Properties

Label 1680.2.bg
Level 1680
Weight 2
Character orbit bg
Rep. character \(\chi_{1680}(961,\cdot)\)
Character field \(\Q(\zeta_{3})\)
Dimension 64
Newform subspaces 22
Sturm bound 768
Trace bound 7

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Defining parameters

Level: \( N \) \(=\) \( 1680 = 2^{4} \cdot 3 \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1680.bg (of order \(3\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 7 \)
Character field: \(\Q(\zeta_{3})\)
Newform subspaces: \( 22 \)
Sturm bound: \(768\)
Trace bound: \(7\)
Distinguishing \(T_p\): \(11\), \(13\), \(17\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(1680, [\chi])\).

Total New Old
Modular forms 816 64 752
Cusp forms 720 64 656
Eisenstein series 96 0 96

Trace form

\( 64q + 4q^{3} + 4q^{7} - 32q^{9} + O(q^{10}) \) \( 64q + 4q^{3} + 4q^{7} - 32q^{9} - 8q^{11} - 4q^{19} - 8q^{23} - 32q^{25} - 8q^{27} - 32q^{29} - 4q^{31} + 16q^{37} - 4q^{39} - 16q^{41} + 24q^{43} + 24q^{49} + 32q^{53} + 32q^{55} + 16q^{57} + 16q^{59} + 16q^{61} + 4q^{63} + 8q^{65} - 12q^{67} - 16q^{71} - 8q^{73} + 4q^{75} - 16q^{77} - 12q^{79} - 32q^{81} - 24q^{87} - 16q^{89} - 44q^{91} - 32q^{97} + 16q^{99} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(1680, [\chi])\) into newform subspaces

Label Dim. \(A\) Field CM Traces $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\)
1680.2.bg.a \(2\) \(13.415\) \(\Q(\sqrt{-3}) \) None \(0\) \(-1\) \(-1\) \(-5\) \(q+(-1+\zeta_{6})q^{3}-\zeta_{6}q^{5}+(-2-\zeta_{6})q^{7}+\cdots\)
1680.2.bg.b \(2\) \(13.415\) \(\Q(\sqrt{-3}) \) None \(0\) \(-1\) \(-1\) \(-4\) \(q+(-1+\zeta_{6})q^{3}-\zeta_{6}q^{5}+(-3+2\zeta_{6})q^{7}+\cdots\)
1680.2.bg.c \(2\) \(13.415\) \(\Q(\sqrt{-3}) \) None \(0\) \(-1\) \(-1\) \(1\) \(q+(-1+\zeta_{6})q^{3}-\zeta_{6}q^{5}+(2-3\zeta_{6})q^{7}+\cdots\)
1680.2.bg.d \(2\) \(13.415\) \(\Q(\sqrt{-3}) \) None \(0\) \(-1\) \(-1\) \(4\) \(q+(-1+\zeta_{6})q^{3}-\zeta_{6}q^{5}+(1+2\zeta_{6})q^{7}+\cdots\)
1680.2.bg.e \(2\) \(13.415\) \(\Q(\sqrt{-3}) \) None \(0\) \(-1\) \(-1\) \(5\) \(q+(-1+\zeta_{6})q^{3}-\zeta_{6}q^{5}+(2+\zeta_{6})q^{7}+\cdots\)
1680.2.bg.f \(2\) \(13.415\) \(\Q(\sqrt{-3}) \) None \(0\) \(-1\) \(1\) \(-5\) \(q+(-1+\zeta_{6})q^{3}+\zeta_{6}q^{5}+(-2-\zeta_{6})q^{7}+\cdots\)
1680.2.bg.g \(2\) \(13.415\) \(\Q(\sqrt{-3}) \) None \(0\) \(-1\) \(1\) \(-4\) \(q+(-1+\zeta_{6})q^{3}+\zeta_{6}q^{5}+(-1-2\zeta_{6})q^{7}+\cdots\)
1680.2.bg.h \(2\) \(13.415\) \(\Q(\sqrt{-3}) \) None \(0\) \(-1\) \(1\) \(-1\) \(q+(-1+\zeta_{6})q^{3}+\zeta_{6}q^{5}+(-2+3\zeta_{6})q^{7}+\cdots\)
1680.2.bg.i \(2\) \(13.415\) \(\Q(\sqrt{-3}) \) None \(0\) \(-1\) \(1\) \(4\) \(q+(-1+\zeta_{6})q^{3}+\zeta_{6}q^{5}+(3-2\zeta_{6})q^{7}+\cdots\)
1680.2.bg.j \(2\) \(13.415\) \(\Q(\sqrt{-3}) \) None \(0\) \(-1\) \(1\) \(5\) \(q+(-1+\zeta_{6})q^{3}+\zeta_{6}q^{5}+(2+\zeta_{6})q^{7}+\cdots\)
1680.2.bg.k \(2\) \(13.415\) \(\Q(\sqrt{-3}) \) None \(0\) \(1\) \(-1\) \(-4\) \(q+(1-\zeta_{6})q^{3}-\zeta_{6}q^{5}+(-3+2\zeta_{6})q^{7}+\cdots\)
1680.2.bg.l \(2\) \(13.415\) \(\Q(\sqrt{-3}) \) None \(0\) \(1\) \(-1\) \(1\) \(q+(1-\zeta_{6})q^{3}-\zeta_{6}q^{5}+(2-3\zeta_{6})q^{7}+\cdots\)
1680.2.bg.m \(2\) \(13.415\) \(\Q(\sqrt{-3}) \) None \(0\) \(1\) \(1\) \(-5\) \(q+(1-\zeta_{6})q^{3}+\zeta_{6}q^{5}+(-2-\zeta_{6})q^{7}+\cdots\)
1680.2.bg.n \(2\) \(13.415\) \(\Q(\sqrt{-3}) \) None \(0\) \(1\) \(1\) \(4\) \(q+(1-\zeta_{6})q^{3}+\zeta_{6}q^{5}+(3-2\zeta_{6})q^{7}+\cdots\)
1680.2.bg.o \(4\) \(13.415\) \(\Q(\zeta_{12})\) None \(0\) \(-2\) \(-2\) \(0\) \(q-\zeta_{12}^{2}q^{3}+(-1+\zeta_{12}^{2})q^{5}+(-\zeta_{12}+\cdots)q^{7}+\cdots\)
1680.2.bg.p \(4\) \(13.415\) \(\Q(\sqrt{2}, \sqrt{-3})\) None \(0\) \(-2\) \(2\) \(2\) \(q+\beta _{2}q^{3}+(1+\beta _{2})q^{5}+(1+\beta _{1}+\beta _{2}+\cdots)q^{7}+\cdots\)
1680.2.bg.q \(4\) \(13.415\) \(\Q(\sqrt{-3}, \sqrt{7})\) None \(0\) \(2\) \(-2\) \(0\) \(q+(1+\beta _{2})q^{3}+\beta _{2}q^{5}+(\beta _{1}+\beta _{3})q^{7}+\cdots\)
1680.2.bg.r \(4\) \(13.415\) \(\Q(\sqrt{2}, \sqrt{-3})\) None \(0\) \(2\) \(-2\) \(2\) \(q+(1+\beta _{2})q^{3}+\beta _{2}q^{5}+(-\beta _{1}-\beta _{2}+\cdots)q^{7}+\cdots\)
1680.2.bg.s \(4\) \(13.415\) \(\Q(\sqrt{-3}, \sqrt{7})\) None \(0\) \(2\) \(2\) \(0\) \(q+(1+\beta _{2})q^{3}-\beta _{2}q^{5}+(\beta _{1}+\beta _{3})q^{7}+\cdots\)
1680.2.bg.t \(4\) \(13.415\) \(\Q(\sqrt{2}, \sqrt{-3})\) None \(0\) \(2\) \(2\) \(2\) \(q+(1+\beta _{1})q^{3}-\beta _{1}q^{5}+(-\beta _{1}+\beta _{3})q^{7}+\cdots\)
1680.2.bg.u \(6\) \(13.415\) 6.0.38363328.2 None \(0\) \(3\) \(-3\) \(2\) \(q+(1+\beta _{3})q^{3}+\beta _{3}q^{5}+(\beta _{1}-\beta _{4})q^{7}+\cdots\)
1680.2.bg.v \(6\) \(13.415\) 6.0.29428272.1 None \(0\) \(3\) \(3\) \(0\) \(q-\beta _{3}q^{3}+(1+\beta _{3})q^{5}+(-\beta _{1}-\beta _{5})q^{7}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(1680, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(1680, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(21, [\chi])\)\(^{\oplus 10}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(28, [\chi])\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(35, [\chi])\)\(^{\oplus 10}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(42, [\chi])\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(56, [\chi])\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(70, [\chi])\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(84, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(105, [\chi])\)\(^{\oplus 5}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(112, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(140, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(168, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(210, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(280, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(336, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(420, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(560, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(840, [\chi])\)\(^{\oplus 2}\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ (\( 1 + T + T^{2} \))(\( 1 + T + T^{2} \))(\( 1 + T + T^{2} \))(\( 1 + T + T^{2} \))(\( 1 + T + T^{2} \))(\( 1 + T + T^{2} \))(\( 1 + T + T^{2} \))(\( 1 + T + T^{2} \))(\( 1 + T + T^{2} \))(\( 1 + T + T^{2} \))(\( 1 - T + T^{2} \))(\( 1 - T + T^{2} \))(\( 1 - T + T^{2} \))(\( 1 - T + T^{2} \))(\( ( 1 + T + T^{2} )^{2} \))(\( ( 1 + T + T^{2} )^{2} \))(\( ( 1 - T + T^{2} )^{2} \))(\( ( 1 - T + T^{2} )^{2} \))(\( ( 1 - T + T^{2} )^{2} \))(\( ( 1 - T + T^{2} )^{2} \))(\( ( 1 - T + T^{2} )^{3} \))(\( ( 1 - T + T^{2} )^{3} \))
$5$ (\( 1 + T + T^{2} \))(\( 1 + T + T^{2} \))(\( 1 + T + T^{2} \))(\( 1 + T + T^{2} \))(\( 1 + T + T^{2} \))(\( 1 - T + T^{2} \))(\( 1 - T + T^{2} \))(\( 1 - T + T^{2} \))(\( 1 - T + T^{2} \))(\( 1 - T + T^{2} \))(\( 1 + T + T^{2} \))(\( 1 + T + T^{2} \))(\( 1 - T + T^{2} \))(\( 1 - T + T^{2} \))(\( ( 1 + T + T^{2} )^{2} \))(\( ( 1 - T + T^{2} )^{2} \))(\( ( 1 + T + T^{2} )^{2} \))(\( ( 1 + T + T^{2} )^{2} \))(\( ( 1 - T + T^{2} )^{2} \))(\( ( 1 - T + T^{2} )^{2} \))(\( ( 1 + T + T^{2} )^{3} \))(\( ( 1 - T + T^{2} )^{3} \))
$7$ (\( 1 + 5 T + 7 T^{2} \))(\( 1 + 4 T + 7 T^{2} \))(\( 1 - T + 7 T^{2} \))(\( 1 - 4 T + 7 T^{2} \))(\( 1 - 5 T + 7 T^{2} \))(\( 1 + 5 T + 7 T^{2} \))(\( 1 + 4 T + 7 T^{2} \))(\( 1 + T + 7 T^{2} \))(\( 1 - 4 T + 7 T^{2} \))(\( 1 - 5 T + 7 T^{2} \))(\( 1 + 4 T + 7 T^{2} \))(\( 1 - T + 7 T^{2} \))(\( 1 + 5 T + 7 T^{2} \))(\( 1 - 4 T + 7 T^{2} \))(\( 1 + 11 T^{2} + 49 T^{4} \))(\( 1 - 2 T - 3 T^{2} - 14 T^{3} + 49 T^{4} \))(\( 1 + 7 T^{2} + 49 T^{4} \))(\( 1 - 2 T - 3 T^{2} - 14 T^{3} + 49 T^{4} \))(\( 1 + 7 T^{2} + 49 T^{4} \))(\( 1 - 2 T - 3 T^{2} - 14 T^{3} + 49 T^{4} \))(\( 1 - 2 T + 8 T^{3} - 98 T^{5} + 343 T^{6} \))(\( 1 - 6 T^{2} + 4 T^{3} - 42 T^{4} + 343 T^{6} \))
$11$ (\( 1 + 2 T - 7 T^{2} + 22 T^{3} + 121 T^{4} \))(\( 1 + T - 10 T^{2} + 11 T^{3} + 121 T^{4} \))(\( 1 - 4 T + 5 T^{2} - 44 T^{3} + 121 T^{4} \))(\( 1 + 5 T + 14 T^{2} + 55 T^{3} + 121 T^{4} \))(\( 1 - 11 T^{2} + 121 T^{4} \))(\( 1 - 2 T - 7 T^{2} - 22 T^{3} + 121 T^{4} \))(\( 1 + T - 10 T^{2} + 11 T^{3} + 121 T^{4} \))(\( 1 + 2 T - 7 T^{2} + 22 T^{3} + 121 T^{4} \))(\( 1 - 3 T - 2 T^{2} - 33 T^{3} + 121 T^{4} \))(\( 1 + 4 T + 5 T^{2} + 44 T^{3} + 121 T^{4} \))(\( 1 + T - 10 T^{2} + 11 T^{3} + 121 T^{4} \))(\( 1 + 6 T + 25 T^{2} + 66 T^{3} + 121 T^{4} \))(\( 1 - 11 T^{2} + 121 T^{4} \))(\( 1 - 3 T - 2 T^{2} - 33 T^{3} + 121 T^{4} \))(\( 1 + 2 T - 16 T^{2} - 4 T^{3} + 235 T^{4} - 44 T^{5} - 1936 T^{6} + 2662 T^{7} + 14641 T^{8} \))(\( 1 - 4 T - 8 T^{2} - 8 T^{3} + 279 T^{4} - 88 T^{5} - 968 T^{6} - 5324 T^{7} + 14641 T^{8} \))(\( 1 - 2 T - 12 T^{2} + 12 T^{3} + 91 T^{4} + 132 T^{5} - 1452 T^{6} - 2662 T^{7} + 14641 T^{8} \))(\( 1 - 20 T^{2} + 279 T^{4} - 2420 T^{6} + 14641 T^{8} \))(\( 1 - 2 T - 12 T^{2} + 12 T^{3} + 91 T^{4} + 132 T^{5} - 1452 T^{6} - 2662 T^{7} + 14641 T^{8} \))(\( 1 - 4 T^{2} - 105 T^{4} - 484 T^{6} + 14641 T^{8} \))(\( 1 + T - 10 T^{2} - 61 T^{3} - 36 T^{4} + 275 T^{5} + 2506 T^{6} + 3025 T^{7} - 4356 T^{8} - 81191 T^{9} - 146410 T^{10} + 161051 T^{11} + 1771561 T^{12} \))(\( 1 + 3 T + 12 T^{2} + 111 T^{3} + 222 T^{4} + 741 T^{5} + 5074 T^{6} + 8151 T^{7} + 26862 T^{8} + 147741 T^{9} + 175692 T^{10} + 483153 T^{11} + 1771561 T^{12} \))
$13$ (\( ( 1 - T + 13 T^{2} )^{2} \))(\( ( 1 + 5 T + 13 T^{2} )^{2} \))(\( ( 1 + 5 T + 13 T^{2} )^{2} \))(\( ( 1 + 5 T + 13 T^{2} )^{2} \))(\( ( 1 + T + 13 T^{2} )^{2} \))(\( ( 1 - 5 T + 13 T^{2} )^{2} \))(\( ( 1 - T + 13 T^{2} )^{2} \))(\( ( 1 - T + 13 T^{2} )^{2} \))(\( ( 1 - T + 13 T^{2} )^{2} \))(\( ( 1 - 7 T + 13 T^{2} )^{2} \))(\( ( 1 - 7 T + 13 T^{2} )^{2} \))(\( ( 1 + 3 T + 13 T^{2} )^{2} \))(\( ( 1 + T + 13 T^{2} )^{2} \))(\( ( 1 - 5 T + 13 T^{2} )^{2} \))(\( ( 1 - 8 T + 39 T^{2} - 104 T^{3} + 169 T^{4} )^{2} \))(\( ( 1 + 6 T + 33 T^{2} + 78 T^{3} + 169 T^{4} )^{2} \))(\( ( 1 + 19 T^{2} + 169 T^{4} )^{2} \))(\( ( 1 + 2 T + 9 T^{2} + 26 T^{3} + 169 T^{4} )^{2} \))(\( ( 1 + 4 T + 23 T^{2} + 52 T^{3} + 169 T^{4} )^{2} \))(\( ( 1 + 2 T + 9 T^{2} + 26 T^{3} + 169 T^{4} )^{2} \))(\( ( 1 - T + 24 T^{2} - 23 T^{3} + 312 T^{4} - 169 T^{5} + 2197 T^{6} )^{2} \))(\( ( 1 + 3 T + 24 T^{2} + 81 T^{3} + 312 T^{4} + 507 T^{5} + 2197 T^{6} )^{2} \))
$17$ (\( 1 - 4 T - T^{2} - 68 T^{3} + 289 T^{4} \))(\( 1 - 4 T - T^{2} - 68 T^{3} + 289 T^{4} \))(\( 1 + 6 T + 19 T^{2} + 102 T^{3} + 289 T^{4} \))(\( 1 - 4 T - T^{2} - 68 T^{3} + 289 T^{4} \))(\( 1 + 2 T - 13 T^{2} + 34 T^{3} + 289 T^{4} \))(\( 1 + 4 T - T^{2} + 68 T^{3} + 289 T^{4} \))(\( 1 - 17 T^{2} + 289 T^{4} \))(\( 1 - 17 T^{2} + 289 T^{4} \))(\( 1 - 17 T^{2} + 289 T^{4} \))(\( 1 - 6 T + 19 T^{2} - 102 T^{3} + 289 T^{4} \))(\( 1 - 4 T - T^{2} - 68 T^{3} + 289 T^{4} \))(\( 1 - 4 T - T^{2} - 68 T^{3} + 289 T^{4} \))(\( 1 + 6 T + 19 T^{2} + 102 T^{3} + 289 T^{4} \))(\( 1 - 17 T^{2} + 289 T^{4} \))(\( 1 + 10 T + 44 T^{2} + 220 T^{3} + 1147 T^{4} + 3740 T^{5} + 12716 T^{6} + 49130 T^{7} + 83521 T^{8} \))(\( 1 - 4 T - 4 T^{2} + 56 T^{3} - 161 T^{4} + 952 T^{5} - 1156 T^{6} - 19652 T^{7} + 83521 T^{8} \))(\( 1 + 2 T - 24 T^{2} - 12 T^{3} + 427 T^{4} - 204 T^{5} - 6936 T^{6} + 9826 T^{7} + 83521 T^{8} \))(\( 1 - 8 T + 16 T^{2} - 112 T^{3} + 927 T^{4} - 1904 T^{5} + 4624 T^{6} - 39304 T^{7} + 83521 T^{8} \))(\( 1 - 6 T - 12 T^{3} + 395 T^{4} - 204 T^{5} - 29478 T^{7} + 83521 T^{8} \))(\( 1 - 16 T^{2} - 33 T^{4} - 4624 T^{6} + 83521 T^{8} \))(\( 1 + 8 T + 7 T^{2} - 40 T^{3} + 296 T^{4} + 944 T^{5} - 2399 T^{6} + 16048 T^{7} + 85544 T^{8} - 196520 T^{9} + 584647 T^{10} + 11358856 T^{11} + 24137569 T^{12} \))(\( 1 + 6 T + 39 T^{2} + 214 T^{3} + 756 T^{4} + 2694 T^{5} + 12725 T^{6} + 45798 T^{7} + 218484 T^{8} + 1051382 T^{9} + 3257319 T^{10} + 8519142 T^{11} + 24137569 T^{12} \))
$19$ (\( ( 1 - 7 T + 19 T^{2} )( 1 + 8 T + 19 T^{2} ) \))(\( 1 - 5 T + 6 T^{2} - 95 T^{3} + 361 T^{4} \))(\( 1 + 5 T + 6 T^{2} + 95 T^{3} + 361 T^{4} \))(\( ( 1 - T + 19 T^{2} )( 1 + 8 T + 19 T^{2} ) \))(\( 1 - 3 T - 10 T^{2} - 57 T^{3} + 361 T^{4} \))(\( 1 - 3 T - 10 T^{2} - 57 T^{3} + 361 T^{4} \))(\( 1 + 3 T - 10 T^{2} + 57 T^{3} + 361 T^{4} \))(\( 1 - 3 T - 10 T^{2} - 57 T^{3} + 361 T^{4} \))(\( ( 1 - T + 19 T^{2} )( 1 + 8 T + 19 T^{2} ) \))(\( 1 - 3 T - 10 T^{2} - 57 T^{3} + 361 T^{4} \))(\( ( 1 - 8 T + 19 T^{2} )( 1 + 7 T + 19 T^{2} ) \))(\( ( 1 - 8 T + 19 T^{2} )( 1 + 7 T + 19 T^{2} ) \))(\( 1 - 5 T + 6 T^{2} - 95 T^{3} + 361 T^{4} \))(\( 1 - 5 T + 6 T^{2} - 95 T^{3} + 361 T^{4} \))(\( 1 - 2 T - 23 T^{2} + 22 T^{3} + 292 T^{4} + 418 T^{5} - 8303 T^{6} - 13718 T^{7} + 130321 T^{8} \))(\( 1 + 2 T - 3 T^{2} - 62 T^{3} - 388 T^{4} - 1178 T^{5} - 1083 T^{6} + 13718 T^{7} + 130321 T^{8} \))(\( 1 - 6 T + 17 T^{2} + 114 T^{3} - 684 T^{4} + 2166 T^{5} + 6137 T^{6} - 41154 T^{7} + 130321 T^{8} \))(\( 1 + 6 T - 3 T^{2} + 6 T^{3} + 476 T^{4} + 114 T^{5} - 1083 T^{6} + 41154 T^{7} + 130321 T^{8} \))(\( ( 1 - 8 T + 19 T^{2} )^{2}( 1 + 7 T + 19 T^{2} )^{2} \))(\( ( 1 - T + 19 T^{2} )^{2}( 1 + 8 T + 19 T^{2} )^{2} \))(\( 1 + T + 5 T^{2} - 6 T^{3} + 91 T^{4} + 509 T^{5} + 13054 T^{6} + 9671 T^{7} + 32851 T^{8} - 41154 T^{9} + 651605 T^{10} + 2476099 T^{11} + 47045881 T^{12} \))(\( ( 1 - 8 T + 19 T^{2} )^{3}( 1 + 7 T + 19 T^{2} )^{3} \))
$23$ (\( 1 - 4 T - 7 T^{2} - 92 T^{3} + 529 T^{4} \))(\( 1 + 3 T - 14 T^{2} + 69 T^{3} + 529 T^{4} \))(\( 1 - 2 T - 19 T^{2} - 46 T^{3} + 529 T^{4} \))(\( 1 - T - 22 T^{2} - 23 T^{3} + 529 T^{4} \))(\( 1 - 6 T + 13 T^{2} - 138 T^{3} + 529 T^{4} \))(\( 1 + 4 T - 7 T^{2} + 92 T^{3} + 529 T^{4} \))(\( 1 - 7 T + 26 T^{2} - 161 T^{3} + 529 T^{4} \))(\( 1 - 23 T^{2} + 529 T^{4} \))(\( 1 + 5 T + 2 T^{2} + 115 T^{3} + 529 T^{4} \))(\( 1 + 2 T - 19 T^{2} + 46 T^{3} + 529 T^{4} \))(\( 1 - T - 22 T^{2} - 23 T^{3} + 529 T^{4} \))(\( 1 + 4 T - 7 T^{2} + 92 T^{3} + 529 T^{4} \))(\( 1 - 6 T + 13 T^{2} - 138 T^{3} + 529 T^{4} \))(\( 1 + 9 T + 58 T^{2} + 207 T^{3} + 529 T^{4} \))(\( 1 + 6 T - 16 T^{2} + 36 T^{3} + 1347 T^{4} + 828 T^{5} - 8464 T^{6} + 73002 T^{7} + 279841 T^{8} \))(\( 1 + 4 T - 16 T^{2} - 56 T^{3} + 127 T^{4} - 1288 T^{5} - 8464 T^{6} + 48668 T^{7} + 279841 T^{8} \))(\( 1 + 2 T - 36 T^{2} - 12 T^{3} + 979 T^{4} - 276 T^{5} - 19044 T^{6} + 24334 T^{7} + 279841 T^{8} \))(\( 1 + 4 T^{2} - 513 T^{4} + 2116 T^{6} + 279841 T^{8} \))(\( 1 + 2 T - 36 T^{2} - 12 T^{3} + 979 T^{4} - 276 T^{5} - 19044 T^{6} + 24334 T^{7} + 279841 T^{8} \))(\( 1 - 28 T^{2} + 255 T^{4} - 14812 T^{6} + 279841 T^{8} \))(\( 1 - 9 T + 42 T^{2} - 135 T^{3} - 348 T^{4} + 7173 T^{5} - 41834 T^{6} + 164979 T^{7} - 184092 T^{8} - 1642545 T^{9} + 11753322 T^{10} - 57927087 T^{11} + 148035889 T^{12} \))(\( 1 + 3 T - 36 T^{2} - 97 T^{3} + 642 T^{4} + 597 T^{5} - 13546 T^{6} + 13731 T^{7} + 339618 T^{8} - 1180199 T^{9} - 10074276 T^{10} + 19309029 T^{11} + 148035889 T^{12} \))
$29$ (\( ( 1 + 29 T^{2} )^{2} \))(\( ( 1 - 8 T + 29 T^{2} )^{2} \))(\( ( 1 + 2 T + 29 T^{2} )^{2} \))(\( ( 1 + 29 T^{2} )^{2} \))(\( ( 1 + 6 T + 29 T^{2} )^{2} \))(\( ( 1 - 4 T + 29 T^{2} )^{2} \))(\( ( 1 + 8 T + 29 T^{2} )^{2} \))(\( ( 1 + 29 T^{2} )^{2} \))(\( ( 1 + 29 T^{2} )^{2} \))(\( ( 1 + 2 T + 29 T^{2} )^{2} \))(\( ( 1 + 8 T + 29 T^{2} )^{2} \))(\( ( 1 + 8 T + 29 T^{2} )^{2} \))(\( ( 1 + 6 T + 29 T^{2} )^{2} \))(\( ( 1 + 29 T^{2} )^{2} \))(\( ( 1 - 2 T + 32 T^{2} - 58 T^{3} + 841 T^{4} )^{2} \))(\( ( 1 - 8 T + 56 T^{2} - 232 T^{3} + 841 T^{4} )^{2} \))(\( ( 1 - 10 T + 76 T^{2} - 290 T^{3} + 841 T^{4} )^{2} \))(\( ( 1 + 4 T + 12 T^{2} + 116 T^{3} + 841 T^{4} )^{2} \))(\( ( 1 - 2 T + 52 T^{2} - 58 T^{3} + 841 T^{4} )^{2} \))(\( ( 1 + 12 T + 76 T^{2} + 348 T^{3} + 841 T^{4} )^{2} \))(\( ( 1 + 9 T^{2} - 256 T^{3} + 261 T^{4} + 24389 T^{6} )^{2} \))(\( ( 1 - 6 T + 81 T^{2} - 300 T^{3} + 2349 T^{4} - 5046 T^{5} + 24389 T^{6} )^{2} \))
$31$ (\( 1 + 5 T - 6 T^{2} + 155 T^{3} + 961 T^{4} \))(\( 1 - 6 T + 5 T^{2} - 186 T^{3} + 961 T^{4} \))(\( 1 + 9 T + 50 T^{2} + 279 T^{3} + 961 T^{4} \))(\( 1 + 2 T - 27 T^{2} + 62 T^{3} + 961 T^{4} \))(\( ( 1 - 11 T + 31 T^{2} )( 1 + 4 T + 31 T^{2} ) \))(\( 1 + T - 30 T^{2} + 31 T^{3} + 961 T^{4} \))(\( 1 + 2 T - 27 T^{2} + 62 T^{3} + 961 T^{4} \))(\( 1 + 9 T + 50 T^{2} + 279 T^{3} + 961 T^{4} \))(\( 1 - 6 T + 5 T^{2} - 186 T^{3} + 961 T^{4} \))(\( ( 1 - 11 T + 31 T^{2} )( 1 + 4 T + 31 T^{2} ) \))(\( 1 - 6 T + 5 T^{2} - 186 T^{3} + 961 T^{4} \))(\( 1 - T - 30 T^{2} - 31 T^{3} + 961 T^{4} \))(\( 1 - 5 T - 6 T^{2} - 155 T^{3} + 961 T^{4} \))(\( 1 + 10 T + 69 T^{2} + 310 T^{3} + 961 T^{4} \))(\( 1 - 6 T - 23 T^{2} + 18 T^{3} + 1404 T^{4} + 558 T^{5} - 22103 T^{6} - 178746 T^{7} + 923521 T^{8} \))(\( 1 + 6 T - 27 T^{2} + 6 T^{3} + 1892 T^{4} + 186 T^{5} - 25947 T^{6} + 178746 T^{7} + 923521 T^{8} \))(\( 1 - 2 T - 31 T^{2} + 54 T^{3} + 140 T^{4} + 1674 T^{5} - 29791 T^{6} - 59582 T^{7} + 923521 T^{8} \))(\( ( 1 + 5 T - 6 T^{2} + 155 T^{3} + 961 T^{4} )^{2} \))(\( ( 1 + 5 T - 6 T^{2} + 155 T^{3} + 961 T^{4} )^{2} \))(\( 1 + 2 T + 13 T^{2} - 142 T^{3} - 1004 T^{4} - 4402 T^{5} + 12493 T^{6} + 59582 T^{7} + 923521 T^{8} \))(\( 1 - 4 T - 58 T^{2} + 132 T^{3} + 2326 T^{4} - 1328 T^{5} - 79754 T^{6} - 41168 T^{7} + 2235286 T^{8} + 3932412 T^{9} - 53564218 T^{10} - 114516604 T^{11} + 887503681 T^{12} \))(\( 1 - 12 T + 78 T^{2} - 204 T^{3} - 1218 T^{4} + 18168 T^{5} - 127402 T^{6} + 563208 T^{7} - 1170498 T^{8} - 6077364 T^{9} + 72034638 T^{10} - 343549812 T^{11} + 887503681 T^{12} \))
$37$ (\( 1 - 5 T - 12 T^{2} - 185 T^{3} + 1369 T^{4} \))(\( 1 + 9 T + 44 T^{2} + 333 T^{3} + 1369 T^{4} \))(\( ( 1 - 10 T + 37 T^{2} )( 1 - T + 37 T^{2} ) \))(\( ( 1 - 10 T + 37 T^{2} )( 1 + 11 T + 37 T^{2} ) \))(\( ( 1 - 10 T + 37 T^{2} )( 1 + 11 T + 37 T^{2} ) \))(\( ( 1 - 11 T + 37 T^{2} )( 1 + 10 T + 37 T^{2} ) \))(\( ( 1 + T + 37 T^{2} )( 1 + 10 T + 37 T^{2} ) \))(\( 1 + 3 T - 28 T^{2} + 111 T^{3} + 1369 T^{4} \))(\( 1 + 3 T - 28 T^{2} + 111 T^{3} + 1369 T^{4} \))(\( 1 - 7 T + 12 T^{2} - 259 T^{3} + 1369 T^{4} \))(\( 1 - 3 T - 28 T^{2} - 111 T^{3} + 1369 T^{4} \))(\( 1 + 7 T + 12 T^{2} + 259 T^{3} + 1369 T^{4} \))(\( 1 - 7 T + 12 T^{2} - 259 T^{3} + 1369 T^{4} \))(\( ( 1 - 11 T + 37 T^{2} )( 1 + 10 T + 37 T^{2} ) \))(\( 1 + 4 T - 35 T^{2} - 92 T^{3} + 640 T^{4} - 3404 T^{5} - 47915 T^{6} + 202612 T^{7} + 1874161 T^{8} \))(\( 1 - 14 T + 75 T^{2} - 658 T^{3} + 6020 T^{4} - 24346 T^{5} + 102675 T^{6} - 709142 T^{7} + 1874161 T^{8} \))(\( 1 - 4 T - 55 T^{2} + 12 T^{3} + 3080 T^{4} + 444 T^{5} - 75295 T^{6} - 202612 T^{7} + 1874161 T^{8} \))(\( 1 - 18 T + 171 T^{2} - 1422 T^{3} + 10052 T^{4} - 52614 T^{5} + 234099 T^{6} - 911754 T^{7} + 1874161 T^{8} \))(\( 1 - 67 T^{2} + 3120 T^{4} - 91723 T^{6} + 1874161 T^{8} \))(\( 1 - 2 T - 53 T^{2} + 34 T^{3} + 1732 T^{4} + 1258 T^{5} - 72557 T^{6} - 101306 T^{7} + 1874161 T^{8} \))(\( 1 + 15 T + 117 T^{2} + 442 T^{3} - 1167 T^{4} - 35733 T^{5} - 291366 T^{6} - 1322121 T^{7} - 1597623 T^{8} + 22388626 T^{9} + 219276837 T^{10} + 1040159355 T^{11} + 2565726409 T^{12} \))(\( 1 + 3 T - 15 T^{2} + 390 T^{3} + 165 T^{4} - 5037 T^{5} + 92954 T^{6} - 186369 T^{7} + 225885 T^{8} + 19754670 T^{9} - 28112415 T^{10} + 208031871 T^{11} + 2565726409 T^{12} \))
$41$ (\( ( 1 - 2 T + 41 T^{2} )^{2} \))(\( ( 1 + 3 T + 41 T^{2} )^{2} \))(\( ( 1 + 8 T + 41 T^{2} )^{2} \))(\( ( 1 - 5 T + 41 T^{2} )^{2} \))(\( ( 1 - 4 T + 41 T^{2} )^{2} \))(\( ( 1 - 6 T + 41 T^{2} )^{2} \))(\( ( 1 + 11 T + 41 T^{2} )^{2} \))(\( ( 1 - 2 T + 41 T^{2} )^{2} \))(\( ( 1 + 3 T + 41 T^{2} )^{2} \))(\( ( 1 + 8 T + 41 T^{2} )^{2} \))(\( ( 1 - 9 T + 41 T^{2} )^{2} \))(\( ( 1 + 6 T + 41 T^{2} )^{2} \))(\( ( 1 - 12 T + 41 T^{2} )^{2} \))(\( ( 1 - 9 T + 41 T^{2} )^{2} \))(\( ( 1 - 2 T + 80 T^{2} - 82 T^{3} + 1681 T^{4} )^{2} \))(\( ( 1 + 4 T + 68 T^{2} + 164 T^{3} + 1681 T^{4} )^{2} \))(\( ( 1 + 6 T + 28 T^{2} + 246 T^{3} + 1681 T^{4} )^{2} \))(\( ( 1 + 8 T + 80 T^{2} + 328 T^{3} + 1681 T^{4} )^{2} \))(\( ( 1 - 2 T + 20 T^{2} - 82 T^{3} + 1681 T^{4} )^{2} \))(\( ( 1 + 64 T^{2} + 1681 T^{4} )^{2} \))(\( ( 1 - 5 T + 21 T^{2} + 208 T^{3} + 861 T^{4} - 8405 T^{5} + 68921 T^{6} )^{2} \))(\( ( 1 + 9 T + 123 T^{2} + 680 T^{3} + 5043 T^{4} + 15129 T^{5} + 68921 T^{6} )^{2} \))
$43$ (\( ( 1 - 9 T + 43 T^{2} )^{2} \))(\( ( 1 - 4 T + 43 T^{2} )^{2} \))(\( ( 1 + T + 43 T^{2} )^{2} \))(\( ( 1 + 12 T + 43 T^{2} )^{2} \))(\( ( 1 - 11 T + 43 T^{2} )^{2} \))(\( ( 1 - T + 43 T^{2} )^{2} \))(\( ( 1 + 8 T + 43 T^{2} )^{2} \))(\( ( 1 + 3 T + 43 T^{2} )^{2} \))(\( ( 1 + 8 T + 43 T^{2} )^{2} \))(\( ( 1 + 5 T + 43 T^{2} )^{2} \))(\( ( 1 - 4 T + 43 T^{2} )^{2} \))(\( ( 1 + T + 43 T^{2} )^{2} \))(\( ( 1 - T + 43 T^{2} )^{2} \))(\( ( 1 + 8 T + 43 T^{2} )^{2} \))(\( ( 1 - 4 T + 63 T^{2} - 172 T^{3} + 1849 T^{4} )^{2} \))(\( ( 1 - 18 T + 165 T^{2} - 774 T^{3} + 1849 T^{4} )^{2} \))(\( ( 1 + 4 T + 27 T^{2} + 172 T^{3} + 1849 T^{4} )^{2} \))(\( ( 1 - 2 T - 11 T^{2} - 86 T^{3} + 1849 T^{4} )^{2} \))(\( ( 1 - 4 T + 83 T^{2} - 172 T^{3} + 1849 T^{4} )^{2} \))(\( ( 1 - 2 T + 69 T^{2} - 86 T^{3} + 1849 T^{4} )^{2} \))(\( ( 1 - 2 T + 108 T^{2} - 136 T^{3} + 4644 T^{4} - 3698 T^{5} + 79507 T^{6} )^{2} \))(\( ( 1 + 90 T^{2} - 88 T^{3} + 3870 T^{4} + 79507 T^{6} )^{2} \))
$47$ (\( 1 + 2 T - 43 T^{2} + 94 T^{3} + 2209 T^{4} \))(\( 1 - T - 46 T^{2} - 47 T^{3} + 2209 T^{4} \))(\( 1 - 6 T - 11 T^{2} - 282 T^{3} + 2209 T^{4} \))(\( 1 + 11 T + 74 T^{2} + 517 T^{3} + 2209 T^{4} \))(\( 1 - 6 T - 11 T^{2} - 282 T^{3} + 2209 T^{4} \))(\( 1 - 10 T + 53 T^{2} - 470 T^{3} + 2209 T^{4} \))(\( 1 + 5 T - 22 T^{2} + 235 T^{3} + 2209 T^{4} \))(\( 1 + 6 T - 11 T^{2} + 282 T^{3} + 2209 T^{4} \))(\( 1 + T - 46 T^{2} + 47 T^{3} + 2209 T^{4} \))(\( 1 - 10 T + 53 T^{2} - 470 T^{3} + 2209 T^{4} \))(\( 1 + 3 T - 38 T^{2} + 141 T^{3} + 2209 T^{4} \))(\( 1 - 2 T - 43 T^{2} - 94 T^{3} + 2209 T^{4} \))(\( 1 - 6 T - 11 T^{2} - 282 T^{3} + 2209 T^{4} \))(\( 1 - 3 T - 38 T^{2} - 141 T^{3} + 2209 T^{4} \))(\( ( 1 - 2 T - 43 T^{2} - 94 T^{3} + 2209 T^{4} )^{2} \))(\( 1 + 4 T + 46 T^{2} - 496 T^{3} - 2061 T^{4} - 23312 T^{5} + 101614 T^{6} + 415292 T^{7} + 4879681 T^{8} \))(\( ( 1 - 6 T - 11 T^{2} - 282 T^{3} + 2209 T^{4} )^{2} \))(\( 1 + 4 T - 50 T^{2} - 112 T^{3} + 1395 T^{4} - 5264 T^{5} - 110450 T^{6} + 415292 T^{7} + 4879681 T^{8} \))(\( 1 + 4 T + 30 T^{2} - 432 T^{3} - 2765 T^{4} - 20304 T^{5} + 66270 T^{6} + 415292 T^{7} + 4879681 T^{8} \))(\( ( 1 + 6 T - 11 T^{2} + 282 T^{3} + 2209 T^{4} )^{2} \))(\( 1 + 11 T - 36 T^{2} - 269 T^{3} + 6292 T^{4} + 18647 T^{5} - 222142 T^{6} + 876409 T^{7} + 13899028 T^{8} - 27928387 T^{9} - 175668516 T^{10} + 2522795077 T^{11} + 10779215329 T^{12} \))(\( 1 - 3 T - 60 T^{2} + 637 T^{3} + 252 T^{4} - 16287 T^{5} + 142082 T^{6} - 765489 T^{7} + 556668 T^{8} + 66135251 T^{9} - 292780860 T^{10} - 688035021 T^{11} + 10779215329 T^{12} \))
$53$ (\( 1 + 12 T + 91 T^{2} + 636 T^{3} + 2809 T^{4} \))(\( 1 - 9 T + 28 T^{2} - 477 T^{3} + 2809 T^{4} \))(\( 1 - 4 T - 37 T^{2} - 212 T^{3} + 2809 T^{4} \))(\( 1 - 9 T + 28 T^{2} - 477 T^{3} + 2809 T^{4} \))(\( 1 - 4 T - 37 T^{2} - 212 T^{3} + 2809 T^{4} \))(\( 1 - 53 T^{2} + 2809 T^{4} \))(\( 1 - 11 T + 68 T^{2} - 583 T^{3} + 2809 T^{4} \))(\( 1 - 53 T^{2} + 2809 T^{4} \))(\( 1 + 5 T - 28 T^{2} + 265 T^{3} + 2809 T^{4} \))(\( 1 - 8 T + 11 T^{2} - 424 T^{3} + 2809 T^{4} \))(\( 1 - T - 52 T^{2} - 53 T^{3} + 2809 T^{4} \))(\( 1 + 4 T - 37 T^{2} + 212 T^{3} + 2809 T^{4} \))(\( 1 - 53 T^{2} + 2809 T^{4} \))(\( 1 - 3 T - 44 T^{2} - 159 T^{3} + 2809 T^{4} \))(\( 1 + 4 T + 14 T^{2} - 416 T^{3} - 3653 T^{4} - 22048 T^{5} + 39326 T^{6} + 595508 T^{7} + 7890481 T^{8} \))(\( 1 + 8 T - 50 T^{2} + 64 T^{3} + 6795 T^{4} + 3392 T^{5} - 140450 T^{6} + 1191016 T^{7} + 7890481 T^{8} \))(\( 1 - 4 T - 66 T^{2} + 96 T^{3} + 3067 T^{4} + 5088 T^{5} - 185394 T^{6} - 595508 T^{7} + 7890481 T^{8} \))(\( 1 - 8 T - 50 T^{2} - 64 T^{3} + 6795 T^{4} - 3392 T^{5} - 140450 T^{6} - 1191016 T^{7} + 7890481 T^{8} \))(\( 1 + 12 T + 30 T^{2} + 96 T^{3} + 2555 T^{4} + 5088 T^{5} + 84270 T^{6} + 1786524 T^{7} + 7890481 T^{8} \))(\( 1 - 34 T^{2} - 1653 T^{4} - 95506 T^{6} + 7890481 T^{8} \))(\( 1 - T - 38 T^{2} - 835 T^{3} - 148 T^{4} + 16847 T^{5} + 372736 T^{6} + 892891 T^{7} - 415732 T^{8} - 124312295 T^{9} - 299838278 T^{10} - 418195493 T^{11} + 22164361129 T^{12} \))(\( 1 - 15 T + 6 T^{2} + 7 T^{3} + 11664 T^{4} - 54399 T^{5} - 166888 T^{6} - 2883147 T^{7} + 32764176 T^{8} + 1042139 T^{9} + 47342886 T^{10} - 6272932395 T^{11} + 22164361129 T^{12} \))
$59$ (\( 1 + 8 T + 5 T^{2} + 472 T^{3} + 3481 T^{4} \))(\( 1 + 4 T - 43 T^{2} + 236 T^{3} + 3481 T^{4} \))(\( 1 - 6 T - 23 T^{2} - 354 T^{3} + 3481 T^{4} \))(\( 1 - 4 T - 43 T^{2} - 236 T^{3} + 3481 T^{4} \))(\( 1 + 14 T + 137 T^{2} + 826 T^{3} + 3481 T^{4} \))(\( 1 - 59 T^{2} + 3481 T^{4} \))(\( 1 - 4 T - 43 T^{2} - 236 T^{3} + 3481 T^{4} \))(\( 1 + 4 T - 43 T^{2} + 236 T^{3} + 3481 T^{4} \))(\( 1 + 4 T - 43 T^{2} + 236 T^{3} + 3481 T^{4} \))(\( 1 - 10 T + 41 T^{2} - 590 T^{3} + 3481 T^{4} \))(\( 1 - 12 T + 85 T^{2} - 708 T^{3} + 3481 T^{4} \))(\( 1 + 8 T + 5 T^{2} + 472 T^{3} + 3481 T^{4} \))(\( 1 + 6 T - 23 T^{2} + 354 T^{3} + 3481 T^{4} \))(\( 1 - 12 T + 85 T^{2} - 708 T^{3} + 3481 T^{4} \))(\( 1 - 10 T - 16 T^{2} + 20 T^{3} + 4075 T^{4} + 1180 T^{5} - 55696 T^{6} - 2053790 T^{7} + 12117361 T^{8} \))(\( 1 - 116 T^{2} + 9975 T^{4} - 403796 T^{6} + 12117361 T^{8} \))(\( 1 - 6 T - 28 T^{2} + 324 T^{3} - 1509 T^{4} + 19116 T^{5} - 97468 T^{6} - 1232274 T^{7} + 12117361 T^{8} \))(\( 1 + 20 T + 200 T^{2} + 1640 T^{3} + 12919 T^{4} + 96760 T^{5} + 696200 T^{6} + 4107580 T^{7} + 12117361 T^{8} \))(\( 1 + 10 T - 36 T^{2} + 180 T^{3} + 9115 T^{4} + 10620 T^{5} - 125316 T^{6} + 2053790 T^{7} + 12117361 T^{8} \))(\( 1 + 12 T + 8 T^{2} + 216 T^{3} + 6519 T^{4} + 12744 T^{5} + 27848 T^{6} + 2464548 T^{7} + 12117361 T^{8} \))(\( 1 - 12 T - 3 T^{2} + 60 T^{3} + 1992 T^{4} + 32316 T^{5} - 533693 T^{6} + 1906644 T^{7} + 6934152 T^{8} + 12322740 T^{9} - 36352083 T^{10} - 8579091588 T^{11} + 42180533641 T^{12} \))(\( 1 - 30 T + 441 T^{2} - 5010 T^{3} + 52200 T^{4} - 489990 T^{5} + 4032079 T^{6} - 28909410 T^{7} + 181708200 T^{8} - 1028948790 T^{9} + 5343756201 T^{10} - 21447728970 T^{11} + 42180533641 T^{12} \))
$61$ (\( ( 1 - 13 T + 61 T^{2} )( 1 - T + 61 T^{2} ) \))(\( 1 - 4 T - 45 T^{2} - 244 T^{3} + 3721 T^{4} \))(\( ( 1 - 13 T + 61 T^{2} )( 1 - T + 61 T^{2} ) \))(\( 1 + 4 T - 45 T^{2} + 244 T^{3} + 3721 T^{4} \))(\( 1 - 6 T - 25 T^{2} - 366 T^{3} + 3721 T^{4} \))(\( 1 - 6 T - 25 T^{2} - 366 T^{3} + 3721 T^{4} \))(\( 1 - 61 T^{2} + 3721 T^{4} \))(\( 1 + 2 T - 57 T^{2} + 122 T^{3} + 3721 T^{4} \))(\( 1 - 8 T + 3 T^{2} - 488 T^{3} + 3721 T^{4} \))(\( 1 - 6 T - 25 T^{2} - 366 T^{3} + 3721 T^{4} \))(\( 1 - 4 T - 45 T^{2} - 244 T^{3} + 3721 T^{4} \))(\( ( 1 - 13 T + 61 T^{2} )( 1 - T + 61 T^{2} ) \))(\( 1 + 2 T - 57 T^{2} + 122 T^{3} + 3721 T^{4} \))(\( 1 + 8 T + 3 T^{2} + 488 T^{3} + 3721 T^{4} \))(\( ( 1 + 4 T - 45 T^{2} + 244 T^{3} + 3721 T^{4} )^{2} \))(\( 1 + 8 T - 2 T^{2} - 448 T^{3} - 3269 T^{4} - 27328 T^{5} - 7442 T^{6} + 1815848 T^{7} + 13845841 T^{8} \))(\( ( 1 + 8 T + 3 T^{2} + 488 T^{3} + 3721 T^{4} )^{2} \))(\( 1 + 24 T + 318 T^{2} + 3264 T^{3} + 27707 T^{4} + 199104 T^{5} + 1183278 T^{6} + 5447544 T^{7} + 13845841 T^{8} \))(\( 1 - 8 T + 38 T^{2} + 768 T^{3} - 6949 T^{4} + 46848 T^{5} + 141398 T^{6} - 1815848 T^{7} + 13845841 T^{8} \))(\( 1 - 8 T - 2 T^{2} + 448 T^{3} - 3269 T^{4} + 27328 T^{5} - 7442 T^{6} - 1815848 T^{7} + 13845841 T^{8} \))(\( 1 + 4 T - 111 T^{2} - 276 T^{3} + 6814 T^{4} + 2692 T^{5} - 435779 T^{6} + 164212 T^{7} + 25354894 T^{8} - 62646756 T^{9} - 1536888351 T^{10} + 3378385204 T^{11} + 51520374361 T^{12} \))(\( ( 1 - 61 T^{2} + 3721 T^{4} )^{3} \))
$67$ (\( 1 - 9 T + 14 T^{2} - 603 T^{3} + 4489 T^{4} \))(\( 1 - 4 T - 51 T^{2} - 268 T^{3} + 4489 T^{4} \))(\( 1 + T - 66 T^{2} + 67 T^{3} + 4489 T^{4} \))(\( 1 + 12 T + 77 T^{2} + 804 T^{3} + 4489 T^{4} \))(\( 1 - 3 T - 58 T^{2} - 201 T^{3} + 4489 T^{4} \))(\( 1 + 15 T + 158 T^{2} + 1005 T^{3} + 4489 T^{4} \))(\( 1 - 67 T^{2} + 4489 T^{4} \))(\( ( 1 - 16 T + 67 T^{2} )( 1 + 11 T + 67 T^{2} ) \))(\( 1 - 67 T^{2} + 4489 T^{4} \))(\( 1 - 3 T - 58 T^{2} - 201 T^{3} + 4489 T^{4} \))(\( 1 - 12 T + 77 T^{2} - 804 T^{3} + 4489 T^{4} \))(\( 1 - 7 T - 18 T^{2} - 469 T^{3} + 4489 T^{4} \))(\( 1 + 7 T - 18 T^{2} + 469 T^{3} + 4489 T^{4} \))(\( 1 - 8 T - 3 T^{2} - 536 T^{3} + 4489 T^{4} \))(\( 1 + 12 T + 49 T^{2} - 468 T^{3} - 5112 T^{4} - 31356 T^{5} + 219961 T^{6} + 3609156 T^{7} + 20151121 T^{8} \))(\( 1 + 2 T + 31 T^{2} - 322 T^{3} - 4028 T^{4} - 21574 T^{5} + 139159 T^{6} + 601526 T^{7} + 20151121 T^{8} \))(\( 1 + 4 T - 115 T^{2} - 12 T^{3} + 11600 T^{4} - 804 T^{5} - 516235 T^{6} + 1203052 T^{7} + 20151121 T^{8} \))(\( 1 + 2 T - 33 T^{2} - 194 T^{3} - 3388 T^{4} - 12998 T^{5} - 148137 T^{6} + 601526 T^{7} + 20151121 T^{8} \))(\( 1 - 4 T - 59 T^{2} + 236 T^{3} + 64 T^{4} + 15812 T^{5} - 264851 T^{6} - 1203052 T^{7} + 20151121 T^{8} \))(\( 1 + 2 T - 113 T^{2} - 34 T^{3} + 8932 T^{4} - 2278 T^{5} - 507257 T^{6} + 601526 T^{7} + 20151121 T^{8} \))(\( 1 + 10 T - 24 T^{2} - 1656 T^{3} - 7412 T^{4} + 58282 T^{5} + 1219138 T^{6} + 3904894 T^{7} - 33272468 T^{8} - 498063528 T^{9} - 483626904 T^{10} + 13501251070 T^{11} + 90458382169 T^{12} \))(\( 1 - 162 T^{2} - 176 T^{3} + 15390 T^{4} + 14256 T^{5} - 1149078 T^{6} + 955152 T^{7} + 69085710 T^{8} - 52934288 T^{9} - 3264481602 T^{10} + 90458382169 T^{12} \))
$71$ (\( ( 1 + 2 T + 71 T^{2} )^{2} \))(\( ( 1 + 2 T + 71 T^{2} )^{2} \))(\( ( 1 + 12 T + 71 T^{2} )^{2} \))(\( ( 1 + 2 T + 71 T^{2} )^{2} \))(\( ( 1 + 71 T^{2} )^{2} \))(\( ( 1 - 6 T + 71 T^{2} )^{2} \))(\( ( 1 - 6 T + 71 T^{2} )^{2} \))(\( ( 1 + 14 T + 71 T^{2} )^{2} \))(\( ( 1 - 6 T + 71 T^{2} )^{2} \))(\( ( 1 + 71 T^{2} )^{2} \))(\( ( 1 - 14 T + 71 T^{2} )^{2} \))(\( ( 1 + 6 T + 71 T^{2} )^{2} \))(\( ( 1 + 12 T + 71 T^{2} )^{2} \))(\( ( 1 - 6 T + 71 T^{2} )^{2} \))(\( ( 1 + 2 T + 116 T^{2} + 142 T^{3} + 5041 T^{4} )^{2} \))(\( ( 1 - 4 T + 144 T^{2} - 284 T^{3} + 5041 T^{4} )^{2} \))(\( ( 1 + 22 T + 256 T^{2} + 1562 T^{3} + 5041 T^{4} )^{2} \))(\( ( 1 + 8 T + 108 T^{2} + 568 T^{3} + 5041 T^{4} )^{2} \))(\( ( 1 - 18 T + 216 T^{2} - 1278 T^{3} + 5041 T^{4} )^{2} \))(\( ( 1 - 20 T^{2} + 5041 T^{4} )^{2} \))(\( ( 1 - 2 T + 67 T^{2} + 304 T^{3} + 4757 T^{4} - 10082 T^{5} + 357911 T^{6} )^{2} \))(\( ( 1 - 12 T + 27 T^{2} + 396 T^{3} + 1917 T^{4} - 60492 T^{5} + 357911 T^{6} )^{2} \))
$73$ (\( 1 + T - 72 T^{2} + 73 T^{3} + 5329 T^{4} \))(\( 1 - 14 T + 123 T^{2} - 1022 T^{3} + 5329 T^{4} \))(\( 1 + 11 T + 48 T^{2} + 803 T^{3} + 5329 T^{4} \))(\( ( 1 - 7 T + 73 T^{2} )( 1 + 17 T + 73 T^{2} ) \))(\( ( 1 - 10 T + 73 T^{2} )( 1 + 17 T + 73 T^{2} ) \))(\( 1 - 3 T - 64 T^{2} - 219 T^{3} + 5329 T^{4} \))(\( 1 - 6 T - 37 T^{2} - 438 T^{3} + 5329 T^{4} \))(\( 1 + T - 72 T^{2} + 73 T^{3} + 5329 T^{4} \))(\( 1 - 14 T + 123 T^{2} - 1022 T^{3} + 5329 T^{4} \))(\( 1 + 15 T + 152 T^{2} + 1095 T^{3} + 5329 T^{4} \))(\( 1 - 14 T + 123 T^{2} - 1022 T^{3} + 5329 T^{4} \))(\( 1 + T - 72 T^{2} + 73 T^{3} + 5329 T^{4} \))(\( 1 + 11 T + 48 T^{2} + 803 T^{3} + 5329 T^{4} \))(\( 1 + 2 T - 69 T^{2} + 146 T^{3} + 5329 T^{4} \))(\( 1 + 8 T - 23 T^{2} - 472 T^{3} - 2432 T^{4} - 34456 T^{5} - 122567 T^{6} + 3112136 T^{7} + 28398241 T^{8} \))(\( 1 - 10 T - 21 T^{2} + 250 T^{3} + 2012 T^{4} + 18250 T^{5} - 111909 T^{6} - 3890170 T^{7} + 28398241 T^{8} \))(\( 1 + 29 T^{2} - 4488 T^{4} + 154541 T^{6} + 28398241 T^{8} \))(\( 1 - 6 T - 101 T^{2} + 54 T^{3} + 10812 T^{4} + 3942 T^{5} - 538229 T^{6} - 2334102 T^{7} + 28398241 T^{8} \))(\( 1 - 20 T + 161 T^{2} - 1860 T^{3} + 22592 T^{4} - 135780 T^{5} + 857969 T^{6} - 7780340 T^{7} + 28398241 T^{8} \))(\( 1 + 10 T - 53 T^{2} + 70 T^{3} + 10780 T^{4} + 5110 T^{5} - 282437 T^{6} + 3890170 T^{7} + 28398241 T^{8} \))(\( 1 - 162 T^{2} + 324 T^{3} + 14418 T^{4} - 26244 T^{5} - 1111534 T^{6} - 1915812 T^{7} + 76833522 T^{8} + 126041508 T^{9} - 4600515042 T^{10} + 151334226289 T^{12} \))(\( ( 1 - 3 T - 36 T^{2} + 887 T^{3} - 2628 T^{4} - 15987 T^{5} + 389017 T^{6} )( 1 + 21 T + 123 T^{2} + 266 T^{3} + 8979 T^{4} + 111909 T^{5} + 389017 T^{6} ) \))
$79$ (\( 1 + 3 T - 70 T^{2} + 237 T^{3} + 6241 T^{4} \))(\( ( 1 - 13 T + 79 T^{2} )( 1 + 17 T + 79 T^{2} ) \))(\( 1 - T - 78 T^{2} - 79 T^{3} + 6241 T^{4} \))(\( 1 + 12 T + 65 T^{2} + 948 T^{3} + 6241 T^{4} \))(\( 1 - 9 T + 2 T^{2} - 711 T^{3} + 6241 T^{4} \))(\( 1 - 9 T + 2 T^{2} - 711 T^{3} + 6241 T^{4} \))(\( 1 + 8 T - 15 T^{2} + 632 T^{3} + 6241 T^{4} \))(\( 1 - 9 T + 2 T^{2} - 711 T^{3} + 6241 T^{4} \))(\( 1 + 16 T + 177 T^{2} + 1264 T^{3} + 6241 T^{4} \))(\( 1 - T - 78 T^{2} - 79 T^{3} + 6241 T^{4} \))(\( ( 1 - 17 T + 79 T^{2} )( 1 + 13 T + 79 T^{2} ) \))(\( 1 + T - 78 T^{2} + 79 T^{3} + 6241 T^{4} \))(\( ( 1 - 4 T + 79 T^{2} )( 1 + 17 T + 79 T^{2} ) \))(\( 1 - 8 T - 15 T^{2} - 632 T^{3} + 6241 T^{4} \))(\( 1 - 6 T - 23 T^{2} + 594 T^{3} - 5604 T^{4} + 46926 T^{5} - 143543 T^{6} - 2958234 T^{7} + 38950081 T^{8} \))(\( 1 - 2 T - 123 T^{2} + 62 T^{3} + 9572 T^{4} + 4898 T^{5} - 767643 T^{6} - 986078 T^{7} + 38950081 T^{8} \))(\( 1 + 6 T - 103 T^{2} - 114 T^{3} + 10236 T^{4} - 9006 T^{5} - 642823 T^{6} + 2958234 T^{7} + 38950081 T^{8} \))(\( 1 + 18 T + 93 T^{2} + 1314 T^{3} + 21524 T^{4} + 103806 T^{5} + 580413 T^{6} + 8874702 T^{7} + 38950081 T^{8} \))(\( ( 1 - 4 T + 79 T^{2} )^{2}( 1 + 17 T + 79 T^{2} )^{2} \))(\( ( 1 - 11 T + 42 T^{2} - 869 T^{3} + 6241 T^{4} )^{2} \))(\( 1 - 18 T + 30 T^{2} - 28 T^{3} + 15834 T^{4} - 50010 T^{5} - 937026 T^{6} - 3950790 T^{7} + 98819994 T^{8} - 13805092 T^{9} + 1168502430 T^{10} - 55387015182 T^{11} + 243087455521 T^{12} \))(\( 1 - 6 T - 114 T^{2} + 932 T^{3} + 5154 T^{4} - 32646 T^{5} - 177042 T^{6} - 2579034 T^{7} + 32166114 T^{8} + 459512348 T^{9} - 4440309234 T^{10} - 18462338394 T^{11} + 243087455521 T^{12} \))
$83$ (\( ( 1 - 18 T + 83 T^{2} )^{2} \))(\( ( 1 - 4 T + 83 T^{2} )^{2} \))(\( ( 1 - 4 T + 83 T^{2} )^{2} \))(\( ( 1 - 12 T + 83 T^{2} )^{2} \))(\( ( 1 + 16 T + 83 T^{2} )^{2} \))(\( ( 1 + 6 T + 83 T^{2} )^{2} \))(\( ( 1 + 8 T + 83 T^{2} )^{2} \))(\( ( 1 - 6 T + 83 T^{2} )^{2} \))(\( ( 1 - 16 T + 83 T^{2} )^{2} \))(\( ( 1 + 8 T + 83 T^{2} )^{2} \))(\( ( 1 + 12 T + 83 T^{2} )^{2} \))(\( ( 1 + 2 T + 83 T^{2} )^{2} \))(\( ( 1 - 12 T + 83 T^{2} )^{2} \))(\( ( 1 + 83 T^{2} )^{2} \))(\( ( 1 + 6 T + 28 T^{2} + 498 T^{3} + 6889 T^{4} )^{2} \))(\( ( 1 + 8 T + 180 T^{2} + 664 T^{3} + 6889 T^{4} )^{2} \))(\( ( 1 - 6 T + 112 T^{2} - 498 T^{3} + 6889 T^{4} )^{2} \))(\( ( 1 + 4 T + 152 T^{2} + 332 T^{3} + 6889 T^{4} )^{2} \))(\( ( 1 + 10 T + 128 T^{2} + 830 T^{3} + 6889 T^{4} )^{2} \))(\( ( 1 - 12 T + 184 T^{2} - 996 T^{3} + 6889 T^{4} )^{2} \))(\( ( 1 + 4 T + 123 T^{2} + 256 T^{3} + 10209 T^{4} + 27556 T^{5} + 571787 T^{6} )^{2} \))(\( ( 1 + 6 T + 243 T^{2} + 948 T^{3} + 20169 T^{4} + 41334 T^{5} + 571787 T^{6} )^{2} \))
$89$ (\( 1 - 4 T - 73 T^{2} - 356 T^{3} + 7921 T^{4} \))(\( 1 - 2 T - 85 T^{2} - 178 T^{3} + 7921 T^{4} \))(\( 1 + 18 T + 235 T^{2} + 1602 T^{3} + 7921 T^{4} \))(\( 1 + 14 T + 107 T^{2} + 1246 T^{3} + 7921 T^{4} \))(\( 1 - 2 T - 85 T^{2} - 178 T^{3} + 7921 T^{4} \))(\( 1 - 89 T^{2} + 7921 T^{4} \))(\( 1 - 10 T + 11 T^{2} - 890 T^{3} + 7921 T^{4} \))(\( 1 - 4 T - 73 T^{2} - 356 T^{3} + 7921 T^{4} \))(\( 1 + 6 T - 53 T^{2} + 534 T^{3} + 7921 T^{4} \))(\( 1 + 2 T - 85 T^{2} + 178 T^{3} + 7921 T^{4} \))(\( 1 - 2 T - 85 T^{2} - 178 T^{3} + 7921 T^{4} \))(\( 1 - 12 T + 55 T^{2} - 1068 T^{3} + 7921 T^{4} \))(\( 1 + 6 T - 53 T^{2} + 534 T^{3} + 7921 T^{4} \))(\( 1 + 6 T - 53 T^{2} + 534 T^{3} + 7921 T^{4} \))(\( 1 + 6 T - 4 T^{2} - 828 T^{3} - 9525 T^{4} - 73692 T^{5} - 31684 T^{6} + 4229814 T^{7} + 62742241 T^{8} \))(\( 1 - 16 T + 32 T^{2} - 736 T^{3} + 19471 T^{4} - 65504 T^{5} + 253472 T^{6} - 11279504 T^{7} + 62742241 T^{8} \))(\( 1 - 2 T + 348 T^{3} - 8261 T^{4} + 30972 T^{5} - 1409938 T^{7} + 62742241 T^{8} \))(\( 1 - 4 T - 148 T^{2} + 56 T^{3} + 17551 T^{4} + 4984 T^{5} - 1172308 T^{6} - 2819876 T^{7} + 62742241 T^{8} \))(\( 1 + 6 T - 88 T^{2} - 324 T^{3} + 4251 T^{4} - 28836 T^{5} - 697048 T^{6} + 4229814 T^{7} + 62742241 T^{8} \))(\( 1 - 12 T - 52 T^{2} - 216 T^{3} + 17679 T^{4} - 19224 T^{5} - 411892 T^{6} - 8459628 T^{7} + 62742241 T^{8} \))(\( 1 + 22 T + 71 T^{2} + 686 T^{3} + 41412 T^{4} + 291554 T^{5} + 35533 T^{6} + 25948306 T^{7} + 328024452 T^{8} + 483608734 T^{9} + 4454699111 T^{10} + 122849307878 T^{11} + 496981290961 T^{12} \))(\( 1 - 177 T^{2} + 584 T^{3} + 15576 T^{4} - 51684 T^{5} - 1293079 T^{6} - 4599876 T^{7} + 123377496 T^{8} + 411701896 T^{9} - 11105376657 T^{10} + 496981290961 T^{12} \))
$97$ (\( ( 1 - 10 T + 97 T^{2} )^{2} \))(\( ( 1 + 8 T + 97 T^{2} )^{2} \))(\( ( 1 + 18 T + 97 T^{2} )^{2} \))(\( ( 1 + 8 T + 97 T^{2} )^{2} \))(\( ( 1 + 2 T + 97 T^{2} )^{2} \))(\( ( 1 - 2 T + 97 T^{2} )^{2} \))(\( ( 1 + 16 T + 97 T^{2} )^{2} \))(\( ( 1 + 14 T + 97 T^{2} )^{2} \))(\( ( 1 - 16 T + 97 T^{2} )^{2} \))(\( ( 1 + 10 T + 97 T^{2} )^{2} \))(\( ( 1 + 16 T + 97 T^{2} )^{2} \))(\( ( 1 + 6 T + 97 T^{2} )^{2} \))(\( ( 1 + 10 T + 97 T^{2} )^{2} \))(\( ( 1 - 8 T + 97 T^{2} )^{2} \))(\( ( 1 - 16 T + 210 T^{2} - 1552 T^{3} + 9409 T^{4} )^{2} \))(\( ( 1 + 16 T + 250 T^{2} + 1552 T^{3} + 9409 T^{4} )^{2} \))(\( ( 1 - 8 T + 97 T^{2} )^{4} \))(\( ( 1 + 122 T^{2} + 9409 T^{4} )^{2} \))(\( ( 1 + 8 T + 98 T^{2} + 776 T^{3} + 9409 T^{4} )^{2} \))(\( ( 1 - 16 T + 186 T^{2} - 1552 T^{3} + 9409 T^{4} )^{2} \))(\( ( 1 - 8 T + 251 T^{2} - 1424 T^{3} + 24347 T^{4} - 75272 T^{5} + 912673 T^{6} )^{2} \))(\( ( 1 - 8 T + 97 T^{2} )^{6} \))
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