Properties

Label 2304.2.d
Level 2304
Weight 2
Character orbit d
Rep. character \(\chi_{2304}(1153,\cdot)\)
Character field \(\Q\)
Dimension 38
Newform subspaces 19
Sturm bound 768
Trace bound 23

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Defining parameters

Level: \( N \) \(=\) \( 2304 = 2^{8} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2304.d (of order \(2\) and degree \(1\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 8 \)
Character field: \(\Q\)
Newform subspaces: \( 19 \)
Sturm bound: \(768\)
Trace bound: \(23\)
Distinguishing \(T_p\): \(5\), \(7\), \(11\), \(17\), \(23\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(2304, [\chi])\).

Total New Old
Modular forms 432 42 390
Cusp forms 336 38 298
Eisenstein series 96 4 92

Trace form

\( 38q + O(q^{10}) \) \( 38q + 4q^{17} - 26q^{25} - 4q^{41} - 10q^{49} + 40q^{65} + 20q^{73} - 20q^{89} - 36q^{97} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(2304, [\chi])\) into newform subspaces

Label Dim. \(A\) Field CM Traces $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\)
2304.2.d.a \(2\) \(18.398\) \(\Q(\sqrt{-1}) \) \(\Q(\sqrt{-3}) \) \(0\) \(0\) \(0\) \(-8\) \(q-4q^{7}-iq^{13}+4iq^{19}+5q^{25}+\cdots\)
2304.2.d.b \(2\) \(18.398\) \(\Q(\sqrt{-1}) \) None \(0\) \(0\) \(0\) \(-8\) \(q+iq^{5}-4q^{7}+iq^{11}+iq^{13}+2q^{17}+\cdots\)
2304.2.d.c \(2\) \(18.398\) \(\Q(\sqrt{-1}) \) None \(0\) \(0\) \(0\) \(-8\) \(q+iq^{5}-4q^{7}-2iq^{11}-iq^{13}+6q^{17}+\cdots\)
2304.2.d.d \(2\) \(18.398\) \(\Q(\sqrt{-1}) \) None \(0\) \(0\) \(0\) \(-4\) \(q-2q^{7}+2iq^{11}-3iq^{13}-6q^{17}+\cdots\)
2304.2.d.e \(2\) \(18.398\) \(\Q(\sqrt{-1}) \) None \(0\) \(0\) \(0\) \(-4\) \(q+iq^{5}-2q^{7}+2iq^{11}+iq^{13}-4q^{17}+\cdots\)
2304.2.d.f \(2\) \(18.398\) \(\Q(\sqrt{-1}) \) None \(0\) \(0\) \(0\) \(-4\) \(q+2iq^{5}-2q^{7}-2iq^{11}-iq^{13}+\cdots\)
2304.2.d.g \(2\) \(18.398\) \(\Q(\sqrt{-1}) \) None \(0\) \(0\) \(0\) \(-4\) \(q+iq^{5}-2q^{7}+2iq^{11}-iq^{13}+4q^{17}+\cdots\)
2304.2.d.h \(2\) \(18.398\) \(\Q(\sqrt{-1}) \) \(\Q(\sqrt{-1}) \) \(0\) \(0\) \(0\) \(0\) \(q+2iq^{5}-3iq^{13}-8q^{17}-11q^{25}+\cdots\)
2304.2.d.i \(2\) \(18.398\) \(\Q(\sqrt{-1}) \) None \(0\) \(0\) \(0\) \(0\) \(q+iq^{5}-2iq^{11}+iq^{13}-2q^{17}+\cdots\)
2304.2.d.j \(2\) \(18.398\) \(\Q(\sqrt{-1}) \) \(\Q(\sqrt{-1}) \) \(0\) \(0\) \(0\) \(0\) \(q+iq^{5}-3iq^{13}-2q^{17}+q^{25}-5iq^{29}+\cdots\)
2304.2.d.k \(2\) \(18.398\) \(\Q(\sqrt{-1}) \) None \(0\) \(0\) \(0\) \(0\) \(q+iq^{5}+2iq^{11}+iq^{13}-2q^{17}+\cdots\)
2304.2.d.l \(2\) \(18.398\) \(\Q(\sqrt{-1}) \) \(\Q(\sqrt{-1}) \) \(0\) \(0\) \(0\) \(0\) \(q+2iq^{5}+3iq^{13}+8q^{17}-11q^{25}+\cdots\)
2304.2.d.m \(2\) \(18.398\) \(\Q(\sqrt{-1}) \) None \(0\) \(0\) \(0\) \(4\) \(q+2q^{7}-2iq^{11}-3iq^{13}-6q^{17}+\cdots\)
2304.2.d.n \(2\) \(18.398\) \(\Q(\sqrt{-1}) \) None \(0\) \(0\) \(0\) \(4\) \(q+iq^{5}+2q^{7}-2iq^{11}+iq^{13}-4q^{17}+\cdots\)
2304.2.d.o \(2\) \(18.398\) \(\Q(\sqrt{-1}) \) None \(0\) \(0\) \(0\) \(4\) \(q+2iq^{5}+2q^{7}+2iq^{11}-iq^{13}+\cdots\)
2304.2.d.p \(2\) \(18.398\) \(\Q(\sqrt{-1}) \) None \(0\) \(0\) \(0\) \(4\) \(q+iq^{5}+2q^{7}-2iq^{11}-iq^{13}+4q^{17}+\cdots\)
2304.2.d.q \(2\) \(18.398\) \(\Q(\sqrt{-1}) \) \(\Q(\sqrt{-3}) \) \(0\) \(0\) \(0\) \(8\) \(q+4q^{7}-iq^{13}-4iq^{19}+5q^{25}+\cdots\)
2304.2.d.r \(2\) \(18.398\) \(\Q(\sqrt{-1}) \) None \(0\) \(0\) \(0\) \(8\) \(q+iq^{5}+4q^{7}-iq^{11}+iq^{13}+2q^{17}+\cdots\)
2304.2.d.s \(2\) \(18.398\) \(\Q(\sqrt{-1}) \) None \(0\) \(0\) \(0\) \(8\) \(q+iq^{5}+4q^{7}+2iq^{11}-iq^{13}+6q^{17}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(2304, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(2304, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(24, [\chi])\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(64, [\chi])\)\(^{\oplus 9}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(72, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(96, [\chi])\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(128, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(192, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(256, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(288, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(384, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(576, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(768, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(1152, [\chi])\)\(^{\oplus 2}\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ 1
$5$ (\( ( 1 - 5 T^{2} )^{2} \))(\( ( 1 - 4 T + 5 T^{2} )( 1 + 4 T + 5 T^{2} ) \))(\( ( 1 - 4 T + 5 T^{2} )( 1 + 4 T + 5 T^{2} ) \))(\( ( 1 - 5 T^{2} )^{2} \))(\( ( 1 - 4 T + 5 T^{2} )( 1 + 4 T + 5 T^{2} ) \))(\( ( 1 - 2 T + 5 T^{2} )( 1 + 2 T + 5 T^{2} ) \))(\( ( 1 - 4 T + 5 T^{2} )( 1 + 4 T + 5 T^{2} ) \))(\( ( 1 - 2 T + 5 T^{2} )( 1 + 2 T + 5 T^{2} ) \))(\( ( 1 - 4 T + 5 T^{2} )( 1 + 4 T + 5 T^{2} ) \))(\( ( 1 - 4 T + 5 T^{2} )( 1 + 4 T + 5 T^{2} ) \))(\( ( 1 - 4 T + 5 T^{2} )( 1 + 4 T + 5 T^{2} ) \))(\( ( 1 - 2 T + 5 T^{2} )( 1 + 2 T + 5 T^{2} ) \))(\( ( 1 - 5 T^{2} )^{2} \))(\( ( 1 - 4 T + 5 T^{2} )( 1 + 4 T + 5 T^{2} ) \))(\( ( 1 - 2 T + 5 T^{2} )( 1 + 2 T + 5 T^{2} ) \))(\( ( 1 - 4 T + 5 T^{2} )( 1 + 4 T + 5 T^{2} ) \))(\( ( 1 - 5 T^{2} )^{2} \))(\( ( 1 - 4 T + 5 T^{2} )( 1 + 4 T + 5 T^{2} ) \))(\( ( 1 - 4 T + 5 T^{2} )( 1 + 4 T + 5 T^{2} ) \))
$7$ (\( ( 1 + 4 T + 7 T^{2} )^{2} \))(\( ( 1 + 4 T + 7 T^{2} )^{2} \))(\( ( 1 + 4 T + 7 T^{2} )^{2} \))(\( ( 1 + 2 T + 7 T^{2} )^{2} \))(\( ( 1 + 2 T + 7 T^{2} )^{2} \))(\( ( 1 + 2 T + 7 T^{2} )^{2} \))(\( ( 1 + 2 T + 7 T^{2} )^{2} \))(\( ( 1 + 7 T^{2} )^{2} \))(\( ( 1 + 7 T^{2} )^{2} \))(\( ( 1 + 7 T^{2} )^{2} \))(\( ( 1 + 7 T^{2} )^{2} \))(\( ( 1 + 7 T^{2} )^{2} \))(\( ( 1 - 2 T + 7 T^{2} )^{2} \))(\( ( 1 - 2 T + 7 T^{2} )^{2} \))(\( ( 1 - 2 T + 7 T^{2} )^{2} \))(\( ( 1 - 2 T + 7 T^{2} )^{2} \))(\( ( 1 - 4 T + 7 T^{2} )^{2} \))(\( ( 1 - 4 T + 7 T^{2} )^{2} \))(\( ( 1 - 4 T + 7 T^{2} )^{2} \))
$11$ (\( ( 1 - 11 T^{2} )^{2} \))(\( 1 - 18 T^{2} + 121 T^{4} \))(\( 1 - 6 T^{2} + 121 T^{4} \))(\( 1 - 6 T^{2} + 121 T^{4} \))(\( 1 - 6 T^{2} + 121 T^{4} \))(\( 1 - 6 T^{2} + 121 T^{4} \))(\( 1 - 6 T^{2} + 121 T^{4} \))(\( ( 1 - 11 T^{2} )^{2} \))(\( 1 - 6 T^{2} + 121 T^{4} \))(\( ( 1 - 11 T^{2} )^{2} \))(\( 1 - 6 T^{2} + 121 T^{4} \))(\( ( 1 - 11 T^{2} )^{2} \))(\( 1 - 6 T^{2} + 121 T^{4} \))(\( 1 - 6 T^{2} + 121 T^{4} \))(\( 1 - 6 T^{2} + 121 T^{4} \))(\( 1 - 6 T^{2} + 121 T^{4} \))(\( ( 1 - 11 T^{2} )^{2} \))(\( 1 - 18 T^{2} + 121 T^{4} \))(\( 1 - 6 T^{2} + 121 T^{4} \))
$13$ (\( 1 - 22 T^{2} + 169 T^{4} \))(\( 1 - 22 T^{2} + 169 T^{4} \))(\( 1 - 22 T^{2} + 169 T^{4} \))(\( ( 1 - 4 T + 13 T^{2} )( 1 + 4 T + 13 T^{2} ) \))(\( 1 - 22 T^{2} + 169 T^{4} \))(\( 1 - 22 T^{2} + 169 T^{4} \))(\( 1 - 22 T^{2} + 169 T^{4} \))(\( ( 1 - 4 T + 13 T^{2} )( 1 + 4 T + 13 T^{2} ) \))(\( 1 - 22 T^{2} + 169 T^{4} \))(\( ( 1 - 4 T + 13 T^{2} )( 1 + 4 T + 13 T^{2} ) \))(\( 1 - 22 T^{2} + 169 T^{4} \))(\( ( 1 - 4 T + 13 T^{2} )( 1 + 4 T + 13 T^{2} ) \))(\( ( 1 - 4 T + 13 T^{2} )( 1 + 4 T + 13 T^{2} ) \))(\( 1 - 22 T^{2} + 169 T^{4} \))(\( 1 - 22 T^{2} + 169 T^{4} \))(\( 1 - 22 T^{2} + 169 T^{4} \))(\( 1 - 22 T^{2} + 169 T^{4} \))(\( 1 - 22 T^{2} + 169 T^{4} \))(\( 1 - 22 T^{2} + 169 T^{4} \))
$17$ (\( ( 1 + 17 T^{2} )^{2} \))(\( ( 1 - 2 T + 17 T^{2} )^{2} \))(\( ( 1 - 6 T + 17 T^{2} )^{2} \))(\( ( 1 + 6 T + 17 T^{2} )^{2} \))(\( ( 1 + 4 T + 17 T^{2} )^{2} \))(\( ( 1 - 2 T + 17 T^{2} )^{2} \))(\( ( 1 - 4 T + 17 T^{2} )^{2} \))(\( ( 1 + 8 T + 17 T^{2} )^{2} \))(\( ( 1 + 2 T + 17 T^{2} )^{2} \))(\( ( 1 + 2 T + 17 T^{2} )^{2} \))(\( ( 1 + 2 T + 17 T^{2} )^{2} \))(\( ( 1 - 8 T + 17 T^{2} )^{2} \))(\( ( 1 + 6 T + 17 T^{2} )^{2} \))(\( ( 1 + 4 T + 17 T^{2} )^{2} \))(\( ( 1 - 2 T + 17 T^{2} )^{2} \))(\( ( 1 - 4 T + 17 T^{2} )^{2} \))(\( ( 1 + 17 T^{2} )^{2} \))(\( ( 1 - 2 T + 17 T^{2} )^{2} \))(\( ( 1 - 6 T + 17 T^{2} )^{2} \))
$19$ (\( 1 + 26 T^{2} + 361 T^{4} \))(\( 1 - 34 T^{2} + 361 T^{4} \))(\( 1 - 22 T^{2} + 361 T^{4} \))(\( ( 1 - 19 T^{2} )^{2} \))(\( 1 - 22 T^{2} + 361 T^{4} \))(\( 1 + 26 T^{2} + 361 T^{4} \))(\( 1 - 22 T^{2} + 361 T^{4} \))(\( ( 1 - 19 T^{2} )^{2} \))(\( 1 - 22 T^{2} + 361 T^{4} \))(\( ( 1 - 19 T^{2} )^{2} \))(\( 1 - 22 T^{2} + 361 T^{4} \))(\( ( 1 - 19 T^{2} )^{2} \))(\( ( 1 - 19 T^{2} )^{2} \))(\( 1 - 22 T^{2} + 361 T^{4} \))(\( 1 + 26 T^{2} + 361 T^{4} \))(\( 1 - 22 T^{2} + 361 T^{4} \))(\( 1 + 26 T^{2} + 361 T^{4} \))(\( 1 - 34 T^{2} + 361 T^{4} \))(\( 1 - 22 T^{2} + 361 T^{4} \))
$23$ (\( ( 1 + 23 T^{2} )^{2} \))(\( ( 1 + 4 T + 23 T^{2} )^{2} \))(\( ( 1 + 23 T^{2} )^{2} \))(\( ( 1 - 4 T + 23 T^{2} )^{2} \))(\( ( 1 + 8 T + 23 T^{2} )^{2} \))(\( ( 1 - 4 T + 23 T^{2} )^{2} \))(\( ( 1 - 8 T + 23 T^{2} )^{2} \))(\( ( 1 + 23 T^{2} )^{2} \))(\( ( 1 + 8 T + 23 T^{2} )^{2} \))(\( ( 1 + 23 T^{2} )^{2} \))(\( ( 1 - 8 T + 23 T^{2} )^{2} \))(\( ( 1 + 23 T^{2} )^{2} \))(\( ( 1 + 4 T + 23 T^{2} )^{2} \))(\( ( 1 - 8 T + 23 T^{2} )^{2} \))(\( ( 1 + 4 T + 23 T^{2} )^{2} \))(\( ( 1 + 8 T + 23 T^{2} )^{2} \))(\( ( 1 + 23 T^{2} )^{2} \))(\( ( 1 - 4 T + 23 T^{2} )^{2} \))(\( ( 1 + 23 T^{2} )^{2} \))
$29$ (\( ( 1 - 29 T^{2} )^{2} \))(\( 1 - 22 T^{2} + 841 T^{4} \))(\( 1 - 54 T^{2} + 841 T^{4} \))(\( ( 1 - 10 T + 29 T^{2} )( 1 + 10 T + 29 T^{2} ) \))(\( 1 - 22 T^{2} + 841 T^{4} \))(\( ( 1 - 29 T^{2} )^{2} \))(\( 1 - 22 T^{2} + 841 T^{4} \))(\( ( 1 - 10 T + 29 T^{2} )( 1 + 10 T + 29 T^{2} ) \))(\( 1 - 22 T^{2} + 841 T^{4} \))(\( ( 1 - 4 T + 29 T^{2} )( 1 + 4 T + 29 T^{2} ) \))(\( 1 - 22 T^{2} + 841 T^{4} \))(\( ( 1 - 10 T + 29 T^{2} )( 1 + 10 T + 29 T^{2} ) \))(\( ( 1 - 10 T + 29 T^{2} )( 1 + 10 T + 29 T^{2} ) \))(\( 1 - 22 T^{2} + 841 T^{4} \))(\( ( 1 - 29 T^{2} )^{2} \))(\( 1 - 22 T^{2} + 841 T^{4} \))(\( ( 1 - 29 T^{2} )^{2} \))(\( 1 - 22 T^{2} + 841 T^{4} \))(\( 1 - 54 T^{2} + 841 T^{4} \))
$31$ (\( ( 1 - 4 T + 31 T^{2} )^{2} \))(\( ( 1 + 31 T^{2} )^{2} \))(\( ( 1 + 4 T + 31 T^{2} )^{2} \))(\( ( 1 - 10 T + 31 T^{2} )^{2} \))(\( ( 1 + 6 T + 31 T^{2} )^{2} \))(\( ( 1 + 6 T + 31 T^{2} )^{2} \))(\( ( 1 + 6 T + 31 T^{2} )^{2} \))(\( ( 1 + 31 T^{2} )^{2} \))(\( ( 1 - 8 T + 31 T^{2} )^{2} \))(\( ( 1 + 31 T^{2} )^{2} \))(\( ( 1 + 8 T + 31 T^{2} )^{2} \))(\( ( 1 + 31 T^{2} )^{2} \))(\( ( 1 + 10 T + 31 T^{2} )^{2} \))(\( ( 1 - 6 T + 31 T^{2} )^{2} \))(\( ( 1 - 6 T + 31 T^{2} )^{2} \))(\( ( 1 - 6 T + 31 T^{2} )^{2} \))(\( ( 1 + 4 T + 31 T^{2} )^{2} \))(\( ( 1 + 31 T^{2} )^{2} \))(\( ( 1 - 4 T + 31 T^{2} )^{2} \))
$37$ (\( 1 + 26 T^{2} + 1369 T^{4} \))(\( 1 + 26 T^{2} + 1369 T^{4} \))(\( ( 1 - 12 T + 37 T^{2} )( 1 + 12 T + 37 T^{2} ) \))(\( ( 1 - 12 T + 37 T^{2} )( 1 + 12 T + 37 T^{2} ) \))(\( ( 1 - 12 T + 37 T^{2} )( 1 + 12 T + 37 T^{2} ) \))(\( ( 1 - 12 T + 37 T^{2} )( 1 + 12 T + 37 T^{2} ) \))(\( ( 1 - 12 T + 37 T^{2} )( 1 + 12 T + 37 T^{2} ) \))(\( ( 1 - 12 T + 37 T^{2} )( 1 + 12 T + 37 T^{2} ) \))(\( 1 - 38 T^{2} + 1369 T^{4} \))(\( ( 1 - 12 T + 37 T^{2} )( 1 + 12 T + 37 T^{2} ) \))(\( 1 - 38 T^{2} + 1369 T^{4} \))(\( ( 1 - 12 T + 37 T^{2} )( 1 + 12 T + 37 T^{2} ) \))(\( ( 1 - 12 T + 37 T^{2} )( 1 + 12 T + 37 T^{2} ) \))(\( ( 1 - 12 T + 37 T^{2} )( 1 + 12 T + 37 T^{2} ) \))(\( ( 1 - 12 T + 37 T^{2} )( 1 + 12 T + 37 T^{2} ) \))(\( ( 1 - 12 T + 37 T^{2} )( 1 + 12 T + 37 T^{2} ) \))(\( 1 + 26 T^{2} + 1369 T^{4} \))(\( 1 + 26 T^{2} + 1369 T^{4} \))(\( ( 1 - 12 T + 37 T^{2} )( 1 + 12 T + 37 T^{2} ) \))
$41$ (\( ( 1 + 41 T^{2} )^{2} \))(\( ( 1 + 6 T + 41 T^{2} )^{2} \))(\( ( 1 - 2 T + 41 T^{2} )^{2} \))(\( ( 1 + 2 T + 41 T^{2} )^{2} \))(\( ( 1 - 12 T + 41 T^{2} )^{2} \))(\( ( 1 - 6 T + 41 T^{2} )^{2} \))(\( ( 1 + 12 T + 41 T^{2} )^{2} \))(\( ( 1 + 8 T + 41 T^{2} )^{2} \))(\( ( 1 + 6 T + 41 T^{2} )^{2} \))(\( ( 1 - 10 T + 41 T^{2} )^{2} \))(\( ( 1 + 6 T + 41 T^{2} )^{2} \))(\( ( 1 - 8 T + 41 T^{2} )^{2} \))(\( ( 1 + 2 T + 41 T^{2} )^{2} \))(\( ( 1 - 12 T + 41 T^{2} )^{2} \))(\( ( 1 - 6 T + 41 T^{2} )^{2} \))(\( ( 1 + 12 T + 41 T^{2} )^{2} \))(\( ( 1 + 41 T^{2} )^{2} \))(\( ( 1 + 6 T + 41 T^{2} )^{2} \))(\( ( 1 - 2 T + 41 T^{2} )^{2} \))
$43$ (\( 1 - 22 T^{2} + 1849 T^{4} \))(\( 1 - 50 T^{2} + 1849 T^{4} \))(\( 1 - 70 T^{2} + 1849 T^{4} \))(\( 1 - 22 T^{2} + 1849 T^{4} \))(\( 1 + 58 T^{2} + 1849 T^{4} \))(\( ( 1 - 43 T^{2} )^{2} \))(\( 1 + 58 T^{2} + 1849 T^{4} \))(\( ( 1 - 43 T^{2} )^{2} \))(\( 1 - 70 T^{2} + 1849 T^{4} \))(\( ( 1 - 43 T^{2} )^{2} \))(\( 1 - 70 T^{2} + 1849 T^{4} \))(\( ( 1 - 43 T^{2} )^{2} \))(\( 1 - 22 T^{2} + 1849 T^{4} \))(\( 1 + 58 T^{2} + 1849 T^{4} \))(\( ( 1 - 43 T^{2} )^{2} \))(\( 1 + 58 T^{2} + 1849 T^{4} \))(\( 1 - 22 T^{2} + 1849 T^{4} \))(\( 1 - 50 T^{2} + 1849 T^{4} \))(\( 1 - 70 T^{2} + 1849 T^{4} \))
$47$ (\( ( 1 + 47 T^{2} )^{2} \))(\( ( 1 + 8 T + 47 T^{2} )^{2} \))(\( ( 1 - 8 T + 47 T^{2} )^{2} \))(\( ( 1 - 12 T + 47 T^{2} )^{2} \))(\( ( 1 - 8 T + 47 T^{2} )^{2} \))(\( ( 1 + 4 T + 47 T^{2} )^{2} \))(\( ( 1 + 8 T + 47 T^{2} )^{2} \))(\( ( 1 + 47 T^{2} )^{2} \))(\( ( 1 + 47 T^{2} )^{2} \))(\( ( 1 + 47 T^{2} )^{2} \))(\( ( 1 + 47 T^{2} )^{2} \))(\( ( 1 + 47 T^{2} )^{2} \))(\( ( 1 + 12 T + 47 T^{2} )^{2} \))(\( ( 1 + 8 T + 47 T^{2} )^{2} \))(\( ( 1 - 4 T + 47 T^{2} )^{2} \))(\( ( 1 - 8 T + 47 T^{2} )^{2} \))(\( ( 1 + 47 T^{2} )^{2} \))(\( ( 1 - 8 T + 47 T^{2} )^{2} \))(\( ( 1 + 8 T + 47 T^{2} )^{2} \))
$53$ (\( ( 1 - 53 T^{2} )^{2} \))(\( 1 - 70 T^{2} + 2809 T^{4} \))(\( 1 - 6 T^{2} + 2809 T^{4} \))(\( 1 + 38 T^{2} + 2809 T^{4} \))(\( 1 - 70 T^{2} + 2809 T^{4} \))(\( ( 1 - 53 T^{2} )^{2} \))(\( 1 - 70 T^{2} + 2809 T^{4} \))(\( ( 1 - 14 T + 53 T^{2} )( 1 + 14 T + 53 T^{2} ) \))(\( 1 - 102 T^{2} + 2809 T^{4} \))(\( ( 1 - 4 T + 53 T^{2} )( 1 + 4 T + 53 T^{2} ) \))(\( 1 - 102 T^{2} + 2809 T^{4} \))(\( ( 1 - 14 T + 53 T^{2} )( 1 + 14 T + 53 T^{2} ) \))(\( 1 + 38 T^{2} + 2809 T^{4} \))(\( 1 - 70 T^{2} + 2809 T^{4} \))(\( ( 1 - 53 T^{2} )^{2} \))(\( 1 - 70 T^{2} + 2809 T^{4} \))(\( ( 1 - 53 T^{2} )^{2} \))(\( 1 - 70 T^{2} + 2809 T^{4} \))(\( 1 - 6 T^{2} + 2809 T^{4} \))
$59$ (\( ( 1 - 59 T^{2} )^{2} \))(\( 1 + 78 T^{2} + 3481 T^{4} \))(\( 1 - 102 T^{2} + 3481 T^{4} \))(\( 1 - 102 T^{2} + 3481 T^{4} \))(\( 1 - 54 T^{2} + 3481 T^{4} \))(\( 1 - 102 T^{2} + 3481 T^{4} \))(\( 1 - 54 T^{2} + 3481 T^{4} \))(\( ( 1 - 59 T^{2} )^{2} \))(\( 1 - 102 T^{2} + 3481 T^{4} \))(\( ( 1 - 59 T^{2} )^{2} \))(\( 1 - 102 T^{2} + 3481 T^{4} \))(\( ( 1 - 59 T^{2} )^{2} \))(\( 1 - 102 T^{2} + 3481 T^{4} \))(\( 1 - 54 T^{2} + 3481 T^{4} \))(\( 1 - 102 T^{2} + 3481 T^{4} \))(\( 1 - 54 T^{2} + 3481 T^{4} \))(\( ( 1 - 59 T^{2} )^{2} \))(\( 1 + 78 T^{2} + 3481 T^{4} \))(\( 1 - 102 T^{2} + 3481 T^{4} \))
$61$ (\( 1 + 74 T^{2} + 3721 T^{4} \))(\( 1 - 118 T^{2} + 3721 T^{4} \))(\( 1 - 86 T^{2} + 3721 T^{4} \))(\( 1 - 118 T^{2} + 3721 T^{4} \))(\( ( 1 - 12 T + 61 T^{2} )( 1 + 12 T + 61 T^{2} ) \))(\( 1 + 74 T^{2} + 3721 T^{4} \))(\( ( 1 - 12 T + 61 T^{2} )( 1 + 12 T + 61 T^{2} ) \))(\( ( 1 - 12 T + 61 T^{2} )( 1 + 12 T + 61 T^{2} ) \))(\( 1 - 118 T^{2} + 3721 T^{4} \))(\( ( 1 - 12 T + 61 T^{2} )( 1 + 12 T + 61 T^{2} ) \))(\( 1 - 118 T^{2} + 3721 T^{4} \))(\( ( 1 - 12 T + 61 T^{2} )( 1 + 12 T + 61 T^{2} ) \))(\( 1 - 118 T^{2} + 3721 T^{4} \))(\( ( 1 - 12 T + 61 T^{2} )( 1 + 12 T + 61 T^{2} ) \))(\( 1 + 74 T^{2} + 3721 T^{4} \))(\( ( 1 - 12 T + 61 T^{2} )( 1 + 12 T + 61 T^{2} ) \))(\( 1 + 74 T^{2} + 3721 T^{4} \))(\( 1 - 118 T^{2} + 3721 T^{4} \))(\( 1 - 86 T^{2} + 3721 T^{4} \))
$67$ (\( 1 + 122 T^{2} + 4489 T^{4} \))(\( 1 - 34 T^{2} + 4489 T^{4} \))(\( 1 - 118 T^{2} + 4489 T^{4} \))(\( 1 - 118 T^{2} + 4489 T^{4} \))(\( 1 - 70 T^{2} + 4489 T^{4} \))(\( 1 - 118 T^{2} + 4489 T^{4} \))(\( 1 - 70 T^{2} + 4489 T^{4} \))(\( ( 1 - 67 T^{2} )^{2} \))(\( 1 - 118 T^{2} + 4489 T^{4} \))(\( ( 1 - 67 T^{2} )^{2} \))(\( 1 - 118 T^{2} + 4489 T^{4} \))(\( ( 1 - 67 T^{2} )^{2} \))(\( 1 - 118 T^{2} + 4489 T^{4} \))(\( 1 - 70 T^{2} + 4489 T^{4} \))(\( 1 - 118 T^{2} + 4489 T^{4} \))(\( 1 - 70 T^{2} + 4489 T^{4} \))(\( 1 + 122 T^{2} + 4489 T^{4} \))(\( 1 - 34 T^{2} + 4489 T^{4} \))(\( 1 - 118 T^{2} + 4489 T^{4} \))
$71$ (\( ( 1 + 71 T^{2} )^{2} \))(\( ( 1 + 12 T + 71 T^{2} )^{2} \))(\( ( 1 - 16 T + 71 T^{2} )^{2} \))(\( ( 1 + 4 T + 71 T^{2} )^{2} \))(\( ( 1 + 71 T^{2} )^{2} \))(\( ( 1 - 12 T + 71 T^{2} )^{2} \))(\( ( 1 + 71 T^{2} )^{2} \))(\( ( 1 + 71 T^{2} )^{2} \))(\( ( 1 - 8 T + 71 T^{2} )^{2} \))(\( ( 1 + 71 T^{2} )^{2} \))(\( ( 1 + 8 T + 71 T^{2} )^{2} \))(\( ( 1 + 71 T^{2} )^{2} \))(\( ( 1 - 4 T + 71 T^{2} )^{2} \))(\( ( 1 + 71 T^{2} )^{2} \))(\( ( 1 + 12 T + 71 T^{2} )^{2} \))(\( ( 1 + 71 T^{2} )^{2} \))(\( ( 1 + 71 T^{2} )^{2} \))(\( ( 1 - 12 T + 71 T^{2} )^{2} \))(\( ( 1 + 16 T + 71 T^{2} )^{2} \))
$73$ (\( ( 1 - 10 T + 73 T^{2} )^{2} \))(\( ( 1 + 14 T + 73 T^{2} )^{2} \))(\( ( 1 - 6 T + 73 T^{2} )^{2} \))(\( ( 1 - 10 T + 73 T^{2} )^{2} \))(\( ( 1 + 2 T + 73 T^{2} )^{2} \))(\( ( 1 - 10 T + 73 T^{2} )^{2} \))(\( ( 1 + 2 T + 73 T^{2} )^{2} \))(\( ( 1 + 6 T + 73 T^{2} )^{2} \))(\( ( 1 + 10 T + 73 T^{2} )^{2} \))(\( ( 1 - 6 T + 73 T^{2} )^{2} \))(\( ( 1 + 10 T + 73 T^{2} )^{2} \))(\( ( 1 + 6 T + 73 T^{2} )^{2} \))(\( ( 1 - 10 T + 73 T^{2} )^{2} \))(\( ( 1 + 2 T + 73 T^{2} )^{2} \))(\( ( 1 - 10 T + 73 T^{2} )^{2} \))(\( ( 1 + 2 T + 73 T^{2} )^{2} \))(\( ( 1 - 10 T + 73 T^{2} )^{2} \))(\( ( 1 + 14 T + 73 T^{2} )^{2} \))(\( ( 1 - 6 T + 73 T^{2} )^{2} \))
$79$ (\( ( 1 - 4 T + 79 T^{2} )^{2} \))(\( ( 1 - 8 T + 79 T^{2} )^{2} \))(\( ( 1 + 4 T + 79 T^{2} )^{2} \))(\( ( 1 + 6 T + 79 T^{2} )^{2} \))(\( ( 1 + 14 T + 79 T^{2} )^{2} \))(\( ( 1 - 10 T + 79 T^{2} )^{2} \))(\( ( 1 + 14 T + 79 T^{2} )^{2} \))(\( ( 1 + 79 T^{2} )^{2} \))(\( ( 1 + 8 T + 79 T^{2} )^{2} \))(\( ( 1 + 79 T^{2} )^{2} \))(\( ( 1 - 8 T + 79 T^{2} )^{2} \))(\( ( 1 + 79 T^{2} )^{2} \))(\( ( 1 - 6 T + 79 T^{2} )^{2} \))(\( ( 1 - 14 T + 79 T^{2} )^{2} \))(\( ( 1 + 10 T + 79 T^{2} )^{2} \))(\( ( 1 - 14 T + 79 T^{2} )^{2} \))(\( ( 1 + 4 T + 79 T^{2} )^{2} \))(\( ( 1 + 8 T + 79 T^{2} )^{2} \))(\( ( 1 - 4 T + 79 T^{2} )^{2} \))
$83$ (\( ( 1 - 83 T^{2} )^{2} \))(\( 1 - 130 T^{2} + 6889 T^{4} \))(\( 1 - 22 T^{2} + 6889 T^{4} \))(\( 1 - 22 T^{2} + 6889 T^{4} \))(\( 1 - 22 T^{2} + 6889 T^{4} \))(\( 1 - 22 T^{2} + 6889 T^{4} \))(\( 1 - 22 T^{2} + 6889 T^{4} \))(\( ( 1 - 83 T^{2} )^{2} \))(\( 1 - 150 T^{2} + 6889 T^{4} \))(\( ( 1 - 83 T^{2} )^{2} \))(\( 1 - 150 T^{2} + 6889 T^{4} \))(\( ( 1 - 83 T^{2} )^{2} \))(\( 1 - 22 T^{2} + 6889 T^{4} \))(\( 1 - 22 T^{2} + 6889 T^{4} \))(\( 1 - 22 T^{2} + 6889 T^{4} \))(\( 1 - 22 T^{2} + 6889 T^{4} \))(\( ( 1 - 83 T^{2} )^{2} \))(\( 1 - 130 T^{2} + 6889 T^{4} \))(\( 1 - 22 T^{2} + 6889 T^{4} \))
$89$ (\( ( 1 + 89 T^{2} )^{2} \))(\( ( 1 + 2 T + 89 T^{2} )^{2} \))(\( ( 1 - 10 T + 89 T^{2} )^{2} \))(\( ( 1 - 2 T + 89 T^{2} )^{2} \))(\( ( 1 + 8 T + 89 T^{2} )^{2} \))(\( ( 1 + 14 T + 89 T^{2} )^{2} \))(\( ( 1 - 8 T + 89 T^{2} )^{2} \))(\( ( 1 - 16 T + 89 T^{2} )^{2} \))(\( ( 1 + 6 T + 89 T^{2} )^{2} \))(\( ( 1 - 10 T + 89 T^{2} )^{2} \))(\( ( 1 + 6 T + 89 T^{2} )^{2} \))(\( ( 1 + 16 T + 89 T^{2} )^{2} \))(\( ( 1 - 2 T + 89 T^{2} )^{2} \))(\( ( 1 + 8 T + 89 T^{2} )^{2} \))(\( ( 1 + 14 T + 89 T^{2} )^{2} \))(\( ( 1 - 8 T + 89 T^{2} )^{2} \))(\( ( 1 + 89 T^{2} )^{2} \))(\( ( 1 + 2 T + 89 T^{2} )^{2} \))(\( ( 1 - 10 T + 89 T^{2} )^{2} \))
$97$ (\( ( 1 - 14 T + 97 T^{2} )^{2} \))(\( ( 1 + 2 T + 97 T^{2} )^{2} \))(\( ( 1 + 14 T + 97 T^{2} )^{2} \))(\( ( 1 + 6 T + 97 T^{2} )^{2} \))(\( ( 1 + 2 T + 97 T^{2} )^{2} \))(\( ( 1 - 10 T + 97 T^{2} )^{2} \))(\( ( 1 + 2 T + 97 T^{2} )^{2} \))(\( ( 1 + 18 T + 97 T^{2} )^{2} \))(\( ( 1 - 2 T + 97 T^{2} )^{2} \))(\( ( 1 - 18 T + 97 T^{2} )^{2} \))(\( ( 1 - 2 T + 97 T^{2} )^{2} \))(\( ( 1 + 18 T + 97 T^{2} )^{2} \))(\( ( 1 + 6 T + 97 T^{2} )^{2} \))(\( ( 1 + 2 T + 97 T^{2} )^{2} \))(\( ( 1 - 10 T + 97 T^{2} )^{2} \))(\( ( 1 + 2 T + 97 T^{2} )^{2} \))(\( ( 1 - 14 T + 97 T^{2} )^{2} \))(\( ( 1 + 2 T + 97 T^{2} )^{2} \))(\( ( 1 + 14 T + 97 T^{2} )^{2} \))
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