Properties

Label 1386.2.k
Level 1386
Weight 2
Character orbit k
Rep. character \(\chi_{1386}(793,\cdot)\)
Character field \(\Q(\zeta_{3})\)
Dimension 64
Newform subspaces 23
Sturm bound 576
Trace bound 11

Related objects

Downloads

Learn more about

Defining parameters

Level: \( N \) \(=\) \( 1386 = 2 \cdot 3^{2} \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1386.k (of order \(3\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 7 \)
Character field: \(\Q(\zeta_{3})\)
Newform subspaces: \( 23 \)
Sturm bound: \(576\)
Trace bound: \(11\)
Distinguishing \(T_p\): \(5\), \(13\), \(17\), \(23\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(1386, [\chi])\).

Total New Old
Modular forms 608 64 544
Cusp forms 544 64 480
Eisenstein series 64 0 64

Trace form

\( 64q - 32q^{4} - 4q^{5} - 8q^{7} + O(q^{10}) \) \( 64q - 32q^{4} - 4q^{5} - 8q^{7} + 8q^{13} + 8q^{14} - 32q^{16} + 12q^{17} + 16q^{19} + 8q^{20} - 16q^{25} - 12q^{26} + 4q^{28} - 8q^{29} - 28q^{31} - 24q^{34} + 28q^{35} - 16q^{37} - 20q^{38} - 40q^{41} + 56q^{43} - 20q^{46} + 8q^{47} + 16q^{49} - 4q^{52} + 4q^{53} - 4q^{56} + 12q^{58} - 20q^{59} - 20q^{61} - 24q^{62} + 64q^{64} + 44q^{65} - 36q^{67} + 12q^{68} - 4q^{70} + 8q^{71} + 48q^{73} + 36q^{74} - 32q^{76} + 4q^{77} + 4q^{79} - 4q^{80} + 8q^{82} - 72q^{83} - 96q^{85} - 4q^{86} - 40q^{89} - 4q^{91} + 40q^{94} + 12q^{95} + 80q^{97} + 48q^{98} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(1386, [\chi])\) into newform subspaces

Label Dim. \(A\) Field CM Traces $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\)
1386.2.k.a \(2\) \(11.067\) \(\Q(\sqrt{-3}) \) None \(-1\) \(0\) \(-4\) \(-5\) \(q-\zeta_{6}q^{2}+(-1+\zeta_{6})q^{4}-4\zeta_{6}q^{5}+\cdots\)
1386.2.k.b \(2\) \(11.067\) \(\Q(\sqrt{-3}) \) None \(-1\) \(0\) \(-3\) \(-1\) \(q-\zeta_{6}q^{2}+(-1+\zeta_{6})q^{4}-3\zeta_{6}q^{5}+\cdots\)
1386.2.k.c \(2\) \(11.067\) \(\Q(\sqrt{-3}) \) None \(-1\) \(0\) \(-3\) \(5\) \(q-\zeta_{6}q^{2}+(-1+\zeta_{6})q^{4}-3\zeta_{6}q^{5}+\cdots\)
1386.2.k.d \(2\) \(11.067\) \(\Q(\sqrt{-3}) \) None \(-1\) \(0\) \(0\) \(-4\) \(q-\zeta_{6}q^{2}+(-1+\zeta_{6})q^{4}+(-3+2\zeta_{6})q^{7}+\cdots\)
1386.2.k.e \(2\) \(11.067\) \(\Q(\sqrt{-3}) \) None \(-1\) \(0\) \(0\) \(-1\) \(q-\zeta_{6}q^{2}+(-1+\zeta_{6})q^{4}+(-2+3\zeta_{6})q^{7}+\cdots\)
1386.2.k.f \(2\) \(11.067\) \(\Q(\sqrt{-3}) \) None \(-1\) \(0\) \(0\) \(4\) \(q-\zeta_{6}q^{2}+(-1+\zeta_{6})q^{4}+(3-2\zeta_{6})q^{7}+\cdots\)
1386.2.k.g \(2\) \(11.067\) \(\Q(\sqrt{-3}) \) None \(-1\) \(0\) \(1\) \(-5\) \(q-\zeta_{6}q^{2}+(-1+\zeta_{6})q^{4}+\zeta_{6}q^{5}+(-2+\cdots)q^{7}+\cdots\)
1386.2.k.h \(2\) \(11.067\) \(\Q(\sqrt{-3}) \) None \(-1\) \(0\) \(2\) \(4\) \(q-\zeta_{6}q^{2}+(-1+\zeta_{6})q^{4}+2\zeta_{6}q^{5}+\cdots\)
1386.2.k.i \(2\) \(11.067\) \(\Q(\sqrt{-3}) \) None \(-1\) \(0\) \(2\) \(4\) \(q-\zeta_{6}q^{2}+(-1+\zeta_{6})q^{4}+2\zeta_{6}q^{5}+\cdots\)
1386.2.k.j \(2\) \(11.067\) \(\Q(\sqrt{-3}) \) None \(-1\) \(0\) \(3\) \(-5\) \(q-\zeta_{6}q^{2}+(-1+\zeta_{6})q^{4}+3\zeta_{6}q^{5}+\cdots\)
1386.2.k.k \(2\) \(11.067\) \(\Q(\sqrt{-3}) \) None \(1\) \(0\) \(-2\) \(4\) \(q+\zeta_{6}q^{2}+(-1+\zeta_{6})q^{4}-2\zeta_{6}q^{5}+\cdots\)
1386.2.k.l \(2\) \(11.067\) \(\Q(\sqrt{-3}) \) None \(1\) \(0\) \(-2\) \(4\) \(q+\zeta_{6}q^{2}+(-1+\zeta_{6})q^{4}-2\zeta_{6}q^{5}+\cdots\)
1386.2.k.m \(2\) \(11.067\) \(\Q(\sqrt{-3}) \) None \(1\) \(0\) \(-1\) \(-5\) \(q+\zeta_{6}q^{2}+(-1+\zeta_{6})q^{4}-\zeta_{6}q^{5}+(-2+\cdots)q^{7}+\cdots\)
1386.2.k.n \(2\) \(11.067\) \(\Q(\sqrt{-3}) \) None \(1\) \(0\) \(0\) \(5\) \(q+\zeta_{6}q^{2}+(-1+\zeta_{6})q^{4}+(2+\zeta_{6})q^{7}+\cdots\)
1386.2.k.o \(2\) \(11.067\) \(\Q(\sqrt{-3}) \) None \(1\) \(0\) \(2\) \(-5\) \(q+\zeta_{6}q^{2}+(-1+\zeta_{6})q^{4}+2\zeta_{6}q^{5}+\cdots\)
1386.2.k.p \(2\) \(11.067\) \(\Q(\sqrt{-3}) \) None \(1\) \(0\) \(3\) \(-1\) \(q+\zeta_{6}q^{2}+(-1+\zeta_{6})q^{4}+3\zeta_{6}q^{5}+\cdots\)
1386.2.k.q \(4\) \(11.067\) \(\Q(\sqrt{2}, \sqrt{-3})\) None \(-2\) \(0\) \(-2\) \(-2\) \(q+(-1-\beta _{2})q^{2}+\beta _{2}q^{4}+(-1+\beta _{1}+\cdots)q^{5}+\cdots\)
1386.2.k.r \(4\) \(11.067\) \(\Q(\sqrt{-3}, \sqrt{7})\) None \(-2\) \(0\) \(0\) \(0\) \(q+(-1-\beta _{2})q^{2}+\beta _{2}q^{4}+\beta _{1}q^{5}-\beta _{1}q^{7}+\cdots\)
1386.2.k.s \(4\) \(11.067\) \(\Q(\sqrt{-3}, \sqrt{7})\) None \(-2\) \(0\) \(2\) \(0\) \(q+(-1-\beta _{2})q^{2}+\beta _{2}q^{4}+(1+\beta _{1}+\beta _{2}+\cdots)q^{5}+\cdots\)
1386.2.k.t \(4\) \(11.067\) \(\Q(\sqrt{2}, \sqrt{-3})\) None \(2\) \(0\) \(-4\) \(-2\) \(q-\beta _{2}q^{2}+(-1-\beta _{2})q^{4}+(-\beta _{1}+2\beta _{2}+\cdots)q^{5}+\cdots\)
1386.2.k.u \(4\) \(11.067\) \(\Q(\sqrt{2}, \sqrt{-3})\) None \(2\) \(0\) \(2\) \(-2\) \(q+(1+\beta _{2})q^{2}+\beta _{2}q^{4}+(1+\beta _{1}+\beta _{2}+\cdots)q^{5}+\cdots\)
1386.2.k.v \(6\) \(11.067\) 6.0.21870000.1 None \(3\) \(0\) \(0\) \(0\) \(q-\beta _{2}q^{2}+(-1-\beta _{2})q^{4}+(\beta _{1}-\beta _{4}+\cdots)q^{5}+\cdots\)
1386.2.k.w \(6\) \(11.067\) 6.0.1156923.1 None \(3\) \(0\) \(0\) \(0\) \(q-\beta _{2}q^{2}+(-1-\beta _{2})q^{4}+\beta _{4}q^{5}+(\beta _{3}+\cdots)q^{7}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(1386, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(1386, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(21, [\chi])\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(42, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(63, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(77, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(126, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(154, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(231, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(462, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(693, [\chi])\)\(^{\oplus 2}\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ (\( 1 + T + T^{2} \))(\( 1 + T + T^{2} \))(\( 1 + T + T^{2} \))(\( 1 + T + T^{2} \))(\( 1 + T + T^{2} \))(\( 1 + T + T^{2} \))(\( 1 + T + T^{2} \))(\( 1 + T + T^{2} \))(\( 1 + T + T^{2} \))(\( 1 + T + T^{2} \))(\( 1 - T + T^{2} \))(\( 1 - T + T^{2} \))(\( 1 - T + T^{2} \))(\( 1 - T + T^{2} \))(\( 1 - T + T^{2} \))(\( 1 - T + T^{2} \))(\( ( 1 + T + T^{2} )^{2} \))(\( ( 1 + T + T^{2} )^{2} \))(\( ( 1 + T + T^{2} )^{2} \))(\( ( 1 - T + T^{2} )^{2} \))(\( ( 1 - T + T^{2} )^{2} \))(\( ( 1 - T + T^{2} )^{3} \))(\( ( 1 - T + T^{2} )^{3} \))
$3$ 1
$5$ (\( 1 + 4 T + 11 T^{2} + 20 T^{3} + 25 T^{4} \))(\( 1 + 3 T + 4 T^{2} + 15 T^{3} + 25 T^{4} \))(\( 1 + 3 T + 4 T^{2} + 15 T^{3} + 25 T^{4} \))(\( 1 - 5 T^{2} + 25 T^{4} \))(\( 1 - 5 T^{2} + 25 T^{4} \))(\( 1 - 5 T^{2} + 25 T^{4} \))(\( 1 - T - 4 T^{2} - 5 T^{3} + 25 T^{4} \))(\( 1 - 2 T - T^{2} - 10 T^{3} + 25 T^{4} \))(\( 1 - 2 T - T^{2} - 10 T^{3} + 25 T^{4} \))(\( 1 - 3 T + 4 T^{2} - 15 T^{3} + 25 T^{4} \))(\( 1 + 2 T - T^{2} + 10 T^{3} + 25 T^{4} \))(\( 1 + 2 T - T^{2} + 10 T^{3} + 25 T^{4} \))(\( 1 + T - 4 T^{2} + 5 T^{3} + 25 T^{4} \))(\( 1 - 5 T^{2} + 25 T^{4} \))(\( 1 - 2 T - T^{2} - 10 T^{3} + 25 T^{4} \))(\( 1 - 3 T + 4 T^{2} - 15 T^{3} + 25 T^{4} \))(\( 1 + 2 T - 5 T^{2} - 2 T^{3} + 36 T^{4} - 10 T^{5} - 125 T^{6} + 250 T^{7} + 625 T^{8} \))(\( 1 - 3 T^{2} - 16 T^{4} - 75 T^{6} + 625 T^{8} \))(\( 1 - 2 T + 12 T^{3} - 29 T^{4} + 60 T^{5} - 250 T^{7} + 625 T^{8} \))(\( 1 + 4 T + 4 T^{2} + 8 T^{3} + 39 T^{4} + 40 T^{5} + 100 T^{6} + 500 T^{7} + 625 T^{8} \))(\( 1 - 2 T - 5 T^{2} + 2 T^{3} + 36 T^{4} + 10 T^{5} - 125 T^{6} - 250 T^{7} + 625 T^{8} \))(\( ( 1 + 20 T^{3} + 125 T^{6} )^{2} \))(\( 1 - 6 T^{2} - 8 T^{3} + 6 T^{4} + 24 T^{5} + 86 T^{6} + 120 T^{7} + 150 T^{8} - 1000 T^{9} - 3750 T^{10} + 15625 T^{12} \))
$7$ (\( 1 + 5 T + 7 T^{2} \))(\( 1 + T + 7 T^{2} \))(\( 1 - 5 T + 7 T^{2} \))(\( 1 + 4 T + 7 T^{2} \))(\( 1 + T + 7 T^{2} \))(\( 1 - 4 T + 7 T^{2} \))(\( 1 + 5 T + 7 T^{2} \))(\( 1 - 4 T + 7 T^{2} \))(\( 1 - 4 T + 7 T^{2} \))(\( 1 + 5 T + 7 T^{2} \))(\( 1 - 4 T + 7 T^{2} \))(\( 1 - 4 T + 7 T^{2} \))(\( 1 + 5 T + 7 T^{2} \))(\( 1 - 5 T + 7 T^{2} \))(\( 1 + 5 T + 7 T^{2} \))(\( 1 + T + 7 T^{2} \))(\( 1 + 2 T - 3 T^{2} + 14 T^{3} + 49 T^{4} \))(\( 1 + 7 T^{2} + 49 T^{4} \))(\( 1 + 7 T^{2} + 49 T^{4} \))(\( 1 + 2 T - 3 T^{2} + 14 T^{3} + 49 T^{4} \))(\( 1 + 2 T - 3 T^{2} + 14 T^{3} + 49 T^{4} \))(\( 1 + 6 T^{2} - 20 T^{3} + 42 T^{4} + 343 T^{6} \))(\( 1 + 12 T^{2} - 4 T^{3} + 84 T^{4} + 343 T^{6} \))
$11$ (\( 1 + T + T^{2} \))(\( 1 - T + T^{2} \))(\( 1 + T + T^{2} \))(\( 1 - T + T^{2} \))(\( 1 + T + T^{2} \))(\( 1 + T + T^{2} \))(\( 1 - T + T^{2} \))(\( 1 + T + T^{2} \))(\( 1 - T + T^{2} \))(\( 1 + T + T^{2} \))(\( 1 + T + T^{2} \))(\( 1 - T + T^{2} \))(\( 1 + T + T^{2} \))(\( 1 + T + T^{2} \))(\( 1 + T + T^{2} \))(\( 1 + T + T^{2} \))(\( ( 1 + T + T^{2} )^{2} \))(\( ( 1 - T + T^{2} )^{2} \))(\( ( 1 - T + T^{2} )^{2} \))(\( ( 1 - T + T^{2} )^{2} \))(\( ( 1 - T + T^{2} )^{2} \))(\( ( 1 + T + T^{2} )^{3} \))(\( ( 1 - T + T^{2} )^{3} \))
$13$ (\( ( 1 + T + 13 T^{2} )^{2} \))(\( ( 1 - 2 T + 13 T^{2} )^{2} \))(\( ( 1 - 2 T + 13 T^{2} )^{2} \))(\( ( 1 + 4 T + 13 T^{2} )^{2} \))(\( ( 1 + T + 13 T^{2} )^{2} \))(\( ( 1 - 4 T + 13 T^{2} )^{2} \))(\( ( 1 - 2 T + 13 T^{2} )^{2} \))(\( ( 1 - 2 T + 13 T^{2} )^{2} \))(\( ( 1 - 2 T + 13 T^{2} )^{2} \))(\( ( 1 - 6 T + 13 T^{2} )^{2} \))(\( ( 1 - 2 T + 13 T^{2} )^{2} \))(\( ( 1 - 2 T + 13 T^{2} )^{2} \))(\( ( 1 - 2 T + 13 T^{2} )^{2} \))(\( ( 1 - 5 T + 13 T^{2} )^{2} \))(\( ( 1 + 7 T + 13 T^{2} )^{2} \))(\( ( 1 - 2 T + 13 T^{2} )^{2} \))(\( ( 1 + 8 T + 34 T^{2} + 104 T^{3} + 169 T^{4} )^{2} \))(\( ( 1 + 4 T + 13 T^{2} )^{4} \))(\( ( 1 - 5 T + 13 T^{2} )^{4} \))(\( ( 1 + 2 T + 19 T^{2} + 26 T^{3} + 169 T^{4} )^{2} \))(\( ( 1 + 8 T + 34 T^{2} + 104 T^{3} + 169 T^{4} )^{2} \))(\( ( 1 + 13 T^{2} )^{6} \))(\( ( 1 + 3 T^{2} + 32 T^{3} + 39 T^{4} + 2197 T^{6} )^{2} \))
$17$ (\( 1 - 2 T - 13 T^{2} - 34 T^{3} + 289 T^{4} \))(\( 1 - 3 T - 8 T^{2} - 51 T^{3} + 289 T^{4} \))(\( 1 - 3 T - 8 T^{2} - 51 T^{3} + 289 T^{4} \))(\( 1 - 3 T - 8 T^{2} - 51 T^{3} + 289 T^{4} \))(\( 1 + 6 T + 19 T^{2} + 102 T^{3} + 289 T^{4} \))(\( 1 + T - 16 T^{2} + 17 T^{3} + 289 T^{4} \))(\( 1 + 5 T + 8 T^{2} + 85 T^{3} + 289 T^{4} \))(\( 1 + T - 16 T^{2} + 17 T^{3} + 289 T^{4} \))(\( 1 - 3 T - 8 T^{2} - 51 T^{3} + 289 T^{4} \))(\( 1 + 5 T + 8 T^{2} + 85 T^{3} + 289 T^{4} \))(\( 1 + 3 T - 8 T^{2} + 51 T^{3} + 289 T^{4} \))(\( 1 - T - 16 T^{2} - 17 T^{3} + 289 T^{4} \))(\( 1 - 5 T + 8 T^{2} - 85 T^{3} + 289 T^{4} \))(\( 1 - 6 T + 19 T^{2} - 102 T^{3} + 289 T^{4} \))(\( 1 - 2 T - 13 T^{2} - 34 T^{3} + 289 T^{4} \))(\( 1 + 3 T - 8 T^{2} + 51 T^{3} + 289 T^{4} \))(\( 1 - 10 T + 49 T^{2} - 170 T^{3} + 612 T^{4} - 2890 T^{5} + 14161 T^{6} - 49130 T^{7} + 83521 T^{8} \))(\( ( 1 + 3 T - 8 T^{2} + 51 T^{3} + 289 T^{4} )^{2} \))(\( ( 1 - 6 T + 19 T^{2} - 102 T^{3} + 289 T^{4} )^{2} \))(\( 1 + 4 T + 10 T^{2} - 112 T^{3} - 525 T^{4} - 1904 T^{5} + 2890 T^{6} + 19652 T^{7} + 83521 T^{8} \))(\( 1 + 10 T + 49 T^{2} + 170 T^{3} + 612 T^{4} + 2890 T^{5} + 14161 T^{6} + 49130 T^{7} + 83521 T^{8} \))(\( 1 - 3 T + 15 T^{2} - 56 T^{3} + 249 T^{4} - 1701 T^{5} + 7982 T^{6} - 28917 T^{7} + 71961 T^{8} - 275128 T^{9} + 1252815 T^{10} - 4259571 T^{11} + 24137569 T^{12} \))(\( ( 1 - T - 16 T^{2} - 17 T^{3} + 289 T^{4} )^{3} \))
$19$ (\( 1 + 6 T + 17 T^{2} + 114 T^{3} + 361 T^{4} \))(\( 1 + 2 T - 15 T^{2} + 38 T^{3} + 361 T^{4} \))(\( 1 + 2 T - 15 T^{2} + 38 T^{3} + 361 T^{4} \))(\( ( 1 - 8 T + 19 T^{2} )( 1 + 7 T + 19 T^{2} ) \))(\( 1 + 2 T - 15 T^{2} + 38 T^{3} + 361 T^{4} \))(\( 1 - 3 T - 10 T^{2} - 57 T^{3} + 361 T^{4} \))(\( 1 - 6 T + 17 T^{2} - 114 T^{3} + 361 T^{4} \))(\( 1 - 3 T - 10 T^{2} - 57 T^{3} + 361 T^{4} \))(\( ( 1 - T + 19 T^{2} )( 1 + 8 T + 19 T^{2} ) \))(\( 1 + 6 T + 17 T^{2} + 114 T^{3} + 361 T^{4} \))(\( ( 1 - T + 19 T^{2} )( 1 + 8 T + 19 T^{2} ) \))(\( 1 - 3 T - 10 T^{2} - 57 T^{3} + 361 T^{4} \))(\( 1 - 6 T + 17 T^{2} - 114 T^{3} + 361 T^{4} \))(\( 1 + 2 T - 15 T^{2} + 38 T^{3} + 361 T^{4} \))(\( 1 - 19 T^{2} + 361 T^{4} \))(\( 1 + 2 T - 15 T^{2} + 38 T^{3} + 361 T^{4} \))(\( 1 - 4 T - 18 T^{2} + 16 T^{3} + 491 T^{4} + 304 T^{5} - 6498 T^{6} - 27436 T^{7} + 130321 T^{8} \))(\( ( 1 - 12 T + 67 T^{2} - 228 T^{3} + 361 T^{4} )( 1 + 12 T + 67 T^{2} + 228 T^{3} + 361 T^{4} ) \))(\( 1 - 6 T - 4 T^{2} - 12 T^{3} + 555 T^{4} - 228 T^{5} - 1444 T^{6} - 41154 T^{7} + 130321 T^{8} \))(\( 1 - 4 T - 24 T^{2} - 8 T^{3} + 935 T^{4} - 152 T^{5} - 8664 T^{6} - 27436 T^{7} + 130321 T^{8} \))(\( 1 - 4 T - 18 T^{2} + 16 T^{3} + 491 T^{4} + 304 T^{5} - 6498 T^{6} - 27436 T^{7} + 130321 T^{8} \))(\( 1 - 9 T + 12 T^{2} + 7 T^{3} + 624 T^{4} - 1317 T^{5} - 7386 T^{6} - 25023 T^{7} + 225264 T^{8} + 48013 T^{9} + 1563852 T^{10} - 22284891 T^{11} + 47045881 T^{12} \))(\( 1 - 3 T - 12 T^{2} + 237 T^{3} - 408 T^{4} - 1839 T^{5} + 24590 T^{6} - 34941 T^{7} - 147288 T^{8} + 1625583 T^{9} - 1563852 T^{10} - 7428297 T^{11} + 47045881 T^{12} \))
$23$ (\( 1 + 2 T - 19 T^{2} + 46 T^{3} + 529 T^{4} \))(\( 1 + 3 T - 14 T^{2} + 69 T^{3} + 529 T^{4} \))(\( 1 - 3 T - 14 T^{2} - 69 T^{3} + 529 T^{4} \))(\( 1 + 3 T - 14 T^{2} + 69 T^{3} + 529 T^{4} \))(\( 1 + 6 T + 13 T^{2} + 138 T^{3} + 529 T^{4} \))(\( 1 + T - 22 T^{2} + 23 T^{3} + 529 T^{4} \))(\( 1 + 7 T + 26 T^{2} + 161 T^{3} + 529 T^{4} \))(\( 1 - T - 22 T^{2} - 23 T^{3} + 529 T^{4} \))(\( 1 - 7 T + 26 T^{2} - 161 T^{3} + 529 T^{4} \))(\( 1 - 5 T + 2 T^{2} - 115 T^{3} + 529 T^{4} \))(\( 1 + 7 T + 26 T^{2} + 161 T^{3} + 529 T^{4} \))(\( 1 + T - 22 T^{2} + 23 T^{3} + 529 T^{4} \))(\( 1 - 7 T + 26 T^{2} - 161 T^{3} + 529 T^{4} \))(\( 1 - 6 T + 13 T^{2} - 138 T^{3} + 529 T^{4} \))(\( 1 + 8 T + 41 T^{2} + 184 T^{3} + 529 T^{4} \))(\( 1 - 3 T - 14 T^{2} - 69 T^{3} + 529 T^{4} \))(\( 1 + 2 T - 25 T^{2} - 34 T^{3} + 220 T^{4} - 782 T^{5} - 13225 T^{6} + 24334 T^{7} + 279841 T^{8} \))(\( 1 - 39 T^{2} + 992 T^{4} - 20631 T^{6} + 279841 T^{8} \))(\( 1 + 2 T - 36 T^{2} - 12 T^{3} + 979 T^{4} - 276 T^{5} - 19044 T^{6} + 24334 T^{7} + 279841 T^{8} \))(\( 1 + 4 T - 16 T^{2} - 56 T^{3} + 127 T^{4} - 1288 T^{5} - 8464 T^{6} + 48668 T^{7} + 279841 T^{8} \))(\( 1 - 2 T - 25 T^{2} + 34 T^{3} + 220 T^{4} + 782 T^{5} - 13225 T^{6} - 24334 T^{7} + 279841 T^{8} \))(\( 1 - 3 T - 33 T^{2} - 28 T^{3} + 651 T^{4} + 2151 T^{5} - 18598 T^{6} + 49473 T^{7} + 344379 T^{8} - 340676 T^{9} - 9234753 T^{10} - 19309029 T^{11} + 148035889 T^{12} \))(\( 1 - 9 T + 9 T^{2} + 138 T^{3} - 9 T^{4} - 3249 T^{5} + 16702 T^{6} - 74727 T^{7} - 4761 T^{8} + 1679046 T^{9} + 2518569 T^{10} - 57927087 T^{11} + 148035889 T^{12} \))
$29$ (\( ( 1 + T + 29 T^{2} )^{2} \))(\( ( 1 + 29 T^{2} )^{2} \))(\( ( 1 - 6 T + 29 T^{2} )^{2} \))(\( ( 1 - 9 T + 29 T^{2} )^{2} \))(\( ( 1 + 9 T + 29 T^{2} )^{2} \))(\( ( 1 - T + 29 T^{2} )^{2} \))(\( ( 1 + 8 T + 29 T^{2} )^{2} \))(\( ( 1 + T + 29 T^{2} )^{2} \))(\( ( 1 + 5 T + 29 T^{2} )^{2} \))(\( ( 1 - 6 T + 29 T^{2} )^{2} \))(\( ( 1 - 5 T + 29 T^{2} )^{2} \))(\( ( 1 - T + 29 T^{2} )^{2} \))(\( ( 1 - 8 T + 29 T^{2} )^{2} \))(\( ( 1 + 3 T + 29 T^{2} )^{2} \))(\( ( 1 - 5 T + 29 T^{2} )^{2} \))(\( ( 1 + 29 T^{2} )^{2} \))(\( ( 1 + 50 T^{2} + 841 T^{4} )^{2} \))(\( ( 1 + 2 T + 29 T^{2} )^{4} \))(\( ( 1 + 2 T + 31 T^{2} + 58 T^{3} + 841 T^{4} )^{2} \))(\( ( 1 - 6 T + 35 T^{2} - 174 T^{3} + 841 T^{4} )^{2} \))(\( ( 1 + 50 T^{2} + 841 T^{4} )^{2} \))(\( ( 1 + 9 T + 39 T^{2} + 174 T^{3} + 1131 T^{4} + 7569 T^{5} + 24389 T^{6} )^{2} \))(\( ( 1 + 9 T + 87 T^{2} + 430 T^{3} + 2523 T^{4} + 7569 T^{5} + 24389 T^{6} )^{2} \))
$31$ (\( ( 1 - 7 T + 31 T^{2} )( 1 + 11 T + 31 T^{2} ) \))(\( 1 + 2 T - 27 T^{2} + 62 T^{3} + 961 T^{4} \))(\( ( 1 - 11 T + 31 T^{2} )( 1 + 7 T + 31 T^{2} ) \))(\( 1 + 2 T - 27 T^{2} + 62 T^{3} + 961 T^{4} \))(\( ( 1 - 11 T + 31 T^{2} )( 1 + 7 T + 31 T^{2} ) \))(\( 1 + 6 T + 5 T^{2} + 186 T^{3} + 961 T^{4} \))(\( 1 + 10 T + 69 T^{2} + 310 T^{3} + 961 T^{4} \))(\( 1 - 2 T - 27 T^{2} - 62 T^{3} + 961 T^{4} \))(\( 1 + 2 T - 27 T^{2} + 62 T^{3} + 961 T^{4} \))(\( ( 1 - 7 T + 31 T^{2} )( 1 + 11 T + 31 T^{2} ) \))(\( 1 + 2 T - 27 T^{2} + 62 T^{3} + 961 T^{4} \))(\( 1 - 2 T - 27 T^{2} - 62 T^{3} + 961 T^{4} \))(\( 1 + 10 T + 69 T^{2} + 310 T^{3} + 961 T^{4} \))(\( 1 + 8 T + 33 T^{2} + 248 T^{3} + 961 T^{4} \))(\( ( 1 - 7 T + 31 T^{2} )( 1 + 11 T + 31 T^{2} ) \))(\( 1 + 2 T - 27 T^{2} + 62 T^{3} + 961 T^{4} \))(\( 1 + 4 T + 22 T^{2} - 272 T^{3} - 1421 T^{4} - 8432 T^{5} + 21142 T^{6} + 119164 T^{7} + 923521 T^{8} \))(\( ( 1 - 11 T + 31 T^{2} )^{2}( 1 + 7 T + 31 T^{2} )^{2} \))(\( ( 1 - 11 T + 31 T^{2} )^{2}( 1 + 7 T + 31 T^{2} )^{2} \))(\( ( 1 - 11 T + 31 T^{2} )^{2}( 1 + 7 T + 31 T^{2} )^{2} \))(\( 1 + 4 T + 22 T^{2} - 272 T^{3} - 1421 T^{4} - 8432 T^{5} + 21142 T^{6} + 119164 T^{7} + 923521 T^{8} \))(\( 1 - 6 T - 9 T^{2} + 90 T^{3} - 450 T^{4} + 2874 T^{5} - 5389 T^{6} + 89094 T^{7} - 432450 T^{8} + 2681190 T^{9} - 8311689 T^{10} - 171774906 T^{11} + 887503681 T^{12} \))(\( 1 + 6 T + 39 T^{2} + 262 T^{3} + 942 T^{4} + 7302 T^{5} + 44307 T^{6} + 226362 T^{7} + 905262 T^{8} + 7805242 T^{9} + 36017319 T^{10} + 171774906 T^{11} + 887503681 T^{12} \))
$37$ (\( 1 - 2 T - 33 T^{2} - 74 T^{3} + 1369 T^{4} \))(\( 1 + 8 T + 27 T^{2} + 296 T^{3} + 1369 T^{4} \))(\( 1 + 2 T - 33 T^{2} + 74 T^{3} + 1369 T^{4} \))(\( 1 - 7 T + 12 T^{2} - 259 T^{3} + 1369 T^{4} \))(\( 1 + 2 T - 33 T^{2} + 74 T^{3} + 1369 T^{4} \))(\( 1 - 3 T - 28 T^{2} - 111 T^{3} + 1369 T^{4} \))(\( 1 - 8 T + 27 T^{2} - 296 T^{3} + 1369 T^{4} \))(\( 1 - 5 T - 12 T^{2} - 185 T^{3} + 1369 T^{4} \))(\( 1 + 3 T - 28 T^{2} + 111 T^{3} + 1369 T^{4} \))(\( 1 - 2 T - 33 T^{2} - 74 T^{3} + 1369 T^{4} \))(\( 1 + 3 T - 28 T^{2} + 111 T^{3} + 1369 T^{4} \))(\( 1 - 5 T - 12 T^{2} - 185 T^{3} + 1369 T^{4} \))(\( 1 - 8 T + 27 T^{2} - 296 T^{3} + 1369 T^{4} \))(\( 1 + 2 T - 33 T^{2} + 74 T^{3} + 1369 T^{4} \))(\( 1 + 4 T - 21 T^{2} + 148 T^{3} + 1369 T^{4} \))(\( 1 + 8 T + 27 T^{2} + 296 T^{3} + 1369 T^{4} \))(\( 1 + 8 T + 6 T^{2} - 128 T^{3} - 373 T^{4} - 4736 T^{5} + 8214 T^{6} + 405224 T^{7} + 1874161 T^{8} \))(\( 1 - 8 T + 2 T^{2} + 96 T^{3} + 107 T^{4} + 3552 T^{5} + 2738 T^{6} - 405224 T^{7} + 1874161 T^{8} \))(\( 1 + 2 T - 64 T^{2} - 12 T^{3} + 3107 T^{4} - 444 T^{5} - 87616 T^{6} + 101306 T^{7} + 1874161 T^{8} \))(\( 1 - 16 T + 120 T^{2} - 992 T^{3} + 7655 T^{4} - 36704 T^{5} + 164280 T^{6} - 810448 T^{7} + 1874161 T^{8} \))(\( 1 + 8 T + 6 T^{2} - 128 T^{3} - 373 T^{4} - 4736 T^{5} + 8214 T^{6} + 405224 T^{7} + 1874161 T^{8} \))(\( 1 + 21 T + 198 T^{2} + 1539 T^{3} + 12636 T^{4} + 84153 T^{5} + 488288 T^{6} + 3113661 T^{7} + 17298684 T^{8} + 77954967 T^{9} + 371083878 T^{10} + 1456223097 T^{11} + 2565726409 T^{12} \))(\( 1 + 9 T - 18 T^{2} - 113 T^{3} + 1620 T^{4} - 3915 T^{5} - 120216 T^{6} - 144855 T^{7} + 2217780 T^{8} - 5723789 T^{9} - 33734898 T^{10} + 624095613 T^{11} + 2565726409 T^{12} \))
$41$ (\( ( 1 - 2 T + 41 T^{2} )^{2} \))(\( ( 1 + 9 T + 41 T^{2} )^{2} \))(\( ( 1 - 3 T + 41 T^{2} )^{2} \))(\( ( 1 - 6 T + 41 T^{2} )^{2} \))(\( ( 1 - 6 T + 41 T^{2} )^{2} \))(\( ( 1 - 6 T + 41 T^{2} )^{2} \))(\( ( 1 - 7 T + 41 T^{2} )^{2} \))(\( ( 1 + 10 T + 41 T^{2} )^{2} \))(\( ( 1 - 6 T + 41 T^{2} )^{2} \))(\( ( 1 + 5 T + 41 T^{2} )^{2} \))(\( ( 1 + 6 T + 41 T^{2} )^{2} \))(\( ( 1 - 10 T + 41 T^{2} )^{2} \))(\( ( 1 + 7 T + 41 T^{2} )^{2} \))(\( ( 1 - 6 T + 41 T^{2} )^{2} \))(\( ( 1 + 4 T + 41 T^{2} )^{2} \))(\( ( 1 - 9 T + 41 T^{2} )^{2} \))(\( ( 1 + 2 T + 51 T^{2} + 82 T^{3} + 1681 T^{4} )^{2} \))(\( ( 1 + 9 T + 41 T^{2} )^{4} \))(\( ( 1 + 6 T + 28 T^{2} + 246 T^{3} + 1681 T^{4} )^{2} \))(\( ( 1 - 8 T + 96 T^{2} - 328 T^{3} + 1681 T^{4} )^{2} \))(\( ( 1 - 2 T + 51 T^{2} - 82 T^{3} + 1681 T^{4} )^{2} \))(\( ( 1 + 12 T + 96 T^{2} + 678 T^{3} + 3936 T^{4} + 20172 T^{5} + 68921 T^{6} )^{2} \))(\( ( 1 + 12 T + 144 T^{2} + 902 T^{3} + 5904 T^{4} + 20172 T^{5} + 68921 T^{6} )^{2} \))
$43$ (\( ( 1 - 4 T + 43 T^{2} )^{2} \))(\( ( 1 + 4 T + 43 T^{2} )^{2} \))(\( ( 1 - 2 T + 43 T^{2} )^{2} \))(\( ( 1 - 11 T + 43 T^{2} )^{2} \))(\( ( 1 + 4 T + 43 T^{2} )^{2} \))(\( ( 1 - T + 43 T^{2} )^{2} \))(\( ( 1 - 4 T + 43 T^{2} )^{2} \))(\( ( 1 - T + 43 T^{2} )^{2} \))(\( ( 1 - 11 T + 43 T^{2} )^{2} \))(\( ( 1 + 10 T + 43 T^{2} )^{2} \))(\( ( 1 - 11 T + 43 T^{2} )^{2} \))(\( ( 1 - T + 43 T^{2} )^{2} \))(\( ( 1 - 4 T + 43 T^{2} )^{2} \))(\( ( 1 + 4 T + 43 T^{2} )^{2} \))(\( ( 1 + 8 T + 43 T^{2} )^{2} \))(\( ( 1 + 4 T + 43 T^{2} )^{2} \))(\( ( 1 + 78 T^{2} + 1849 T^{4} )^{2} \))(\( ( 1 - 8 T + 74 T^{2} - 344 T^{3} + 1849 T^{4} )^{2} \))(\( ( 1 + 4 T + 43 T^{2} )^{4} \))(\( ( 1 + 54 T^{2} + 1849 T^{4} )^{2} \))(\( ( 1 + 78 T^{2} + 1849 T^{4} )^{2} \))(\( ( 1 - 3 T - 3 T^{2} + 146 T^{3} - 129 T^{4} - 5547 T^{5} + 79507 T^{6} )^{2} \))(\( ( 1 - 9 T + 117 T^{2} - 610 T^{3} + 5031 T^{4} - 16641 T^{5} + 79507 T^{6} )^{2} \))
$47$ (\( 1 - 2 T - 43 T^{2} - 94 T^{3} + 2209 T^{4} \))(\( 1 + 3 T - 38 T^{2} + 141 T^{3} + 2209 T^{4} \))(\( 1 + 9 T + 34 T^{2} + 423 T^{3} + 2209 T^{4} \))(\( 1 + 3 T - 38 T^{2} + 141 T^{3} + 2209 T^{4} \))(\( 1 + 6 T - 11 T^{2} + 282 T^{3} + 2209 T^{4} \))(\( 1 + T - 46 T^{2} + 47 T^{3} + 2209 T^{4} \))(\( 1 - T - 46 T^{2} - 47 T^{3} + 2209 T^{4} \))(\( 1 + 7 T + 2 T^{2} + 329 T^{3} + 2209 T^{4} \))(\( 1 - 7 T + 2 T^{2} - 329 T^{3} + 2209 T^{4} \))(\( 1 - 9 T + 34 T^{2} - 423 T^{3} + 2209 T^{4} \))(\( 1 + 7 T + 2 T^{2} + 329 T^{3} + 2209 T^{4} \))(\( 1 - 7 T + 2 T^{2} - 329 T^{3} + 2209 T^{4} \))(\( 1 + T - 46 T^{2} + 47 T^{3} + 2209 T^{4} \))(\( 1 - 6 T - 11 T^{2} - 282 T^{3} + 2209 T^{4} \))(\( 1 - 2 T - 43 T^{2} - 94 T^{3} + 2209 T^{4} \))(\( 1 - 3 T - 38 T^{2} - 141 T^{3} + 2209 T^{4} \))(\( 1 - 10 T - T^{2} - 70 T^{3} + 3292 T^{4} - 3290 T^{5} - 2209 T^{6} - 1038230 T^{7} + 4879681 T^{8} \))(\( 1 - 8 T + 17 T^{2} + 376 T^{3} - 3008 T^{4} + 17672 T^{5} + 37553 T^{6} - 830584 T^{7} + 4879681 T^{8} \))(\( 1 - 16 T + 126 T^{2} - 576 T^{3} + 2659 T^{4} - 27072 T^{5} + 278334 T^{6} - 1661168 T^{7} + 4879681 T^{8} \))(\( 1 + 4 T - 10 T^{2} - 272 T^{3} - 2285 T^{4} - 12784 T^{5} - 22090 T^{6} + 415292 T^{7} + 4879681 T^{8} \))(\( 1 + 10 T - T^{2} + 70 T^{3} + 3292 T^{4} + 3290 T^{5} - 2209 T^{6} + 1038230 T^{7} + 4879681 T^{8} \))(\( 1 - 3 T - 105 T^{2} + 204 T^{3} + 6819 T^{4} - 5601 T^{5} - 348518 T^{6} - 263247 T^{7} + 15063171 T^{8} + 21179892 T^{9} - 512366505 T^{10} - 688035021 T^{11} + 10779215329 T^{12} \))(\( 1 + 15 T + 33 T^{2} + 98 T^{3} + 6975 T^{4} + 29487 T^{5} - 87106 T^{6} + 1385889 T^{7} + 15407775 T^{8} + 10174654 T^{9} + 161029473 T^{10} + 3440175105 T^{11} + 10779215329 T^{12} \))
$53$ (\( 1 + 12 T + 91 T^{2} + 636 T^{3} + 2809 T^{4} \))(\( 1 + 6 T - 17 T^{2} + 318 T^{3} + 2809 T^{4} \))(\( 1 - 6 T - 17 T^{2} - 318 T^{3} + 2809 T^{4} \))(\( 1 - 53 T^{2} + 2809 T^{4} \))(\( 1 - 53 T^{2} + 2809 T^{4} \))(\( 1 - 53 T^{2} + 2809 T^{4} \))(\( 1 + 6 T - 17 T^{2} + 318 T^{3} + 2809 T^{4} \))(\( 1 + 12 T + 91 T^{2} + 636 T^{3} + 2809 T^{4} \))(\( 1 - 4 T - 37 T^{2} - 212 T^{3} + 2809 T^{4} \))(\( 1 - 2 T - 49 T^{2} - 106 T^{3} + 2809 T^{4} \))(\( 1 + 4 T - 37 T^{2} + 212 T^{3} + 2809 T^{4} \))(\( 1 - 12 T + 91 T^{2} - 636 T^{3} + 2809 T^{4} \))(\( 1 - 6 T - 17 T^{2} - 318 T^{3} + 2809 T^{4} \))(\( 1 + 12 T + 91 T^{2} + 636 T^{3} + 2809 T^{4} \))(\( 1 + 6 T - 17 T^{2} + 318 T^{3} + 2809 T^{4} \))(\( 1 - 6 T - 17 T^{2} - 318 T^{3} + 2809 T^{4} \))(\( 1 - 16 T + 94 T^{2} - 896 T^{3} + 9867 T^{4} - 47488 T^{5} + 264046 T^{6} - 2382032 T^{7} + 7890481 T^{8} \))(\( ( 1 - 4 T - 37 T^{2} - 212 T^{3} + 2809 T^{4} )^{2} \))(\( 1 + 2 T - 96 T^{2} - 12 T^{3} + 6979 T^{4} - 636 T^{5} - 269664 T^{6} + 297754 T^{7} + 7890481 T^{8} \))(\( 1 + 4 T + 4 T^{2} - 376 T^{3} - 3513 T^{4} - 19928 T^{5} + 11236 T^{6} + 595508 T^{7} + 7890481 T^{8} \))(\( 1 + 16 T + 94 T^{2} + 896 T^{3} + 9867 T^{4} + 47488 T^{5} + 264046 T^{6} + 2382032 T^{7} + 7890481 T^{8} \))(\( ( 1 - 8 T + 11 T^{2} - 424 T^{3} + 2809 T^{4} )^{3} \))(\( 1 - 123 T^{2} - 64 T^{3} + 8610 T^{4} + 3936 T^{5} - 503059 T^{6} + 208608 T^{7} + 24185490 T^{8} - 9528128 T^{9} - 970529163 T^{10} + 22164361129 T^{12} \))
$59$ (\( 1 - 9 T + 22 T^{2} - 531 T^{3} + 3481 T^{4} \))(\( 1 + 6 T - 23 T^{2} + 354 T^{3} + 3481 T^{4} \))(\( 1 + 12 T + 85 T^{2} + 708 T^{3} + 3481 T^{4} \))(\( 1 - 9 T + 22 T^{2} - 531 T^{3} + 3481 T^{4} \))(\( 1 + 3 T - 50 T^{2} + 177 T^{3} + 3481 T^{4} \))(\( 1 - 7 T - 10 T^{2} - 413 T^{3} + 3481 T^{4} \))(\( 1 + 6 T - 23 T^{2} + 354 T^{3} + 3481 T^{4} \))(\( 1 - 3 T - 50 T^{2} - 177 T^{3} + 3481 T^{4} \))(\( 1 + 11 T + 62 T^{2} + 649 T^{3} + 3481 T^{4} \))(\( 1 + 12 T + 85 T^{2} + 708 T^{3} + 3481 T^{4} \))(\( 1 - 11 T + 62 T^{2} - 649 T^{3} + 3481 T^{4} \))(\( 1 + 3 T - 50 T^{2} + 177 T^{3} + 3481 T^{4} \))(\( 1 - 6 T - 23 T^{2} - 354 T^{3} + 3481 T^{4} \))(\( 1 + 3 T - 50 T^{2} + 177 T^{3} + 3481 T^{4} \))(\( 1 - 3 T - 50 T^{2} - 177 T^{3} + 3481 T^{4} \))(\( 1 - 6 T - 23 T^{2} - 354 T^{3} + 3481 T^{4} \))(\( 1 + 4 T - 74 T^{2} - 112 T^{3} + 3675 T^{4} - 6608 T^{5} - 257594 T^{6} + 821516 T^{7} + 12117361 T^{8} \))(\( 1 + 8 T + 42 T^{2} - 768 T^{3} - 6773 T^{4} - 45312 T^{5} + 146202 T^{6} + 1643032 T^{7} + 12117361 T^{8} \))(\( 1 - 4 T - 99 T^{2} + 12 T^{3} + 8800 T^{4} + 708 T^{5} - 344619 T^{6} - 821516 T^{7} + 12117361 T^{8} \))(\( 1 + 14 T + 31 T^{2} + 658 T^{3} + 12180 T^{4} + 38822 T^{5} + 107911 T^{6} + 2875306 T^{7} + 12117361 T^{8} \))(\( 1 - 4 T - 74 T^{2} + 112 T^{3} + 3675 T^{4} + 6608 T^{5} - 257594 T^{6} - 821516 T^{7} + 12117361 T^{8} \))(\( 1 + 9 T - 48 T^{2} - 867 T^{3} - 96 T^{4} + 25137 T^{5} + 175174 T^{6} + 1483083 T^{7} - 334176 T^{8} - 178063593 T^{9} - 581633328 T^{10} + 6434318691 T^{11} + 42180533641 T^{12} \))(\( 1 - 9 T - 141 T^{3} - 864 T^{4} + 30879 T^{5} - 171866 T^{6} + 1821861 T^{7} - 3007584 T^{8} - 28958439 T^{9} - 6434318691 T^{11} + 42180533641 T^{12} \))
$61$ (\( 1 - 5 T - 36 T^{2} - 305 T^{3} + 3721 T^{4} \))(\( 1 + 5 T - 36 T^{2} + 305 T^{3} + 3721 T^{4} \))(\( 1 + 5 T - 36 T^{2} + 305 T^{3} + 3721 T^{4} \))(\( 1 - 10 T + 39 T^{2} - 610 T^{3} + 3721 T^{4} \))(\( 1 + 11 T + 60 T^{2} + 671 T^{3} + 3721 T^{4} \))(\( 1 + 6 T - 25 T^{2} + 366 T^{3} + 3721 T^{4} \))(\( ( 1 - 13 T + 61 T^{2} )( 1 + 14 T + 61 T^{2} ) \))(\( ( 1 - 13 T + 61 T^{2} )( 1 - T + 61 T^{2} ) \))(\( 1 + 10 T + 39 T^{2} + 610 T^{3} + 3721 T^{4} \))(\( 1 - 5 T - 36 T^{2} - 305 T^{3} + 3721 T^{4} \))(\( 1 + 10 T + 39 T^{2} + 610 T^{3} + 3721 T^{4} \))(\( ( 1 - 13 T + 61 T^{2} )( 1 - T + 61 T^{2} ) \))(\( ( 1 - 13 T + 61 T^{2} )( 1 + 14 T + 61 T^{2} ) \))(\( 1 - 7 T - 12 T^{2} - 427 T^{3} + 3721 T^{4} \))(\( ( 1 - 13 T + 61 T^{2} )( 1 + 14 T + 61 T^{2} ) \))(\( 1 + 5 T - 36 T^{2} + 305 T^{3} + 3721 T^{4} \))(\( 1 - 6 T - 93 T^{2} - 42 T^{3} + 10724 T^{4} - 2562 T^{5} - 346053 T^{6} - 1361886 T^{7} + 13845841 T^{8} \))(\( 1 + 8 T - 11 T^{2} - 376 T^{3} - 2000 T^{4} - 22936 T^{5} - 40931 T^{6} + 1815848 T^{7} + 13845841 T^{8} \))(\( 1 + 18 T + 149 T^{2} + 954 T^{3} + 7140 T^{4} + 58194 T^{5} + 554429 T^{6} + 4085658 T^{7} + 13845841 T^{8} \))(\( 1 + 18 T + 129 T^{2} + 1314 T^{3} + 14540 T^{4} + 80154 T^{5} + 480009 T^{6} + 4085658 T^{7} + 13845841 T^{8} \))(\( 1 - 6 T - 93 T^{2} - 42 T^{3} + 10724 T^{4} - 2562 T^{5} - 346053 T^{6} - 1361886 T^{7} + 13845841 T^{8} \))(\( 1 - 6 T - 84 T^{2} + 940 T^{3} + 2100 T^{4} - 34086 T^{5} + 130506 T^{6} - 2079246 T^{7} + 7814100 T^{8} + 213362140 T^{9} - 1163050644 T^{10} - 5067577806 T^{11} + 51520374361 T^{12} \))(\( 1 - 6 T - 18 T^{2} + 688 T^{3} - 3414 T^{4} - 10122 T^{5} + 398166 T^{6} - 617442 T^{7} - 12703494 T^{8} + 156162928 T^{9} - 249225138 T^{10} - 5067577806 T^{11} + 51520374361 T^{12} \))
$67$ (\( 1 - 9 T + 14 T^{2} - 603 T^{3} + 4489 T^{4} \))(\( ( 1 - 5 T + 67 T^{2} )( 1 + 16 T + 67 T^{2} ) \))(\( ( 1 - 11 T + 67 T^{2} )( 1 + 16 T + 67 T^{2} ) \))(\( 1 - 4 T - 51 T^{2} - 268 T^{3} + 4489 T^{4} \))(\( ( 1 - 5 T + 67 T^{2} )( 1 + 16 T + 67 T^{2} ) \))(\( 1 - 4 T - 51 T^{2} - 268 T^{3} + 4489 T^{4} \))(\( 1 + 3 T - 58 T^{2} + 201 T^{3} + 4489 T^{4} \))(\( 1 + 12 T + 77 T^{2} + 804 T^{3} + 4489 T^{4} \))(\( 1 - 4 T - 51 T^{2} - 268 T^{3} + 4489 T^{4} \))(\( ( 1 - 11 T + 67 T^{2} )( 1 + 16 T + 67 T^{2} ) \))(\( 1 - 4 T - 51 T^{2} - 268 T^{3} + 4489 T^{4} \))(\( 1 + 12 T + 77 T^{2} + 804 T^{3} + 4489 T^{4} \))(\( 1 + 3 T - 58 T^{2} + 201 T^{3} + 4489 T^{4} \))(\( 1 - 13 T + 102 T^{2} - 871 T^{3} + 4489 T^{4} \))(\( 1 + 9 T + 14 T^{2} + 603 T^{3} + 4489 T^{4} \))(\( ( 1 - 5 T + 67 T^{2} )( 1 + 16 T + 67 T^{2} ) \))(\( 1 - 10 T + 13 T^{2} + 470 T^{3} - 3620 T^{4} + 31490 T^{5} + 58357 T^{6} - 3007630 T^{7} + 20151121 T^{8} \))(\( 1 - 6 T + 5 T^{2} + 618 T^{3} - 5940 T^{4} + 41406 T^{5} + 22445 T^{6} - 1804578 T^{7} + 20151121 T^{8} \))(\( 1 - 8 T - 23 T^{2} + 376 T^{3} - 1208 T^{4} + 25192 T^{5} - 103247 T^{6} - 2406104 T^{7} + 20151121 T^{8} \))(\( 1 + 14 T + 31 T^{2} + 434 T^{3} + 9604 T^{4} + 29078 T^{5} + 139159 T^{6} + 4210682 T^{7} + 20151121 T^{8} \))(\( 1 - 10 T + 13 T^{2} + 470 T^{3} - 3620 T^{4} + 31490 T^{5} + 58357 T^{6} - 3007630 T^{7} + 20151121 T^{8} \))(\( 1 + 6 T - 102 T^{2} - 356 T^{3} + 6246 T^{4} - 2322 T^{5} - 485562 T^{6} - 155574 T^{7} + 28038294 T^{8} - 107071628 T^{9} - 2055414342 T^{10} + 8100750642 T^{11} + 90458382169 T^{12} \))(\( 1 + 6 T - 150 T^{2} - 324 T^{3} + 17814 T^{4} + 11310 T^{5} - 1359610 T^{6} + 757770 T^{7} + 79967046 T^{8} - 97447212 T^{9} - 3022668150 T^{10} + 8100750642 T^{11} + 90458382169 T^{12} \))
$71$ (\( ( 1 + 4 T + 71 T^{2} )^{2} \))(\( ( 1 + 71 T^{2} )^{2} \))(\( ( 1 + 12 T + 71 T^{2} )^{2} \))(\( ( 1 + 3 T + 71 T^{2} )^{2} \))(\( ( 1 + 71 T^{2} )^{2} \))(\( ( 1 - 15 T + 71 T^{2} )^{2} \))(\( ( 1 + 8 T + 71 T^{2} )^{2} \))(\( ( 1 - 5 T + 71 T^{2} )^{2} \))(\( ( 1 + 5 T + 71 T^{2} )^{2} \))(\( ( 1 + 4 T + 71 T^{2} )^{2} \))(\( ( 1 - 5 T + 71 T^{2} )^{2} \))(\( ( 1 + 5 T + 71 T^{2} )^{2} \))(\( ( 1 - 8 T + 71 T^{2} )^{2} \))(\( ( 1 - 12 T + 71 T^{2} )^{2} \))(\( ( 1 - 2 T + 71 T^{2} )^{2} \))(\( ( 1 + 71 T^{2} )^{2} \))(\( ( 1 - 4 T + 18 T^{2} - 284 T^{3} + 5041 T^{4} )^{2} \))(\( ( 1 - 16 T + 178 T^{2} - 1136 T^{3} + 5041 T^{4} )^{2} \))(\( ( 1 + 14 T + 184 T^{2} + 994 T^{3} + 5041 T^{4} )^{2} \))(\( ( 1 - 8 T + 108 T^{2} - 568 T^{3} + 5041 T^{4} )^{2} \))(\( ( 1 + 4 T + 18 T^{2} + 284 T^{3} + 5041 T^{4} )^{2} \))(\( ( 1 + 9 T + 105 T^{2} + 1170 T^{3} + 7455 T^{4} + 45369 T^{5} + 357911 T^{6} )^{2} \))(\( ( 1 + 3 T + 189 T^{2} + 362 T^{3} + 13419 T^{4} + 15123 T^{5} + 357911 T^{6} )^{2} \))
$73$ (\( 1 - 2 T - 69 T^{2} - 146 T^{3} + 5329 T^{4} \))(\( 1 + 2 T - 69 T^{2} + 146 T^{3} + 5329 T^{4} \))(\( 1 - 16 T + 183 T^{2} - 1168 T^{3} + 5329 T^{4} \))(\( 1 - 4 T - 57 T^{2} - 292 T^{3} + 5329 T^{4} \))(\( 1 + 2 T - 69 T^{2} + 146 T^{3} + 5329 T^{4} \))(\( 1 + 12 T + 71 T^{2} + 876 T^{3} + 5329 T^{4} \))(\( ( 1 - 7 T + 73 T^{2} )( 1 + 17 T + 73 T^{2} ) \))(\( 1 - 8 T - 9 T^{2} - 584 T^{3} + 5329 T^{4} \))(\( 1 - 8 T - 9 T^{2} - 584 T^{3} + 5329 T^{4} \))(\( 1 + 12 T + 71 T^{2} + 876 T^{3} + 5329 T^{4} \))(\( 1 - 8 T - 9 T^{2} - 584 T^{3} + 5329 T^{4} \))(\( 1 - 8 T - 9 T^{2} - 584 T^{3} + 5329 T^{4} \))(\( ( 1 - 7 T + 73 T^{2} )( 1 + 17 T + 73 T^{2} ) \))(\( ( 1 - 17 T + 73 T^{2} )( 1 + 7 T + 73 T^{2} ) \))(\( 1 + 4 T - 57 T^{2} + 292 T^{3} + 5329 T^{4} \))(\( 1 + 2 T - 69 T^{2} + 146 T^{3} + 5329 T^{4} \))(\( 1 - 4 T - 126 T^{2} + 16 T^{3} + 13667 T^{4} + 1168 T^{5} - 671454 T^{6} - 1556068 T^{7} + 28398241 T^{8} \))(\( 1 - 20 T + 182 T^{2} - 1440 T^{3} + 12995 T^{4} - 105120 T^{5} + 969878 T^{6} - 7780340 T^{7} + 28398241 T^{8} \))(\( 1 + 6 T - 112 T^{2} + 12 T^{3} + 13947 T^{4} + 876 T^{5} - 596848 T^{6} + 2334102 T^{7} + 28398241 T^{8} \))(\( 1 - 16 T + 48 T^{2} - 992 T^{3} + 19247 T^{4} - 72416 T^{5} + 255792 T^{6} - 6224272 T^{7} + 28398241 T^{8} \))(\( 1 - 4 T - 126 T^{2} + 16 T^{3} + 13667 T^{4} + 1168 T^{5} - 671454 T^{6} - 1556068 T^{7} + 28398241 T^{8} \))(\( 1 - 99 T^{2} - 960 T^{3} + 2574 T^{4} + 47520 T^{5} + 292961 T^{6} + 3468960 T^{7} + 13716846 T^{8} - 373456320 T^{9} - 2811425859 T^{10} + 151334226289 T^{12} \))(\( 1 - 123 T^{2} - 384 T^{3} + 6150 T^{4} + 23616 T^{5} - 289519 T^{6} + 1723968 T^{7} + 32773350 T^{8} - 149382528 T^{9} - 3492983643 T^{10} + 151334226289 T^{12} \))
$79$ (\( 1 - 15 T + 146 T^{2} - 1185 T^{3} + 6241 T^{4} \))(\( ( 1 - 17 T + 79 T^{2} )( 1 + 4 T + 79 T^{2} ) \))(\( ( 1 + 4 T + 79 T^{2} )( 1 + 13 T + 79 T^{2} ) \))(\( 1 - 16 T + 177 T^{2} - 1264 T^{3} + 6241 T^{4} \))(\( 1 + 5 T - 54 T^{2} + 395 T^{3} + 6241 T^{4} \))(\( 1 - 79 T^{2} + 6241 T^{4} \))(\( 1 - 9 T + 2 T^{2} - 711 T^{3} + 6241 T^{4} \))(\( 1 - 79 T^{2} + 6241 T^{4} \))(\( 1 - 8 T - 15 T^{2} - 632 T^{3} + 6241 T^{4} \))(\( 1 - T - 78 T^{2} - 79 T^{3} + 6241 T^{4} \))(\( 1 - 8 T - 15 T^{2} - 632 T^{3} + 6241 T^{4} \))(\( 1 - 79 T^{2} + 6241 T^{4} \))(\( 1 - 9 T + 2 T^{2} - 711 T^{3} + 6241 T^{4} \))(\( 1 - T - 78 T^{2} - 79 T^{3} + 6241 T^{4} \))(\( 1 + 9 T + 2 T^{2} + 711 T^{3} + 6241 T^{4} \))(\( ( 1 - 17 T + 79 T^{2} )( 1 + 4 T + 79 T^{2} ) \))(\( 1 + 18 T + 103 T^{2} + 1134 T^{3} + 17004 T^{4} + 89586 T^{5} + 642823 T^{6} + 8874702 T^{7} + 38950081 T^{8} \))(\( 1 + 16 T + 41 T^{2} + 912 T^{3} + 19760 T^{4} + 72048 T^{5} + 255881 T^{6} + 7888624 T^{7} + 38950081 T^{8} \))(\( 1 - 151 T^{2} + 16560 T^{4} - 942391 T^{6} + 38950081 T^{8} \))(\( 1 - 18 T + 103 T^{2} - 1134 T^{3} + 17004 T^{4} - 89586 T^{5} + 642823 T^{6} - 8874702 T^{7} + 38950081 T^{8} \))(\( 1 + 18 T + 103 T^{2} + 1134 T^{3} + 17004 T^{4} + 89586 T^{5} + 642823 T^{6} + 8874702 T^{7} + 38950081 T^{8} \))(\( 1 + 12 T - 126 T^{2} - 520 T^{3} + 29010 T^{4} + 70572 T^{5} - 2137326 T^{6} + 5575188 T^{7} + 181051410 T^{8} - 256380280 T^{9} - 4907710206 T^{10} + 36924676788 T^{11} + 243087455521 T^{12} \))(\( 1 + 12 T - 60 T^{2} - 608 T^{3} + 7164 T^{4} - 180 T^{5} - 865506 T^{6} - 14220 T^{7} + 44710524 T^{8} - 299767712 T^{9} - 2337004860 T^{10} + 36924676788 T^{11} + 243087455521 T^{12} \))
$83$ (\( ( 1 - 6 T + 83 T^{2} )^{2} \))(\( ( 1 - 9 T + 83 T^{2} )^{2} \))(\( ( 1 + 9 T + 83 T^{2} )^{2} \))(\( ( 1 + 83 T^{2} )^{2} \))(\( ( 1 - 6 T + 83 T^{2} )^{2} \))(\( ( 1 - 16 T + 83 T^{2} )^{2} \))(\( ( 1 + 15 T + 83 T^{2} )^{2} \))(\( ( 1 + 6 T + 83 T^{2} )^{2} \))(\( ( 1 - 14 T + 83 T^{2} )^{2} \))(\( ( 1 + T + 83 T^{2} )^{2} \))(\( ( 1 + 14 T + 83 T^{2} )^{2} \))(\( ( 1 - 6 T + 83 T^{2} )^{2} \))(\( ( 1 - 15 T + 83 T^{2} )^{2} \))(\( ( 1 + 6 T + 83 T^{2} )^{2} \))(\( ( 1 + 6 T + 83 T^{2} )^{2} \))(\( ( 1 + 9 T + 83 T^{2} )^{2} \))(\( ( 1 + 14 T + 207 T^{2} + 1162 T^{3} + 6889 T^{4} )^{2} \))(\( ( 1 + 10 T + 79 T^{2} + 830 T^{3} + 6889 T^{4} )^{2} \))(\( ( 1 + 16 T + 202 T^{2} + 1328 T^{3} + 6889 T^{4} )^{2} \))(\( ( 1 + 4 T - 30 T^{2} + 332 T^{3} + 6889 T^{4} )^{2} \))(\( ( 1 - 14 T + 207 T^{2} - 1162 T^{3} + 6889 T^{4} )^{2} \))(\( ( 1 + 18 T + 282 T^{2} + 2824 T^{3} + 23406 T^{4} + 124002 T^{5} + 571787 T^{6} )^{2} \))(\( ( 1 - 6 T + 138 T^{2} - 1184 T^{3} + 11454 T^{4} - 41334 T^{5} + 571787 T^{6} )^{2} \))
$89$ (\( 1 - 6 T - 53 T^{2} - 534 T^{3} + 7921 T^{4} \))(\( 1 + 12 T + 55 T^{2} + 1068 T^{3} + 7921 T^{4} \))(\( 1 + 6 T - 53 T^{2} + 534 T^{3} + 7921 T^{4} \))(\( 1 - 89 T^{2} + 7921 T^{4} \))(\( 1 + 18 T + 235 T^{2} + 1602 T^{3} + 7921 T^{4} \))(\( 1 + 8 T - 25 T^{2} + 712 T^{3} + 7921 T^{4} \))(\( 1 - 12 T + 55 T^{2} - 1068 T^{3} + 7921 T^{4} \))(\( 1 - 6 T - 53 T^{2} - 534 T^{3} + 7921 T^{4} \))(\( 1 + 2 T - 85 T^{2} + 178 T^{3} + 7921 T^{4} \))(\( 1 - 6 T - 53 T^{2} - 534 T^{3} + 7921 T^{4} \))(\( 1 - 2 T - 85 T^{2} - 178 T^{3} + 7921 T^{4} \))(\( 1 + 6 T - 53 T^{2} + 534 T^{3} + 7921 T^{4} \))(\( 1 + 12 T + 55 T^{2} + 1068 T^{3} + 7921 T^{4} \))(\( 1 - 6 T - 53 T^{2} - 534 T^{3} + 7921 T^{4} \))(\( 1 - 6 T - 53 T^{2} - 534 T^{3} + 7921 T^{4} \))(\( 1 - 12 T + 55 T^{2} - 1068 T^{3} + 7921 T^{4} \))(\( 1 + 8 T - 58 T^{2} - 448 T^{3} + 1267 T^{4} - 39872 T^{5} - 459418 T^{6} + 5639752 T^{7} + 62742241 T^{8} \))(\( 1 - 16 T + 42 T^{2} - 576 T^{3} + 15091 T^{4} - 51264 T^{5} + 332682 T^{6} - 11279504 T^{7} + 62742241 T^{8} \))(\( 1 + 8 T - 18 T^{2} - 768 T^{3} - 6893 T^{4} - 68352 T^{5} - 142578 T^{6} + 5639752 T^{7} + 62742241 T^{8} \))(\( 1 - 8 T - 58 T^{2} + 448 T^{3} + 1267 T^{4} + 39872 T^{5} - 459418 T^{6} - 5639752 T^{7} + 62742241 T^{8} \))(\( 1 - 8 T - 58 T^{2} + 448 T^{3} + 1267 T^{4} + 39872 T^{5} - 459418 T^{6} - 5639752 T^{7} + 62742241 T^{8} \))(\( 1 + 12 T - 111 T^{2} - 1180 T^{3} + 16890 T^{4} + 90972 T^{5} - 1023511 T^{6} + 8096508 T^{7} + 133785690 T^{8} - 831863420 T^{9} - 6964388751 T^{10} + 67008713388 T^{11} + 496981290961 T^{12} \))(\( 1 + 36 T + 633 T^{2} + 8396 T^{3} + 102066 T^{4} + 1165620 T^{5} + 11868377 T^{6} + 103740180 T^{7} + 808464786 T^{8} + 5918919724 T^{9} + 39715838553 T^{10} + 201026140164 T^{11} + 496981290961 T^{12} \))
$97$ (\( ( 1 + 5 T + 97 T^{2} )^{2} \))(\( ( 1 - 5 T + 97 T^{2} )^{2} \))(\( ( 1 - 17 T + 97 T^{2} )^{2} \))(\( ( 1 + T + 97 T^{2} )^{2} \))(\( ( 1 + 13 T + 97 T^{2} )^{2} \))(\( ( 1 - 7 T + 97 T^{2} )^{2} \))(\( ( 1 - 13 T + 97 T^{2} )^{2} \))(\( ( 1 - 7 T + 97 T^{2} )^{2} \))(\( ( 1 - 15 T + 97 T^{2} )^{2} \))(\( ( 1 - 9 T + 97 T^{2} )^{2} \))(\( ( 1 - 15 T + 97 T^{2} )^{2} \))(\( ( 1 - 7 T + 97 T^{2} )^{2} \))(\( ( 1 - 13 T + 97 T^{2} )^{2} \))(\( ( 1 + 13 T + 97 T^{2} )^{2} \))(\( ( 1 - 7 T + 97 T^{2} )^{2} \))(\( ( 1 - 5 T + 97 T^{2} )^{2} \))(\( ( 1 + 26 T + 355 T^{2} + 2522 T^{3} + 9409 T^{4} )^{2} \))(\( ( 1 + 2 T + 83 T^{2} + 194 T^{3} + 9409 T^{4} )^{2} \))(\( ( 1 + 22 T + 287 T^{2} + 2134 T^{3} + 9409 T^{4} )^{2} \))(\( ( 1 + 2 T + 187 T^{2} + 194 T^{3} + 9409 T^{4} )^{2} \))(\( ( 1 + 26 T + 355 T^{2} + 2522 T^{3} + 9409 T^{4} )^{2} \))(\( ( 1 - 7 T + 97 T^{2} )^{6} \))(\( ( 1 - 9 T + 210 T^{2} - 1753 T^{3} + 20370 T^{4} - 84681 T^{5} + 912673 T^{6} )^{2} \))
show more
show less