 8 T + 102 T^{2} - 536 T^{3} + 4489 T^{4} \))
$71$ (\( 1 + 14 T + 71 T^{2} \))(\( 1 - 12 T + 71 T^{2} \))(\( 1 - 4 T + 71 T^{2} \))(\( 1 - 8 T + 71 T^{2} \))(\( 1 - 2 T + 71 T^{2} \))(\( 1 - 6 T + 71 T^{2} \))(\( 1 + 71 T^{2} \))(\( 1 - 12 T + 71 T^{2} \))(\( 1 + 8 T + 71 T^{2} \))(\( 1 - 14 T + 71 T^{2} \))(\( 1 - 14 T + 71 T^{2} \))(\( 1 + 8 T + 71 T^{2} \))(\( 1 + 12 T + 71 T^{2} \))(\( 1 + 4 T + 71 T^{2} \))(\( 1 + 6 T + 71 T^{2} \))(\( 1 + 2 T + 71 T^{2} \))(\( 1 - 8 T + 71 T^{2} \))(\( 1 + 12 T + 71 T^{2} \))(\( 1 + 71 T^{2} \))(\( 1 + 14 T + 71 T^{2} \))(\( 1 - 12 T + 166 T^{2} - 852 T^{3} + 5041 T^{4} \))(\( 1 + 12 T + 166 T^{2} + 852 T^{3} + 5041 T^{4} \))
$73$ (\( 1 + 2 T + 73 T^{2} \))(\( 1 + 14 T + 73 T^{2} \))(\( 1 - 2 T + 73 T^{2} \))(\( 1 + 6 T + 73 T^{2} \))(\( 1 + 2 T + 73 T^{2} \))(\( 1 + 10 T + 73 T^{2} \))(\( 1 + 6 T + 73 T^{2} \))(\( 1 - 2 T + 73 T^{2} \))(\( 1 - 10 T + 73 T^{2} \))(\( 1 - 6 T + 73 T^{2} \))(\( 1 + 2 T + 73 T^{2} \))(\( 1 + 6 T + 73 T^{2} \))(\( 1 + 14 T + 73 T^{2} \))(\( 1 - 2 T + 73 T^{2} \))(\( 1 + 10 T + 73 T^{2} \))(\( 1 + 2 T + 73 T^{2} \))(\( 1 - 10 T + 73 T^{2} \))(\( 1 - 2 T + 73 T^{2} \))(\( 1 + 6 T + 73 T^{2} \))(\( 1 - 6 T + 73 T^{2} \))(\( 1 - 12 T + 134 T^{2} - 876 T^{3} + 5329 T^{4} \))(\( 1 - 12 T + 134 T^{2} - 876 T^{3} + 5329 T^{4} \))
$79$ (\( 1 + 12 T + 79 T^{2} \))(\( 1 - 8 T + 79 T^{2} \))(\( 1 - 8 T + 79 T^{2} \))(\( 1 + 8 T + 79 T^{2} \))(\( 1 + 4 T + 79 T^{2} \))(\( 1 + 4 T + 79 T^{2} \))(\( 1 + 16 T + 79 T^{2} \))(\( 1 + 79 T^{2} \))(\( 1 + 79 T^{2} \))(\( 1 - 4 T + 79 T^{2} \))(\( 1 - 12 T + 79 T^{2} \))(\( 1 - 8 T + 79 T^{2} \))(\( 1 + 8 T + 79 T^{2} \))(\( 1 + 8 T + 79 T^{2} \))(\( 1 - 4 T + 79 T^{2} \))(\( 1 - 4 T + 79 T^{2} \))(\( 1 + 79 T^{2} \))(\( 1 + 79 T^{2} \))(\( 1 - 16 T + 79 T^{2} \))(\( 1 + 4 T + 79 T^{2} \))(\( 1 + 8 T + 126 T^{2} + 632 T^{3} + 6241 T^{4} \))(\( 1 - 8 T + 126 T^{2} - 632 T^{3} + 6241 T^{4} \))
$83$ (\( 1 + 4 T + 83 T^{2} \))(\( 1 - 12 T + 83 T^{2} \))(\( 1 + 4 T + 83 T^{2} \))(\( 1 + 12 T + 83 T^{2} \))(\( 1 - 12 T + 83 T^{2} \))(\( 1 - 12 T + 83 T^{2} \))(\( 1 - 12 T + 83 T^{2} \))(\( 1 + 12 T + 83 T^{2} \))(\( 1 + 4 T + 83 T^{2} \))(\( 1 - 12 T + 83 T^{2} \))(\( 1 - 4 T + 83 T^{2} \))(\( 1 - 12 T + 83 T^{2} \))(\( 1 + 12 T + 83 T^{2} \))(\( 1 - 4 T + 83 T^{2} \))(\( 1 + 12 T + 83 T^{2} \))(\( 1 + 12 T + 83 T^{2} \))(\( 1 - 4 T + 83 T^{2} \))(\( 1 - 12 T + 83 T^{2} \))(\( 1 + 12 T + 83 T^{2} \))(\( 1 + 12 T + 83 T^{2} \))(\( ( 1 - 4 T + 83 T^{2} )^{2} \))(\( ( 1 + 4 T + 83 T^{2} )^{2} \))
$89$ (\( 1 + 89 T^{2} \))(\( 1 + 2 T + 89 T^{2} \))(\( 1 - 6 T + 89 T^{2} \))(\( 1 + 14 T + 89 T^{2} \))(\( 1 + 89 T^{2} \))(\( 1 - 12 T + 89 T^{2} \))(\( 1 + 14 T + 89 T^{2} \))(\( 1 - 6 T + 89 T^{2} \))(\( 1 + 6 T + 89 T^{2} \))(\( 1 - 4 T + 89 T^{2} \))(\( 1 + 89 T^{2} \))(\( 1 + 14 T + 89 T^{2} \))(\( 1 + 2 T + 89 T^{2} \))(\( 1 - 6 T + 89 T^{2} \))(\( 1 - 12 T + 89 T^{2} \))(\( 1 + 89 T^{2} \))(\( 1 + 6 T + 89 T^{2} \))(\( 1 - 6 T + 89 T^{2} \))(\( 1 + 14 T + 89 T^{2} \))(\( 1 - 4 T + 89 T^{2} \))(\( 1 + 166 T^{2} + 7921 T^{4} \))(\( 1 + 166 T^{2} + 7921 T^{4} \))
$97$ (\( 1 + 2 T + 97 T^{2} \))(\( 1 - 10 T + 97 T^{2} \))(\( 1 - 10 T + 97 T^{2} \))(\( 1 + 6 T + 97 T^{2} \))(\( 1 - 6 T + 97 T^{2} \))(\( 1 + 10 T + 97 T^{2} \))(\( 1 - 18 T + 97 T^{2} \))(\( 1 - 2 T + 97 T^{2} \))(\( 1 + 14 T + 97 T^{2} \))(\( 1 + 2 T + 97 T^{2} \))(\( 1 + 2 T + 97 T^{2} \))(\( 1 + 6 T + 97 T^{2} \))(\( 1 - 10 T + 97 T^{2} \))(\( 1 - 10 T + 97 T^{2} \))(\( 1 + 10 T + 97 T^{2} \))(\( 1 - 6 T + 97 T^{2} \))(\( 1 + 14 T + 97 T^{2} \))(\( 1 - 2 T + 97 T^{2} \))(\( 1 - 18 T + 97 T^{2} \))(\( 1 + 2 T + 97 T^{2} \))(\( 1 + 4 T + 150 T^{2} + 388 T^{3} + 9409 T^{4} \))(\( 1 + 4 T + 150 T^{2} + 388 T^{3} + 9409 T^{4} \))
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