Properties

Label 2304.2.a
Level $2304$
Weight $2$
Character orbit 2304.a
Rep. character $\chi_{2304}(1,\cdot)$
Character field $\Q$
Dimension $38$
Newform subspaces $26$
Sturm bound $768$
Trace bound $19$

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Defining parameters

Level: \( N \) \(=\) \( 2304 = 2^{8} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2304.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 26 \)
Sturm bound: \(768\)
Trace bound: \(19\)
Distinguishing \(T_p\): \(5\), \(7\), \(11\), \(13\), \(19\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(2304))\).

Total New Old
Modular forms 432 42 390
Cusp forms 337 38 299
Eisenstein series 95 4 91

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.
\(+\)\(+\)\(+\)\(6\)
\(+\)\(-\)\(-\)\(12\)
\(-\)\(+\)\(-\)\(10\)
\(-\)\(-\)\(+\)\(10\)
Plus space\(+\)\(16\)
Minus space\(-\)\(22\)

Trace form

\( 38q + O(q^{10}) \) \( 38q + 4q^{17} + 34q^{25} - 4q^{41} + 54q^{49} + 52q^{73} + 12q^{89} + 28q^{97} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(2304))\) into newform subspaces

Label Dim. \(A\) Field CM Traces A-L signs $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\) 2 3
2304.2.a.a \(1\) \(18.398\) \(\Q\) \(\Q(\sqrt{-1}) \) \(0\) \(0\) \(-4\) \(0\) \(-\) \(-\) \(q-4q^{5}+4q^{13}+2q^{17}+11q^{25}+\cdots\)
2304.2.a.b \(1\) \(18.398\) \(\Q\) None \(0\) \(0\) \(-2\) \(-2\) \(-\) \(-\) \(q-2q^{5}-2q^{7}+4q^{13}+2q^{17}+4q^{19}+\cdots\)
2304.2.a.c \(1\) \(18.398\) \(\Q\) \(\Q(\sqrt{-1}) \) \(0\) \(0\) \(-2\) \(0\) \(+\) \(+\) \(q-2q^{5}-4q^{13}+8q^{17}-q^{25}+10q^{29}+\cdots\)
2304.2.a.d \(1\) \(18.398\) \(\Q\) \(\Q(\sqrt{-1}) \) \(0\) \(0\) \(-2\) \(0\) \(-\) \(+\) \(q-2q^{5}+4q^{13}-8q^{17}-q^{25}+10q^{29}+\cdots\)
2304.2.a.e \(1\) \(18.398\) \(\Q\) None \(0\) \(0\) \(-2\) \(2\) \(+\) \(-\) \(q-2q^{5}+2q^{7}+4q^{13}+2q^{17}-4q^{19}+\cdots\)
2304.2.a.f \(1\) \(18.398\) \(\Q\) None \(0\) \(0\) \(0\) \(-4\) \(+\) \(-\) \(q-4q^{7}-4q^{11}+4q^{13}+2q^{17}-4q^{19}+\cdots\)
2304.2.a.g \(1\) \(18.398\) \(\Q\) None \(0\) \(0\) \(0\) \(-4\) \(-\) \(-\) \(q-4q^{7}+4q^{11}-4q^{13}+2q^{17}+4q^{19}+\cdots\)
2304.2.a.h \(1\) \(18.398\) \(\Q\) \(\Q(\sqrt{-2}) \) \(0\) \(0\) \(0\) \(0\) \(-\) \(-\) \(q-6q^{11}+6q^{17}+2q^{19}-5q^{25}+\cdots\)
2304.2.a.i \(1\) \(18.398\) \(\Q\) \(\Q(\sqrt{-2}) \) \(0\) \(0\) \(0\) \(0\) \(+\) \(-\) \(q+6q^{11}+6q^{17}-2q^{19}-5q^{25}+\cdots\)
2304.2.a.j \(1\) \(18.398\) \(\Q\) None \(0\) \(0\) \(0\) \(4\) \(-\) \(-\) \(q+4q^{7}-4q^{11}-4q^{13}+2q^{17}-4q^{19}+\cdots\)
2304.2.a.k \(1\) \(18.398\) \(\Q\) None \(0\) \(0\) \(0\) \(4\) \(+\) \(-\) \(q+4q^{7}+4q^{11}+4q^{13}+2q^{17}+4q^{19}+\cdots\)
2304.2.a.l \(1\) \(18.398\) \(\Q\) None \(0\) \(0\) \(2\) \(-2\) \(-\) \(-\) \(q+2q^{5}-2q^{7}-4q^{13}+2q^{17}-4q^{19}+\cdots\)
2304.2.a.m \(1\) \(18.398\) \(\Q\) \(\Q(\sqrt{-1}) \) \(0\) \(0\) \(2\) \(0\) \(+\) \(+\) \(q+2q^{5}-4q^{13}-8q^{17}-q^{25}-10q^{29}+\cdots\)
2304.2.a.n \(1\) \(18.398\) \(\Q\) \(\Q(\sqrt{-1}) \) \(0\) \(0\) \(2\) \(0\) \(-\) \(+\) \(q+2q^{5}+4q^{13}+8q^{17}-q^{25}-10q^{29}+\cdots\)
2304.2.a.o \(1\) \(18.398\) \(\Q\) None \(0\) \(0\) \(2\) \(2\) \(+\) \(-\) \(q+2q^{5}+2q^{7}-4q^{13}+2q^{17}+4q^{19}+\cdots\)
2304.2.a.p \(1\) \(18.398\) \(\Q\) \(\Q(\sqrt{-1}) \) \(0\) \(0\) \(4\) \(0\) \(+\) \(-\) \(q+4q^{5}-4q^{13}+2q^{17}+11q^{25}+\cdots\)
2304.2.a.q \(2\) \(18.398\) \(\Q(\sqrt{2}) \) \(\Q(\sqrt{-6}) \) \(0\) \(0\) \(0\) \(-4\) \(+\) \(+\) \(q+\beta q^{5}-2q^{7}-2\beta q^{11}+3q^{25}-\beta q^{29}+\cdots\)
2304.2.a.r \(2\) \(18.398\) \(\Q(\sqrt{2}) \) None \(0\) \(0\) \(0\) \(0\) \(+\) \(-\) \(q+\beta q^{5}-\beta q^{7}-4q^{11}+2\beta q^{13}+2q^{17}+\cdots\)
2304.2.a.s \(2\) \(18.398\) \(\Q(\sqrt{3}) \) None \(0\) \(0\) \(0\) \(0\) \(-\) \(-\) \(q+\beta q^{5}-\beta q^{7}-6q^{17}-4q^{19}-2\beta q^{23}+\cdots\)
2304.2.a.t \(2\) \(18.398\) \(\Q(\sqrt{2}) \) \(\Q(\sqrt{-2}) \) \(0\) \(0\) \(0\) \(0\) \(-\) \(-\) \(q+\beta q^{11}-6q^{17}-3\beta q^{19}-5q^{25}+\cdots\)
2304.2.a.u \(2\) \(18.398\) \(\Q(\sqrt{3}) \) None \(0\) \(0\) \(0\) \(0\) \(+\) \(-\) \(q+\beta q^{5}+\beta q^{7}-6q^{17}+4q^{19}+2\beta q^{23}+\cdots\)
2304.2.a.v \(2\) \(18.398\) \(\Q(\sqrt{3}) \) \(\Q(\sqrt{-3}) \) \(0\) \(0\) \(0\) \(0\) \(+\) \(+\) \(q+\beta q^{7}-2\beta q^{13}-8q^{19}-5q^{25}+\cdots\)
2304.2.a.w \(2\) \(18.398\) \(\Q(\sqrt{3}) \) \(\Q(\sqrt{-3}) \) \(0\) \(0\) \(0\) \(0\) \(-\) \(+\) \(q-\beta q^{7}-2\beta q^{13}+8q^{19}-5q^{25}+\cdots\)
2304.2.a.x \(2\) \(18.398\) \(\Q(\sqrt{2}) \) None \(0\) \(0\) \(0\) \(0\) \(+\) \(-\) \(q+\beta q^{5}+\beta q^{7}+4q^{11}+2\beta q^{13}+2q^{17}+\cdots\)
2304.2.a.y \(2\) \(18.398\) \(\Q(\sqrt{2}) \) \(\Q(\sqrt{-6}) \) \(0\) \(0\) \(0\) \(4\) \(-\) \(+\) \(q+\beta q^{5}+2q^{7}+2\beta q^{11}+3q^{25}-\beta q^{29}+\cdots\)
2304.2.a.z \(4\) \(18.398\) \(\Q(\zeta_{24})^+\) \(\Q(\sqrt{-6}) \) \(0\) \(0\) \(0\) \(0\) \(-\) \(+\) \(q+\beta _{2}q^{5}+\beta _{1}q^{7}+\beta _{3}q^{11}+7q^{25}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(2304))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(2304)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(24))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(32))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(36))\)\(^{\oplus 7}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(48))\)\(^{\oplus 10}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(64))\)\(^{\oplus 9}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(72))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(96))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(128))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(144))\)\(^{\oplus 5}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(192))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(256))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(288))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(384))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(576))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(768))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(1152))\)\(^{\oplus 2}\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ 1
$5$ (\( 1 + 4 T + 5 T^{2} \))(\( 1 + 2 T + 5 T^{2} \))(\( 1 + 2 T + 5 T^{2} \))(\( 1 + 2 T + 5 T^{2} \))(\( 1 + 2 T + 5 T^{2} \))(\( 1 + 5 T^{2} \))(\( 1 + 5 T^{2} \))(\( 1 + 5 T^{2} \))(\( 1 + 5 T^{2} \))(\( 1 + 5 T^{2} \))(\( 1 + 5 T^{2} \))(\( 1 - 2 T + 5 T^{2} \))(\( 1 - 2 T + 5 T^{2} \))(\( 1 - 2 T + 5 T^{2} \))(\( 1 - 2 T + 5 T^{2} \))(\( 1 - 4 T + 5 T^{2} \))(\( 1 + 2 T^{2} + 25 T^{4} \))(\( 1 + 2 T^{2} + 25 T^{4} \))(\( 1 - 2 T^{2} + 25 T^{4} \))(\( ( 1 + 5 T^{2} )^{2} \))(\( 1 - 2 T^{2} + 25 T^{4} \))(\( ( 1 + 5 T^{2} )^{2} \))(\( ( 1 + 5 T^{2} )^{2} \))(\( 1 + 2 T^{2} + 25 T^{4} \))(\( 1 + 2 T^{2} + 25 T^{4} \))(\( ( 1 - 2 T^{2} + 25 T^{4} )^{2} \))
$7$ (\( 1 + 7 T^{2} \))(\( 1 + 2 T + 7 T^{2} \))(\( 1 + 7 T^{2} \))(\( 1 + 7 T^{2} \))(\( 1 - 2 T + 7 T^{2} \))(\( 1 + 4 T + 7 T^{2} \))(\( 1 + 4 T + 7 T^{2} \))(\( 1 + 7 T^{2} \))(\( 1 + 7 T^{2} \))(\( 1 - 4 T + 7 T^{2} \))(\( 1 - 4 T + 7 T^{2} \))(\( 1 + 2 T + 7 T^{2} \))(\( 1 + 7 T^{2} \))(\( 1 + 7 T^{2} \))(\( 1 - 2 T + 7 T^{2} \))(\( 1 + 7 T^{2} \))(\( ( 1 + 2 T + 7 T^{2} )^{2} \))(\( 1 + 6 T^{2} + 49 T^{4} \))(\( 1 + 2 T^{2} + 49 T^{4} \))(\( ( 1 + 7 T^{2} )^{2} \))(\( 1 + 2 T^{2} + 49 T^{4} \))(\( 1 + 2 T^{2} + 49 T^{4} \))(\( 1 + 2 T^{2} + 49 T^{4} \))(\( 1 + 6 T^{2} + 49 T^{4} \))(\( ( 1 - 2 T + 7 T^{2} )^{2} \))(\( ( 1 - 10 T^{2} + 49 T^{4} )^{2} \))
$11$ (\( 1 + 11 T^{2} \))(\( 1 + 11 T^{2} \))(\( 1 + 11 T^{2} \))(\( 1 + 11 T^{2} \))(\( 1 + 11 T^{2} \))(\( 1 + 4 T + 11 T^{2} \))(\( 1 - 4 T + 11 T^{2} \))(\( 1 + 6 T + 11 T^{2} \))(\( 1 - 6 T + 11 T^{2} \))(\( 1 + 4 T + 11 T^{2} \))(\( 1 - 4 T + 11 T^{2} \))(\( 1 + 11 T^{2} \))(\( 1 + 11 T^{2} \))(\( 1 + 11 T^{2} \))(\( 1 + 11 T^{2} \))(\( 1 + 11 T^{2} \))(\( 1 - 10 T^{2} + 121 T^{4} \))(\( ( 1 + 4 T + 11 T^{2} )^{2} \))(\( ( 1 + 11 T^{2} )^{2} \))(\( 1 + 14 T^{2} + 121 T^{4} \))(\( ( 1 + 11 T^{2} )^{2} \))(\( ( 1 + 11 T^{2} )^{2} \))(\( ( 1 + 11 T^{2} )^{2} \))(\( ( 1 - 4 T + 11 T^{2} )^{2} \))(\( 1 - 10 T^{2} + 121 T^{4} \))(\( ( 1 - 10 T^{2} + 121 T^{4} )^{2} \))
$13$ (\( 1 - 4 T + 13 T^{2} \))(\( 1 - 4 T + 13 T^{2} \))(\( 1 + 4 T + 13 T^{2} \))(\( 1 - 4 T + 13 T^{2} \))(\( 1 - 4 T + 13 T^{2} \))(\( 1 - 4 T + 13 T^{2} \))(\( 1 + 4 T + 13 T^{2} \))(\( 1 + 13 T^{2} \))(\( 1 + 13 T^{2} \))(\( 1 + 4 T + 13 T^{2} \))(\( 1 - 4 T + 13 T^{2} \))(\( 1 + 4 T + 13 T^{2} \))(\( 1 + 4 T + 13 T^{2} \))(\( 1 - 4 T + 13 T^{2} \))(\( 1 + 4 T + 13 T^{2} \))(\( 1 + 4 T + 13 T^{2} \))(\( ( 1 + 13 T^{2} )^{2} \))(\( 1 - 6 T^{2} + 169 T^{4} \))(\( ( 1 + 13 T^{2} )^{2} \))(\( ( 1 + 13 T^{2} )^{2} \))(\( ( 1 + 13 T^{2} )^{2} \))(\( 1 - 22 T^{2} + 169 T^{4} \))(\( 1 - 22 T^{2} + 169 T^{4} \))(\( 1 - 6 T^{2} + 169 T^{4} \))(\( ( 1 + 13 T^{2} )^{2} \))(\( ( 1 + 13 T^{2} )^{4} \))
$17$ (\( 1 - 2 T + 17 T^{2} \))(\( 1 - 2 T + 17 T^{2} \))(\( 1 - 8 T + 17 T^{2} \))(\( 1 + 8 T + 17 T^{2} \))(\( 1 - 2 T + 17 T^{2} \))(\( 1 - 2 T + 17 T^{2} \))(\( 1 - 2 T + 17 T^{2} \))(\( 1 - 6 T + 17 T^{2} \))(\( 1 - 6 T + 17 T^{2} \))(\( 1 - 2 T + 17 T^{2} \))(\( 1 - 2 T + 17 T^{2} \))(\( 1 - 2 T + 17 T^{2} \))(\( 1 + 8 T + 17 T^{2} \))(\( 1 - 8 T + 17 T^{2} \))(\( 1 - 2 T + 17 T^{2} \))(\( 1 - 2 T + 17 T^{2} \))(\( ( 1 + 17 T^{2} )^{2} \))(\( ( 1 - 2 T + 17 T^{2} )^{2} \))(\( ( 1 + 6 T + 17 T^{2} )^{2} \))(\( ( 1 + 6 T + 17 T^{2} )^{2} \))(\( ( 1 + 6 T + 17 T^{2} )^{2} \))(\( ( 1 + 17 T^{2} )^{2} \))(\( ( 1 + 17 T^{2} )^{2} \))(\( ( 1 - 2 T + 17 T^{2} )^{2} \))(\( ( 1 + 17 T^{2} )^{2} \))(\( ( 1 + 17 T^{2} )^{4} \))
$19$ (\( 1 + 19 T^{2} \))(\( 1 - 4 T + 19 T^{2} \))(\( 1 + 19 T^{2} \))(\( 1 + 19 T^{2} \))(\( 1 + 4 T + 19 T^{2} \))(\( 1 + 4 T + 19 T^{2} \))(\( 1 - 4 T + 19 T^{2} \))(\( 1 - 2 T + 19 T^{2} \))(\( 1 + 2 T + 19 T^{2} \))(\( 1 + 4 T + 19 T^{2} \))(\( 1 - 4 T + 19 T^{2} \))(\( 1 + 4 T + 19 T^{2} \))(\( 1 + 19 T^{2} \))(\( 1 + 19 T^{2} \))(\( 1 - 4 T + 19 T^{2} \))(\( 1 + 19 T^{2} \))(\( ( 1 + 19 T^{2} )^{2} \))(\( ( 1 - 4 T + 19 T^{2} )^{2} \))(\( ( 1 + 4 T + 19 T^{2} )^{2} \))(\( 1 - 34 T^{2} + 361 T^{4} \))(\( ( 1 - 4 T + 19 T^{2} )^{2} \))(\( ( 1 + 8 T + 19 T^{2} )^{2} \))(\( ( 1 - 8 T + 19 T^{2} )^{2} \))(\( ( 1 + 4 T + 19 T^{2} )^{2} \))(\( ( 1 + 19 T^{2} )^{2} \))(\( ( 1 + 19 T^{2} )^{4} \))
$23$ (\( 1 + 23 T^{2} \))(\( 1 + 4 T + 23 T^{2} \))(\( 1 + 23 T^{2} \))(\( 1 + 23 T^{2} \))(\( 1 - 4 T + 23 T^{2} \))(\( 1 - 8 T + 23 T^{2} \))(\( 1 - 8 T + 23 T^{2} \))(\( 1 + 23 T^{2} \))(\( 1 + 23 T^{2} \))(\( 1 + 8 T + 23 T^{2} \))(\( 1 + 8 T + 23 T^{2} \))(\( 1 + 4 T + 23 T^{2} \))(\( 1 + 23 T^{2} \))(\( 1 + 23 T^{2} \))(\( 1 - 4 T + 23 T^{2} \))(\( 1 + 23 T^{2} \))(\( ( 1 + 23 T^{2} )^{2} \))(\( 1 + 14 T^{2} + 529 T^{4} \))(\( 1 - 2 T^{2} + 529 T^{4} \))(\( ( 1 + 23 T^{2} )^{2} \))(\( 1 - 2 T^{2} + 529 T^{4} \))(\( ( 1 + 23 T^{2} )^{2} \))(\( ( 1 + 23 T^{2} )^{2} \))(\( 1 + 14 T^{2} + 529 T^{4} \))(\( ( 1 + 23 T^{2} )^{2} \))(\( ( 1 + 23 T^{2} )^{4} \))
$29$ (\( 1 + 4 T + 29 T^{2} \))(\( 1 + 6 T + 29 T^{2} \))(\( 1 - 10 T + 29 T^{2} \))(\( 1 - 10 T + 29 T^{2} \))(\( 1 + 6 T + 29 T^{2} \))(\( 1 - 8 T + 29 T^{2} \))(\( 1 + 8 T + 29 T^{2} \))(\( 1 + 29 T^{2} \))(\( 1 + 29 T^{2} \))(\( 1 + 8 T + 29 T^{2} \))(\( 1 - 8 T + 29 T^{2} \))(\( 1 - 6 T + 29 T^{2} \))(\( 1 + 10 T + 29 T^{2} \))(\( 1 + 10 T + 29 T^{2} \))(\( 1 - 6 T + 29 T^{2} \))(\( 1 - 4 T + 29 T^{2} \))(\( 1 + 50 T^{2} + 841 T^{4} \))(\( 1 + 50 T^{2} + 841 T^{4} \))(\( 1 + 46 T^{2} + 841 T^{4} \))(\( ( 1 + 29 T^{2} )^{2} \))(\( 1 + 46 T^{2} + 841 T^{4} \))(\( ( 1 + 29 T^{2} )^{2} \))(\( ( 1 + 29 T^{2} )^{2} \))(\( 1 + 50 T^{2} + 841 T^{4} \))(\( 1 + 50 T^{2} + 841 T^{4} \))(\( ( 1 - 50 T^{2} + 841 T^{4} )^{2} \))
$31$ (\( 1 + 31 T^{2} \))(\( 1 + 2 T + 31 T^{2} \))(\( 1 + 31 T^{2} \))(\( 1 + 31 T^{2} \))(\( 1 - 2 T + 31 T^{2} \))(\( 1 + 4 T + 31 T^{2} \))(\( 1 + 4 T + 31 T^{2} \))(\( 1 + 31 T^{2} \))(\( 1 + 31 T^{2} \))(\( 1 - 4 T + 31 T^{2} \))(\( 1 - 4 T + 31 T^{2} \))(\( 1 + 2 T + 31 T^{2} \))(\( 1 + 31 T^{2} \))(\( 1 + 31 T^{2} \))(\( 1 - 2 T + 31 T^{2} \))(\( 1 + 31 T^{2} \))(\( ( 1 + 10 T + 31 T^{2} )^{2} \))(\( 1 - 10 T^{2} + 961 T^{4} \))(\( 1 + 50 T^{2} + 961 T^{4} \))(\( ( 1 + 31 T^{2} )^{2} \))(\( 1 + 50 T^{2} + 961 T^{4} \))(\( 1 - 46 T^{2} + 961 T^{4} \))(\( 1 - 46 T^{2} + 961 T^{4} \))(\( 1 - 10 T^{2} + 961 T^{4} \))(\( ( 1 - 10 T + 31 T^{2} )^{2} \))(\( ( 1 + 38 T^{2} + 961 T^{4} )^{2} \))
$37$ (\( 1 + 12 T + 37 T^{2} \))(\( 1 - 8 T + 37 T^{2} \))(\( 1 + 12 T + 37 T^{2} \))(\( 1 - 12 T + 37 T^{2} \))(\( 1 - 8 T + 37 T^{2} \))(\( 1 + 4 T + 37 T^{2} \))(\( 1 - 4 T + 37 T^{2} \))(\( 1 + 37 T^{2} \))(\( 1 + 37 T^{2} \))(\( 1 - 4 T + 37 T^{2} \))(\( 1 + 4 T + 37 T^{2} \))(\( 1 + 8 T + 37 T^{2} \))(\( 1 + 12 T + 37 T^{2} \))(\( 1 - 12 T + 37 T^{2} \))(\( 1 + 8 T + 37 T^{2} \))(\( 1 - 12 T + 37 T^{2} \))(\( ( 1 + 37 T^{2} )^{2} \))(\( ( 1 + 37 T^{2} )^{2} \))(\( 1 + 26 T^{2} + 1369 T^{4} \))(\( ( 1 + 37 T^{2} )^{2} \))(\( 1 + 26 T^{2} + 1369 T^{4} \))(\( 1 + 26 T^{2} + 1369 T^{4} \))(\( 1 + 26 T^{2} + 1369 T^{4} \))(\( ( 1 + 37 T^{2} )^{2} \))(\( ( 1 + 37 T^{2} )^{2} \))(\( ( 1 + 37 T^{2} )^{4} \))
$41$ (\( 1 - 10 T + 41 T^{2} \))(\( 1 - 2 T + 41 T^{2} \))(\( 1 + 8 T + 41 T^{2} \))(\( 1 - 8 T + 41 T^{2} \))(\( 1 - 2 T + 41 T^{2} \))(\( 1 + 6 T + 41 T^{2} \))(\( 1 + 6 T + 41 T^{2} \))(\( 1 + 6 T + 41 T^{2} \))(\( 1 + 6 T + 41 T^{2} \))(\( 1 + 6 T + 41 T^{2} \))(\( 1 + 6 T + 41 T^{2} \))(\( 1 - 2 T + 41 T^{2} \))(\( 1 - 8 T + 41 T^{2} \))(\( 1 + 8 T + 41 T^{2} \))(\( 1 - 2 T + 41 T^{2} \))(\( 1 - 10 T + 41 T^{2} \))(\( ( 1 + 41 T^{2} )^{2} \))(\( ( 1 - 10 T + 41 T^{2} )^{2} \))(\( ( 1 + 6 T + 41 T^{2} )^{2} \))(\( ( 1 + 6 T + 41 T^{2} )^{2} \))(\( ( 1 + 6 T + 41 T^{2} )^{2} \))(\( ( 1 + 41 T^{2} )^{2} \))(\( ( 1 + 41 T^{2} )^{2} \))(\( ( 1 - 10 T + 41 T^{2} )^{2} \))(\( ( 1 + 41 T^{2} )^{2} \))(\( ( 1 + 41 T^{2} )^{4} \))
$43$ (\( 1 + 43 T^{2} \))(\( 1 - 4 T + 43 T^{2} \))(\( 1 + 43 T^{2} \))(\( 1 + 43 T^{2} \))(\( 1 + 4 T + 43 T^{2} \))(\( 1 - 4 T + 43 T^{2} \))(\( 1 + 4 T + 43 T^{2} \))(\( 1 + 10 T + 43 T^{2} \))(\( 1 - 10 T + 43 T^{2} \))(\( 1 - 4 T + 43 T^{2} \))(\( 1 + 4 T + 43 T^{2} \))(\( 1 + 4 T + 43 T^{2} \))(\( 1 + 43 T^{2} \))(\( 1 + 43 T^{2} \))(\( 1 - 4 T + 43 T^{2} \))(\( 1 + 43 T^{2} \))(\( ( 1 + 43 T^{2} )^{2} \))(\( ( 1 - 12 T + 43 T^{2} )^{2} \))(\( ( 1 + 4 T + 43 T^{2} )^{2} \))(\( 1 + 14 T^{2} + 1849 T^{4} \))(\( ( 1 - 4 T + 43 T^{2} )^{2} \))(\( ( 1 + 8 T + 43 T^{2} )^{2} \))(\( ( 1 - 8 T + 43 T^{2} )^{2} \))(\( ( 1 + 12 T + 43 T^{2} )^{2} \))(\( ( 1 + 43 T^{2} )^{2} \))(\( ( 1 + 43 T^{2} )^{4} \))
$47$ (\( 1 + 47 T^{2} \))(\( 1 + 12 T + 47 T^{2} \))(\( 1 + 47 T^{2} \))(\( 1 + 47 T^{2} \))(\( 1 - 12 T + 47 T^{2} \))(\( 1 - 8 T + 47 T^{2} \))(\( 1 - 8 T + 47 T^{2} \))(\( 1 + 47 T^{2} \))(\( 1 + 47 T^{2} \))(\( 1 + 8 T + 47 T^{2} \))(\( 1 + 8 T + 47 T^{2} \))(\( 1 + 12 T + 47 T^{2} \))(\( 1 + 47 T^{2} \))(\( 1 + 47 T^{2} \))(\( 1 - 12 T + 47 T^{2} \))(\( 1 + 47 T^{2} \))(\( ( 1 + 47 T^{2} )^{2} \))(\( 1 + 62 T^{2} + 2209 T^{4} \))(\( 1 + 46 T^{2} + 2209 T^{4} \))(\( ( 1 + 47 T^{2} )^{2} \))(\( 1 + 46 T^{2} + 2209 T^{4} \))(\( ( 1 + 47 T^{2} )^{2} \))(\( ( 1 + 47 T^{2} )^{2} \))(\( 1 + 62 T^{2} + 2209 T^{4} \))(\( ( 1 + 47 T^{2} )^{2} \))(\( ( 1 + 47 T^{2} )^{4} \))
$53$ (\( 1 + 4 T + 53 T^{2} \))(\( 1 + 6 T + 53 T^{2} \))(\( 1 + 14 T + 53 T^{2} \))(\( 1 + 14 T + 53 T^{2} \))(\( 1 + 6 T + 53 T^{2} \))(\( 1 - 8 T + 53 T^{2} \))(\( 1 + 8 T + 53 T^{2} \))(\( 1 + 53 T^{2} \))(\( 1 + 53 T^{2} \))(\( 1 + 8 T + 53 T^{2} \))(\( 1 - 8 T + 53 T^{2} \))(\( 1 - 6 T + 53 T^{2} \))(\( 1 - 14 T + 53 T^{2} \))(\( 1 - 14 T + 53 T^{2} \))(\( 1 - 6 T + 53 T^{2} \))(\( 1 - 4 T + 53 T^{2} \))(\( 1 - 94 T^{2} + 2809 T^{4} \))(\( 1 + 98 T^{2} + 2809 T^{4} \))(\( 1 + 94 T^{2} + 2809 T^{4} \))(\( ( 1 + 53 T^{2} )^{2} \))(\( 1 + 94 T^{2} + 2809 T^{4} \))(\( ( 1 + 53 T^{2} )^{2} \))(\( ( 1 + 53 T^{2} )^{2} \))(\( 1 + 98 T^{2} + 2809 T^{4} \))(\( 1 - 94 T^{2} + 2809 T^{4} \))(\( ( 1 + 94 T^{2} + 2809 T^{4} )^{2} \))
$59$ (\( 1 + 59 T^{2} \))(\( 1 - 4 T + 59 T^{2} \))(\( 1 + 59 T^{2} \))(\( 1 + 59 T^{2} \))(\( 1 + 4 T + 59 T^{2} \))(\( 1 - 12 T + 59 T^{2} \))(\( 1 + 12 T + 59 T^{2} \))(\( 1 + 6 T + 59 T^{2} \))(\( 1 - 6 T + 59 T^{2} \))(\( 1 - 12 T + 59 T^{2} \))(\( 1 + 12 T + 59 T^{2} \))(\( 1 + 4 T + 59 T^{2} \))(\( 1 + 59 T^{2} \))(\( 1 + 59 T^{2} \))(\( 1 - 4 T + 59 T^{2} \))(\( 1 + 59 T^{2} \))(\( 1 - 10 T^{2} + 3481 T^{4} \))(\( ( 1 - 4 T + 59 T^{2} )^{2} \))(\( ( 1 + 12 T + 59 T^{2} )^{2} \))(\( 1 - 82 T^{2} + 3481 T^{4} \))(\( ( 1 - 12 T + 59 T^{2} )^{2} \))(\( ( 1 + 59 T^{2} )^{2} \))(\( ( 1 + 59 T^{2} )^{2} \))(\( ( 1 + 4 T + 59 T^{2} )^{2} \))(\( 1 - 10 T^{2} + 3481 T^{4} \))(\( ( 1 - 10 T^{2} + 3481 T^{4} )^{2} \))
$61$ (\( 1 + 12 T + 61 T^{2} \))(\( 1 + 61 T^{2} \))(\( 1 + 12 T + 61 T^{2} \))(\( 1 - 12 T + 61 T^{2} \))(\( 1 + 61 T^{2} \))(\( 1 - 12 T + 61 T^{2} \))(\( 1 + 12 T + 61 T^{2} \))(\( 1 + 61 T^{2} \))(\( 1 + 61 T^{2} \))(\( 1 + 12 T + 61 T^{2} \))(\( 1 - 12 T + 61 T^{2} \))(\( 1 + 61 T^{2} \))(\( 1 + 12 T + 61 T^{2} \))(\( 1 - 12 T + 61 T^{2} \))(\( 1 + 61 T^{2} \))(\( 1 - 12 T + 61 T^{2} \))(\( ( 1 + 61 T^{2} )^{2} \))(\( 1 - 6 T^{2} + 3721 T^{4} \))(\( 1 + 74 T^{2} + 3721 T^{4} \))(\( ( 1 + 61 T^{2} )^{2} \))(\( 1 + 74 T^{2} + 3721 T^{4} \))(\( 1 + 74 T^{2} + 3721 T^{4} \))(\( 1 + 74 T^{2} + 3721 T^{4} \))(\( 1 - 6 T^{2} + 3721 T^{4} \))(\( ( 1 + 61 T^{2} )^{2} \))(\( ( 1 + 61 T^{2} )^{4} \))
$67$ (\( 1 + 67 T^{2} \))(\( 1 + 12 T + 67 T^{2} \))(\( 1 + 67 T^{2} \))(\( 1 + 67 T^{2} \))(\( 1 - 12 T + 67 T^{2} \))(\( 1 - 12 T + 67 T^{2} \))(\( 1 + 12 T + 67 T^{2} \))(\( 1 + 14 T + 67 T^{2} \))(\( 1 - 14 T + 67 T^{2} \))(\( 1 - 12 T + 67 T^{2} \))(\( 1 + 12 T + 67 T^{2} \))(\( 1 - 12 T + 67 T^{2} \))(\( 1 + 67 T^{2} \))(\( 1 + 67 T^{2} \))(\( 1 + 12 T + 67 T^{2} \))(\( 1 + 67 T^{2} \))(\( ( 1 + 67 T^{2} )^{2} \))(\( ( 1 - 4 T + 67 T^{2} )^{2} \))(\( ( 1 - 4 T + 67 T^{2} )^{2} \))(\( 1 + 62 T^{2} + 4489 T^{4} \))(\( ( 1 + 4 T + 67 T^{2} )^{2} \))(\( ( 1 + 16 T + 67 T^{2} )^{2} \))(\( ( 1 - 16 T + 67 T^{2} )^{2} \))(\( ( 1 + 4 T + 67 T^{2} )^{2} \))(\( ( 1 + 67 T^{2} )^{2} \))(\( ( 1 + 67 T^{2} )^{4} \))
$71$ (\( 1 + 71 T^{2} \))(\( 1 + 12 T + 71 T^{2} \))(\( 1 + 71 T^{2} \))(\( 1 + 71 T^{2} \))(\( 1 - 12 T + 71 T^{2} \))(\( 1 + 8 T + 71 T^{2} \))(\( 1 + 8 T + 71 T^{2} \))(\( 1 + 71 T^{2} \))(\( 1 + 71 T^{2} \))(\( 1 - 8 T + 71 T^{2} \))(\( 1 - 8 T + 71 T^{2} \))(\( 1 + 12 T + 71 T^{2} \))(\( 1 + 71 T^{2} \))(\( 1 + 71 T^{2} \))(\( 1 - 12 T + 71 T^{2} \))(\( 1 + 71 T^{2} \))(\( ( 1 + 71 T^{2} )^{2} \))(\( 1 + 110 T^{2} + 5041 T^{4} \))(\( 1 + 94 T^{2} + 5041 T^{4} \))(\( ( 1 + 71 T^{2} )^{2} \))(\( 1 + 94 T^{2} + 5041 T^{4} \))(\( ( 1 + 71 T^{2} )^{2} \))(\( ( 1 + 71 T^{2} )^{2} \))(\( 1 + 110 T^{2} + 5041 T^{4} \))(\( ( 1 + 71 T^{2} )^{2} \))(\( ( 1 + 71 T^{2} )^{4} \))
$73$ (\( 1 + 6 T + 73 T^{2} \))(\( 1 - 6 T + 73 T^{2} \))(\( 1 - 6 T + 73 T^{2} \))(\( 1 - 6 T + 73 T^{2} \))(\( 1 - 6 T + 73 T^{2} \))(\( 1 + 6 T + 73 T^{2} \))(\( 1 + 6 T + 73 T^{2} \))(\( 1 + 2 T + 73 T^{2} \))(\( 1 + 2 T + 73 T^{2} \))(\( 1 + 6 T + 73 T^{2} \))(\( 1 + 6 T + 73 T^{2} \))(\( 1 - 6 T + 73 T^{2} \))(\( 1 - 6 T + 73 T^{2} \))(\( 1 - 6 T + 73 T^{2} \))(\( 1 - 6 T + 73 T^{2} \))(\( 1 + 6 T + 73 T^{2} \))(\( ( 1 + 14 T + 73 T^{2} )^{2} \))(\( ( 1 - 2 T + 73 T^{2} )^{2} \))(\( ( 1 + 2 T + 73 T^{2} )^{2} \))(\( ( 1 - 2 T + 73 T^{2} )^{2} \))(\( ( 1 + 2 T + 73 T^{2} )^{2} \))(\( ( 1 - 10 T + 73 T^{2} )^{2} \))(\( ( 1 - 10 T + 73 T^{2} )^{2} \))(\( ( 1 - 2 T + 73 T^{2} )^{2} \))(\( ( 1 + 14 T + 73 T^{2} )^{2} \))(\( ( 1 - 14 T + 73 T^{2} )^{4} \))
$79$ (\( 1 + 79 T^{2} \))(\( 1 + 10 T + 79 T^{2} \))(\( 1 + 79 T^{2} \))(\( 1 + 79 T^{2} \))(\( 1 - 10 T + 79 T^{2} \))(\( 1 + 4 T + 79 T^{2} \))(\( 1 + 4 T + 79 T^{2} \))(\( 1 + 79 T^{2} \))(\( 1 + 79 T^{2} \))(\( 1 - 4 T + 79 T^{2} \))(\( 1 - 4 T + 79 T^{2} \))(\( 1 + 10 T + 79 T^{2} \))(\( 1 + 79 T^{2} \))(\( 1 + 79 T^{2} \))(\( 1 - 10 T + 79 T^{2} \))(\( 1 + 79 T^{2} \))(\( ( 1 + 10 T + 79 T^{2} )^{2} \))(\( 1 + 86 T^{2} + 6241 T^{4} \))(\( 1 + 50 T^{2} + 6241 T^{4} \))(\( ( 1 + 79 T^{2} )^{2} \))(\( 1 + 50 T^{2} + 6241 T^{4} \))(\( 1 - 142 T^{2} + 6241 T^{4} \))(\( 1 - 142 T^{2} + 6241 T^{4} \))(\( 1 + 86 T^{2} + 6241 T^{4} \))(\( ( 1 - 10 T + 79 T^{2} )^{2} \))(\( ( 1 - 58 T^{2} + 6241 T^{4} )^{2} \))
$83$ (\( 1 + 83 T^{2} \))(\( 1 + 16 T + 83 T^{2} \))(\( 1 + 83 T^{2} \))(\( 1 + 83 T^{2} \))(\( 1 - 16 T + 83 T^{2} \))(\( 1 - 4 T + 83 T^{2} \))(\( 1 + 4 T + 83 T^{2} \))(\( 1 + 18 T + 83 T^{2} \))(\( 1 - 18 T + 83 T^{2} \))(\( 1 - 4 T + 83 T^{2} \))(\( 1 + 4 T + 83 T^{2} \))(\( 1 - 16 T + 83 T^{2} \))(\( 1 + 83 T^{2} \))(\( 1 + 83 T^{2} \))(\( 1 + 16 T + 83 T^{2} \))(\( 1 + 83 T^{2} \))(\( 1 + 134 T^{2} + 6889 T^{4} \))(\( ( 1 - 4 T + 83 T^{2} )^{2} \))(\( ( 1 + 83 T^{2} )^{2} \))(\( 1 + 158 T^{2} + 6889 T^{4} \))(\( ( 1 + 83 T^{2} )^{2} \))(\( ( 1 + 83 T^{2} )^{2} \))(\( ( 1 + 83 T^{2} )^{2} \))(\( ( 1 + 4 T + 83 T^{2} )^{2} \))(\( 1 + 134 T^{2} + 6889 T^{4} \))(\( ( 1 + 134 T^{2} + 6889 T^{4} )^{2} \))
$89$ (\( 1 + 10 T + 89 T^{2} \))(\( 1 + 10 T + 89 T^{2} \))(\( 1 + 16 T + 89 T^{2} \))(\( 1 - 16 T + 89 T^{2} \))(\( 1 + 10 T + 89 T^{2} \))(\( 1 - 6 T + 89 T^{2} \))(\( 1 - 6 T + 89 T^{2} \))(\( 1 - 18 T + 89 T^{2} \))(\( 1 - 18 T + 89 T^{2} \))(\( 1 - 6 T + 89 T^{2} \))(\( 1 - 6 T + 89 T^{2} \))(\( 1 + 10 T + 89 T^{2} \))(\( 1 - 16 T + 89 T^{2} \))(\( 1 + 16 T + 89 T^{2} \))(\( 1 + 10 T + 89 T^{2} \))(\( 1 + 10 T + 89 T^{2} \))(\( ( 1 + 89 T^{2} )^{2} \))(\( ( 1 - 6 T + 89 T^{2} )^{2} \))(\( ( 1 - 6 T + 89 T^{2} )^{2} \))(\( ( 1 + 18 T + 89 T^{2} )^{2} \))(\( ( 1 - 6 T + 89 T^{2} )^{2} \))(\( ( 1 + 89 T^{2} )^{2} \))(\( ( 1 + 89 T^{2} )^{2} \))(\( ( 1 - 6 T + 89 T^{2} )^{2} \))(\( ( 1 + 89 T^{2} )^{2} \))(\( ( 1 + 89 T^{2} )^{4} \))
$97$ (\( 1 + 18 T + 97 T^{2} \))(\( 1 + 2 T + 97 T^{2} \))(\( 1 - 18 T + 97 T^{2} \))(\( 1 - 18 T + 97 T^{2} \))(\( 1 + 2 T + 97 T^{2} \))(\( 1 + 2 T + 97 T^{2} \))(\( 1 + 2 T + 97 T^{2} \))(\( 1 - 10 T + 97 T^{2} \))(\( 1 - 10 T + 97 T^{2} \))(\( 1 + 2 T + 97 T^{2} \))(\( 1 + 2 T + 97 T^{2} \))(\( 1 + 2 T + 97 T^{2} \))(\( 1 - 18 T + 97 T^{2} \))(\( 1 - 18 T + 97 T^{2} \))(\( 1 + 2 T + 97 T^{2} \))(\( 1 + 18 T + 97 T^{2} \))(\( ( 1 - 2 T + 97 T^{2} )^{2} \))(\( ( 1 - 14 T + 97 T^{2} )^{2} \))(\( ( 1 + 2 T + 97 T^{2} )^{2} \))(\( ( 1 + 10 T + 97 T^{2} )^{2} \))(\( ( 1 + 2 T + 97 T^{2} )^{2} \))(\( ( 1 + 14 T + 97 T^{2} )^{2} \))(\( ( 1 + 14 T + 97 T^{2} )^{2} \))(\( ( 1 - 14 T + 97 T^{2} )^{2} \))(\( ( 1 - 2 T + 97 T^{2} )^{2} \))(\( ( 1 - 2 T + 97 T^{2} )^{4} \))
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