Properties

Label 4032.2.c
Level 4032
Weight 2
Character orbit c
Rep. character \(\chi_{4032}(2017,\cdot)\)
Character field \(\Q\)
Dimension 60
Newform subspaces 18
Sturm bound 1536
Trace bound 25

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Defining parameters

Level: \( N \) \(=\) \( 4032 = 2^{6} \cdot 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4032.c (of order \(2\) and degree \(1\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 8 \)
Character field: \(\Q\)
Newform subspaces: \( 18 \)
Sturm bound: \(1536\)
Trace bound: \(25\)
Distinguishing \(T_p\): \(5\), \(11\), \(13\), \(17\), \(23\), \(31\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(4032, [\chi])\).

Total New Old
Modular forms 816 60 756
Cusp forms 720 60 660
Eisenstein series 96 0 96

Trace form

\( 60q + O(q^{10}) \) \( 60q + 24q^{17} - 84q^{25} - 72q^{41} + 60q^{49} + 24q^{73} + 24q^{89} + 24q^{97} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(4032, [\chi])\) into newform subspaces

Label Dim. \(A\) Field CM Traces $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\)
4032.2.c.a \(2\) \(32.196\) \(\Q(\sqrt{-1}) \) None \(0\) \(0\) \(0\) \(-2\) \(q+iq^{5}-q^{7}+2iq^{11}-iq^{13}-2q^{17}+\cdots\)
4032.2.c.b \(2\) \(32.196\) \(\Q(\sqrt{-1}) \) None \(0\) \(0\) \(0\) \(-2\) \(q+2iq^{5}-q^{7}+iq^{11}-2iq^{13}-2q^{17}+\cdots\)
4032.2.c.c \(2\) \(32.196\) \(\Q(\sqrt{-1}) \) None \(0\) \(0\) \(0\) \(-2\) \(q+iq^{5}-q^{7}-iq^{11}-iq^{13}-2q^{17}+\cdots\)
4032.2.c.d \(2\) \(32.196\) \(\Q(\sqrt{-1}) \) None \(0\) \(0\) \(0\) \(-2\) \(q+iq^{5}-q^{7}+2iq^{11}+iq^{13}+2q^{17}+\cdots\)
4032.2.c.e \(2\) \(32.196\) \(\Q(\sqrt{-1}) \) None \(0\) \(0\) \(0\) \(-2\) \(q+iq^{5}-q^{7}-iq^{11}+3iq^{13}+6q^{17}+\cdots\)
4032.2.c.f \(2\) \(32.196\) \(\Q(\sqrt{-1}) \) None \(0\) \(0\) \(0\) \(2\) \(q+2iq^{5}+q^{7}-iq^{11}-2iq^{13}-2q^{17}+\cdots\)
4032.2.c.g \(2\) \(32.196\) \(\Q(\sqrt{-1}) \) None \(0\) \(0\) \(0\) \(2\) \(q+iq^{5}+q^{7}+iq^{11}-iq^{13}-2q^{17}+\cdots\)
4032.2.c.h \(2\) \(32.196\) \(\Q(\sqrt{-1}) \) None \(0\) \(0\) \(0\) \(2\) \(q+iq^{5}+q^{7}-2iq^{11}-iq^{13}-2q^{17}+\cdots\)
4032.2.c.i \(2\) \(32.196\) \(\Q(\sqrt{-1}) \) None \(0\) \(0\) \(0\) \(2\) \(q+iq^{5}+q^{7}-2iq^{11}+iq^{13}+2q^{17}+\cdots\)
4032.2.c.j \(2\) \(32.196\) \(\Q(\sqrt{-1}) \) None \(0\) \(0\) \(0\) \(2\) \(q+iq^{5}+q^{7}+iq^{11}+3iq^{13}+6q^{17}+\cdots\)
4032.2.c.k \(4\) \(32.196\) \(\Q(\zeta_{12})\) None \(0\) \(0\) \(0\) \(-4\) \(q-\zeta_{12}q^{5}-q^{7}-2\zeta_{12}q^{11}+(-2\zeta_{12}+\cdots)q^{13}+\cdots\)
4032.2.c.l \(4\) \(32.196\) \(\Q(\zeta_{12})\) None \(0\) \(0\) \(0\) \(-4\) \(q-\zeta_{12}^{3}q^{5}-q^{7}+(-\zeta_{12}^{2}-\zeta_{12}^{3})q^{11}+\cdots\)
4032.2.c.m \(4\) \(32.196\) \(\Q(\zeta_{12})\) None \(0\) \(0\) \(0\) \(-4\) \(q-\zeta_{12}^{3}q^{5}-q^{7}+(\zeta_{12}^{2}+3\zeta_{12}^{3})q^{11}+\cdots\)
4032.2.c.n \(4\) \(32.196\) \(\Q(\zeta_{12})\) None \(0\) \(0\) \(0\) \(4\) \(q-\zeta_{12}q^{5}+q^{7}+2\zeta_{12}q^{11}+(-2\zeta_{12}+\cdots)q^{13}+\cdots\)
4032.2.c.o \(4\) \(32.196\) \(\Q(\zeta_{12})\) None \(0\) \(0\) \(0\) \(4\) \(q-\zeta_{12}^{3}q^{5}+q^{7}+(\zeta_{12}^{2}+\zeta_{12}^{3})q^{11}+\cdots\)
4032.2.c.p \(4\) \(32.196\) \(\Q(\zeta_{12})\) None \(0\) \(0\) \(0\) \(4\) \(q-\zeta_{12}^{3}q^{5}+q^{7}+(-\zeta_{12}^{2}-3\zeta_{12}^{3})q^{11}+\cdots\)
4032.2.c.q \(8\) \(32.196\) 8.0.897122304.10 None \(0\) \(0\) \(0\) \(-8\) \(q-\beta _{5}q^{5}-q^{7}+\beta _{5}q^{11}+(-\beta _{4}-\beta _{6}+\cdots)q^{13}+\cdots\)
4032.2.c.r \(8\) \(32.196\) 8.0.897122304.10 None \(0\) \(0\) \(0\) \(8\) \(q-\beta _{5}q^{5}+q^{7}-\beta _{5}q^{11}+(-\beta _{4}-\beta _{6}+\cdots)q^{13}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(4032, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(4032, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(24, [\chi])\)\(^{\oplus 16}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(56, [\chi])\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(64, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(72, [\chi])\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(96, [\chi])\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(168, [\chi])\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(192, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(224, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(288, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(448, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(504, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(576, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(672, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(1344, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(2016, [\chi])\)\(^{\oplus 2}\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ 1
$5$ (\( ( 1 - 4 T + 5 T^{2} )( 1 + 4 T + 5 T^{2} ) \))(\( ( 1 - 2 T + 5 T^{2} )( 1 + 2 T + 5 T^{2} ) \))(\( ( 1 - 4 T + 5 T^{2} )( 1 + 4 T + 5 T^{2} ) \))(\( ( 1 - 4 T + 5 T^{2} )( 1 + 4 T + 5 T^{2} ) \))(\( ( 1 - 4 T + 5 T^{2} )( 1 + 4 T + 5 T^{2} ) \))(\( ( 1 - 2 T + 5 T^{2} )( 1 + 2 T + 5 T^{2} ) \))(\( ( 1 - 4 T + 5 T^{2} )( 1 + 4 T + 5 T^{2} ) \))(\( ( 1 - 4 T + 5 T^{2} )( 1 + 4 T + 5 T^{2} ) \))(\( ( 1 - 4 T + 5 T^{2} )( 1 + 4 T + 5 T^{2} ) \))(\( ( 1 - 4 T + 5 T^{2} )( 1 + 4 T + 5 T^{2} ) \))(\( 1 - 12 T^{2} + 74 T^{4} - 300 T^{6} + 625 T^{8} \))(\( 1 - 12 T^{2} + 74 T^{4} - 300 T^{6} + 625 T^{8} \))(\( 1 - 12 T^{2} + 74 T^{4} - 300 T^{6} + 625 T^{8} \))(\( 1 - 12 T^{2} + 74 T^{4} - 300 T^{6} + 625 T^{8} \))(\( 1 - 12 T^{2} + 74 T^{4} - 300 T^{6} + 625 T^{8} \))(\( 1 - 12 T^{2} + 74 T^{4} - 300 T^{6} + 625 T^{8} \))(\( ( 1 + 2 T^{4} + 625 T^{8} )^{2} \))(\( ( 1 + 2 T^{4} + 625 T^{8} )^{2} \))
$7$ (\( ( 1 + T )^{2} \))(\( ( 1 + T )^{2} \))(\( ( 1 + T )^{2} \))(\( ( 1 + T )^{2} \))(\( ( 1 + T )^{2} \))(\( ( 1 - T )^{2} \))(\( ( 1 - T )^{2} \))(\( ( 1 - T )^{2} \))(\( ( 1 - T )^{2} \))(\( ( 1 - T )^{2} \))(\( ( 1 + T )^{4} \))(\( ( 1 + T )^{4} \))(\( ( 1 + T )^{4} \))(\( ( 1 - T )^{4} \))(\( ( 1 - T )^{4} \))(\( ( 1 - T )^{4} \))(\( ( 1 + T )^{8} \))(\( ( 1 - T )^{8} \))
$11$ (\( 1 - 6 T^{2} + 121 T^{4} \))(\( 1 - 18 T^{2} + 121 T^{4} \))(\( 1 - 18 T^{2} + 121 T^{4} \))(\( 1 - 6 T^{2} + 121 T^{4} \))(\( 1 - 18 T^{2} + 121 T^{4} \))(\( 1 - 18 T^{2} + 121 T^{4} \))(\( 1 - 18 T^{2} + 121 T^{4} \))(\( 1 - 6 T^{2} + 121 T^{4} \))(\( 1 - 6 T^{2} + 121 T^{4} \))(\( 1 - 18 T^{2} + 121 T^{4} \))(\( 1 - 12 T^{2} + 86 T^{4} - 1452 T^{6} + 14641 T^{8} \))(\( 1 - 36 T^{2} + 554 T^{4} - 4356 T^{6} + 14641 T^{8} \))(\( 1 + 12 T^{2} + 170 T^{4} + 1452 T^{6} + 14641 T^{8} \))(\( 1 - 12 T^{2} + 86 T^{4} - 1452 T^{6} + 14641 T^{8} \))(\( 1 - 36 T^{2} + 554 T^{4} - 4356 T^{6} + 14641 T^{8} \))(\( 1 + 12 T^{2} + 170 T^{4} + 1452 T^{6} + 14641 T^{8} \))(\( ( 1 - 24 T^{2} + 338 T^{4} - 2904 T^{6} + 14641 T^{8} )^{2} \))(\( ( 1 - 24 T^{2} + 338 T^{4} - 2904 T^{6} + 14641 T^{8} )^{2} \))
$13$ (\( 1 - 22 T^{2} + 169 T^{4} \))(\( ( 1 - 6 T + 13 T^{2} )( 1 + 6 T + 13 T^{2} ) \))(\( 1 - 22 T^{2} + 169 T^{4} \))(\( 1 - 22 T^{2} + 169 T^{4} \))(\( ( 1 - 4 T + 13 T^{2} )( 1 + 4 T + 13 T^{2} ) \))(\( ( 1 - 6 T + 13 T^{2} )( 1 + 6 T + 13 T^{2} ) \))(\( 1 - 22 T^{2} + 169 T^{4} \))(\( 1 - 22 T^{2} + 169 T^{4} \))(\( 1 - 22 T^{2} + 169 T^{4} \))(\( ( 1 - 4 T + 13 T^{2} )( 1 + 4 T + 13 T^{2} ) \))(\( 1 + 4 T^{2} + 42 T^{4} + 676 T^{6} + 28561 T^{8} \))(\( ( 1 - 6 T + 13 T^{2} )^{2}( 1 + 6 T + 13 T^{2} )^{2} \))(\( ( 1 - 13 T^{2} )^{4} \))(\( 1 + 4 T^{2} + 42 T^{4} + 676 T^{6} + 28561 T^{8} \))(\( ( 1 - 6 T + 13 T^{2} )^{2}( 1 + 6 T + 13 T^{2} )^{2} \))(\( ( 1 - 13 T^{2} )^{4} \))(\( ( 1 - 20 T^{2} + 246 T^{4} - 3380 T^{6} + 28561 T^{8} )^{2} \))(\( ( 1 - 20 T^{2} + 246 T^{4} - 3380 T^{6} + 28561 T^{8} )^{2} \))
$17$ (\( ( 1 + 2 T + 17 T^{2} )^{2} \))(\( ( 1 + 2 T + 17 T^{2} )^{2} \))(\( ( 1 + 2 T + 17 T^{2} )^{2} \))(\( ( 1 - 2 T + 17 T^{2} )^{2} \))(\( ( 1 - 6 T + 17 T^{2} )^{2} \))(\( ( 1 + 2 T + 17 T^{2} )^{2} \))(\( ( 1 + 2 T + 17 T^{2} )^{2} \))(\( ( 1 + 2 T + 17 T^{2} )^{2} \))(\( ( 1 - 2 T + 17 T^{2} )^{2} \))(\( ( 1 - 6 T + 17 T^{2} )^{2} \))(\( ( 1 + 2 T + 17 T^{2} )^{4} \))(\( ( 1 - 2 T + 8 T^{2} - 34 T^{3} + 289 T^{4} )^{2} \))(\( ( 1 - 6 T + 40 T^{2} - 102 T^{3} + 289 T^{4} )^{2} \))(\( ( 1 + 2 T + 17 T^{2} )^{4} \))(\( ( 1 - 2 T + 8 T^{2} - 34 T^{3} + 289 T^{4} )^{2} \))(\( ( 1 - 6 T + 40 T^{2} - 102 T^{3} + 289 T^{4} )^{2} \))(\( ( 1 + 24 T^{2} + 290 T^{4} + 6936 T^{6} + 83521 T^{8} )^{2} \))(\( ( 1 + 24 T^{2} + 290 T^{4} + 6936 T^{6} + 83521 T^{8} )^{2} \))
$19$ (\( ( 1 - 19 T^{2} )^{2} \))(\( 1 - 2 T^{2} + 361 T^{4} \))(\( ( 1 - 19 T^{2} )^{2} \))(\( ( 1 - 19 T^{2} )^{2} \))(\( 1 + 26 T^{2} + 361 T^{4} \))(\( 1 - 2 T^{2} + 361 T^{4} \))(\( ( 1 - 19 T^{2} )^{2} \))(\( ( 1 - 19 T^{2} )^{2} \))(\( ( 1 - 19 T^{2} )^{2} \))(\( 1 + 26 T^{2} + 361 T^{4} \))(\( 1 - 52 T^{2} + 1290 T^{4} - 18772 T^{6} + 130321 T^{8} \))(\( ( 1 - 8 T + 19 T^{2} )^{2}( 1 + 8 T + 19 T^{2} )^{2} \))(\( 1 - 20 T^{2} + 54 T^{4} - 7220 T^{6} + 130321 T^{8} \))(\( 1 - 52 T^{2} + 1290 T^{4} - 18772 T^{6} + 130321 T^{8} \))(\( ( 1 - 8 T + 19 T^{2} )^{2}( 1 + 8 T + 19 T^{2} )^{2} \))(\( 1 - 20 T^{2} + 54 T^{4} - 7220 T^{6} + 130321 T^{8} \))(\( ( 1 - 8 T + 19 T^{2} )^{4}( 1 + 8 T + 19 T^{2} )^{4} \))(\( ( 1 - 8 T + 19 T^{2} )^{4}( 1 + 8 T + 19 T^{2} )^{4} \))
$23$ (\( ( 1 + 6 T + 23 T^{2} )^{2} \))(\( ( 1 + 23 T^{2} )^{2} \))(\( ( 1 + 23 T^{2} )^{2} \))(\( ( 1 - 6 T + 23 T^{2} )^{2} \))(\( ( 1 - 8 T + 23 T^{2} )^{2} \))(\( ( 1 + 23 T^{2} )^{2} \))(\( ( 1 + 23 T^{2} )^{2} \))(\( ( 1 - 6 T + 23 T^{2} )^{2} \))(\( ( 1 + 6 T + 23 T^{2} )^{2} \))(\( ( 1 + 8 T + 23 T^{2} )^{2} \))(\( ( 1 + 34 T^{2} + 529 T^{4} )^{2} \))(\( ( 1 - 6 T + 52 T^{2} - 138 T^{3} + 529 T^{4} )^{2} \))(\( ( 1 + 14 T + 92 T^{2} + 322 T^{3} + 529 T^{4} )^{2} \))(\( ( 1 + 34 T^{2} + 529 T^{4} )^{2} \))(\( ( 1 + 6 T + 52 T^{2} + 138 T^{3} + 529 T^{4} )^{2} \))(\( ( 1 - 14 T + 92 T^{2} - 322 T^{3} + 529 T^{4} )^{2} \))(\( ( 1 + 32 T^{2} + 882 T^{4} + 16928 T^{6} + 279841 T^{8} )^{2} \))(\( ( 1 + 32 T^{2} + 882 T^{4} + 16928 T^{6} + 279841 T^{8} )^{2} \))
$29$ (\( 1 - 54 T^{2} + 841 T^{4} \))(\( 1 + 6 T^{2} + 841 T^{4} \))(\( 1 + 6 T^{2} + 841 T^{4} \))(\( 1 - 54 T^{2} + 841 T^{4} \))(\( ( 1 - 29 T^{2} )^{2} \))(\( 1 + 6 T^{2} + 841 T^{4} \))(\( 1 + 6 T^{2} + 841 T^{4} \))(\( 1 - 54 T^{2} + 841 T^{4} \))(\( 1 - 54 T^{2} + 841 T^{4} \))(\( ( 1 - 29 T^{2} )^{2} \))(\( 1 - 84 T^{2} + 3254 T^{4} - 70644 T^{6} + 707281 T^{8} \))(\( 1 - 60 T^{2} + 1814 T^{4} - 50460 T^{6} + 707281 T^{8} \))(\( ( 1 - 46 T^{2} + 841 T^{4} )^{2} \))(\( 1 - 84 T^{2} + 3254 T^{4} - 70644 T^{6} + 707281 T^{8} \))(\( 1 - 60 T^{2} + 1814 T^{4} - 50460 T^{6} + 707281 T^{8} \))(\( ( 1 - 46 T^{2} + 841 T^{4} )^{2} \))(\( ( 1 - 4 T + 14 T^{2} - 116 T^{3} + 841 T^{4} )^{2}( 1 + 4 T + 14 T^{2} + 116 T^{3} + 841 T^{4} )^{2} \))(\( ( 1 - 4 T + 14 T^{2} - 116 T^{3} + 841 T^{4} )^{2}( 1 + 4 T + 14 T^{2} + 116 T^{3} + 841 T^{4} )^{2} \))
$31$ (\( ( 1 + 4 T + 31 T^{2} )^{2} \))(\( ( 1 - 8 T + 31 T^{2} )^{2} \))(\( ( 1 - 8 T + 31 T^{2} )^{2} \))(\( ( 1 + 4 T + 31 T^{2} )^{2} \))(\( ( 1 + 8 T + 31 T^{2} )^{2} \))(\( ( 1 + 8 T + 31 T^{2} )^{2} \))(\( ( 1 + 8 T + 31 T^{2} )^{2} \))(\( ( 1 - 4 T + 31 T^{2} )^{2} \))(\( ( 1 - 4 T + 31 T^{2} )^{2} \))(\( ( 1 - 8 T + 31 T^{2} )^{2} \))(\( ( 1 + 4 T + 31 T^{2} )^{4} \))(\( ( 1 - 2 T + 31 T^{2} )^{4} \))(\( ( 1 + 4 T + 18 T^{2} + 124 T^{3} + 961 T^{4} )^{2} \))(\( ( 1 - 4 T + 31 T^{2} )^{4} \))(\( ( 1 + 2 T + 31 T^{2} )^{4} \))(\( ( 1 - 4 T + 18 T^{2} - 124 T^{3} + 961 T^{4} )^{2} \))(\( ( 1 - 2 T + 31 T^{2} )^{8} \))(\( ( 1 + 2 T + 31 T^{2} )^{8} \))
$37$ (\( 1 - 58 T^{2} + 1369 T^{4} \))(\( 1 - 10 T^{2} + 1369 T^{4} \))(\( 1 - 58 T^{2} + 1369 T^{4} \))(\( 1 - 58 T^{2} + 1369 T^{4} \))(\( 1 - 58 T^{2} + 1369 T^{4} \))(\( 1 - 10 T^{2} + 1369 T^{4} \))(\( 1 - 58 T^{2} + 1369 T^{4} \))(\( 1 - 58 T^{2} + 1369 T^{4} \))(\( 1 - 58 T^{2} + 1369 T^{4} \))(\( 1 - 58 T^{2} + 1369 T^{4} \))(\( 1 - 116 T^{2} + 5910 T^{4} - 158804 T^{6} + 1874161 T^{8} \))(\( ( 1 - 12 T + 37 T^{2} )^{2}( 1 + 12 T + 37 T^{2} )^{2} \))(\( ( 1 + 26 T^{2} + 1369 T^{4} )^{2} \))(\( 1 - 116 T^{2} + 5910 T^{4} - 158804 T^{6} + 1874161 T^{8} \))(\( ( 1 - 12 T + 37 T^{2} )^{2}( 1 + 12 T + 37 T^{2} )^{2} \))(\( ( 1 + 26 T^{2} + 1369 T^{4} )^{2} \))(\( ( 1 + 4 T^{2} - 330 T^{4} + 5476 T^{6} + 1874161 T^{8} )^{2} \))(\( ( 1 + 4 T^{2} - 330 T^{4} + 5476 T^{6} + 1874161 T^{8} )^{2} \))
$41$ (\( ( 1 + 10 T + 41 T^{2} )^{2} \))(\( ( 1 + 10 T + 41 T^{2} )^{2} \))(\( ( 1 - 2 T + 41 T^{2} )^{2} \))(\( ( 1 - 10 T + 41 T^{2} )^{2} \))(\( ( 1 + 6 T + 41 T^{2} )^{2} \))(\( ( 1 + 10 T + 41 T^{2} )^{2} \))(\( ( 1 - 2 T + 41 T^{2} )^{2} \))(\( ( 1 + 10 T + 41 T^{2} )^{2} \))(\( ( 1 - 10 T + 41 T^{2} )^{2} \))(\( ( 1 + 6 T + 41 T^{2} )^{2} \))(\( ( 1 - 2 T + 41 T^{2} )^{4} \))(\( ( 1 + 2 T + 8 T^{2} + 82 T^{3} + 1681 T^{4} )^{2} \))(\( ( 1 + 6 T + 88 T^{2} + 246 T^{3} + 1681 T^{4} )^{2} \))(\( ( 1 - 2 T + 41 T^{2} )^{4} \))(\( ( 1 + 2 T + 8 T^{2} + 82 T^{3} + 1681 T^{4} )^{2} \))(\( ( 1 + 6 T + 88 T^{2} + 246 T^{3} + 1681 T^{4} )^{2} \))(\( ( 1 + 120 T^{2} + 6530 T^{4} + 201720 T^{6} + 2825761 T^{8} )^{2} \))(\( ( 1 + 120 T^{2} + 6530 T^{4} + 201720 T^{6} + 2825761 T^{8} )^{2} \))
$43$ (\( 1 - 82 T^{2} + 1849 T^{4} \))(\( 1 - 82 T^{2} + 1849 T^{4} \))(\( 1 + 14 T^{2} + 1849 T^{4} \))(\( 1 - 82 T^{2} + 1849 T^{4} \))(\( 1 - 50 T^{2} + 1849 T^{4} \))(\( 1 - 82 T^{2} + 1849 T^{4} \))(\( 1 + 14 T^{2} + 1849 T^{4} \))(\( 1 - 82 T^{2} + 1849 T^{4} \))(\( 1 - 82 T^{2} + 1849 T^{4} \))(\( 1 - 50 T^{2} + 1849 T^{4} \))(\( 1 - 140 T^{2} + 8406 T^{4} - 258860 T^{6} + 3418801 T^{8} \))(\( 1 - 68 T^{2} + 4086 T^{4} - 125732 T^{6} + 3418801 T^{8} \))(\( 1 - 4 T^{2} - 3210 T^{4} - 7396 T^{6} + 3418801 T^{8} \))(\( 1 - 140 T^{2} + 8406 T^{4} - 258860 T^{6} + 3418801 T^{8} \))(\( 1 - 68 T^{2} + 4086 T^{4} - 125732 T^{6} + 3418801 T^{8} \))(\( 1 - 4 T^{2} - 3210 T^{4} - 7396 T^{6} + 3418801 T^{8} \))(\( ( 1 - 68 T^{2} + 4086 T^{4} - 125732 T^{6} + 3418801 T^{8} )^{2} \))(\( ( 1 - 68 T^{2} + 4086 T^{4} - 125732 T^{6} + 3418801 T^{8} )^{2} \))
$47$ (\( ( 1 - 8 T + 47 T^{2} )^{2} \))(\( ( 1 - 8 T + 47 T^{2} )^{2} \))(\( ( 1 + 4 T + 47 T^{2} )^{2} \))(\( ( 1 + 8 T + 47 T^{2} )^{2} \))(\( ( 1 - 12 T + 47 T^{2} )^{2} \))(\( ( 1 + 8 T + 47 T^{2} )^{2} \))(\( ( 1 - 4 T + 47 T^{2} )^{2} \))(\( ( 1 + 8 T + 47 T^{2} )^{2} \))(\( ( 1 - 8 T + 47 T^{2} )^{2} \))(\( ( 1 + 12 T + 47 T^{2} )^{2} \))(\( ( 1 + 8 T + 62 T^{2} + 376 T^{3} + 2209 T^{4} )^{2} \))(\( ( 1 + 4 T + 47 T^{2} )^{4} \))(\( ( 1 + 46 T^{2} + 2209 T^{4} )^{2} \))(\( ( 1 - 8 T + 62 T^{2} - 376 T^{3} + 2209 T^{4} )^{2} \))(\( ( 1 - 4 T + 47 T^{2} )^{4} \))(\( ( 1 + 46 T^{2} + 2209 T^{4} )^{2} \))(\( ( 1 + 60 T^{2} + 4550 T^{4} + 132540 T^{6} + 4879681 T^{8} )^{2} \))(\( ( 1 + 60 T^{2} + 4550 T^{4} + 132540 T^{6} + 4879681 T^{8} )^{2} \))
$53$ (\( 1 - 70 T^{2} + 2809 T^{4} \))(\( ( 1 - 53 T^{2} )^{2} \))(\( 1 + 38 T^{2} + 2809 T^{4} \))(\( 1 - 70 T^{2} + 2809 T^{4} \))(\( 1 + 38 T^{2} + 2809 T^{4} \))(\( ( 1 - 53 T^{2} )^{2} \))(\( 1 + 38 T^{2} + 2809 T^{4} \))(\( 1 - 70 T^{2} + 2809 T^{4} \))(\( 1 - 70 T^{2} + 2809 T^{4} \))(\( 1 + 38 T^{2} + 2809 T^{4} \))(\( ( 1 + 38 T^{2} + 2809 T^{4} )^{2} \))(\( ( 1 - 94 T^{2} + 2809 T^{4} )^{2} \))(\( ( 1 - 94 T^{2} + 2809 T^{4} )^{2} \))(\( ( 1 + 38 T^{2} + 2809 T^{4} )^{2} \))(\( ( 1 - 94 T^{2} + 2809 T^{4} )^{2} \))(\( ( 1 - 94 T^{2} + 2809 T^{4} )^{2} \))(\( ( 1 - 53 T^{2} )^{8} \))(\( ( 1 - 53 T^{2} )^{8} \))
$59$ (\( 1 - 102 T^{2} + 3481 T^{4} \))(\( 1 - 18 T^{2} + 3481 T^{4} \))(\( 1 - 102 T^{2} + 3481 T^{4} \))(\( 1 - 102 T^{2} + 3481 T^{4} \))(\( 1 - 102 T^{2} + 3481 T^{4} \))(\( 1 - 18 T^{2} + 3481 T^{4} \))(\( 1 - 102 T^{2} + 3481 T^{4} \))(\( 1 - 102 T^{2} + 3481 T^{4} \))(\( 1 - 102 T^{2} + 3481 T^{4} \))(\( 1 - 102 T^{2} + 3481 T^{4} \))(\( 1 - 84 T^{2} + 8426 T^{4} - 292404 T^{6} + 12117361 T^{8} \))(\( ( 1 - 54 T^{2} + 3481 T^{4} )^{2} \))(\( 1 - 108 T^{2} + 6806 T^{4} - 375948 T^{6} + 12117361 T^{8} \))(\( 1 - 84 T^{2} + 8426 T^{4} - 292404 T^{6} + 12117361 T^{8} \))(\( ( 1 - 54 T^{2} + 3481 T^{4} )^{2} \))(\( 1 - 108 T^{2} + 6806 T^{4} - 375948 T^{6} + 12117361 T^{8} \))(\( ( 1 - 108 T^{2} + 9110 T^{4} - 375948 T^{6} + 12117361 T^{8} )^{2} \))(\( ( 1 - 108 T^{2} + 9110 T^{4} - 375948 T^{6} + 12117361 T^{8} )^{2} \))
$61$ (\( ( 1 - 12 T + 61 T^{2} )( 1 + 12 T + 61 T^{2} ) \))(\( 1 - 106 T^{2} + 3721 T^{4} \))(\( 1 - 118 T^{2} + 3721 T^{4} \))(\( ( 1 - 12 T + 61 T^{2} )( 1 + 12 T + 61 T^{2} ) \))(\( ( 1 - 12 T + 61 T^{2} )( 1 + 12 T + 61 T^{2} ) \))(\( 1 - 106 T^{2} + 3721 T^{4} \))(\( 1 - 118 T^{2} + 3721 T^{4} \))(\( ( 1 - 12 T + 61 T^{2} )( 1 + 12 T + 61 T^{2} ) \))(\( ( 1 - 12 T + 61 T^{2} )( 1 + 12 T + 61 T^{2} ) \))(\( ( 1 - 12 T + 61 T^{2} )( 1 + 12 T + 61 T^{2} ) \))(\( 1 + 52 T^{2} + 7530 T^{4} + 193492 T^{6} + 13845841 T^{8} \))(\( 1 - 20 T^{2} + 5814 T^{4} - 74420 T^{6} + 13845841 T^{8} \))(\( 1 - 212 T^{2} + 18486 T^{4} - 788852 T^{6} + 13845841 T^{8} \))(\( 1 + 52 T^{2} + 7530 T^{4} + 193492 T^{6} + 13845841 T^{8} \))(\( 1 - 20 T^{2} + 5814 T^{4} - 74420 T^{6} + 13845841 T^{8} \))(\( 1 - 212 T^{2} + 18486 T^{4} - 788852 T^{6} + 13845841 T^{8} \))(\( ( 1 - 212 T^{2} + 18486 T^{4} - 788852 T^{6} + 13845841 T^{8} )^{2} \))(\( ( 1 - 212 T^{2} + 18486 T^{4} - 788852 T^{6} + 13845841 T^{8} )^{2} \))
$67$ (\( 1 - 130 T^{2} + 4489 T^{4} \))(\( 1 - 130 T^{2} + 4489 T^{4} \))(\( 1 - 130 T^{2} + 4489 T^{4} \))(\( 1 - 130 T^{2} + 4489 T^{4} \))(\( 1 - 130 T^{2} + 4489 T^{4} \))(\( 1 - 130 T^{2} + 4489 T^{4} \))(\( 1 - 130 T^{2} + 4489 T^{4} \))(\( 1 - 130 T^{2} + 4489 T^{4} \))(\( 1 - 130 T^{2} + 4489 T^{4} \))(\( 1 - 130 T^{2} + 4489 T^{4} \))(\( ( 1 - 70 T^{2} + 4489 T^{4} )^{2} \))(\( 1 - 20 T^{2} + 2166 T^{4} - 89780 T^{6} + 20151121 T^{8} \))(\( 1 - 212 T^{2} + 19446 T^{4} - 951668 T^{6} + 20151121 T^{8} \))(\( ( 1 - 70 T^{2} + 4489 T^{4} )^{2} \))(\( 1 - 20 T^{2} + 2166 T^{4} - 89780 T^{6} + 20151121 T^{8} \))(\( 1 - 212 T^{2} + 19446 T^{4} - 951668 T^{6} + 20151121 T^{8} \))(\( ( 1 - 34 T^{2} + 4489 T^{4} )^{4} \))(\( ( 1 - 34 T^{2} + 4489 T^{4} )^{4} \))
$71$ (\( ( 1 - 2 T + 71 T^{2} )^{2} \))(\( ( 1 - 8 T + 71 T^{2} )^{2} \))(\( ( 1 - 8 T + 71 T^{2} )^{2} \))(\( ( 1 + 2 T + 71 T^{2} )^{2} \))(\( ( 1 + 71 T^{2} )^{2} \))(\( ( 1 + 8 T + 71 T^{2} )^{2} \))(\( ( 1 + 8 T + 71 T^{2} )^{2} \))(\( ( 1 + 2 T + 71 T^{2} )^{2} \))(\( ( 1 - 2 T + 71 T^{2} )^{2} \))(\( ( 1 + 71 T^{2} )^{2} \))(\( ( 1 + 8 T + 110 T^{2} + 568 T^{3} + 5041 T^{4} )^{2} \))(\( ( 1 - 10 T + 92 T^{2} - 710 T^{3} + 5041 T^{4} )^{2} \))(\( ( 1 + 18 T + 196 T^{2} + 1278 T^{3} + 5041 T^{4} )^{2} \))(\( ( 1 - 8 T + 110 T^{2} - 568 T^{3} + 5041 T^{4} )^{2} \))(\( ( 1 + 10 T + 92 T^{2} + 710 T^{3} + 5041 T^{4} )^{2} \))(\( ( 1 - 18 T + 196 T^{2} - 1278 T^{3} + 5041 T^{4} )^{2} \))(\( ( 1 + 96 T^{2} + 12338 T^{4} + 483936 T^{6} + 25411681 T^{8} )^{2} \))(\( ( 1 + 96 T^{2} + 12338 T^{4} + 483936 T^{6} + 25411681 T^{8} )^{2} \))
$73$ (\( ( 1 + 6 T + 73 T^{2} )^{2} \))(\( ( 1 + 6 T + 73 T^{2} )^{2} \))(\( ( 1 - 6 T + 73 T^{2} )^{2} \))(\( ( 1 + 6 T + 73 T^{2} )^{2} \))(\( ( 1 - 6 T + 73 T^{2} )^{2} \))(\( ( 1 + 6 T + 73 T^{2} )^{2} \))(\( ( 1 - 6 T + 73 T^{2} )^{2} \))(\( ( 1 + 6 T + 73 T^{2} )^{2} \))(\( ( 1 + 6 T + 73 T^{2} )^{2} \))(\( ( 1 - 6 T + 73 T^{2} )^{2} \))(\( ( 1 - 12 T + 134 T^{2} - 876 T^{3} + 5329 T^{4} )^{2} \))(\( ( 1 + 24 T + 278 T^{2} + 1752 T^{3} + 5329 T^{4} )^{2} \))(\( ( 1 + 38 T^{2} + 5329 T^{4} )^{2} \))(\( ( 1 - 12 T + 134 T^{2} - 876 T^{3} + 5329 T^{4} )^{2} \))(\( ( 1 + 24 T + 278 T^{2} + 1752 T^{3} + 5329 T^{4} )^{2} \))(\( ( 1 + 38 T^{2} + 5329 T^{4} )^{2} \))(\( ( 1 - 12 T + 134 T^{2} - 876 T^{3} + 5329 T^{4} )^{4} \))(\( ( 1 - 12 T + 134 T^{2} - 876 T^{3} + 5329 T^{4} )^{4} \))
$79$ (\( ( 1 + 8 T + 79 T^{2} )^{2} \))(\( ( 1 + 8 T + 79 T^{2} )^{2} \))(\( ( 1 - 16 T + 79 T^{2} )^{2} \))(\( ( 1 + 8 T + 79 T^{2} )^{2} \))(\( ( 1 + 79 T^{2} )^{2} \))(\( ( 1 - 8 T + 79 T^{2} )^{2} \))(\( ( 1 + 16 T + 79 T^{2} )^{2} \))(\( ( 1 - 8 T + 79 T^{2} )^{2} \))(\( ( 1 - 8 T + 79 T^{2} )^{2} \))(\( ( 1 + 79 T^{2} )^{2} \))(\( ( 1 - 8 T + 126 T^{2} - 632 T^{3} + 6241 T^{4} )^{2} \))(\( ( 1 + 4 T + 54 T^{2} + 316 T^{3} + 6241 T^{4} )^{2} \))(\( ( 1 + 12 T + 182 T^{2} + 948 T^{3} + 6241 T^{4} )^{2} \))(\( ( 1 + 8 T + 126 T^{2} + 632 T^{3} + 6241 T^{4} )^{2} \))(\( ( 1 - 4 T + 54 T^{2} - 316 T^{3} + 6241 T^{4} )^{2} \))(\( ( 1 - 12 T + 182 T^{2} - 948 T^{3} + 6241 T^{4} )^{2} \))(\( ( 1 - 8 T + 126 T^{2} - 632 T^{3} + 6241 T^{4} )^{4} \))(\( ( 1 + 8 T + 126 T^{2} + 632 T^{3} + 6241 T^{4} )^{4} \))
$83$ (\( 1 - 22 T^{2} + 6889 T^{4} \))(\( 1 - 130 T^{2} + 6889 T^{4} \))(\( ( 1 - 83 T^{2} )^{2} \))(\( 1 - 22 T^{2} + 6889 T^{4} \))(\( 1 - 102 T^{2} + 6889 T^{4} \))(\( 1 - 130 T^{2} + 6889 T^{4} \))(\( ( 1 - 83 T^{2} )^{2} \))(\( 1 - 22 T^{2} + 6889 T^{4} \))(\( 1 - 22 T^{2} + 6889 T^{4} \))(\( 1 - 102 T^{2} + 6889 T^{4} \))(\( 1 - 164 T^{2} + 17802 T^{4} - 1129796 T^{6} + 47458321 T^{8} \))(\( 1 - 236 T^{2} + 25974 T^{4} - 1625804 T^{6} + 47458321 T^{8} \))(\( 1 - 108 T^{2} + 14966 T^{4} - 744012 T^{6} + 47458321 T^{8} \))(\( 1 - 164 T^{2} + 17802 T^{4} - 1129796 T^{6} + 47458321 T^{8} \))(\( 1 - 236 T^{2} + 25974 T^{4} - 1625804 T^{6} + 47458321 T^{8} \))(\( 1 - 108 T^{2} + 14966 T^{4} - 744012 T^{6} + 47458321 T^{8} \))(\( ( 1 - 92 T^{2} + 8982 T^{4} - 633788 T^{6} + 47458321 T^{8} )^{2} \))(\( ( 1 - 92 T^{2} + 8982 T^{4} - 633788 T^{6} + 47458321 T^{8} )^{2} \))
$89$ (\( ( 1 - 2 T + 89 T^{2} )^{2} \))(\( ( 1 + 10 T + 89 T^{2} )^{2} \))(\( ( 1 - 14 T + 89 T^{2} )^{2} \))(\( ( 1 + 2 T + 89 T^{2} )^{2} \))(\( ( 1 - 6 T + 89 T^{2} )^{2} \))(\( ( 1 + 10 T + 89 T^{2} )^{2} \))(\( ( 1 - 14 T + 89 T^{2} )^{2} \))(\( ( 1 - 2 T + 89 T^{2} )^{2} \))(\( ( 1 + 2 T + 89 T^{2} )^{2} \))(\( ( 1 - 6 T + 89 T^{2} )^{2} \))(\( ( 1 - 4 T - 10 T^{2} - 356 T^{3} + 7921 T^{4} )^{2} \))(\( ( 1 + 14 T + 200 T^{2} + 1246 T^{3} + 7921 T^{4} )^{2} \))(\( ( 1 - 6 T + 184 T^{2} - 534 T^{3} + 7921 T^{4} )^{2} \))(\( ( 1 - 4 T - 10 T^{2} - 356 T^{3} + 7921 T^{4} )^{2} \))(\( ( 1 + 14 T + 200 T^{2} + 1246 T^{3} + 7921 T^{4} )^{2} \))(\( ( 1 - 6 T + 184 T^{2} - 534 T^{3} + 7921 T^{4} )^{2} \))(\( ( 1 + 120 T^{2} + 5570 T^{4} + 950520 T^{6} + 62742241 T^{8} )^{2} \))(\( ( 1 + 120 T^{2} + 5570 T^{4} + 950520 T^{6} + 62742241 T^{8} )^{2} \))
$97$ (\( ( 1 - 2 T + 97 T^{2} )^{2} \))(\( ( 1 - 2 T + 97 T^{2} )^{2} \))(\( ( 1 + 10 T + 97 T^{2} )^{2} \))(\( ( 1 - 2 T + 97 T^{2} )^{2} \))(\( ( 1 + 10 T + 97 T^{2} )^{2} \))(\( ( 1 - 2 T + 97 T^{2} )^{2} \))(\( ( 1 + 10 T + 97 T^{2} )^{2} \))(\( ( 1 - 2 T + 97 T^{2} )^{2} \))(\( ( 1 - 2 T + 97 T^{2} )^{2} \))(\( ( 1 + 10 T + 97 T^{2} )^{2} \))(\( ( 1 - 4 T + 150 T^{2} - 388 T^{3} + 9409 T^{4} )^{2} \))(\( ( 1 - 16 T + 150 T^{2} - 1552 T^{3} + 9409 T^{4} )^{2} \))(\( ( 1 + 8 T + 102 T^{2} + 776 T^{3} + 9409 T^{4} )^{2} \))(\( ( 1 - 4 T + 150 T^{2} - 388 T^{3} + 9409 T^{4} )^{2} \))(\( ( 1 - 16 T + 150 T^{2} - 1552 T^{3} + 9409 T^{4} )^{2} \))(\( ( 1 + 8 T + 102 T^{2} + 776 T^{3} + 9409 T^{4} )^{2} \))(\( ( 1 - 4 T + 150 T^{2} - 388 T^{3} + 9409 T^{4} )^{4} \))(\( ( 1 - 4 T + 150 T^{2} - 388 T^{3} + 9409 T^{4} )^{4} \))
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