Properties

 Label 2100.2.q Level $2100$ Weight $2$ Character orbit 2100.q Rep. character $\chi_{2100}(1201,\cdot)$ Character field $\Q(\zeta_{3})$ Dimension $50$ Newform subspaces $13$ Sturm bound $960$ Trace bound $13$

Related objects

Defining parameters

 Level: $$N$$ $$=$$ $$2100 = 2^{2} \cdot 3 \cdot 5^{2} \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2100.q (of order $$3$$ and degree $$2$$) Character conductor: $$\operatorname{cond}(\chi)$$ $$=$$ $$7$$ Character field: $$\Q(\zeta_{3})$$ Newform subspaces: $$13$$ Sturm bound: $$960$$ Trace bound: $$13$$ Distinguishing $$T_p$$: $$11$$, $$13$$

Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(2100, [\chi])$$.

Total New Old
Modular forms 1032 50 982
Cusp forms 888 50 838
Eisenstein series 144 0 144

Trace form

 $$50q + q^{3} + q^{7} - 25q^{9} + O(q^{10})$$ $$50q + q^{3} + q^{7} - 25q^{9} + 2q^{11} - 6q^{13} + 3q^{19} - 6q^{21} + 8q^{23} - 2q^{27} + 8q^{29} - 11q^{31} + 6q^{33} - 19q^{37} - q^{39} - 12q^{41} - 18q^{43} + 14q^{47} - 7q^{49} - 12q^{53} + 18q^{57} - 12q^{59} - 4q^{61} + 7q^{63} + 19q^{67} + 32q^{69} - 4q^{71} + 15q^{73} + 28q^{77} - 13q^{79} - 25q^{81} + 52q^{83} - 4q^{87} - 32q^{89} - 57q^{91} - 5q^{93} - 84q^{97} - 4q^{99} + O(q^{100})$$

Decomposition of $$S_{2}^{\mathrm{new}}(2100, [\chi])$$ into newform subspaces

Label Dim. $$A$$ Field CM Traces $q$-expansion
$$a_2$$ $$a_3$$ $$a_5$$ $$a_7$$
2100.2.q.a $$2$$ $$16.769$$ $$\Q(\sqrt{-3})$$ None $$0$$ $$-1$$ $$0$$ $$-5$$ $$q+(-1+\zeta_{6})q^{3}+(-2-\zeta_{6})q^{7}-\zeta_{6}q^{9}+\cdots$$
2100.2.q.b $$2$$ $$16.769$$ $$\Q(\sqrt{-3})$$ None $$0$$ $$-1$$ $$0$$ $$-1$$ $$q+(-1+\zeta_{6})q^{3}+(-2+3\zeta_{6})q^{7}-\zeta_{6}q^{9}+\cdots$$
2100.2.q.c $$2$$ $$16.769$$ $$\Q(\sqrt{-3})$$ None $$0$$ $$-1$$ $$0$$ $$4$$ $$q+(-1+\zeta_{6})q^{3}+(3-2\zeta_{6})q^{7}-\zeta_{6}q^{9}+\cdots$$
2100.2.q.d $$2$$ $$16.769$$ $$\Q(\sqrt{-3})$$ None $$0$$ $$-1$$ $$0$$ $$5$$ $$q+(-1+\zeta_{6})q^{3}+(2+\zeta_{6})q^{7}-\zeta_{6}q^{9}+\cdots$$
2100.2.q.e $$2$$ $$16.769$$ $$\Q(\sqrt{-3})$$ None $$0$$ $$1$$ $$0$$ $$-4$$ $$q+(1-\zeta_{6})q^{3}+(-3+2\zeta_{6})q^{7}-\zeta_{6}q^{9}+\cdots$$
2100.2.q.f $$4$$ $$16.769$$ $$\Q(\sqrt{2}, \sqrt{-3})$$ None $$0$$ $$-2$$ $$0$$ $$-2$$ $$q+(-1-\beta _{1})q^{3}+(-1-\beta _{1}-\beta _{2}+\beta _{3})q^{7}+\cdots$$
2100.2.q.g $$4$$ $$16.769$$ $$\Q(\sqrt{-3}, \sqrt{7})$$ None $$0$$ $$-2$$ $$0$$ $$0$$ $$q+(-1-\beta _{2})q^{3}-\beta _{1}q^{7}+\beta _{2}q^{9}+(3+\cdots)q^{11}+\cdots$$
2100.2.q.h $$4$$ $$16.769$$ $$\Q(\sqrt{-3}, \sqrt{7})$$ None $$0$$ $$2$$ $$0$$ $$0$$ $$q+(1+\beta _{2})q^{3}+(-\beta _{1}-\beta _{3})q^{7}+\beta _{2}q^{9}+\cdots$$
2100.2.q.i $$4$$ $$16.769$$ $$\Q(\sqrt{-3}, \sqrt{7})$$ None $$0$$ $$2$$ $$0$$ $$0$$ $$q+(1+\beta _{2})q^{3}+\beta _{1}q^{7}+\beta _{2}q^{9}+(3+\beta _{1}+\cdots)q^{11}+\cdots$$
2100.2.q.j $$4$$ $$16.769$$ $$\Q(\sqrt{2}, \sqrt{-3})$$ None $$0$$ $$2$$ $$0$$ $$2$$ $$q+(1+\beta _{1})q^{3}+(1+\beta _{1}+\beta _{2}-\beta _{3})q^{7}+\cdots$$
2100.2.q.k $$4$$ $$16.769$$ $$\Q(\sqrt{2}, \sqrt{-3})$$ None $$0$$ $$2$$ $$0$$ $$2$$ $$q+(1+\beta _{1})q^{3}+(-\beta _{1}-\beta _{3})q^{7}+\beta _{1}q^{9}+\cdots$$
2100.2.q.l $$8$$ $$16.769$$ 8.0.$$\cdots$$.3 None $$0$$ $$-4$$ $$0$$ $$-2$$ $$q+\beta _{4}q^{3}+(-\beta _{1}+\beta _{6})q^{7}+(-1-\beta _{4}+\cdots)q^{9}+\cdots$$
2100.2.q.m $$8$$ $$16.769$$ 8.0.$$\cdots$$.3 None $$0$$ $$4$$ $$0$$ $$2$$ $$q-\beta _{4}q^{3}+(\beta _{1}-\beta _{6})q^{7}+(-1-\beta _{4}+\cdots)q^{9}+\cdots$$

Decomposition of $$S_{2}^{\mathrm{old}}(2100, [\chi])$$ into lower level spaces

$$S_{2}^{\mathrm{old}}(2100, [\chi]) \cong$$ $$S_{2}^{\mathrm{new}}(21, [\chi])$$$$^{\oplus 9}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(28, [\chi])$$$$^{\oplus 6}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(35, [\chi])$$$$^{\oplus 12}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(42, [\chi])$$$$^{\oplus 6}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(70, [\chi])$$$$^{\oplus 8}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(84, [\chi])$$$$^{\oplus 3}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(105, [\chi])$$$$^{\oplus 6}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(140, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(175, [\chi])$$$$^{\oplus 6}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(210, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(350, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(420, [\chi])$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(525, [\chi])$$$$^{\oplus 3}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(700, [\chi])$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(1050, [\chi])$$$$^{\oplus 2}$$

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ ($$1 + T + T^{2}$$)($$1 + T + T^{2}$$)($$1 + T + T^{2}$$)($$1 + T + T^{2}$$)($$1 - T + T^{2}$$)($$( 1 + T + T^{2} )^{2}$$)($$( 1 + T + T^{2} )^{2}$$)($$( 1 - T + T^{2} )^{2}$$)($$( 1 - T + T^{2} )^{2}$$)($$( 1 - T + T^{2} )^{2}$$)($$( 1 - T + T^{2} )^{2}$$)($$( 1 + T + T^{2} )^{4}$$)($$( 1 - T + T^{2} )^{4}$$)
$5$ 1
$7$ ($$1 + 5 T + 7 T^{2}$$)($$1 + T + 7 T^{2}$$)($$1 - 4 T + 7 T^{2}$$)($$1 - 5 T + 7 T^{2}$$)($$1 + 4 T + 7 T^{2}$$)($$1 + 2 T - 3 T^{2} + 14 T^{3} + 49 T^{4}$$)($$1 + 7 T^{2} + 49 T^{4}$$)($$1 + 7 T^{2} + 49 T^{4}$$)($$1 + 7 T^{2} + 49 T^{4}$$)($$1 - 2 T - 3 T^{2} - 14 T^{3} + 49 T^{4}$$)($$1 - 2 T - 3 T^{2} - 14 T^{3} + 49 T^{4}$$)($$1 + 2 T + 4 T^{2} - 22 T^{3} - 83 T^{4} - 154 T^{5} + 196 T^{6} + 686 T^{7} + 2401 T^{8}$$)($$1 - 2 T + 4 T^{2} + 22 T^{3} - 83 T^{4} + 154 T^{5} + 196 T^{6} - 686 T^{7} + 2401 T^{8}$$)
$11$ ($$1 - 2 T - 7 T^{2} - 22 T^{3} + 121 T^{4}$$)($$1 + 2 T - 7 T^{2} + 22 T^{3} + 121 T^{4}$$)($$1 + 2 T - 7 T^{2} + 22 T^{3} + 121 T^{4}$$)($$1 - 4 T + 5 T^{2} - 44 T^{3} + 121 T^{4}$$)($$1 + 2 T - 7 T^{2} + 22 T^{3} + 121 T^{4}$$)($$1 - 4 T^{2} - 105 T^{4} - 484 T^{6} + 14641 T^{8}$$)($$1 - 6 T + 12 T^{2} - 12 T^{3} + 59 T^{4} - 132 T^{5} + 1452 T^{6} - 7986 T^{7} + 14641 T^{8}$$)($$1 + 2 T - 12 T^{2} - 12 T^{3} + 91 T^{4} - 132 T^{5} - 1452 T^{6} + 2662 T^{7} + 14641 T^{8}$$)($$1 - 6 T + 12 T^{2} - 12 T^{3} + 59 T^{4} - 132 T^{5} + 1452 T^{6} - 7986 T^{7} + 14641 T^{8}$$)($$1 - 4 T^{2} - 105 T^{4} - 484 T^{6} + 14641 T^{8}$$)($$1 - 4 T^{2} - 105 T^{4} - 484 T^{6} + 14641 T^{8}$$)($$1 + 4 T - 22 T^{2} - 68 T^{3} + 396 T^{4} + 526 T^{5} - 6648 T^{6} - 1364 T^{7} + 89791 T^{8} - 15004 T^{9} - 804408 T^{10} + 700106 T^{11} + 5797836 T^{12} - 10951468 T^{13} - 38974342 T^{14} + 77948684 T^{15} + 214358881 T^{16}$$)($$1 + 4 T - 22 T^{2} - 68 T^{3} + 396 T^{4} + 526 T^{5} - 6648 T^{6} - 1364 T^{7} + 89791 T^{8} - 15004 T^{9} - 804408 T^{10} + 700106 T^{11} + 5797836 T^{12} - 10951468 T^{13} - 38974342 T^{14} + 77948684 T^{15} + 214358881 T^{16}$$)
$13$ ($$( 1 + T + 13 T^{2} )^{2}$$)($$( 1 - 3 T + 13 T^{2} )^{2}$$)($$( 1 + 2 T + 13 T^{2} )^{2}$$)($$( 1 + 7 T + 13 T^{2} )^{2}$$)($$( 1 - 2 T + 13 T^{2} )^{2}$$)($$( 1 - 5 T + 13 T^{2} )^{4}$$)($$( 1 + T + 13 T^{2} )^{4}$$)($$( 1 + 19 T^{2} + 169 T^{4} )^{2}$$)($$( 1 - T + 13 T^{2} )^{4}$$)($$( 1 + 5 T + 13 T^{2} )^{4}$$)($$( 1 - 2 T + 9 T^{2} - 26 T^{3} + 169 T^{4} )^{2}$$)($$( 1 + 2 T + 28 T^{2} + 78 T^{3} + 465 T^{4} + 1014 T^{5} + 4732 T^{6} + 4394 T^{7} + 28561 T^{8} )^{2}$$)($$( 1 - 2 T + 28 T^{2} - 78 T^{3} + 465 T^{4} - 1014 T^{5} + 4732 T^{6} - 4394 T^{7} + 28561 T^{8} )^{2}$$)
$17$ ($$1 + 4 T - T^{2} + 68 T^{3} + 289 T^{4}$$)($$1 - 8 T + 47 T^{2} - 136 T^{3} + 289 T^{4}$$)($$1 + 2 T - 13 T^{2} + 34 T^{3} + 289 T^{4}$$)($$1 + 6 T + 19 T^{2} + 102 T^{3} + 289 T^{4}$$)($$1 - 2 T - 13 T^{2} - 34 T^{3} + 289 T^{4}$$)($$1 - 16 T^{2} - 33 T^{4} - 4624 T^{6} + 83521 T^{8}$$)($$1 - 2 T - 24 T^{2} + 12 T^{3} + 427 T^{4} + 204 T^{5} - 6936 T^{6} - 9826 T^{7} + 83521 T^{8}$$)($$1 - 2 T - 24 T^{2} + 12 T^{3} + 427 T^{4} + 204 T^{5} - 6936 T^{6} - 9826 T^{7} + 83521 T^{8}$$)($$1 + 2 T - 24 T^{2} - 12 T^{3} + 427 T^{4} - 204 T^{5} - 6936 T^{6} + 9826 T^{7} + 83521 T^{8}$$)($$1 - 16 T^{2} - 33 T^{4} - 4624 T^{6} + 83521 T^{8}$$)($$1 - 16 T^{2} - 33 T^{4} - 4624 T^{6} + 83521 T^{8}$$)($$1 - 2 T - 40 T^{2} + 144 T^{3} + 788 T^{4} - 3412 T^{5} - 5360 T^{6} + 32022 T^{7} + 6583 T^{8} + 544374 T^{9} - 1549040 T^{10} - 16763156 T^{11} + 65814548 T^{12} + 204459408 T^{13} - 965502760 T^{14} - 820677346 T^{15} + 6975757441 T^{16}$$)($$1 + 2 T - 40 T^{2} - 144 T^{3} + 788 T^{4} + 3412 T^{5} - 5360 T^{6} - 32022 T^{7} + 6583 T^{8} - 544374 T^{9} - 1549040 T^{10} + 16763156 T^{11} + 65814548 T^{12} - 204459408 T^{13} - 965502760 T^{14} + 820677346 T^{15} + 6975757441 T^{16}$$)
$19$ ($$( 1 - 8 T + 19 T^{2} )( 1 + 7 T + 19 T^{2} )$$)($$( 1 - 8 T + 19 T^{2} )( 1 + 7 T + 19 T^{2} )$$)($$1 + 4 T - 3 T^{2} + 76 T^{3} + 361 T^{4}$$)($$1 + 3 T - 10 T^{2} + 57 T^{3} + 361 T^{4}$$)($$1 + 4 T - 3 T^{2} + 76 T^{3} + 361 T^{4}$$)($$1 - 2 T - 17 T^{2} + 34 T^{3} + 4 T^{4} + 646 T^{5} - 6137 T^{6} - 13718 T^{7} + 130321 T^{8}$$)($$1 - 4 T - 19 T^{2} + 12 T^{3} + 560 T^{4} + 228 T^{5} - 6859 T^{6} - 27436 T^{7} + 130321 T^{8}$$)($$1 + 6 T + 17 T^{2} - 114 T^{3} - 684 T^{4} - 2166 T^{5} + 6137 T^{6} + 41154 T^{7} + 130321 T^{8}$$)($$1 - 4 T - 19 T^{2} + 12 T^{3} + 560 T^{4} + 228 T^{5} - 6859 T^{6} - 27436 T^{7} + 130321 T^{8}$$)($$1 - 2 T - 17 T^{2} + 34 T^{3} + 4 T^{4} + 646 T^{5} - 6137 T^{6} - 13718 T^{7} + 130321 T^{8}$$)($$( 1 - 8 T + 19 T^{2} )^{2}( 1 + T + 19 T^{2} )^{2}$$)($$1 + 4 T - 18 T^{2} + 112 T^{3} + 697 T^{4} - 2652 T^{5} + 9806 T^{6} + 65960 T^{7} - 161964 T^{8} + 1253240 T^{9} + 3539966 T^{10} - 18190068 T^{11} + 90833737 T^{12} + 277323088 T^{13} - 846825858 T^{14} + 3575486956 T^{15} + 16983563041 T^{16}$$)($$1 + 4 T - 18 T^{2} + 112 T^{3} + 697 T^{4} - 2652 T^{5} + 9806 T^{6} + 65960 T^{7} - 161964 T^{8} + 1253240 T^{9} + 3539966 T^{10} - 18190068 T^{11} + 90833737 T^{12} + 277323088 T^{13} - 846825858 T^{14} + 3575486956 T^{15} + 16983563041 T^{16}$$)
$23$ ($$1 - 4 T - 7 T^{2} - 92 T^{3} + 529 T^{4}$$)($$1 - 8 T + 41 T^{2} - 184 T^{3} + 529 T^{4}$$)($$1 - 8 T + 41 T^{2} - 184 T^{3} + 529 T^{4}$$)($$1 + 2 T - 19 T^{2} + 46 T^{3} + 529 T^{4}$$)($$1 + 8 T + 41 T^{2} + 184 T^{3} + 529 T^{4}$$)($$( 1 + 6 T + 13 T^{2} + 138 T^{3} + 529 T^{4} )^{2}$$)($$( 1 + 2 T - 19 T^{2} + 46 T^{3} + 529 T^{4} )^{2}$$)($$1 + 2 T - 36 T^{2} - 12 T^{3} + 979 T^{4} - 276 T^{5} - 19044 T^{6} + 24334 T^{7} + 279841 T^{8}$$)($$( 1 - 2 T - 19 T^{2} - 46 T^{3} + 529 T^{4} )^{2}$$)($$( 1 - 6 T + 13 T^{2} - 138 T^{3} + 529 T^{4} )^{2}$$)($$1 - 28 T^{2} + 255 T^{4} - 14812 T^{6} + 279841 T^{8}$$)($$1 + 6 T - 44 T^{2} - 272 T^{3} + 1468 T^{4} + 6616 T^{5} - 39952 T^{6} - 50582 T^{7} + 1119911 T^{8} - 1163386 T^{9} - 21134608 T^{10} + 80496872 T^{11} + 410806588 T^{12} - 1750685296 T^{13} - 6513579116 T^{14} + 20428952682 T^{15} + 78310985281 T^{16}$$)($$1 - 6 T - 44 T^{2} + 272 T^{3} + 1468 T^{4} - 6616 T^{5} - 39952 T^{6} + 50582 T^{7} + 1119911 T^{8} + 1163386 T^{9} - 21134608 T^{10} - 80496872 T^{11} + 410806588 T^{12} + 1750685296 T^{13} - 6513579116 T^{14} - 20428952682 T^{15} + 78310985281 T^{16}$$)
$29$ ($$( 1 + 29 T^{2} )^{2}$$)($$( 1 - 4 T + 29 T^{2} )^{2}$$)($$( 1 - 4 T + 29 T^{2} )^{2}$$)($$( 1 + 2 T + 29 T^{2} )^{2}$$)($$( 1 - 4 T + 29 T^{2} )^{2}$$)($$( 1 - 14 T^{2} + 841 T^{4} )^{2}$$)($$( 1 - 4 T + 34 T^{2} - 116 T^{3} + 841 T^{4} )^{2}$$)($$( 1 - 10 T + 76 T^{2} - 290 T^{3} + 841 T^{4} )^{2}$$)($$( 1 - 4 T + 34 T^{2} - 116 T^{3} + 841 T^{4} )^{2}$$)($$( 1 - 14 T^{2} + 841 T^{4} )^{2}$$)($$( 1 + 12 T + 76 T^{2} + 348 T^{3} + 841 T^{4} )^{2}$$)($$( 1 + 6 T + 50 T^{2} + 300 T^{3} + 2316 T^{4} + 8700 T^{5} + 42050 T^{6} + 146334 T^{7} + 707281 T^{8} )^{2}$$)($$( 1 + 6 T + 50 T^{2} + 300 T^{3} + 2316 T^{4} + 8700 T^{5} + 42050 T^{6} + 146334 T^{7} + 707281 T^{8} )^{2}$$)
$31$ ($$1 - 5 T - 6 T^{2} - 155 T^{3} + 961 T^{4}$$)($$1 + 3 T - 22 T^{2} + 93 T^{3} + 961 T^{4}$$)($$1 + 3 T - 22 T^{2} + 93 T^{3} + 961 T^{4}$$)($$( 1 - 4 T + 31 T^{2} )( 1 + 11 T + 31 T^{2} )$$)($$1 + 3 T - 22 T^{2} + 93 T^{3} + 961 T^{4}$$)($$( 1 - 11 T + 31 T^{2} )^{2}( 1 + 7 T + 31 T^{2} )^{2}$$)($$( 1 - 7 T + 31 T^{2} )^{2}( 1 + 11 T + 31 T^{2} )^{2}$$)($$1 + 2 T - 31 T^{2} - 54 T^{3} + 140 T^{4} - 1674 T^{5} - 29791 T^{6} + 59582 T^{7} + 923521 T^{8}$$)($$( 1 - 7 T + 31 T^{2} )^{2}( 1 + 11 T + 31 T^{2} )^{2}$$)($$( 1 - 11 T + 31 T^{2} )^{2}( 1 + 7 T + 31 T^{2} )^{2}$$)($$1 - 2 T + 13 T^{2} + 142 T^{3} - 1004 T^{4} + 4402 T^{5} + 12493 T^{6} - 59582 T^{7} + 923521 T^{8}$$)($$1 - 22 T^{2} + 400 T^{3} - 59 T^{4} - 10600 T^{5} + 70338 T^{6} + 164200 T^{7} - 2333796 T^{8} + 5090200 T^{9} + 67594818 T^{10} - 315784600 T^{11} - 54487739 T^{12} + 11451660400 T^{13} - 19525080982 T^{14} + 852891037441 T^{16}$$)($$1 - 22 T^{2} + 400 T^{3} - 59 T^{4} - 10600 T^{5} + 70338 T^{6} + 164200 T^{7} - 2333796 T^{8} + 5090200 T^{9} + 67594818 T^{10} - 315784600 T^{11} - 54487739 T^{12} + 11451660400 T^{13} - 19525080982 T^{14} + 852891037441 T^{16}$$)
$37$ ($$1 + 5 T - 12 T^{2} + 185 T^{3} + 1369 T^{4}$$)($$( 1 - 10 T + 37 T^{2} )( 1 + 11 T + 37 T^{2} )$$)($$1 - 9 T + 44 T^{2} - 333 T^{3} + 1369 T^{4}$$)($$1 + 7 T + 12 T^{2} + 259 T^{3} + 1369 T^{4}$$)($$1 + 9 T + 44 T^{2} + 333 T^{3} + 1369 T^{4}$$)($$( 1 + 5 T - 12 T^{2} + 185 T^{3} + 1369 T^{4} )^{2}$$)($$1 - 6 T - 19 T^{2} + 114 T^{3} + 324 T^{4} + 4218 T^{5} - 26011 T^{6} - 303918 T^{7} + 1874161 T^{8}$$)($$1 + 4 T - 55 T^{2} - 12 T^{3} + 3080 T^{4} - 444 T^{5} - 75295 T^{6} + 202612 T^{7} + 1874161 T^{8}$$)($$1 + 6 T - 19 T^{2} - 114 T^{3} + 324 T^{4} - 4218 T^{5} - 26011 T^{6} + 303918 T^{7} + 1874161 T^{8}$$)($$( 1 - 5 T - 12 T^{2} - 185 T^{3} + 1369 T^{4} )^{2}$$)($$1 + 2 T - 53 T^{2} - 34 T^{3} + 1732 T^{4} - 1258 T^{5} - 72557 T^{6} + 101306 T^{7} + 1874161 T^{8}$$)($$1 + 4 T - 108 T^{2} - 292 T^{3} + 7187 T^{4} + 10530 T^{5} - 383576 T^{6} - 128186 T^{7} + 16685028 T^{8} - 4742882 T^{9} - 525115544 T^{10} + 533376090 T^{11} + 13469595107 T^{12} - 20248435444 T^{13} - 277098452172 T^{14} + 379727508532 T^{15} + 3512479453921 T^{16}$$)($$1 - 4 T - 108 T^{2} + 292 T^{3} + 7187 T^{4} - 10530 T^{5} - 383576 T^{6} + 128186 T^{7} + 16685028 T^{8} + 4742882 T^{9} - 525115544 T^{10} - 533376090 T^{11} + 13469595107 T^{12} + 20248435444 T^{13} - 277098452172 T^{14} - 379727508532 T^{15} + 3512479453921 T^{16}$$)
$41$ ($$( 1 - 2 T + 41 T^{2} )^{2}$$)($$( 1 - 6 T + 41 T^{2} )^{2}$$)($$( 1 - 6 T + 41 T^{2} )^{2}$$)($$( 1 + 8 T + 41 T^{2} )^{2}$$)($$( 1 - 6 T + 41 T^{2} )^{2}$$)($$( 1 + 12 T + 100 T^{2} + 492 T^{3} + 1681 T^{4} )^{2}$$)($$( 1 + 6 T + 84 T^{2} + 246 T^{3} + 1681 T^{4} )^{2}$$)($$( 1 + 6 T + 28 T^{2} + 246 T^{3} + 1681 T^{4} )^{2}$$)($$( 1 + 6 T + 84 T^{2} + 246 T^{3} + 1681 T^{4} )^{2}$$)($$( 1 + 12 T + 100 T^{2} + 492 T^{3} + 1681 T^{4} )^{2}$$)($$( 1 + 64 T^{2} + 1681 T^{4} )^{2}$$)($$( 1 - 12 T + 206 T^{2} - 1514 T^{3} + 13536 T^{4} - 62074 T^{5} + 346286 T^{6} - 827052 T^{7} + 2825761 T^{8} )^{2}$$)($$( 1 - 12 T + 206 T^{2} - 1514 T^{3} + 13536 T^{4} - 62074 T^{5} + 346286 T^{6} - 827052 T^{7} + 2825761 T^{8} )^{2}$$)
$43$ ($$( 1 - 9 T + 43 T^{2} )^{2}$$)($$( 1 + 11 T + 43 T^{2} )^{2}$$)($$( 1 + T + 43 T^{2} )^{2}$$)($$( 1 + 5 T + 43 T^{2} )^{2}$$)($$( 1 - T + 43 T^{2} )^{2}$$)($$( 1 + 8 T + 30 T^{2} + 344 T^{3} + 1849 T^{4} )^{2}$$)($$( 1 - 4 T + 62 T^{2} - 172 T^{3} + 1849 T^{4} )^{2}$$)($$( 1 + 4 T + 27 T^{2} + 172 T^{3} + 1849 T^{4} )^{2}$$)($$( 1 + 4 T + 62 T^{2} + 172 T^{3} + 1849 T^{4} )^{2}$$)($$( 1 - 8 T + 30 T^{2} - 344 T^{3} + 1849 T^{4} )^{2}$$)($$( 1 - 2 T + 69 T^{2} - 86 T^{3} + 1849 T^{4} )^{2}$$)($$( 1 - 16 T + 160 T^{2} - 1402 T^{3} + 10561 T^{4} - 60286 T^{5} + 295840 T^{6} - 1272112 T^{7} + 3418801 T^{8} )^{2}$$)($$( 1 + 16 T + 160 T^{2} + 1402 T^{3} + 10561 T^{4} + 60286 T^{5} + 295840 T^{6} + 1272112 T^{7} + 3418801 T^{8} )^{2}$$)
$47$ ($$1 + 2 T - 43 T^{2} + 94 T^{3} + 2209 T^{4}$$)($$1 - 6 T - 11 T^{2} - 282 T^{3} + 2209 T^{4}$$)($$1 - 6 T - 11 T^{2} - 282 T^{3} + 2209 T^{4}$$)($$1 - 10 T + 53 T^{2} - 470 T^{3} + 2209 T^{4}$$)($$1 + 6 T - 11 T^{2} + 282 T^{3} + 2209 T^{4}$$)($$1 + 12 T + 32 T^{2} + 216 T^{3} + 3567 T^{4} + 10152 T^{5} + 70688 T^{6} + 1245876 T^{7} + 4879681 T^{8}$$)($$1 + 10 T + 44 T^{2} - 380 T^{3} - 3773 T^{4} - 17860 T^{5} + 97196 T^{6} + 1038230 T^{7} + 4879681 T^{8}$$)($$( 1 - 6 T - 11 T^{2} - 282 T^{3} + 2209 T^{4} )^{2}$$)($$1 - 10 T + 44 T^{2} + 380 T^{3} - 3773 T^{4} + 17860 T^{5} + 97196 T^{6} - 1038230 T^{7} + 4879681 T^{8}$$)($$1 - 12 T + 32 T^{2} - 216 T^{3} + 3567 T^{4} - 10152 T^{5} + 70688 T^{6} - 1245876 T^{7} + 4879681 T^{8}$$)($$( 1 + 6 T - 11 T^{2} + 282 T^{3} + 2209 T^{4} )^{2}$$)($$1 + 2 T - 64 T^{2} + 84 T^{3} + 1982 T^{4} - 7370 T^{5} + 122536 T^{6} + 409218 T^{7} - 7587185 T^{8} + 19233246 T^{9} + 270682024 T^{10} - 765175510 T^{11} + 9671527742 T^{12} + 19264980588 T^{13} - 689869781056 T^{14} + 1013246240926 T^{15} + 23811286661761 T^{16}$$)($$1 - 2 T - 64 T^{2} - 84 T^{3} + 1982 T^{4} + 7370 T^{5} + 122536 T^{6} - 409218 T^{7} - 7587185 T^{8} - 19233246 T^{9} + 270682024 T^{10} + 765175510 T^{11} + 9671527742 T^{12} - 19264980588 T^{13} - 689869781056 T^{14} - 1013246240926 T^{15} + 23811286661761 T^{16}$$)
$53$ ($$1 - 12 T + 91 T^{2} - 636 T^{3} + 2809 T^{4}$$)($$1 + 12 T + 91 T^{2} + 636 T^{3} + 2809 T^{4}$$)($$1 + 2 T - 49 T^{2} + 106 T^{3} + 2809 T^{4}$$)($$1 + 8 T + 11 T^{2} + 424 T^{3} + 2809 T^{4}$$)($$1 - 2 T - 49 T^{2} - 106 T^{3} + 2809 T^{4}$$)($$1 - 88 T^{2} + 4935 T^{4} - 247192 T^{6} + 7890481 T^{8}$$)($$1 - 10 T - 24 T^{2} - 180 T^{3} + 7267 T^{4} - 9540 T^{5} - 67416 T^{6} - 1488770 T^{7} + 7890481 T^{8}$$)($$1 + 4 T - 66 T^{2} - 96 T^{3} + 3067 T^{4} - 5088 T^{5} - 185394 T^{6} + 595508 T^{7} + 7890481 T^{8}$$)($$1 + 10 T - 24 T^{2} + 180 T^{3} + 7267 T^{4} + 9540 T^{5} - 67416 T^{6} + 1488770 T^{7} + 7890481 T^{8}$$)($$1 - 88 T^{2} + 4935 T^{4} - 247192 T^{6} + 7890481 T^{8}$$)($$1 - 34 T^{2} - 1653 T^{4} - 95506 T^{6} + 7890481 T^{8}$$)($$1 + 20 T + 68 T^{2} - 200 T^{3} + 12154 T^{4} + 129900 T^{5} + 85936 T^{6} + 2744060 T^{7} + 61991603 T^{8} + 145435180 T^{9} + 241394224 T^{10} + 19339122300 T^{11} + 95900906074 T^{12} - 83639098600 T^{13} + 1507176556772 T^{14} + 23494222796740 T^{15} + 62259690411361 T^{16}$$)($$1 - 20 T + 68 T^{2} + 200 T^{3} + 12154 T^{4} - 129900 T^{5} + 85936 T^{6} - 2744060 T^{7} + 61991603 T^{8} - 145435180 T^{9} + 241394224 T^{10} - 19339122300 T^{11} + 95900906074 T^{12} + 83639098600 T^{13} + 1507176556772 T^{14} - 23494222796740 T^{15} + 62259690411361 T^{16}$$)
$59$ ($$1 - 8 T + 5 T^{2} - 472 T^{3} + 3481 T^{4}$$)($$1 + 4 T - 43 T^{2} + 236 T^{3} + 3481 T^{4}$$)($$1 - 6 T - 23 T^{2} - 354 T^{3} + 3481 T^{4}$$)($$1 + 10 T + 41 T^{2} + 590 T^{3} + 3481 T^{4}$$)($$1 - 6 T - 23 T^{2} - 354 T^{3} + 3481 T^{4}$$)($$1 + 12 T + 8 T^{2} + 216 T^{3} + 6519 T^{4} + 12744 T^{5} + 27848 T^{6} + 2464548 T^{7} + 12117361 T^{8}$$)($$1 + 14 T + 92 T^{2} - 196 T^{3} - 4229 T^{4} - 11564 T^{5} + 320252 T^{6} + 2875306 T^{7} + 12117361 T^{8}$$)($$1 + 6 T - 28 T^{2} - 324 T^{3} - 1509 T^{4} - 19116 T^{5} - 97468 T^{6} + 1232274 T^{7} + 12117361 T^{8}$$)($$1 + 14 T + 92 T^{2} - 196 T^{3} - 4229 T^{4} - 11564 T^{5} + 320252 T^{6} + 2875306 T^{7} + 12117361 T^{8}$$)($$1 + 12 T + 8 T^{2} + 216 T^{3} + 6519 T^{4} + 12744 T^{5} + 27848 T^{6} + 2464548 T^{7} + 12117361 T^{8}$$)($$1 - 12 T + 8 T^{2} - 216 T^{3} + 6519 T^{4} - 12744 T^{5} + 27848 T^{6} - 2464548 T^{7} + 12117361 T^{8}$$)($$1 - 14 T + 98 T^{2} - 2364 T^{3} + 23888 T^{4} - 127648 T^{5} + 2057308 T^{6} - 17027622 T^{7} + 74320339 T^{8} - 1004629698 T^{9} + 7161489148 T^{10} - 26216218592 T^{11} + 289459519568 T^{12} - 1690081042836 T^{13} + 4133692296818 T^{14} - 34841120787466 T^{15} + 146830437604321 T^{16}$$)($$1 - 14 T + 98 T^{2} - 2364 T^{3} + 23888 T^{4} - 127648 T^{5} + 2057308 T^{6} - 17027622 T^{7} + 74320339 T^{8} - 1004629698 T^{9} + 7161489148 T^{10} - 26216218592 T^{11} + 289459519568 T^{12} - 1690081042836 T^{13} + 4133692296818 T^{14} - 34841120787466 T^{15} + 146830437604321 T^{16}$$)
$61$ ($$( 1 - 13 T + 61 T^{2} )( 1 - T + 61 T^{2} )$$)($$1 - 6 T - 25 T^{2} - 366 T^{3} + 3721 T^{4}$$)($$( 1 - 14 T + 61 T^{2} )( 1 + 13 T + 61 T^{2} )$$)($$1 - 6 T - 25 T^{2} - 366 T^{3} + 3721 T^{4}$$)($$( 1 - 14 T + 61 T^{2} )( 1 + 13 T + 61 T^{2} )$$)($$( 1 - 14 T + 61 T^{2} )^{2}( 1 + 13 T + 61 T^{2} )^{2}$$)($$1 - 2 T - 7 T^{2} + 222 T^{3} - 3844 T^{4} + 13542 T^{5} - 26047 T^{6} - 453962 T^{7} + 13845841 T^{8}$$)($$( 1 + 8 T + 3 T^{2} + 488 T^{3} + 3721 T^{4} )^{2}$$)($$1 - 2 T - 7 T^{2} + 222 T^{3} - 3844 T^{4} + 13542 T^{5} - 26047 T^{6} - 453962 T^{7} + 13845841 T^{8}$$)($$( 1 - 14 T + 61 T^{2} )^{2}( 1 + 13 T + 61 T^{2} )^{2}$$)($$1 - 8 T - 2 T^{2} + 448 T^{3} - 3269 T^{4} + 27328 T^{5} - 7442 T^{6} - 1815848 T^{7} + 13845841 T^{8}$$)($$1 + 16 T + 48 T^{2} - 408 T^{3} - 2562 T^{4} - 7524 T^{5} - 100912 T^{6} - 1615000 T^{7} - 17451933 T^{8} - 98515000 T^{9} - 375493552 T^{10} - 1707805044 T^{11} - 35473044642 T^{12} - 344595290808 T^{13} + 2472977969328 T^{14} + 50283885376336 T^{15} + 191707312997281 T^{16}$$)($$1 + 16 T + 48 T^{2} - 408 T^{3} - 2562 T^{4} - 7524 T^{5} - 100912 T^{6} - 1615000 T^{7} - 17451933 T^{8} - 98515000 T^{9} - 375493552 T^{10} - 1707805044 T^{11} - 35473044642 T^{12} - 344595290808 T^{13} + 2472977969328 T^{14} + 50283885376336 T^{15} + 191707312997281 T^{16}$$)
$67$ ($$1 - 9 T + 14 T^{2} - 603 T^{3} + 4489 T^{4}$$)($$1 - 13 T + 102 T^{2} - 871 T^{3} + 4489 T^{4}$$)($$1 + 12 T + 77 T^{2} + 804 T^{3} + 4489 T^{4}$$)($$1 - 3 T - 58 T^{2} - 201 T^{3} + 4489 T^{4}$$)($$1 - 12 T + 77 T^{2} - 804 T^{3} + 4489 T^{4}$$)($$1 - 2 T + 31 T^{2} + 322 T^{3} - 4028 T^{4} + 21574 T^{5} + 139159 T^{6} - 601526 T^{7} + 20151121 T^{8}$$)($$1 - 71 T^{2} + 552 T^{4} - 318719 T^{6} + 20151121 T^{8}$$)($$1 + 4 T - 115 T^{2} - 12 T^{3} + 11600 T^{4} - 804 T^{5} - 516235 T^{6} + 1203052 T^{7} + 20151121 T^{8}$$)($$1 - 71 T^{2} + 552 T^{4} - 318719 T^{6} + 20151121 T^{8}$$)($$1 + 2 T + 31 T^{2} - 322 T^{3} - 4028 T^{4} - 21574 T^{5} + 139159 T^{6} + 601526 T^{7} + 20151121 T^{8}$$)($$1 + 2 T - 113 T^{2} - 34 T^{3} + 8932 T^{4} - 2278 T^{5} - 507257 T^{6} + 601526 T^{7} + 20151121 T^{8}$$)($$1 + 18 T + 20 T^{2} - 1396 T^{3} - 1157 T^{4} + 130618 T^{5} + 724920 T^{6} - 842608 T^{7} - 21438852 T^{8} - 56454736 T^{9} + 3254165880 T^{10} + 39285061534 T^{11} - 23314846997 T^{12} - 1884774649372 T^{13} + 1809167643380 T^{14} + 109092808895814 T^{15} + 406067677556641 T^{16}$$)($$1 - 18 T + 20 T^{2} + 1396 T^{3} - 1157 T^{4} - 130618 T^{5} + 724920 T^{6} + 842608 T^{7} - 21438852 T^{8} + 56454736 T^{9} + 3254165880 T^{10} - 39285061534 T^{11} - 23314846997 T^{12} + 1884774649372 T^{13} + 1809167643380 T^{14} - 109092808895814 T^{15} + 406067677556641 T^{16}$$)
$71$ ($$( 1 - 2 T + 71 T^{2} )^{2}$$)($$( 1 + 10 T + 71 T^{2} )^{2}$$)($$( 1 - 10 T + 71 T^{2} )^{2}$$)($$( 1 + 71 T^{2} )^{2}$$)($$( 1 - 10 T + 71 T^{2} )^{2}$$)($$( 1 + 70 T^{2} + 5041 T^{4} )^{2}$$)($$( 1 + 4 T + 118 T^{2} + 284 T^{3} + 5041 T^{4} )^{2}$$)($$( 1 - 22 T + 256 T^{2} - 1562 T^{3} + 5041 T^{4} )^{2}$$)($$( 1 + 4 T + 118 T^{2} + 284 T^{3} + 5041 T^{4} )^{2}$$)($$( 1 + 70 T^{2} + 5041 T^{4} )^{2}$$)($$( 1 - 20 T^{2} + 5041 T^{4} )^{2}$$)($$( 1 + 14 T + 158 T^{2} + 444 T^{3} + 3552 T^{4} + 31524 T^{5} + 796478 T^{6} + 5010754 T^{7} + 25411681 T^{8} )^{2}$$)($$( 1 + 14 T + 158 T^{2} + 444 T^{3} + 3552 T^{4} + 31524 T^{5} + 796478 T^{6} + 5010754 T^{7} + 25411681 T^{8} )^{2}$$)
$73$ ($$1 - T - 72 T^{2} - 73 T^{3} + 5329 T^{4}$$)($$1 + 11 T + 48 T^{2} + 803 T^{3} + 5329 T^{4}$$)($$1 + T - 72 T^{2} + 73 T^{3} + 5329 T^{4}$$)($$1 - 15 T + 152 T^{2} - 1095 T^{3} + 5329 T^{4}$$)($$1 - T - 72 T^{2} - 73 T^{3} + 5329 T^{4}$$)($$1 + 10 T + T^{2} - 470 T^{3} - 2828 T^{4} - 34310 T^{5} + 5329 T^{6} + 3890170 T^{7} + 28398241 T^{8}$$)($$1 + 2 T - 115 T^{2} - 54 T^{3} + 8540 T^{4} - 3942 T^{5} - 612835 T^{6} + 778034 T^{7} + 28398241 T^{8}$$)($$1 + 29 T^{2} - 4488 T^{4} + 154541 T^{6} + 28398241 T^{8}$$)($$1 - 2 T - 115 T^{2} + 54 T^{3} + 8540 T^{4} + 3942 T^{5} - 612835 T^{6} - 778034 T^{7} + 28398241 T^{8}$$)($$1 - 10 T + T^{2} + 470 T^{3} - 2828 T^{4} + 34310 T^{5} + 5329 T^{6} - 3890170 T^{7} + 28398241 T^{8}$$)($$1 - 10 T - 53 T^{2} - 70 T^{3} + 10780 T^{4} - 5110 T^{5} - 282437 T^{6} - 3890170 T^{7} + 28398241 T^{8}$$)($$1 - 8 T - 144 T^{2} + 1148 T^{3} + 10427 T^{4} - 49662 T^{5} - 1117916 T^{6} + 363682 T^{7} + 112617972 T^{8} + 26548786 T^{9} - 5957374364 T^{10} - 19319362254 T^{11} + 296108458907 T^{12} + 2379886188764 T^{13} - 21792128585616 T^{14} - 88379188152776 T^{15} + 806460091894081 T^{16}$$)($$1 + 8 T - 144 T^{2} - 1148 T^{3} + 10427 T^{4} + 49662 T^{5} - 1117916 T^{6} - 363682 T^{7} + 112617972 T^{8} - 26548786 T^{9} - 5957374364 T^{10} + 19319362254 T^{11} + 296108458907 T^{12} - 2379886188764 T^{13} - 21792128585616 T^{14} + 88379188152776 T^{15} + 806460091894081 T^{16}$$)
$79$ ($$1 - 3 T - 70 T^{2} - 237 T^{3} + 6241 T^{4}$$)($$1 - 3 T - 70 T^{2} - 237 T^{3} + 6241 T^{4}$$)($$1 + 7 T - 30 T^{2} + 553 T^{3} + 6241 T^{4}$$)($$1 + T - 78 T^{2} + 79 T^{3} + 6241 T^{4}$$)($$1 + 7 T - 30 T^{2} + 553 T^{3} + 6241 T^{4}$$)($$1 + 10 T - 65 T^{2} + 70 T^{3} + 13084 T^{4} + 5530 T^{5} - 405665 T^{6} + 4930390 T^{7} + 38950081 T^{8}$$)($$1 - 8 T - 103 T^{2} - 72 T^{3} + 16592 T^{4} - 5688 T^{5} - 642823 T^{6} - 3944312 T^{7} + 38950081 T^{8}$$)($$1 - 6 T - 103 T^{2} + 114 T^{3} + 10236 T^{4} + 9006 T^{5} - 642823 T^{6} - 2958234 T^{7} + 38950081 T^{8}$$)($$1 - 8 T - 103 T^{2} - 72 T^{3} + 16592 T^{4} - 5688 T^{5} - 642823 T^{6} - 3944312 T^{7} + 38950081 T^{8}$$)($$1 + 10 T - 65 T^{2} + 70 T^{3} + 13084 T^{4} + 5530 T^{5} - 405665 T^{6} + 4930390 T^{7} + 38950081 T^{8}$$)($$( 1 + 11 T + 42 T^{2} + 869 T^{3} + 6241 T^{4} )^{2}$$)($$1 - 8 T - 18 T^{2} - 632 T^{3} + 6121 T^{4} + 6180 T^{5} + 974510 T^{6} - 7889020 T^{7} - 17559900 T^{8} - 623232580 T^{9} + 6081916910 T^{10} + 3046981020 T^{11} + 238413445801 T^{12} - 1944699644168 T^{13} - 4375574199378 T^{14} - 153631271889272 T^{15} + 1517108809906561 T^{16}$$)($$1 - 8 T - 18 T^{2} - 632 T^{3} + 6121 T^{4} + 6180 T^{5} + 974510 T^{6} - 7889020 T^{7} - 17559900 T^{8} - 623232580 T^{9} + 6081916910 T^{10} + 3046981020 T^{11} + 238413445801 T^{12} - 1944699644168 T^{13} - 4375574199378 T^{14} - 153631271889272 T^{15} + 1517108809906561 T^{16}$$)
$83$ ($$( 1 - 18 T + 83 T^{2} )^{2}$$)($$( 1 + 2 T + 83 T^{2} )^{2}$$)($$( 1 - 18 T + 83 T^{2} )^{2}$$)($$( 1 + 8 T + 83 T^{2} )^{2}$$)($$( 1 + 18 T + 83 T^{2} )^{2}$$)($$( 1 - 12 T + 184 T^{2} - 996 T^{3} + 6889 T^{4} )^{2}$$)($$( 1 + 2 T + 160 T^{2} + 166 T^{3} + 6889 T^{4} )^{2}$$)($$( 1 - 6 T + 112 T^{2} - 498 T^{3} + 6889 T^{4} )^{2}$$)($$( 1 - 2 T + 160 T^{2} - 166 T^{3} + 6889 T^{4} )^{2}$$)($$( 1 + 12 T + 184 T^{2} + 996 T^{3} + 6889 T^{4} )^{2}$$)($$( 1 - 12 T + 184 T^{2} - 996 T^{3} + 6889 T^{4} )^{2}$$)($$( 1 + 10 T + 200 T^{2} + 1808 T^{3} + 24704 T^{4} + 150064 T^{5} + 1377800 T^{6} + 5717870 T^{7} + 47458321 T^{8} )^{2}$$)($$( 1 - 10 T + 200 T^{2} - 1808 T^{3} + 24704 T^{4} - 150064 T^{5} + 1377800 T^{6} - 5717870 T^{7} + 47458321 T^{8} )^{2}$$)
$89$ ($$1 - 4 T - 73 T^{2} - 356 T^{3} + 7921 T^{4}$$)($$1 - 89 T^{2} + 7921 T^{4}$$)($$1 + 10 T + 11 T^{2} + 890 T^{3} + 7921 T^{4}$$)($$1 + 2 T - 85 T^{2} + 178 T^{3} + 7921 T^{4}$$)($$1 + 10 T + 11 T^{2} + 890 T^{3} + 7921 T^{4}$$)($$1 + 24 T + 272 T^{2} + 3024 T^{3} + 33231 T^{4} + 269136 T^{5} + 2154512 T^{6} + 16919256 T^{7} + 62742241 T^{8}$$)($$1 - 2 T - 112 T^{2} + 124 T^{3} + 5179 T^{4} + 11036 T^{5} - 887152 T^{6} - 1409938 T^{7} + 62742241 T^{8}$$)($$1 - 2 T + 348 T^{3} - 8261 T^{4} + 30972 T^{5} - 1409938 T^{7} + 62742241 T^{8}$$)($$1 - 2 T - 112 T^{2} + 124 T^{3} + 5179 T^{4} + 11036 T^{5} - 887152 T^{6} - 1409938 T^{7} + 62742241 T^{8}$$)($$1 + 24 T + 272 T^{2} + 3024 T^{3} + 33231 T^{4} + 269136 T^{5} + 2154512 T^{6} + 16919256 T^{7} + 62742241 T^{8}$$)($$1 - 12 T - 52 T^{2} - 216 T^{3} + 17679 T^{4} - 19224 T^{5} - 411892 T^{6} - 8459628 T^{7} + 62742241 T^{8}$$)($$1 - 8 T + 2 T^{2} - 1724 T^{3} + 19584 T^{4} - 80534 T^{5} + 2467704 T^{6} - 22993064 T^{7} + 80738407 T^{8} - 2046382696 T^{9} + 19546683384 T^{10} - 56773973446 T^{11} + 1228744047744 T^{12} - 9626918490076 T^{13} + 993962581922 T^{14} - 353850679164232 T^{15} + 3936588805702081 T^{16}$$)($$1 - 8 T + 2 T^{2} - 1724 T^{3} + 19584 T^{4} - 80534 T^{5} + 2467704 T^{6} - 22993064 T^{7} + 80738407 T^{8} - 2046382696 T^{9} + 19546683384 T^{10} - 56773973446 T^{11} + 1228744047744 T^{12} - 9626918490076 T^{13} + 993962581922 T^{14} - 353850679164232 T^{15} + 3936588805702081 T^{16}$$)
$97$ ($$( 1 + 10 T + 97 T^{2} )^{2}$$)($$( 1 + 10 T + 97 T^{2} )^{2}$$)($$( 1 - 5 T + 97 T^{2} )^{2}$$)($$( 1 - 10 T + 97 T^{2} )^{2}$$)($$( 1 + 5 T + 97 T^{2} )^{2}$$)($$( 1 + 2 T - 93 T^{2} + 194 T^{3} + 9409 T^{4} )^{2}$$)($$( 1 + 18 T + 247 T^{2} + 1746 T^{3} + 9409 T^{4} )^{2}$$)($$( 1 + 8 T + 97 T^{2} )^{4}$$)($$( 1 - 18 T + 247 T^{2} - 1746 T^{3} + 9409 T^{4} )^{2}$$)($$( 1 - 2 T - 93 T^{2} - 194 T^{3} + 9409 T^{4} )^{2}$$)($$( 1 + 16 T + 186 T^{2} + 1552 T^{3} + 9409 T^{4} )^{2}$$)($$( 1 + 10 T + 340 T^{2} + 2590 T^{3} + 47986 T^{4} + 251230 T^{5} + 3199060 T^{6} + 9126730 T^{7} + 88529281 T^{8} )^{2}$$)($$( 1 - 10 T + 340 T^{2} - 2590 T^{3} + 47986 T^{4} - 251230 T^{5} + 3199060 T^{6} - 9126730 T^{7} + 88529281 T^{8} )^{2}$$)