Properties

Label 2268.2.j
Level 2268
Weight 2
Character orbit j
Rep. character \(\chi_{2268}(757,\cdot)\)
Character field \(\Q(\zeta_{3})\)
Dimension 48
Newform subspaces 18
Sturm bound 864
Trace bound 13

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Defining parameters

Level: \( N \) \(=\) \( 2268 = 2^{2} \cdot 3^{4} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2268.j (of order \(3\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 9 \)
Character field: \(\Q(\zeta_{3})\)
Newform subspaces: \( 18 \)
Sturm bound: \(864\)
Trace bound: \(13\)
Distinguishing \(T_p\): \(5\), \(11\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(2268, [\chi])\).

Total New Old
Modular forms 936 48 888
Cusp forms 792 48 744
Eisenstein series 144 0 144

Trace form

\( 48q + O(q^{10}) \) \( 48q - 24q^{19} - 36q^{25} - 12q^{31} + 24q^{37} + 12q^{43} - 24q^{49} + 24q^{55} - 24q^{73} - 12q^{79} - 24q^{85} + 24q^{91} - 12q^{97} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(2268, [\chi])\) into newform subspaces

Label Dim. \(A\) Field CM Traces $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\)
2268.2.j.a \(2\) \(18.110\) \(\Q(\sqrt{-3}) \) None \(0\) \(0\) \(-4\) \(1\) \(q-4\zeta_{6}q^{5}+(1-\zeta_{6})q^{7}+(-2+2\zeta_{6})q^{11}+\cdots\)
2268.2.j.b \(2\) \(18.110\) \(\Q(\sqrt{-3}) \) None \(0\) \(0\) \(-3\) \(-1\) \(q-3\zeta_{6}q^{5}+(-1+\zeta_{6})q^{7}+(-3+3\zeta_{6})q^{11}+\cdots\)
2268.2.j.c \(2\) \(18.110\) \(\Q(\sqrt{-3}) \) None \(0\) \(0\) \(-3\) \(-1\) \(q-3\zeta_{6}q^{5}+(-1+\zeta_{6})q^{7}-2\zeta_{6}q^{13}+\cdots\)
2268.2.j.d \(2\) \(18.110\) \(\Q(\sqrt{-3}) \) None \(0\) \(0\) \(-2\) \(1\) \(q-2\zeta_{6}q^{5}+(1-\zeta_{6})q^{7}+(-1+\zeta_{6})q^{11}+\cdots\)
2268.2.j.e \(2\) \(18.110\) \(\Q(\sqrt{-3}) \) None \(0\) \(0\) \(-1\) \(1\) \(q-\zeta_{6}q^{5}+(1-\zeta_{6})q^{7}+(-2+2\zeta_{6})q^{11}+\cdots\)
2268.2.j.f \(2\) \(18.110\) \(\Q(\sqrt{-3}) \) None \(0\) \(0\) \(0\) \(-1\) \(q+(-1+\zeta_{6})q^{7}+(-6+6\zeta_{6})q^{11}+\cdots\)
2268.2.j.g \(2\) \(18.110\) \(\Q(\sqrt{-3}) \) None \(0\) \(0\) \(0\) \(-1\) \(q+(-1+\zeta_{6})q^{7}+(-3+3\zeta_{6})q^{11}+\cdots\)
2268.2.j.h \(2\) \(18.110\) \(\Q(\sqrt{-3}) \) None \(0\) \(0\) \(0\) \(-1\) \(q+(-1+\zeta_{6})q^{7}+(3-3\zeta_{6})q^{11}+4\zeta_{6}q^{13}+\cdots\)
2268.2.j.i \(2\) \(18.110\) \(\Q(\sqrt{-3}) \) None \(0\) \(0\) \(0\) \(-1\) \(q+(-1+\zeta_{6})q^{7}+(6-6\zeta_{6})q^{11}-2\zeta_{6}q^{13}+\cdots\)
2268.2.j.j \(2\) \(18.110\) \(\Q(\sqrt{-3}) \) None \(0\) \(0\) \(1\) \(1\) \(q+\zeta_{6}q^{5}+(1-\zeta_{6})q^{7}+(2-2\zeta_{6})q^{11}+\cdots\)
2268.2.j.k \(2\) \(18.110\) \(\Q(\sqrt{-3}) \) None \(0\) \(0\) \(2\) \(1\) \(q+2\zeta_{6}q^{5}+(1-\zeta_{6})q^{7}+(1-\zeta_{6})q^{11}+\cdots\)
2268.2.j.l \(2\) \(18.110\) \(\Q(\sqrt{-3}) \) None \(0\) \(0\) \(3\) \(-1\) \(q+3\zeta_{6}q^{5}+(-1+\zeta_{6})q^{7}-2\zeta_{6}q^{13}+\cdots\)
2268.2.j.m \(2\) \(18.110\) \(\Q(\sqrt{-3}) \) None \(0\) \(0\) \(3\) \(-1\) \(q+3\zeta_{6}q^{5}+(-1+\zeta_{6})q^{7}+(3-3\zeta_{6})q^{11}+\cdots\)
2268.2.j.n \(2\) \(18.110\) \(\Q(\sqrt{-3}) \) None \(0\) \(0\) \(4\) \(1\) \(q+4\zeta_{6}q^{5}+(1-\zeta_{6})q^{7}+(2-2\zeta_{6})q^{11}+\cdots\)
2268.2.j.o \(4\) \(18.110\) \(\Q(\zeta_{12})\) None \(0\) \(0\) \(0\) \(-2\) \(q-\zeta_{12}^{2}q^{5}+(-1+\zeta_{12})q^{7}+(2\zeta_{12}^{2}+\cdots)q^{11}+\cdots\)
2268.2.j.p \(4\) \(18.110\) \(\Q(\zeta_{12})\) None \(0\) \(0\) \(0\) \(-2\) \(q-2\zeta_{12}^{2}q^{5}+(-1+\zeta_{12})q^{7}+(\zeta_{12}^{2}+\cdots)q^{11}+\cdots\)
2268.2.j.q \(4\) \(18.110\) \(\Q(\sqrt{-3}, \sqrt{13})\) None \(0\) \(0\) \(0\) \(2\) \(q-\beta _{2}q^{5}+(1-\beta _{1})q^{7}+(-\beta _{2}-\beta _{3})q^{11}+\cdots\)
2268.2.j.r \(8\) \(18.110\) 8.0.2702336256.1 None \(0\) \(0\) \(0\) \(4\) \(q+(\beta _{3}-\beta _{6})q^{5}+(1-\beta _{2})q^{7}+(\beta _{4}+\beta _{5}+\cdots)q^{11}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(2268, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(2268, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(18, [\chi])\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(36, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(54, [\chi])\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(63, [\chi])\)\(^{\oplus 9}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(81, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(108, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(126, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(162, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(189, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(252, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(324, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(378, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(567, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(756, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(1134, [\chi])\)\(^{\oplus 2}\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ 1
$5$ (\( 1 + 4 T + 11 T^{2} + 20 T^{3} + 25 T^{4} \))(\( 1 + 3 T + 4 T^{2} + 15 T^{3} + 25 T^{4} \))(\( 1 + 3 T + 4 T^{2} + 15 T^{3} + 25 T^{4} \))(\( 1 + 2 T - T^{2} + 10 T^{3} + 25 T^{4} \))(\( 1 + T - 4 T^{2} + 5 T^{3} + 25 T^{4} \))(\( 1 - 5 T^{2} + 25 T^{4} \))(\( 1 - 5 T^{2} + 25 T^{4} \))(\( 1 - 5 T^{2} + 25 T^{4} \))(\( 1 - 5 T^{2} + 25 T^{4} \))(\( 1 - T - 4 T^{2} - 5 T^{3} + 25 T^{4} \))(\( 1 - 2 T - T^{2} - 10 T^{3} + 25 T^{4} \))(\( 1 - 3 T + 4 T^{2} - 15 T^{3} + 25 T^{4} \))(\( 1 - 3 T + 4 T^{2} - 15 T^{3} + 25 T^{4} \))(\( 1 - 4 T + 11 T^{2} - 20 T^{3} + 25 T^{4} \))(\( 1 - 7 T^{2} + 24 T^{4} - 175 T^{6} + 625 T^{8} \))(\( 1 + 2 T^{2} - 21 T^{4} + 50 T^{6} + 625 T^{8} \))(\( 1 + 3 T^{2} - 16 T^{4} + 75 T^{6} + 625 T^{8} \))(\( ( 1 - 9 T^{2} + 25 T^{4} )^{2}( 1 + 9 T^{2} + 56 T^{4} + 225 T^{6} + 625 T^{8} ) \))
$7$ (\( 1 - T + T^{2} \))(\( 1 + T + T^{2} \))(\( 1 + T + T^{2} \))(\( 1 - T + T^{2} \))(\( 1 - T + T^{2} \))(\( 1 + T + T^{2} \))(\( 1 + T + T^{2} \))(\( 1 + T + T^{2} \))(\( 1 + T + T^{2} \))(\( 1 - T + T^{2} \))(\( 1 - T + T^{2} \))(\( 1 + T + T^{2} \))(\( 1 + T + T^{2} \))(\( 1 - T + T^{2} \))(\( ( 1 + T + T^{2} )^{2} \))(\( ( 1 + T + T^{2} )^{2} \))(\( ( 1 - T + T^{2} )^{2} \))(\( ( 1 - T + T^{2} )^{4} \))
$11$ (\( 1 + 2 T - 7 T^{2} + 22 T^{3} + 121 T^{4} \))(\( 1 + 3 T - 2 T^{2} + 33 T^{3} + 121 T^{4} \))(\( 1 - 11 T^{2} + 121 T^{4} \))(\( 1 + T - 10 T^{2} + 11 T^{3} + 121 T^{4} \))(\( 1 + 2 T - 7 T^{2} + 22 T^{3} + 121 T^{4} \))(\( 1 + 6 T + 25 T^{2} + 66 T^{3} + 121 T^{4} \))(\( 1 + 3 T - 2 T^{2} + 33 T^{3} + 121 T^{4} \))(\( 1 - 3 T - 2 T^{2} - 33 T^{3} + 121 T^{4} \))(\( 1 - 6 T + 25 T^{2} - 66 T^{3} + 121 T^{4} \))(\( 1 - 2 T - 7 T^{2} - 22 T^{3} + 121 T^{4} \))(\( 1 - T - 10 T^{2} - 11 T^{3} + 121 T^{4} \))(\( 1 - 11 T^{2} + 121 T^{4} \))(\( 1 - 3 T - 2 T^{2} - 33 T^{3} + 121 T^{4} \))(\( 1 - 2 T - 7 T^{2} - 22 T^{3} + 121 T^{4} \))(\( 1 - 10 T^{2} - 21 T^{4} - 1210 T^{6} + 14641 T^{8} \))(\( 1 - 19 T^{2} + 240 T^{4} - 2299 T^{6} + 14641 T^{8} \))(\( 1 - 9 T^{2} - 40 T^{4} - 1089 T^{6} + 14641 T^{8} \))(\( 1 - 21 T^{2} + 217 T^{4} + 378 T^{6} - 17826 T^{8} + 45738 T^{10} + 3177097 T^{12} - 37202781 T^{14} + 214358881 T^{16} \))
$13$ (\( 1 - 6 T + 23 T^{2} - 78 T^{3} + 169 T^{4} \))(\( ( 1 - 5 T + 13 T^{2} )( 1 + 7 T + 13 T^{2} ) \))(\( ( 1 - 5 T + 13 T^{2} )( 1 + 7 T + 13 T^{2} ) \))(\( 1 - 13 T^{2} + 169 T^{4} \))(\( 1 - 13 T^{2} + 169 T^{4} \))(\( ( 1 - 5 T + 13 T^{2} )( 1 + 7 T + 13 T^{2} ) \))(\( 1 - 4 T + 3 T^{2} - 52 T^{3} + 169 T^{4} \))(\( 1 - 4 T + 3 T^{2} - 52 T^{3} + 169 T^{4} \))(\( ( 1 - 5 T + 13 T^{2} )( 1 + 7 T + 13 T^{2} ) \))(\( 1 - 13 T^{2} + 169 T^{4} \))(\( 1 - 13 T^{2} + 169 T^{4} \))(\( ( 1 - 5 T + 13 T^{2} )( 1 + 7 T + 13 T^{2} ) \))(\( ( 1 - 5 T + 13 T^{2} )( 1 + 7 T + 13 T^{2} ) \))(\( 1 - 6 T + 23 T^{2} - 78 T^{3} + 169 T^{4} \))(\( ( 1 - 2 T + 13 T^{2} )^{2}( 1 + 7 T + 13 T^{2} )^{2} \))(\( ( 1 - 4 T + 3 T^{2} - 52 T^{3} + 169 T^{4} )^{2} \))(\( ( 1 + 6 T + 23 T^{2} + 78 T^{3} + 169 T^{4} )^{2} \))(\( ( 1 - 3 T - 5 T^{2} + 36 T^{3} - 90 T^{4} + 468 T^{5} - 845 T^{6} - 6591 T^{7} + 28561 T^{8} )^{2} \))
$17$ (\( ( 1 + 4 T + 17 T^{2} )^{2} \))(\( ( 1 + 6 T + 17 T^{2} )^{2} \))(\( ( 1 - 3 T + 17 T^{2} )^{2} \))(\( ( 1 + 2 T + 17 T^{2} )^{2} \))(\( ( 1 - 5 T + 17 T^{2} )^{2} \))(\( ( 1 + 17 T^{2} )^{2} \))(\( ( 1 + 6 T + 17 T^{2} )^{2} \))(\( ( 1 - 6 T + 17 T^{2} )^{2} \))(\( ( 1 + 17 T^{2} )^{2} \))(\( ( 1 + 5 T + 17 T^{2} )^{2} \))(\( ( 1 - 2 T + 17 T^{2} )^{2} \))(\( ( 1 + 3 T + 17 T^{2} )^{2} \))(\( ( 1 - 6 T + 17 T^{2} )^{2} \))(\( ( 1 - 4 T + 17 T^{2} )^{2} \))(\( ( 1 + 31 T^{2} + 289 T^{4} )^{2} \))(\( ( 1 + 22 T^{2} + 289 T^{4} )^{2} \))(\( ( 1 - 18 T^{2} + 289 T^{4} )^{2} \))(\( ( 1 + 45 T^{2} + 956 T^{4} + 13005 T^{6} + 83521 T^{8} )^{2} \))
$19$ (\( ( 1 + 4 T + 19 T^{2} )^{2} \))(\( ( 1 - 5 T + 19 T^{2} )^{2} \))(\( ( 1 + 4 T + 19 T^{2} )^{2} \))(\( ( 1 - 2 T + 19 T^{2} )^{2} \))(\( ( 1 - 2 T + 19 T^{2} )^{2} \))(\( ( 1 + 4 T + 19 T^{2} )^{2} \))(\( ( 1 + 4 T + 19 T^{2} )^{2} \))(\( ( 1 + 4 T + 19 T^{2} )^{2} \))(\( ( 1 + 4 T + 19 T^{2} )^{2} \))(\( ( 1 - 2 T + 19 T^{2} )^{2} \))(\( ( 1 - 2 T + 19 T^{2} )^{2} \))(\( ( 1 + 4 T + 19 T^{2} )^{2} \))(\( ( 1 - 5 T + 19 T^{2} )^{2} \))(\( ( 1 + 4 T + 19 T^{2} )^{2} \))(\( ( 1 - 2 T + 19 T^{2} )^{4} \))(\( ( 1 - 2 T + 19 T^{2} )^{4} \))(\( ( 1 + T + 19 T^{2} )^{4} \))(\( ( 1 + 2 T - 18 T^{2} + 38 T^{3} + 361 T^{4} )^{4} \))
$23$ (\( 1 + 2 T - 19 T^{2} + 46 T^{3} + 529 T^{4} \))(\( 1 - 9 T + 58 T^{2} - 207 T^{3} + 529 T^{4} \))(\( 1 + 6 T + 13 T^{2} + 138 T^{3} + 529 T^{4} \))(\( 1 + 4 T - 7 T^{2} + 92 T^{3} + 529 T^{4} \))(\( 1 + 2 T - 19 T^{2} + 46 T^{3} + 529 T^{4} \))(\( 1 + 6 T + 13 T^{2} + 138 T^{3} + 529 T^{4} \))(\( 1 - 23 T^{2} + 529 T^{4} \))(\( 1 - 23 T^{2} + 529 T^{4} \))(\( 1 - 6 T + 13 T^{2} - 138 T^{3} + 529 T^{4} \))(\( 1 - 2 T - 19 T^{2} - 46 T^{3} + 529 T^{4} \))(\( 1 - 4 T - 7 T^{2} - 92 T^{3} + 529 T^{4} \))(\( 1 - 6 T + 13 T^{2} - 138 T^{3} + 529 T^{4} \))(\( 1 + 9 T + 58 T^{2} + 207 T^{3} + 529 T^{4} \))(\( 1 - 2 T - 19 T^{2} - 46 T^{3} + 529 T^{4} \))(\( 1 - 34 T^{2} + 627 T^{4} - 17986 T^{6} + 279841 T^{8} \))(\( 1 - 34 T^{2} + 627 T^{4} - 17986 T^{6} + 279841 T^{8} \))(\( 1 - 33 T^{2} + 560 T^{4} - 17457 T^{6} + 279841 T^{8} \))(\( ( 1 + 30 T^{2} + 371 T^{4} + 15870 T^{6} + 279841 T^{8} )^{2} \))
$29$ (\( 1 - 2 T - 25 T^{2} - 58 T^{3} + 841 T^{4} \))(\( 1 + 6 T + 7 T^{2} + 174 T^{3} + 841 T^{4} \))(\( 1 - 29 T^{2} + 841 T^{4} \))(\( 1 + 2 T - 25 T^{2} + 58 T^{3} + 841 T^{4} \))(\( 1 + 10 T + 71 T^{2} + 290 T^{3} + 841 T^{4} \))(\( 1 - 6 T + 7 T^{2} - 174 T^{3} + 841 T^{4} \))(\( 1 - 6 T + 7 T^{2} - 174 T^{3} + 841 T^{4} \))(\( 1 + 6 T + 7 T^{2} + 174 T^{3} + 841 T^{4} \))(\( 1 + 6 T + 7 T^{2} + 174 T^{3} + 841 T^{4} \))(\( 1 - 10 T + 71 T^{2} - 290 T^{3} + 841 T^{4} \))(\( 1 - 2 T - 25 T^{2} - 58 T^{3} + 841 T^{4} \))(\( 1 - 29 T^{2} + 841 T^{4} \))(\( 1 - 6 T + 7 T^{2} - 174 T^{3} + 841 T^{4} \))(\( 1 + 2 T - 25 T^{2} + 58 T^{3} + 841 T^{4} \))(\( 1 - 55 T^{2} + 2184 T^{4} - 46255 T^{6} + 707281 T^{8} \))(\( 1 - 10 T^{2} - 741 T^{4} - 8410 T^{6} + 707281 T^{8} \))(\( 1 - 6 T^{2} - 805 T^{4} - 5046 T^{6} + 707281 T^{8} \))(\( 1 - 33 T^{2} - 167 T^{4} + 14058 T^{6} - 45594 T^{8} + 11822778 T^{10} - 118115927 T^{12} - 19629169593 T^{14} + 500246412961 T^{16} \))
$31$ (\( 1 - 31 T^{2} + 961 T^{4} \))(\( 1 - T - 30 T^{2} - 31 T^{3} + 961 T^{4} \))(\( 1 - 10 T + 69 T^{2} - 310 T^{3} + 961 T^{4} \))(\( 1 + 6 T + 5 T^{2} + 186 T^{3} + 961 T^{4} \))(\( 1 - 31 T^{2} + 961 T^{4} \))(\( 1 + 8 T + 33 T^{2} + 248 T^{3} + 961 T^{4} \))(\( 1 + 2 T - 27 T^{2} + 62 T^{3} + 961 T^{4} \))(\( 1 + 2 T - 27 T^{2} + 62 T^{3} + 961 T^{4} \))(\( 1 + 8 T + 33 T^{2} + 248 T^{3} + 961 T^{4} \))(\( 1 - 31 T^{2} + 961 T^{4} \))(\( 1 + 6 T + 5 T^{2} + 186 T^{3} + 961 T^{4} \))(\( 1 - 10 T + 69 T^{2} - 310 T^{3} + 961 T^{4} \))(\( 1 - T - 30 T^{2} - 31 T^{3} + 961 T^{4} \))(\( 1 - 31 T^{2} + 961 T^{4} \))(\( ( 1 + 2 T - 27 T^{2} + 62 T^{3} + 961 T^{4} )^{2} \))(\( ( 1 + 2 T - 27 T^{2} + 62 T^{3} + 961 T^{4} )^{2} \))(\( ( 1 + 9 T + 50 T^{2} + 279 T^{3} + 961 T^{4} )^{2} \))(\( ( 1 - 6 T + 5 T^{2} - 186 T^{3} + 961 T^{4} )^{4} \))
$37$ (\( ( 1 - 2 T + 37 T^{2} )^{2} \))(\( ( 1 - 11 T + 37 T^{2} )^{2} \))(\( ( 1 + 7 T + 37 T^{2} )^{2} \))(\( ( 1 - 5 T + 37 T^{2} )^{2} \))(\( ( 1 - 5 T + 37 T^{2} )^{2} \))(\( ( 1 - 2 T + 37 T^{2} )^{2} \))(\( ( 1 + 7 T + 37 T^{2} )^{2} \))(\( ( 1 + 7 T + 37 T^{2} )^{2} \))(\( ( 1 - 2 T + 37 T^{2} )^{2} \))(\( ( 1 - 5 T + 37 T^{2} )^{2} \))(\( ( 1 - 5 T + 37 T^{2} )^{2} \))(\( ( 1 + 7 T + 37 T^{2} )^{2} \))(\( ( 1 - 11 T + 37 T^{2} )^{2} \))(\( ( 1 - 2 T + 37 T^{2} )^{2} \))(\( ( 1 + 7 T + 37 T^{2} )^{4} \))(\( ( 1 - 11 T + 37 T^{2} )^{4} \))(\( ( 1 + T + 37 T^{2} )^{4} \))(\( ( 1 + 8 T + 33 T^{2} + 296 T^{3} + 1369 T^{4} )^{4} \))
$41$ (\( 1 - 41 T^{2} + 1681 T^{4} \))(\( 1 + 3 T - 32 T^{2} + 123 T^{3} + 1681 T^{4} \))(\( 1 + 9 T + 40 T^{2} + 369 T^{3} + 1681 T^{4} \))(\( 1 + 6 T - 5 T^{2} + 246 T^{3} + 1681 T^{4} \))(\( 1 + 3 T - 32 T^{2} + 123 T^{3} + 1681 T^{4} \))(\( 1 - 12 T + 103 T^{2} - 492 T^{3} + 1681 T^{4} \))(\( 1 + 12 T + 103 T^{2} + 492 T^{3} + 1681 T^{4} \))(\( 1 - 12 T + 103 T^{2} - 492 T^{3} + 1681 T^{4} \))(\( 1 + 12 T + 103 T^{2} + 492 T^{3} + 1681 T^{4} \))(\( 1 - 3 T - 32 T^{2} - 123 T^{3} + 1681 T^{4} \))(\( 1 - 6 T - 5 T^{2} - 246 T^{3} + 1681 T^{4} \))(\( 1 - 9 T + 40 T^{2} - 369 T^{3} + 1681 T^{4} \))(\( 1 - 3 T - 32 T^{2} - 123 T^{3} + 1681 T^{4} \))(\( 1 - 41 T^{2} + 1681 T^{4} \))(\( 1 - 34 T^{2} - 525 T^{4} - 57154 T^{6} + 2825761 T^{8} \))(\( 1 - 70 T^{2} + 3219 T^{4} - 117670 T^{6} + 2825761 T^{8} \))(\( 1 + 35 T^{2} - 456 T^{4} + 58835 T^{6} + 2825761 T^{8} \))(\( ( 1 - 34 T^{2} - 525 T^{4} - 57154 T^{6} + 2825761 T^{8} )^{2} \))
$43$ (\( 1 - 4 T - 27 T^{2} - 172 T^{3} + 1849 T^{4} \))(\( 1 - 4 T - 27 T^{2} - 172 T^{3} + 1849 T^{4} \))(\( ( 1 - 8 T + 43 T^{2} )( 1 + 13 T + 43 T^{2} ) \))(\( ( 1 - 8 T + 43 T^{2} )( 1 + 13 T + 43 T^{2} ) \))(\( 1 - 7 T + 6 T^{2} - 301 T^{3} + 1849 T^{4} \))(\( 1 - 4 T - 27 T^{2} - 172 T^{3} + 1849 T^{4} \))(\( 1 - 7 T + 6 T^{2} - 301 T^{3} + 1849 T^{4} \))(\( 1 - 7 T + 6 T^{2} - 301 T^{3} + 1849 T^{4} \))(\( 1 - 4 T - 27 T^{2} - 172 T^{3} + 1849 T^{4} \))(\( 1 - 7 T + 6 T^{2} - 301 T^{3} + 1849 T^{4} \))(\( ( 1 - 8 T + 43 T^{2} )( 1 + 13 T + 43 T^{2} ) \))(\( ( 1 - 8 T + 43 T^{2} )( 1 + 13 T + 43 T^{2} ) \))(\( 1 - 4 T - 27 T^{2} - 172 T^{3} + 1849 T^{4} \))(\( 1 - 4 T - 27 T^{2} - 172 T^{3} + 1849 T^{4} \))(\( ( 1 - 5 T + 43 T^{2} )^{2}( 1 + 13 T + 43 T^{2} )^{2} \))(\( ( 1 - T - 42 T^{2} - 43 T^{3} + 1849 T^{4} )^{2} \))(\( ( 1 - 5 T + 43 T^{2} )^{2}( 1 + 13 T + 43 T^{2} )^{2} \))(\( ( 1 - 5 T - 53 T^{2} + 40 T^{3} + 3160 T^{4} + 1720 T^{5} - 97997 T^{6} - 397535 T^{7} + 3418801 T^{8} )^{2} \))
$47$ (\( 1 + 12 T + 97 T^{2} + 564 T^{3} + 2209 T^{4} \))(\( 1 - 12 T + 97 T^{2} - 564 T^{3} + 2209 T^{4} \))(\( 1 + 3 T - 38 T^{2} + 141 T^{3} + 2209 T^{4} \))(\( 1 - 47 T^{2} + 2209 T^{4} \))(\( 1 - 3 T - 38 T^{2} - 141 T^{3} + 2209 T^{4} \))(\( 1 - 12 T + 97 T^{2} - 564 T^{3} + 2209 T^{4} \))(\( 1 + 6 T - 11 T^{2} + 282 T^{3} + 2209 T^{4} \))(\( 1 - 6 T - 11 T^{2} - 282 T^{3} + 2209 T^{4} \))(\( 1 + 12 T + 97 T^{2} + 564 T^{3} + 2209 T^{4} \))(\( 1 + 3 T - 38 T^{2} + 141 T^{3} + 2209 T^{4} \))(\( 1 - 47 T^{2} + 2209 T^{4} \))(\( 1 - 3 T - 38 T^{2} - 141 T^{3} + 2209 T^{4} \))(\( 1 + 12 T + 97 T^{2} + 564 T^{3} + 2209 T^{4} \))(\( 1 - 12 T + 97 T^{2} - 564 T^{3} + 2209 T^{4} \))(\( 1 + 14 T^{2} - 2013 T^{4} + 30926 T^{6} + 4879681 T^{8} \))(\( ( 1 - 47 T^{2} + 2209 T^{4} )^{2} \))(\( ( 1 - 47 T^{2} + 2209 T^{4} )^{2} \))(\( ( 1 - 82 T^{2} + 4515 T^{4} - 181138 T^{6} + 4879681 T^{8} )^{2} \))
$53$ (\( ( 1 + 6 T + 53 T^{2} )^{2} \))(\( ( 1 + 53 T^{2} )^{2} \))(\( ( 1 - 6 T + 53 T^{2} )^{2} \))(\( ( 1 + 9 T + 53 T^{2} )^{2} \))(\( ( 1 - 6 T + 53 T^{2} )^{2} \))(\( ( 1 - 6 T + 53 T^{2} )^{2} \))(\( ( 1 + 3 T + 53 T^{2} )^{2} \))(\( ( 1 - 3 T + 53 T^{2} )^{2} \))(\( ( 1 + 6 T + 53 T^{2} )^{2} \))(\( ( 1 + 6 T + 53 T^{2} )^{2} \))(\( ( 1 - 9 T + 53 T^{2} )^{2} \))(\( ( 1 + 6 T + 53 T^{2} )^{2} \))(\( ( 1 + 53 T^{2} )^{2} \))(\( ( 1 - 6 T + 53 T^{2} )^{2} \))(\( ( 1 - 86 T^{2} + 2809 T^{4} )^{2} \))(\( ( 1 + 103 T^{2} + 2809 T^{4} )^{2} \))(\( ( 1 + 53 T^{2} )^{4} \))(\( ( 1 + 89 T^{2} + 4392 T^{4} + 250001 T^{6} + 7890481 T^{8} )^{2} \))
$59$ (\( 1 - 8 T + 5 T^{2} - 472 T^{3} + 3481 T^{4} \))(\( 1 - 59 T^{2} + 3481 T^{4} \))(\( 1 - 9 T + 22 T^{2} - 531 T^{3} + 3481 T^{4} \))(\( 1 + 2 T - 55 T^{2} + 118 T^{3} + 3481 T^{4} \))(\( 1 + T - 58 T^{2} + 59 T^{3} + 3481 T^{4} \))(\( 1 - 59 T^{2} + 3481 T^{4} \))(\( 1 + 6 T - 23 T^{2} + 354 T^{3} + 3481 T^{4} \))(\( 1 - 6 T - 23 T^{2} - 354 T^{3} + 3481 T^{4} \))(\( 1 - 59 T^{2} + 3481 T^{4} \))(\( 1 - T - 58 T^{2} - 59 T^{3} + 3481 T^{4} \))(\( 1 - 2 T - 55 T^{2} - 118 T^{3} + 3481 T^{4} \))(\( 1 + 9 T + 22 T^{2} + 531 T^{3} + 3481 T^{4} \))(\( 1 - 59 T^{2} + 3481 T^{4} \))(\( 1 + 8 T + 5 T^{2} + 472 T^{3} + 3481 T^{4} \))(\( 1 - 10 T^{2} - 3381 T^{4} - 34810 T^{6} + 12117361 T^{8} \))(\( 1 - 10 T^{2} - 3381 T^{4} - 34810 T^{6} + 12117361 T^{8} \))(\( 1 + 90 T^{2} + 4619 T^{4} + 313290 T^{6} + 12117361 T^{8} \))(\( ( 1 - 42 T^{2} - 1717 T^{4} - 146202 T^{6} + 12117361 T^{8} )^{2} \))
$61$ (\( 1 + 6 T - 25 T^{2} + 366 T^{3} + 3721 T^{4} \))(\( 1 + 8 T + 3 T^{2} + 488 T^{3} + 3721 T^{4} \))(\( 1 + 8 T + 3 T^{2} + 488 T^{3} + 3721 T^{4} \))(\( 1 - 61 T^{2} + 3721 T^{4} \))(\( 1 - 6 T - 25 T^{2} - 366 T^{3} + 3721 T^{4} \))(\( 1 - 10 T + 39 T^{2} - 610 T^{3} + 3721 T^{4} \))(\( 1 + 2 T - 57 T^{2} + 122 T^{3} + 3721 T^{4} \))(\( 1 + 2 T - 57 T^{2} + 122 T^{3} + 3721 T^{4} \))(\( 1 - 10 T + 39 T^{2} - 610 T^{3} + 3721 T^{4} \))(\( 1 - 6 T - 25 T^{2} - 366 T^{3} + 3721 T^{4} \))(\( 1 - 61 T^{2} + 3721 T^{4} \))(\( 1 + 8 T + 3 T^{2} + 488 T^{3} + 3721 T^{4} \))(\( 1 + 8 T + 3 T^{2} + 488 T^{3} + 3721 T^{4} \))(\( 1 + 6 T - 25 T^{2} + 366 T^{3} + 3721 T^{4} \))(\( ( 1 - 14 T + 61 T^{2} )^{2}( 1 + 13 T + 61 T^{2} )^{2} \))(\( ( 1 + 8 T + 3 T^{2} + 488 T^{3} + 3721 T^{4} )^{2} \))(\( ( 1 - 61 T^{2} + 3721 T^{4} )^{2} \))(\( ( 1 - 15 T + 61 T^{2} )^{4}( 1 + 15 T + 164 T^{2} + 915 T^{3} + 3721 T^{4} )^{2} \))
$67$ (\( 1 - 8 T - 3 T^{2} - 536 T^{3} + 4489 T^{4} \))(\( 1 - 10 T + 33 T^{2} - 670 T^{3} + 4489 T^{4} \))(\( 1 + 8 T - 3 T^{2} + 536 T^{3} + 4489 T^{4} \))(\( 1 + 7 T - 18 T^{2} + 469 T^{3} + 4489 T^{4} \))(\( 1 + 4 T - 51 T^{2} + 268 T^{3} + 4489 T^{4} \))(\( 1 + 8 T - 3 T^{2} + 536 T^{3} + 4489 T^{4} \))(\( 1 - 13 T + 102 T^{2} - 871 T^{3} + 4489 T^{4} \))(\( 1 - 13 T + 102 T^{2} - 871 T^{3} + 4489 T^{4} \))(\( 1 + 8 T - 3 T^{2} + 536 T^{3} + 4489 T^{4} \))(\( 1 + 4 T - 51 T^{2} + 268 T^{3} + 4489 T^{4} \))(\( 1 + 7 T - 18 T^{2} + 469 T^{3} + 4489 T^{4} \))(\( 1 + 8 T - 3 T^{2} + 536 T^{3} + 4489 T^{4} \))(\( 1 - 10 T + 33 T^{2} - 670 T^{3} + 4489 T^{4} \))(\( 1 - 8 T - 3 T^{2} - 536 T^{3} + 4489 T^{4} \))(\( ( 1 + 2 T - 63 T^{2} + 134 T^{3} + 4489 T^{4} )^{2} \))(\( ( 1 - 5 T + 67 T^{2} )^{2}( 1 + 16 T + 67 T^{2} )^{2} \))(\( ( 1 - 2 T - 63 T^{2} - 134 T^{3} + 4489 T^{4} )^{2} \))(\( ( 1 - 7 T - 83 T^{2} + 14 T^{3} + 9652 T^{4} + 938 T^{5} - 372587 T^{6} - 2105341 T^{7} + 20151121 T^{8} )^{2} \))
$71$ (\( ( 1 - 14 T + 71 T^{2} )^{2} \))(\( ( 1 - 3 T + 71 T^{2} )^{2} \))(\( ( 1 + 12 T + 71 T^{2} )^{2} \))(\( ( 1 + 11 T + 71 T^{2} )^{2} \))(\( ( 1 - 8 T + 71 T^{2} )^{2} \))(\( ( 1 + 6 T + 71 T^{2} )^{2} \))(\( ( 1 + 9 T + 71 T^{2} )^{2} \))(\( ( 1 - 9 T + 71 T^{2} )^{2} \))(\( ( 1 - 6 T + 71 T^{2} )^{2} \))(\( ( 1 + 8 T + 71 T^{2} )^{2} \))(\( ( 1 - 11 T + 71 T^{2} )^{2} \))(\( ( 1 - 12 T + 71 T^{2} )^{2} \))(\( ( 1 + 3 T + 71 T^{2} )^{2} \))(\( ( 1 + 14 T + 71 T^{2} )^{2} \))(\( ( 1 + 71 T^{2} )^{4} \))(\( ( 1 + 115 T^{2} + 5041 T^{4} )^{2} \))(\( ( 1 + 129 T^{2} + 5041 T^{4} )^{2} \))(\( ( 1 + 201 T^{2} + 19484 T^{4} + 1013241 T^{6} + 25411681 T^{8} )^{2} \))
$73$ (\( ( 1 + 2 T + 73 T^{2} )^{2} \))(\( ( 1 - 8 T + 73 T^{2} )^{2} \))(\( ( 1 + 10 T + 73 T^{2} )^{2} \))(\( ( 1 - 4 T + 73 T^{2} )^{2} \))(\( ( 1 - 10 T + 73 T^{2} )^{2} \))(\( ( 1 + 10 T + 73 T^{2} )^{2} \))(\( ( 1 - 8 T + 73 T^{2} )^{2} \))(\( ( 1 - 8 T + 73 T^{2} )^{2} \))(\( ( 1 + 10 T + 73 T^{2} )^{2} \))(\( ( 1 - 10 T + 73 T^{2} )^{2} \))(\( ( 1 - 4 T + 73 T^{2} )^{2} \))(\( ( 1 + 10 T + 73 T^{2} )^{2} \))(\( ( 1 - 8 T + 73 T^{2} )^{2} \))(\( ( 1 + 2 T + 73 T^{2} )^{2} \))(\( ( 1 - 11 T + 73 T^{2} )^{4} \))(\( ( 1 + 16 T + 73 T^{2} )^{4} \))(\( ( 1 - 4 T + 73 T^{2} )^{4} \))(\( ( 1 + 13 T + 174 T^{2} + 949 T^{3} + 5329 T^{4} )^{4} \))
$79$ (\( 1 + 12 T + 65 T^{2} + 948 T^{3} + 6241 T^{4} \))(\( ( 1 - 17 T + 79 T^{2} )( 1 + 13 T + 79 T^{2} ) \))(\( 1 + 5 T - 54 T^{2} + 395 T^{3} + 6241 T^{4} \))(\( 1 + 15 T + 146 T^{2} + 1185 T^{3} + 6241 T^{4} \))(\( 1 - 3 T - 70 T^{2} - 237 T^{3} + 6241 T^{4} \))(\( ( 1 - 17 T + 79 T^{2} )( 1 + 13 T + 79 T^{2} ) \))(\( 1 - T - 78 T^{2} - 79 T^{3} + 6241 T^{4} \))(\( 1 - T - 78 T^{2} - 79 T^{3} + 6241 T^{4} \))(\( ( 1 - 17 T + 79 T^{2} )( 1 + 13 T + 79 T^{2} ) \))(\( 1 - 3 T - 70 T^{2} - 237 T^{3} + 6241 T^{4} \))(\( 1 + 15 T + 146 T^{2} + 1185 T^{3} + 6241 T^{4} \))(\( 1 + 5 T - 54 T^{2} + 395 T^{3} + 6241 T^{4} \))(\( ( 1 - 17 T + 79 T^{2} )( 1 + 13 T + 79 T^{2} ) \))(\( 1 + 12 T + 65 T^{2} + 948 T^{3} + 6241 T^{4} \))(\( ( 1 + 14 T + 117 T^{2} + 1106 T^{3} + 6241 T^{4} )^{2} \))(\( ( 1 - 17 T + 79 T^{2} )^{2}( 1 + 4 T + 79 T^{2} )^{2} \))(\( ( 1 - 79 T^{2} + 6241 T^{4} )^{2} \))(\( ( 1 - 15 T + 25 T^{2} - 630 T^{3} + 15984 T^{4} - 49770 T^{5} + 156025 T^{6} - 7395585 T^{7} + 38950081 T^{8} )^{2} \))
$83$ (\( 1 - 4 T - 67 T^{2} - 332 T^{3} + 6889 T^{4} \))(\( 1 + 6 T - 47 T^{2} + 498 T^{3} + 6889 T^{4} \))(\( 1 + 9 T - 2 T^{2} + 747 T^{3} + 6889 T^{4} \))(\( 1 + 10 T + 17 T^{2} + 830 T^{3} + 6889 T^{4} \))(\( 1 - 13 T + 86 T^{2} - 1079 T^{3} + 6889 T^{4} \))(\( 1 + 12 T + 61 T^{2} + 996 T^{3} + 6889 T^{4} \))(\( 1 - 83 T^{2} + 6889 T^{4} \))(\( 1 - 83 T^{2} + 6889 T^{4} \))(\( 1 - 12 T + 61 T^{2} - 996 T^{3} + 6889 T^{4} \))(\( 1 + 13 T + 86 T^{2} + 1079 T^{3} + 6889 T^{4} \))(\( 1 - 10 T + 17 T^{2} - 830 T^{3} + 6889 T^{4} \))(\( 1 - 9 T - 2 T^{2} - 747 T^{3} + 6889 T^{4} \))(\( 1 - 6 T - 47 T^{2} - 498 T^{3} + 6889 T^{4} \))(\( 1 + 4 T - 67 T^{2} + 332 T^{3} + 6889 T^{4} \))(\( 1 - 154 T^{2} + 16827 T^{4} - 1060906 T^{6} + 47458321 T^{8} \))(\( 1 + 134 T^{2} + 11067 T^{4} + 923126 T^{6} + 47458321 T^{8} \))(\( 1 - 114 T^{2} + 6107 T^{4} - 785346 T^{6} + 47458321 T^{8} \))(\( 1 - 240 T^{2} + 31474 T^{4} - 2963520 T^{6} + 238303155 T^{8} - 20415689280 T^{10} + 1493703195154 T^{12} - 78465689608560 T^{14} + 2252292232139041 T^{16} \))
$89$ (\( ( 1 + 89 T^{2} )^{2} \))(\( ( 1 - 3 T + 89 T^{2} )^{2} \))(\( ( 1 - 18 T + 89 T^{2} )^{2} \))(\( ( 1 + 12 T + 89 T^{2} )^{2} \))(\( ( 1 + 6 T + 89 T^{2} )^{2} \))(\( ( 1 + 12 T + 89 T^{2} )^{2} \))(\( ( 1 + 12 T + 89 T^{2} )^{2} \))(\( ( 1 - 12 T + 89 T^{2} )^{2} \))(\( ( 1 - 12 T + 89 T^{2} )^{2} \))(\( ( 1 - 6 T + 89 T^{2} )^{2} \))(\( ( 1 - 12 T + 89 T^{2} )^{2} \))(\( ( 1 + 18 T + 89 T^{2} )^{2} \))(\( ( 1 + 3 T + 89 T^{2} )^{2} \))(\( ( 1 + 89 T^{2} )^{2} \))(\( ( 1 + 175 T^{2} + 7921 T^{4} )^{2} \))(\( ( 1 + 130 T^{2} + 7921 T^{4} )^{2} \))(\( ( 1 + 61 T^{2} + 7921 T^{4} )^{2} \))(\( ( 1 + 89 T^{2} + 2304 T^{4} + 704969 T^{6} + 62742241 T^{8} )^{2} \))
$97$ (\( 1 - 2 T - 93 T^{2} - 194 T^{3} + 9409 T^{4} \))(\( 1 + 8 T - 33 T^{2} + 776 T^{3} + 9409 T^{4} \))(\( 1 + 8 T - 33 T^{2} + 776 T^{3} + 9409 T^{4} \))(\( 1 + 10 T + 3 T^{2} + 970 T^{3} + 9409 T^{4} \))(\( ( 1 - 19 T + 97 T^{2} )( 1 + 5 T + 97 T^{2} ) \))(\( 1 - 10 T + 3 T^{2} - 970 T^{3} + 9409 T^{4} \))(\( 1 - 4 T - 81 T^{2} - 388 T^{3} + 9409 T^{4} \))(\( 1 - 4 T - 81 T^{2} - 388 T^{3} + 9409 T^{4} \))(\( 1 - 10 T + 3 T^{2} - 970 T^{3} + 9409 T^{4} \))(\( ( 1 - 19 T + 97 T^{2} )( 1 + 5 T + 97 T^{2} ) \))(\( 1 + 10 T + 3 T^{2} + 970 T^{3} + 9409 T^{4} \))(\( 1 + 8 T - 33 T^{2} + 776 T^{3} + 9409 T^{4} \))(\( 1 + 8 T - 33 T^{2} + 776 T^{3} + 9409 T^{4} \))(\( 1 - 2 T - 93 T^{2} - 194 T^{3} + 9409 T^{4} \))(\( ( 1 + 2 T - 93 T^{2} + 194 T^{3} + 9409 T^{4} )^{2} \))(\( ( 1 + 2 T - 93 T^{2} + 194 T^{3} + 9409 T^{4} )^{2} \))(\( ( 1 - 8 T - 33 T^{2} - 776 T^{3} + 9409 T^{4} )^{2} \))(\( ( 1 + 14 T + 10 T^{2} - 112 T^{3} + 6175 T^{4} - 10864 T^{5} + 94090 T^{6} + 12777422 T^{7} + 88529281 T^{8} )^{2} \))
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