Properties

Label 3024.2.b
Level $3024$
Weight $2$
Character orbit 3024.b
Rep. character $\chi_{3024}(1567,\cdot)$
Character field $\Q$
Dimension $64$
Newform subspaces $22$
Sturm bound $1152$
Trace bound $47$

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Defining parameters

Level: \( N \) \(=\) \( 3024 = 2^{4} \cdot 3^{3} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3024.b (of order \(2\) and degree \(1\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 28 \)
Character field: \(\Q\)
Newform subspaces: \( 22 \)
Sturm bound: \(1152\)
Trace bound: \(47\)
Distinguishing \(T_p\): \(5\), \(11\), \(13\), \(17\), \(19\), \(29\), \(47\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(3024, [\chi])\).

Total New Old
Modular forms 612 64 548
Cusp forms 540 64 476
Eisenstein series 72 0 72

Trace form

\( 64q + O(q^{10}) \) \( 64q - 88q^{25} - 8q^{37} - 8q^{49} - 24q^{85} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(3024, [\chi])\) into newform subspaces

Label Dim. \(A\) Field CM Traces $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\)
3024.2.b.a \(2\) \(24.147\) \(\Q(\sqrt{-3}) \) \(\Q(\sqrt{-3}) \) \(0\) \(0\) \(0\) \(-5\) \(q+(-2-\zeta_{6})q^{7}+(-1+2\zeta_{6})q^{13}+\cdots\)
3024.2.b.b \(2\) \(24.147\) \(\Q(\sqrt{-3}) \) None \(0\) \(0\) \(0\) \(-4\) \(q-\zeta_{6}q^{5}+(-2+\zeta_{6})q^{7}+2\zeta_{6}q^{11}+\cdots\)
3024.2.b.c \(2\) \(24.147\) \(\Q(\sqrt{-3}) \) None \(0\) \(0\) \(0\) \(-4\) \(q-\zeta_{6}q^{5}+(-2+\zeta_{6})q^{7}-2\zeta_{6}q^{11}+\cdots\)
3024.2.b.d \(2\) \(24.147\) \(\Q(\sqrt{-3}) \) None \(0\) \(0\) \(0\) \(-4\) \(q-\zeta_{6}q^{5}+(-2-\zeta_{6})q^{7}+2\zeta_{6}q^{11}+\cdots\)
3024.2.b.e \(2\) \(24.147\) \(\Q(\sqrt{-3}) \) None \(0\) \(0\) \(0\) \(-4\) \(q-\zeta_{6}q^{5}+(-2-\zeta_{6})q^{7}-2\zeta_{6}q^{11}+\cdots\)
3024.2.b.f \(2\) \(24.147\) \(\Q(\sqrt{-3}) \) \(\Q(\sqrt{-3}) \) \(0\) \(0\) \(0\) \(-1\) \(q+(-1+\zeta_{6})q^{7}+(1-2\zeta_{6})q^{13}-7q^{19}+\cdots\)
3024.2.b.g \(2\) \(24.147\) \(\Q(\sqrt{-3}) \) None \(0\) \(0\) \(0\) \(-1\) \(q+(1-2\zeta_{6})q^{5}+(-2+3\zeta_{6})q^{7}+(3+\cdots)q^{11}+\cdots\)
3024.2.b.h \(2\) \(24.147\) \(\Q(\sqrt{-3}) \) None \(0\) \(0\) \(0\) \(-1\) \(q+(1-2\zeta_{6})q^{5}+(1-3\zeta_{6})q^{7}+(3-6\zeta_{6})q^{11}+\cdots\)
3024.2.b.i \(2\) \(24.147\) \(\Q(\sqrt{-3}) \) None \(0\) \(0\) \(0\) \(1\) \(q+(1-2\zeta_{6})q^{5}+(2-3\zeta_{6})q^{7}+(-3+\cdots)q^{11}+\cdots\)
3024.2.b.j \(2\) \(24.147\) \(\Q(\sqrt{-3}) \) None \(0\) \(0\) \(0\) \(1\) \(q+(1-2\zeta_{6})q^{5}+(-1+3\zeta_{6})q^{7}+(-3+\cdots)q^{11}+\cdots\)
3024.2.b.k \(2\) \(24.147\) \(\Q(\sqrt{-3}) \) \(\Q(\sqrt{-3}) \) \(0\) \(0\) \(0\) \(1\) \(q+\zeta_{6}q^{7}+(-1+2\zeta_{6})q^{13}+7q^{19}+\cdots\)
3024.2.b.l \(2\) \(24.147\) \(\Q(\sqrt{-3}) \) None \(0\) \(0\) \(0\) \(4\) \(q-\zeta_{6}q^{5}+(2-\zeta_{6})q^{7}+2\zeta_{6}q^{11}+2\zeta_{6}q^{13}+\cdots\)
3024.2.b.m \(2\) \(24.147\) \(\Q(\sqrt{-3}) \) None \(0\) \(0\) \(0\) \(4\) \(q-\zeta_{6}q^{5}+(2-\zeta_{6})q^{7}-2\zeta_{6}q^{11}-2\zeta_{6}q^{13}+\cdots\)
3024.2.b.n \(2\) \(24.147\) \(\Q(\sqrt{-3}) \) None \(0\) \(0\) \(0\) \(4\) \(q-\zeta_{6}q^{5}+(2+\zeta_{6})q^{7}+2\zeta_{6}q^{11}-2\zeta_{6}q^{13}+\cdots\)
3024.2.b.o \(2\) \(24.147\) \(\Q(\sqrt{-3}) \) None \(0\) \(0\) \(0\) \(4\) \(q-\zeta_{6}q^{5}+(2+\zeta_{6})q^{7}-2\zeta_{6}q^{11}+2\zeta_{6}q^{13}+\cdots\)
3024.2.b.p \(2\) \(24.147\) \(\Q(\sqrt{-3}) \) \(\Q(\sqrt{-3}) \) \(0\) \(0\) \(0\) \(5\) \(q+(3-\zeta_{6})q^{7}+(1-2\zeta_{6})q^{13}-q^{19}+\cdots\)
3024.2.b.q \(4\) \(24.147\) \(\Q(\sqrt{-6}, \sqrt{7})\) \(\Q(\sqrt{-21}) \) \(0\) \(0\) \(0\) \(0\) \(q+\beta _{1}q^{5}-\beta _{2}q^{7}-\beta _{1}q^{11}+(\beta _{1}-\beta _{3})q^{17}+\cdots\)
3024.2.b.r \(4\) \(24.147\) \(\Q(\sqrt{-3}, \sqrt{7})\) None \(0\) \(0\) \(0\) \(0\) \(q+2\beta _{2}q^{5}+\beta _{1}q^{7}-\beta _{3}q^{13}+\beta _{2}q^{17}+\cdots\)
3024.2.b.s \(4\) \(24.147\) \(\Q(\sqrt{-3}, \sqrt{7})\) None \(0\) \(0\) \(0\) \(0\) \(q+2\beta _{2}q^{5}+\beta _{1}q^{7}+\beta _{3}q^{13}+\beta _{2}q^{17}+\cdots\)
3024.2.b.t \(4\) \(24.147\) \(\Q(\sqrt{-6}, \sqrt{7})\) \(\Q(\sqrt{-21}) \) \(0\) \(0\) \(0\) \(0\) \(q+\beta _{1}q^{5}+\beta _{2}q^{7}+\beta _{1}q^{11}+(\beta _{1}-\beta _{3})q^{17}+\cdots\)
3024.2.b.u \(8\) \(24.147\) 8.0.\(\cdots\).13 None \(0\) \(0\) \(0\) \(-8\) \(q+\beta _{1}q^{5}+(-1-\beta _{5})q^{7}+\beta _{2}q^{11}+\cdots\)
3024.2.b.v \(8\) \(24.147\) 8.0.\(\cdots\).13 None \(0\) \(0\) \(0\) \(8\) \(q-\beta _{1}q^{5}+(1-\beta _{5})q^{7}+\beta _{2}q^{11}+(-\beta _{3}+\cdots)q^{13}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(3024, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(3024, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(28, [\chi])\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(84, [\chi])\)\(^{\oplus 9}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(112, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(252, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(336, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(756, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(1008, [\chi])\)\(^{\oplus 2}\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ 1
$5$ (\( ( 1 - 5 T^{2} )^{2} \))(\( 1 - 7 T^{2} + 25 T^{4} \))(\( 1 - 7 T^{2} + 25 T^{4} \))(\( 1 - 7 T^{2} + 25 T^{4} \))(\( 1 - 7 T^{2} + 25 T^{4} \))(\( ( 1 - 5 T^{2} )^{2} \))(\( 1 - 7 T^{2} + 25 T^{4} \))(\( 1 - 7 T^{2} + 25 T^{4} \))(\( 1 - 7 T^{2} + 25 T^{4} \))(\( 1 - 7 T^{2} + 25 T^{4} \))(\( ( 1 - 5 T^{2} )^{2} \))(\( 1 - 7 T^{2} + 25 T^{4} \))(\( 1 - 7 T^{2} + 25 T^{4} \))(\( 1 - 7 T^{2} + 25 T^{4} \))(\( 1 - 7 T^{2} + 25 T^{4} \))(\( ( 1 - 5 T^{2} )^{2} \))(\( 1 + 4 T^{2} - 9 T^{4} + 100 T^{6} + 625 T^{8} \))(\( ( 1 + 2 T^{2} + 25 T^{4} )^{2} \))(\( ( 1 + 2 T^{2} + 25 T^{4} )^{2} \))(\( 1 + 4 T^{2} - 9 T^{4} + 100 T^{6} + 625 T^{8} \))(\( ( 1 - 2 T^{2} + 33 T^{4} - 50 T^{6} + 625 T^{8} )^{2} \))(\( ( 1 - 2 T^{2} + 33 T^{4} - 50 T^{6} + 625 T^{8} )^{2} \))
$7$ (\( 1 + 5 T + 7 T^{2} \))(\( 1 + 4 T + 7 T^{2} \))(\( 1 + 4 T + 7 T^{2} \))(\( 1 + 4 T + 7 T^{2} \))(\( 1 + 4 T + 7 T^{2} \))(\( 1 + T + 7 T^{2} \))(\( 1 + T + 7 T^{2} \))(\( 1 + T + 7 T^{2} \))(\( 1 - T + 7 T^{2} \))(\( 1 - T + 7 T^{2} \))(\( 1 - T + 7 T^{2} \))(\( 1 - 4 T + 7 T^{2} \))(\( 1 - 4 T + 7 T^{2} \))(\( 1 - 4 T + 7 T^{2} \))(\( 1 - 4 T + 7 T^{2} \))(\( 1 - 5 T + 7 T^{2} \))(\( ( 1 - 7 T^{2} )^{2} \))(\( ( 1 - 7 T^{2} )^{2} \))(\( ( 1 - 7 T^{2} )^{2} \))(\( ( 1 - 7 T^{2} )^{2} \))(\( ( 1 + 2 T + 7 T^{2} )^{4} \))(\( ( 1 - 2 T + 7 T^{2} )^{4} \))
$11$ (\( ( 1 - 11 T^{2} )^{2} \))(\( 1 - 10 T^{2} + 121 T^{4} \))(\( 1 - 10 T^{2} + 121 T^{4} \))(\( 1 - 10 T^{2} + 121 T^{4} \))(\( 1 - 10 T^{2} + 121 T^{4} \))(\( ( 1 - 11 T^{2} )^{2} \))(\( 1 + 5 T^{2} + 121 T^{4} \))(\( 1 + 5 T^{2} + 121 T^{4} \))(\( 1 + 5 T^{2} + 121 T^{4} \))(\( 1 + 5 T^{2} + 121 T^{4} \))(\( ( 1 - 11 T^{2} )^{2} \))(\( 1 - 10 T^{2} + 121 T^{4} \))(\( 1 - 10 T^{2} + 121 T^{4} \))(\( 1 - 10 T^{2} + 121 T^{4} \))(\( 1 - 10 T^{2} + 121 T^{4} \))(\( ( 1 - 11 T^{2} )^{2} \))(\( 1 - 20 T^{2} + 279 T^{4} - 2420 T^{6} + 14641 T^{8} \))(\( ( 1 - 11 T^{2} )^{4} \))(\( ( 1 - 11 T^{2} )^{4} \))(\( 1 - 20 T^{2} + 279 T^{4} - 2420 T^{6} + 14641 T^{8} \))(\( ( 1 - 14 T^{2} + 129 T^{4} - 1694 T^{6} + 14641 T^{8} )^{2} \))(\( ( 1 - 14 T^{2} + 129 T^{4} - 1694 T^{6} + 14641 T^{8} )^{2} \))
$13$ (\( ( 1 - 7 T + 13 T^{2} )( 1 + 7 T + 13 T^{2} ) \))(\( 1 - 14 T^{2} + 169 T^{4} \))(\( 1 - 14 T^{2} + 169 T^{4} \))(\( 1 - 14 T^{2} + 169 T^{4} \))(\( 1 - 14 T^{2} + 169 T^{4} \))(\( ( 1 - 5 T + 13 T^{2} )( 1 + 5 T + 13 T^{2} ) \))(\( 1 - 14 T^{2} + 169 T^{4} \))(\( 1 - 14 T^{2} + 169 T^{4} \))(\( 1 - 14 T^{2} + 169 T^{4} \))(\( 1 - 14 T^{2} + 169 T^{4} \))(\( ( 1 - 5 T + 13 T^{2} )( 1 + 5 T + 13 T^{2} ) \))(\( 1 - 14 T^{2} + 169 T^{4} \))(\( 1 - 14 T^{2} + 169 T^{4} \))(\( 1 - 14 T^{2} + 169 T^{4} \))(\( 1 - 14 T^{2} + 169 T^{4} \))(\( ( 1 - 7 T + 13 T^{2} )( 1 + 7 T + 13 T^{2} ) \))(\( ( 1 - 13 T^{2} )^{4} \))(\( ( 1 - 5 T^{2} + 169 T^{4} )^{2} \))(\( ( 1 - 5 T^{2} + 169 T^{4} )^{2} \))(\( ( 1 - 13 T^{2} )^{4} \))(\( ( 1 - 16 T^{2} + 114 T^{4} - 2704 T^{6} + 28561 T^{8} )^{2} \))(\( ( 1 - 16 T^{2} + 114 T^{4} - 2704 T^{6} + 28561 T^{8} )^{2} \))
$17$ (\( ( 1 - 17 T^{2} )^{2} \))(\( 1 - 31 T^{2} + 289 T^{4} \))(\( 1 - 7 T^{2} + 289 T^{4} \))(\( 1 - 31 T^{2} + 289 T^{4} \))(\( 1 - 7 T^{2} + 289 T^{4} \))(\( ( 1 - 17 T^{2} )^{2} \))(\( 1 - 22 T^{2} + 289 T^{4} \))(\( 1 - 22 T^{2} + 289 T^{4} \))(\( 1 - 22 T^{2} + 289 T^{4} \))(\( 1 - 22 T^{2} + 289 T^{4} \))(\( ( 1 - 17 T^{2} )^{2} \))(\( 1 - 7 T^{2} + 289 T^{4} \))(\( 1 - 31 T^{2} + 289 T^{4} \))(\( 1 - 7 T^{2} + 289 T^{4} \))(\( 1 - 31 T^{2} + 289 T^{4} \))(\( ( 1 - 17 T^{2} )^{2} \))(\( ( 1 + 20 T^{2} + 289 T^{4} )^{2} \))(\( ( 1 - 31 T^{2} + 289 T^{4} )^{2} \))(\( ( 1 - 31 T^{2} + 289 T^{4} )^{2} \))(\( ( 1 + 20 T^{2} + 289 T^{4} )^{2} \))(\( ( 1 - 32 T^{2} + 762 T^{4} - 9248 T^{6} + 83521 T^{8} )^{2} \))(\( ( 1 - 32 T^{2} + 762 T^{4} - 9248 T^{6} + 83521 T^{8} )^{2} \))
$19$ (\( ( 1 - T + 19 T^{2} )^{2} \))(\( ( 1 - 2 T + 19 T^{2} )^{2} \))(\( ( 1 - 2 T + 19 T^{2} )^{2} \))(\( ( 1 - 2 T + 19 T^{2} )^{2} \))(\( ( 1 - 2 T + 19 T^{2} )^{2} \))(\( ( 1 + 7 T + 19 T^{2} )^{2} \))(\( ( 1 - 8 T + 19 T^{2} )^{2} \))(\( ( 1 - 8 T + 19 T^{2} )^{2} \))(\( ( 1 + 8 T + 19 T^{2} )^{2} \))(\( ( 1 + 8 T + 19 T^{2} )^{2} \))(\( ( 1 - 7 T + 19 T^{2} )^{2} \))(\( ( 1 + 2 T + 19 T^{2} )^{2} \))(\( ( 1 + 2 T + 19 T^{2} )^{2} \))(\( ( 1 + 2 T + 19 T^{2} )^{2} \))(\( ( 1 + 2 T + 19 T^{2} )^{2} \))(\( ( 1 + T + 19 T^{2} )^{2} \))(\( ( 1 + 12 T + 67 T^{2} + 228 T^{3} + 361 T^{4} )^{2} \))(\( ( 1 + 10 T^{2} + 361 T^{4} )^{2} \))(\( ( 1 + 10 T^{2} + 361 T^{4} )^{2} \))(\( ( 1 - 12 T + 67 T^{2} - 228 T^{3} + 361 T^{4} )^{2} \))(\( ( 1 - 2 T + 21 T^{2} - 38 T^{3} + 361 T^{4} )^{4} \))(\( ( 1 + 2 T + 21 T^{2} + 38 T^{3} + 361 T^{4} )^{4} \))
$23$ (\( ( 1 - 23 T^{2} )^{2} \))(\( ( 1 - 23 T^{2} )^{2} \))(\( 1 + 2 T^{2} + 529 T^{4} \))(\( ( 1 - 23 T^{2} )^{2} \))(\( 1 + 2 T^{2} + 529 T^{4} \))(\( ( 1 - 23 T^{2} )^{2} \))(\( 1 + 2 T^{2} + 529 T^{4} \))(\( 1 + 2 T^{2} + 529 T^{4} \))(\( 1 + 2 T^{2} + 529 T^{4} \))(\( 1 + 2 T^{2} + 529 T^{4} \))(\( ( 1 - 23 T^{2} )^{2} \))(\( 1 + 2 T^{2} + 529 T^{4} \))(\( ( 1 - 23 T^{2} )^{2} \))(\( 1 + 2 T^{2} + 529 T^{4} \))(\( ( 1 - 23 T^{2} )^{2} \))(\( ( 1 - 23 T^{2} )^{2} \))(\( 1 + 4 T^{2} - 513 T^{4} + 2116 T^{6} + 279841 T^{8} \))(\( ( 1 - 25 T^{2} + 529 T^{4} )^{2} \))(\( ( 1 - 25 T^{2} + 529 T^{4} )^{2} \))(\( 1 + 4 T^{2} - 513 T^{4} + 2116 T^{6} + 279841 T^{8} \))(\( ( 1 - 14 T^{2} - 351 T^{4} - 7406 T^{6} + 279841 T^{8} )^{2} \))(\( ( 1 - 14 T^{2} - 351 T^{4} - 7406 T^{6} + 279841 T^{8} )^{2} \))
$29$ (\( ( 1 + 29 T^{2} )^{2} \))(\( ( 1 + 6 T + 29 T^{2} )^{2} \))(\( ( 1 + 6 T + 29 T^{2} )^{2} \))(\( ( 1 - 6 T + 29 T^{2} )^{2} \))(\( ( 1 - 6 T + 29 T^{2} )^{2} \))(\( ( 1 + 29 T^{2} )^{2} \))(\( ( 1 + 6 T + 29 T^{2} )^{2} \))(\( ( 1 - 6 T + 29 T^{2} )^{2} \))(\( ( 1 + 6 T + 29 T^{2} )^{2} \))(\( ( 1 - 6 T + 29 T^{2} )^{2} \))(\( ( 1 + 29 T^{2} )^{2} \))(\( ( 1 + 6 T + 29 T^{2} )^{2} \))(\( ( 1 + 6 T + 29 T^{2} )^{2} \))(\( ( 1 - 6 T + 29 T^{2} )^{2} \))(\( ( 1 - 6 T + 29 T^{2} )^{2} \))(\( ( 1 + 29 T^{2} )^{2} \))(\( ( 1 + 29 T^{2} )^{4} \))(\( ( 1 - 5 T^{2} + 841 T^{4} )^{2} \))(\( ( 1 - 5 T^{2} + 841 T^{4} )^{2} \))(\( ( 1 + 29 T^{2} )^{4} \))(\( ( 1 + 8 T^{2} + 1050 T^{4} + 6728 T^{6} + 707281 T^{8} )^{2} \))(\( ( 1 + 8 T^{2} + 1050 T^{4} + 6728 T^{6} + 707281 T^{8} )^{2} \))
$31$ (\( ( 1 + 4 T + 31 T^{2} )^{2} \))(\( ( 1 + 2 T + 31 T^{2} )^{2} \))(\( ( 1 - 10 T + 31 T^{2} )^{2} \))(\( ( 1 + 2 T + 31 T^{2} )^{2} \))(\( ( 1 - 10 T + 31 T^{2} )^{2} \))(\( ( 1 - 4 T + 31 T^{2} )^{2} \))(\( ( 1 + 5 T + 31 T^{2} )^{2} \))(\( ( 1 + 5 T + 31 T^{2} )^{2} \))(\( ( 1 - 5 T + 31 T^{2} )^{2} \))(\( ( 1 - 5 T + 31 T^{2} )^{2} \))(\( ( 1 + 4 T + 31 T^{2} )^{2} \))(\( ( 1 + 10 T + 31 T^{2} )^{2} \))(\( ( 1 - 2 T + 31 T^{2} )^{2} \))(\( ( 1 + 10 T + 31 T^{2} )^{2} \))(\( ( 1 - 2 T + 31 T^{2} )^{2} \))(\( ( 1 - 4 T + 31 T^{2} )^{2} \))(\( ( 1 + 6 T + 43 T^{2} + 186 T^{3} + 961 T^{4} )^{2} \))(\( ( 1 + 55 T^{2} + 961 T^{4} )^{2} \))(\( ( 1 + 55 T^{2} + 961 T^{4} )^{2} \))(\( ( 1 - 6 T + 43 T^{2} - 186 T^{3} + 961 T^{4} )^{2} \))(\( ( 1 + 2 T + 45 T^{2} + 62 T^{3} + 961 T^{4} )^{4} \))(\( ( 1 - 2 T + 45 T^{2} - 62 T^{3} + 961 T^{4} )^{4} \))
$37$ (\( ( 1 - T + 37 T^{2} )^{2} \))(\( ( 1 - T + 37 T^{2} )^{2} \))(\( ( 1 + 11 T + 37 T^{2} )^{2} \))(\( ( 1 - T + 37 T^{2} )^{2} \))(\( ( 1 + 11 T + 37 T^{2} )^{2} \))(\( ( 1 + 11 T + 37 T^{2} )^{2} \))(\( ( 1 - 4 T + 37 T^{2} )^{2} \))(\( ( 1 - 4 T + 37 T^{2} )^{2} \))(\( ( 1 - 4 T + 37 T^{2} )^{2} \))(\( ( 1 - 4 T + 37 T^{2} )^{2} \))(\( ( 1 + 11 T + 37 T^{2} )^{2} \))(\( ( 1 + 11 T + 37 T^{2} )^{2} \))(\( ( 1 - T + 37 T^{2} )^{2} \))(\( ( 1 + 11 T + 37 T^{2} )^{2} \))(\( ( 1 - T + 37 T^{2} )^{2} \))(\( ( 1 - T + 37 T^{2} )^{2} \))(\( ( 1 - 8 T + 27 T^{2} - 296 T^{3} + 1369 T^{4} )^{2} \))(\( ( 1 - 4 T + 37 T^{2} )^{4} \))(\( ( 1 - 4 T + 37 T^{2} )^{4} \))(\( ( 1 - 8 T + 27 T^{2} - 296 T^{3} + 1369 T^{4} )^{2} \))(\( ( 1 - T + 37 T^{2} )^{8} \))(\( ( 1 - T + 37 T^{2} )^{8} \))
$41$ (\( ( 1 - 41 T^{2} )^{2} \))(\( 1 - 55 T^{2} + 1681 T^{4} \))(\( 1 - 79 T^{2} + 1681 T^{4} \))(\( 1 - 55 T^{2} + 1681 T^{4} \))(\( 1 - 79 T^{2} + 1681 T^{4} \))(\( ( 1 - 41 T^{2} )^{2} \))(\( 1 - 34 T^{2} + 1681 T^{4} \))(\( 1 - 34 T^{2} + 1681 T^{4} \))(\( 1 - 34 T^{2} + 1681 T^{4} \))(\( 1 - 34 T^{2} + 1681 T^{4} \))(\( ( 1 - 41 T^{2} )^{2} \))(\( 1 - 79 T^{2} + 1681 T^{4} \))(\( 1 - 55 T^{2} + 1681 T^{4} \))(\( 1 - 79 T^{2} + 1681 T^{4} \))(\( 1 - 55 T^{2} + 1681 T^{4} \))(\( ( 1 - 41 T^{2} )^{2} \))(\( 1 - 68 T^{2} + 2943 T^{4} - 114308 T^{6} + 2825761 T^{8} \))(\( ( 1 - 34 T^{2} + 1681 T^{4} )^{2} \))(\( ( 1 - 34 T^{2} + 1681 T^{4} )^{2} \))(\( 1 - 68 T^{2} + 2943 T^{4} - 114308 T^{6} + 2825761 T^{8} \))(\( ( 1 - 50 T^{2} + 3825 T^{4} - 84050 T^{6} + 2825761 T^{8} )^{2} \))(\( ( 1 - 50 T^{2} + 3825 T^{4} - 84050 T^{6} + 2825761 T^{8} )^{2} \))
$43$ (\( ( 1 - 8 T + 43 T^{2} )( 1 + 8 T + 43 T^{2} ) \))(\( ( 1 - 13 T + 43 T^{2} )( 1 + 13 T + 43 T^{2} ) \))(\( 1 - 59 T^{2} + 1849 T^{4} \))(\( ( 1 - 13 T + 43 T^{2} )( 1 + 13 T + 43 T^{2} ) \))(\( 1 - 59 T^{2} + 1849 T^{4} \))(\( ( 1 - 8 T + 43 T^{2} )( 1 + 8 T + 43 T^{2} ) \))(\( 1 - 74 T^{2} + 1849 T^{4} \))(\( 1 - 74 T^{2} + 1849 T^{4} \))(\( 1 - 74 T^{2} + 1849 T^{4} \))(\( 1 - 74 T^{2} + 1849 T^{4} \))(\( ( 1 - 8 T + 43 T^{2} )( 1 + 8 T + 43 T^{2} ) \))(\( 1 - 59 T^{2} + 1849 T^{4} \))(\( ( 1 - 13 T + 43 T^{2} )( 1 + 13 T + 43 T^{2} ) \))(\( 1 - 59 T^{2} + 1849 T^{4} \))(\( ( 1 - 13 T + 43 T^{2} )( 1 + 13 T + 43 T^{2} ) \))(\( ( 1 - 8 T + 43 T^{2} )( 1 + 8 T + 43 T^{2} ) \))(\( ( 1 - 43 T^{2} )^{4} \))(\( ( 1 - 13 T + 43 T^{2} )^{2}( 1 + 13 T + 43 T^{2} )^{2} \))(\( ( 1 - 13 T + 43 T^{2} )^{2}( 1 + 13 T + 43 T^{2} )^{2} \))(\( ( 1 - 43 T^{2} )^{4} \))(\( ( 1 - 64 T^{2} + 3570 T^{4} - 118336 T^{6} + 3418801 T^{8} )^{2} \))(\( ( 1 - 64 T^{2} + 3570 T^{4} - 118336 T^{6} + 3418801 T^{8} )^{2} \))
$47$ (\( ( 1 + 47 T^{2} )^{2} \))(\( ( 1 + 3 T + 47 T^{2} )^{2} \))(\( ( 1 - 9 T + 47 T^{2} )^{2} \))(\( ( 1 - 3 T + 47 T^{2} )^{2} \))(\( ( 1 + 9 T + 47 T^{2} )^{2} \))(\( ( 1 + 47 T^{2} )^{2} \))(\( ( 1 - 6 T + 47 T^{2} )^{2} \))(\( ( 1 + 6 T + 47 T^{2} )^{2} \))(\( ( 1 + 6 T + 47 T^{2} )^{2} \))(\( ( 1 - 6 T + 47 T^{2} )^{2} \))(\( ( 1 + 47 T^{2} )^{2} \))(\( ( 1 + 9 T + 47 T^{2} )^{2} \))(\( ( 1 - 3 T + 47 T^{2} )^{2} \))(\( ( 1 - 9 T + 47 T^{2} )^{2} \))(\( ( 1 + 3 T + 47 T^{2} )^{2} \))(\( ( 1 + 47 T^{2} )^{2} \))(\( ( 1 + 47 T^{2} )^{4} \))(\( ( 1 + 6 T + 47 T^{2} )^{4} \))(\( ( 1 - 6 T + 47 T^{2} )^{4} \))(\( ( 1 + 47 T^{2} )^{4} \))(\( ( 1 + 8 T^{2} - 1398 T^{4} + 17672 T^{6} + 4879681 T^{8} )^{2} \))(\( ( 1 + 8 T^{2} - 1398 T^{4} + 17672 T^{6} + 4879681 T^{8} )^{2} \))
$53$ (\( ( 1 + 53 T^{2} )^{2} \))(\( ( 1 + 12 T + 53 T^{2} )^{2} \))(\( ( 1 - 12 T + 53 T^{2} )^{2} \))(\( ( 1 - 12 T + 53 T^{2} )^{2} \))(\( ( 1 + 12 T + 53 T^{2} )^{2} \))(\( ( 1 + 53 T^{2} )^{2} \))(\( ( 1 + 3 T + 53 T^{2} )^{2} \))(\( ( 1 - 3 T + 53 T^{2} )^{2} \))(\( ( 1 + 3 T + 53 T^{2} )^{2} \))(\( ( 1 - 3 T + 53 T^{2} )^{2} \))(\( ( 1 + 53 T^{2} )^{2} \))(\( ( 1 - 12 T + 53 T^{2} )^{2} \))(\( ( 1 + 12 T + 53 T^{2} )^{2} \))(\( ( 1 + 12 T + 53 T^{2} )^{2} \))(\( ( 1 - 12 T + 53 T^{2} )^{2} \))(\( ( 1 + 53 T^{2} )^{2} \))(\( ( 1 + 53 T^{2} )^{4} \))(\( ( 1 + 43 T^{2} + 2809 T^{4} )^{2} \))(\( ( 1 + 43 T^{2} + 2809 T^{4} )^{2} \))(\( ( 1 + 53 T^{2} )^{4} \))(\( ( 1 - 4 T^{2} + 3030 T^{4} - 11236 T^{6} + 7890481 T^{8} )^{2} \))(\( ( 1 - 4 T^{2} + 3030 T^{4} - 11236 T^{6} + 7890481 T^{8} )^{2} \))
$59$ (\( ( 1 + 59 T^{2} )^{2} \))(\( ( 1 - 3 T + 59 T^{2} )^{2} \))(\( ( 1 - 3 T + 59 T^{2} )^{2} \))(\( ( 1 + 3 T + 59 T^{2} )^{2} \))(\( ( 1 + 3 T + 59 T^{2} )^{2} \))(\( ( 1 + 59 T^{2} )^{2} \))(\( ( 1 - 12 T + 59 T^{2} )^{2} \))(\( ( 1 + 12 T + 59 T^{2} )^{2} \))(\( ( 1 + 12 T + 59 T^{2} )^{2} \))(\( ( 1 - 12 T + 59 T^{2} )^{2} \))(\( ( 1 + 59 T^{2} )^{2} \))(\( ( 1 + 3 T + 59 T^{2} )^{2} \))(\( ( 1 + 3 T + 59 T^{2} )^{2} \))(\( ( 1 - 3 T + 59 T^{2} )^{2} \))(\( ( 1 - 3 T + 59 T^{2} )^{2} \))(\( ( 1 + 59 T^{2} )^{2} \))(\( ( 1 + 59 T^{2} )^{4} \))(\( ( 1 + 3 T + 59 T^{2} )^{4} \))(\( ( 1 - 3 T + 59 T^{2} )^{4} \))(\( ( 1 + 59 T^{2} )^{4} \))(\( ( 1 + 128 T^{2} + 10410 T^{4} + 445568 T^{6} + 12117361 T^{8} )^{2} \))(\( ( 1 + 128 T^{2} + 10410 T^{4} + 445568 T^{6} + 12117361 T^{8} )^{2} \))
$61$ (\( ( 1 - 13 T + 61 T^{2} )( 1 + 13 T + 61 T^{2} ) \))(\( 1 - 14 T^{2} + 3721 T^{4} \))(\( 1 - 110 T^{2} + 3721 T^{4} \))(\( 1 - 14 T^{2} + 3721 T^{4} \))(\( 1 - 110 T^{2} + 3721 T^{4} \))(\( ( 1 - T + 61 T^{2} )( 1 + T + 61 T^{2} ) \))(\( 1 + 70 T^{2} + 3721 T^{4} \))(\( 1 + 70 T^{2} + 3721 T^{4} \))(\( 1 + 70 T^{2} + 3721 T^{4} \))(\( 1 + 70 T^{2} + 3721 T^{4} \))(\( ( 1 - T + 61 T^{2} )( 1 + T + 61 T^{2} ) \))(\( 1 - 110 T^{2} + 3721 T^{4} \))(\( 1 - 14 T^{2} + 3721 T^{4} \))(\( 1 - 110 T^{2} + 3721 T^{4} \))(\( 1 - 14 T^{2} + 3721 T^{4} \))(\( ( 1 - 13 T + 61 T^{2} )( 1 + 13 T + 61 T^{2} ) \))(\( ( 1 - 61 T^{2} )^{4} \))(\( ( 1 - 38 T^{2} + 3721 T^{4} )^{2} \))(\( ( 1 - 38 T^{2} + 3721 T^{4} )^{2} \))(\( ( 1 - 61 T^{2} )^{4} \))(\( ( 1 - 110 T^{2} + 3721 T^{4} )^{4} \))(\( ( 1 - 110 T^{2} + 3721 T^{4} )^{4} \))
$67$ (\( ( 1 - 11 T + 67 T^{2} )( 1 + 11 T + 67 T^{2} ) \))(\( 1 - 26 T^{2} + 4489 T^{4} \))(\( ( 1 - 16 T + 67 T^{2} )( 1 + 16 T + 67 T^{2} ) \))(\( 1 - 26 T^{2} + 4489 T^{4} \))(\( ( 1 - 16 T + 67 T^{2} )( 1 + 16 T + 67 T^{2} ) \))(\( ( 1 - 5 T + 67 T^{2} )( 1 + 5 T + 67 T^{2} ) \))(\( ( 1 - 16 T + 67 T^{2} )( 1 + 16 T + 67 T^{2} ) \))(\( ( 1 - 16 T + 67 T^{2} )( 1 + 16 T + 67 T^{2} ) \))(\( ( 1 - 16 T + 67 T^{2} )( 1 + 16 T + 67 T^{2} ) \))(\( ( 1 - 16 T + 67 T^{2} )( 1 + 16 T + 67 T^{2} ) \))(\( ( 1 - 5 T + 67 T^{2} )( 1 + 5 T + 67 T^{2} ) \))(\( ( 1 - 16 T + 67 T^{2} )( 1 + 16 T + 67 T^{2} ) \))(\( 1 - 26 T^{2} + 4489 T^{4} \))(\( ( 1 - 16 T + 67 T^{2} )( 1 + 16 T + 67 T^{2} ) \))(\( 1 - 26 T^{2} + 4489 T^{4} \))(\( ( 1 - 11 T + 67 T^{2} )( 1 + 11 T + 67 T^{2} ) \))(\( ( 1 - 67 T^{2} )^{4} \))(\( ( 1 - 11 T + 67 T^{2} )^{2}( 1 + 11 T + 67 T^{2} )^{2} \))(\( ( 1 - 11 T + 67 T^{2} )^{2}( 1 + 11 T + 67 T^{2} )^{2} \))(\( ( 1 - 67 T^{2} )^{4} \))(\( ( 1 - 136 T^{2} + 11010 T^{4} - 610504 T^{6} + 20151121 T^{8} )^{2} \))(\( ( 1 - 136 T^{2} + 11010 T^{4} - 610504 T^{6} + 20151121 T^{8} )^{2} \))
$71$ (\( ( 1 - 71 T^{2} )^{2} \))(\( 1 + 50 T^{2} + 5041 T^{4} \))(\( 1 + 50 T^{2} + 5041 T^{4} \))(\( 1 + 50 T^{2} + 5041 T^{4} \))(\( 1 + 50 T^{2} + 5041 T^{4} \))(\( ( 1 - 71 T^{2} )^{2} \))(\( 1 - 130 T^{2} + 5041 T^{4} \))(\( 1 - 130 T^{2} + 5041 T^{4} \))(\( 1 - 130 T^{2} + 5041 T^{4} \))(\( 1 - 130 T^{2} + 5041 T^{4} \))(\( ( 1 - 71 T^{2} )^{2} \))(\( 1 + 50 T^{2} + 5041 T^{4} \))(\( 1 + 50 T^{2} + 5041 T^{4} \))(\( 1 + 50 T^{2} + 5041 T^{4} \))(\( 1 + 50 T^{2} + 5041 T^{4} \))(\( ( 1 - 71 T^{2} )^{2} \))(\( 1 + 100 T^{2} + 4959 T^{4} + 504100 T^{6} + 25411681 T^{8} \))(\( ( 1 - 121 T^{2} + 5041 T^{4} )^{2} \))(\( ( 1 - 121 T^{2} + 5041 T^{4} )^{2} \))(\( 1 + 100 T^{2} + 4959 T^{4} + 504100 T^{6} + 25411681 T^{8} \))(\( ( 1 - 266 T^{2} + 27753 T^{4} - 1340906 T^{6} + 25411681 T^{8} )^{2} \))(\( ( 1 - 266 T^{2} + 27753 T^{4} - 1340906 T^{6} + 25411681 T^{8} )^{2} \))
$73$ (\( ( 1 - 17 T + 73 T^{2} )( 1 + 17 T + 73 T^{2} ) \))(\( ( 1 - 73 T^{2} )^{2} \))(\( 1 - 98 T^{2} + 5329 T^{4} \))(\( ( 1 - 73 T^{2} )^{2} \))(\( 1 - 98 T^{2} + 5329 T^{4} \))(\( ( 1 - 7 T + 73 T^{2} )( 1 + 7 T + 73 T^{2} ) \))(\( ( 1 - 17 T + 73 T^{2} )( 1 + 17 T + 73 T^{2} ) \))(\( ( 1 - 17 T + 73 T^{2} )( 1 + 17 T + 73 T^{2} ) \))(\( ( 1 - 17 T + 73 T^{2} )( 1 + 17 T + 73 T^{2} ) \))(\( ( 1 - 17 T + 73 T^{2} )( 1 + 17 T + 73 T^{2} ) \))(\( ( 1 - 7 T + 73 T^{2} )( 1 + 7 T + 73 T^{2} ) \))(\( 1 - 98 T^{2} + 5329 T^{4} \))(\( ( 1 - 73 T^{2} )^{2} \))(\( 1 - 98 T^{2} + 5329 T^{4} \))(\( ( 1 - 73 T^{2} )^{2} \))(\( ( 1 - 17 T + 73 T^{2} )( 1 + 17 T + 73 T^{2} ) \))(\( ( 1 - 73 T^{2} )^{4} \))(\( ( 1 - 62 T^{2} + 5329 T^{4} )^{2} \))(\( ( 1 - 62 T^{2} + 5329 T^{4} )^{2} \))(\( ( 1 - 73 T^{2} )^{4} \))(\( ( 1 - 88 T^{2} + 2226 T^{4} - 468952 T^{6} + 28398241 T^{8} )^{2} \))(\( ( 1 - 88 T^{2} + 2226 T^{4} - 468952 T^{6} + 28398241 T^{8} )^{2} \))
$79$ (\( ( 1 - 13 T + 79 T^{2} )( 1 + 13 T + 79 T^{2} ) \))(\( 1 - 155 T^{2} + 6241 T^{4} \))(\( ( 1 - 13 T + 79 T^{2} )( 1 + 13 T + 79 T^{2} ) \))(\( 1 - 155 T^{2} + 6241 T^{4} \))(\( ( 1 - 13 T + 79 T^{2} )( 1 + 13 T + 79 T^{2} ) \))(\( ( 1 - 17 T + 79 T^{2} )( 1 + 17 T + 79 T^{2} ) \))(\( 1 - 146 T^{2} + 6241 T^{4} \))(\( 1 - 146 T^{2} + 6241 T^{4} \))(\( 1 - 146 T^{2} + 6241 T^{4} \))(\( 1 - 146 T^{2} + 6241 T^{4} \))(\( ( 1 - 17 T + 79 T^{2} )( 1 + 17 T + 79 T^{2} ) \))(\( ( 1 - 13 T + 79 T^{2} )( 1 + 13 T + 79 T^{2} ) \))(\( 1 - 155 T^{2} + 6241 T^{4} \))(\( ( 1 - 13 T + 79 T^{2} )( 1 + 13 T + 79 T^{2} ) \))(\( 1 - 155 T^{2} + 6241 T^{4} \))(\( ( 1 - 13 T + 79 T^{2} )( 1 + 13 T + 79 T^{2} ) \))(\( ( 1 - 79 T^{2} )^{4} \))(\( ( 1 - 110 T^{2} + 6241 T^{4} )^{2} \))(\( ( 1 - 110 T^{2} + 6241 T^{4} )^{2} \))(\( ( 1 - 79 T^{2} )^{4} \))(\( ( 1 - 208 T^{2} + 22146 T^{4} - 1298128 T^{6} + 38950081 T^{8} )^{2} \))(\( ( 1 - 208 T^{2} + 22146 T^{4} - 1298128 T^{6} + 38950081 T^{8} )^{2} \))
$83$ (\( ( 1 + 83 T^{2} )^{2} \))(\( ( 1 + 9 T + 83 T^{2} )^{2} \))(\( ( 1 - 15 T + 83 T^{2} )^{2} \))(\( ( 1 - 9 T + 83 T^{2} )^{2} \))(\( ( 1 + 15 T + 83 T^{2} )^{2} \))(\( ( 1 + 83 T^{2} )^{2} \))(\( ( 1 - 15 T + 83 T^{2} )^{2} \))(\( ( 1 + 15 T + 83 T^{2} )^{2} \))(\( ( 1 + 15 T + 83 T^{2} )^{2} \))(\( ( 1 - 15 T + 83 T^{2} )^{2} \))(\( ( 1 + 83 T^{2} )^{2} \))(\( ( 1 + 15 T + 83 T^{2} )^{2} \))(\( ( 1 - 9 T + 83 T^{2} )^{2} \))(\( ( 1 - 15 T + 83 T^{2} )^{2} \))(\( ( 1 + 9 T + 83 T^{2} )^{2} \))(\( ( 1 + 83 T^{2} )^{2} \))(\( ( 1 + 83 T^{2} )^{4} \))(\( ( 1 - 12 T + 83 T^{2} )^{4} \))(\( ( 1 + 12 T + 83 T^{2} )^{4} \))(\( ( 1 + 83 T^{2} )^{4} \))(\( ( 1 + 152 T^{2} + 13722 T^{4} + 1047128 T^{6} + 47458321 T^{8} )^{2} \))(\( ( 1 + 152 T^{2} + 13722 T^{4} + 1047128 T^{6} + 47458321 T^{8} )^{2} \))
$89$ (\( ( 1 - 89 T^{2} )^{2} \))(\( 1 - 130 T^{2} + 7921 T^{4} \))(\( 1 - 130 T^{2} + 7921 T^{4} \))(\( 1 - 130 T^{2} + 7921 T^{4} \))(\( 1 - 130 T^{2} + 7921 T^{4} \))(\( ( 1 - 89 T^{2} )^{2} \))(\( 1 - 70 T^{2} + 7921 T^{4} \))(\( 1 - 70 T^{2} + 7921 T^{4} \))(\( 1 - 70 T^{2} + 7921 T^{4} \))(\( 1 - 70 T^{2} + 7921 T^{4} \))(\( ( 1 - 89 T^{2} )^{2} \))(\( 1 - 130 T^{2} + 7921 T^{4} \))(\( 1 - 130 T^{2} + 7921 T^{4} \))(\( 1 - 130 T^{2} + 7921 T^{4} \))(\( 1 - 130 T^{2} + 7921 T^{4} \))(\( ( 1 - 89 T^{2} )^{2} \))(\( 1 + 172 T^{2} + 21663 T^{4} + 1362412 T^{6} + 62742241 T^{8} \))(\( ( 1 - 151 T^{2} + 7921 T^{4} )^{2} \))(\( ( 1 - 151 T^{2} + 7921 T^{4} )^{2} \))(\( 1 + 172 T^{2} + 21663 T^{4} + 1362412 T^{6} + 62742241 T^{8} \))(\( ( 1 - 146 T^{2} + 13233 T^{4} - 1156466 T^{6} + 62742241 T^{8} )^{2} \))(\( ( 1 - 146 T^{2} + 13233 T^{4} - 1156466 T^{6} + 62742241 T^{8} )^{2} \))
$97$ (\( ( 1 - 5 T + 97 T^{2} )( 1 + 5 T + 97 T^{2} ) \))(\( 1 - 182 T^{2} + 9409 T^{4} \))(\( 1 + 106 T^{2} + 9409 T^{4} \))(\( 1 - 182 T^{2} + 9409 T^{4} \))(\( 1 + 106 T^{2} + 9409 T^{4} \))(\( ( 1 - 19 T + 97 T^{2} )( 1 + 19 T + 97 T^{2} ) \))(\( 1 - 119 T^{2} + 9409 T^{4} \))(\( 1 - 119 T^{2} + 9409 T^{4} \))(\( 1 - 119 T^{2} + 9409 T^{4} \))(\( 1 - 119 T^{2} + 9409 T^{4} \))(\( ( 1 - 19 T + 97 T^{2} )( 1 + 19 T + 97 T^{2} ) \))(\( 1 + 106 T^{2} + 9409 T^{4} \))(\( 1 - 182 T^{2} + 9409 T^{4} \))(\( 1 + 106 T^{2} + 9409 T^{4} \))(\( 1 - 182 T^{2} + 9409 T^{4} \))(\( ( 1 - 5 T + 97 T^{2} )( 1 + 5 T + 97 T^{2} ) \))(\( ( 1 - 97 T^{2} )^{4} \))(\( ( 1 - 110 T^{2} + 9409 T^{4} )^{2} \))(\( ( 1 - 110 T^{2} + 9409 T^{4} )^{2} \))(\( ( 1 - 97 T^{2} )^{4} \))(\( ( 1 - 172 T^{2} + 21606 T^{4} - 1618348 T^{6} + 88529281 T^{8} )^{2} \))(\( ( 1 - 172 T^{2} + 21606 T^{4} - 1618348 T^{6} + 88529281 T^{8} )^{2} \))
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