# Properties

 Label 810.2.i Level $810$ Weight $2$ Character orbit 810.i Rep. character $\chi_{810}(109,\cdot)$ Character field $\Q(\zeta_{6})$ Dimension $48$ Newform subspaces $9$ Sturm bound $324$ Trace bound $11$

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$810 = 2 \cdot 3^{4} \cdot 5$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 810.i (of order $$6$$ and degree $$2$$) Character conductor: $$\operatorname{cond}(\chi)$$ $$=$$ $$45$$ Character field: $$\Q(\zeta_{6})$$ Newform subspaces: $$9$$ Sturm bound: $$324$$ Trace bound: $$11$$ Distinguishing $$T_p$$: $$7$$, $$11$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(810, [\chi])$$.

Total New Old
Modular forms 372 48 324
Cusp forms 276 48 228
Eisenstein series 96 0 96

## Trace form

 $$48q + 24q^{4} + O(q^{10})$$ $$48q + 24q^{4} - 24q^{16} - 12q^{25} + 24q^{31} + 24q^{46} + 36q^{49} + 84q^{55} - 12q^{61} - 48q^{64} - 30q^{70} + 84q^{79} + 6q^{85} + 48q^{91} + 12q^{94} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(810, [\chi])$$ into newform subspaces

Label Dim. $$A$$ Field CM Traces $q$-expansion
$$a_2$$ $$a_3$$ $$a_5$$ $$a_7$$
810.2.i.a $$4$$ $$6.468$$ $$\Q(\zeta_{12})$$ None $$0$$ $$0$$ $$-4$$ $$0$$ $$q-\zeta_{12}q^{2}+\zeta_{12}^{2}q^{4}+(\zeta_{12}-2\zeta_{12}^{2}+\cdots)q^{5}+\cdots$$
810.2.i.b $$4$$ $$6.468$$ $$\Q(\zeta_{12})$$ None $$0$$ $$0$$ $$-4$$ $$0$$ $$q+\zeta_{12}q^{2}+\zeta_{12}^{2}q^{4}+(-\zeta_{12}-2\zeta_{12}^{2}+\cdots)q^{5}+\cdots$$
810.2.i.c $$4$$ $$6.468$$ $$\Q(\zeta_{12})$$ None $$0$$ $$0$$ $$-2$$ $$0$$ $$q+\zeta_{12}q^{2}+\zeta_{12}^{2}q^{4}+(2\zeta_{12}-\zeta_{12}^{2}+\cdots)q^{5}+\cdots$$
810.2.i.d $$4$$ $$6.468$$ $$\Q(\zeta_{12})$$ None $$0$$ $$0$$ $$2$$ $$0$$ $$q-\zeta_{12}q^{2}+\zeta_{12}^{2}q^{4}+(-2\zeta_{12}+\zeta_{12}^{2}+\cdots)q^{5}+\cdots$$
810.2.i.e $$4$$ $$6.468$$ $$\Q(\zeta_{12})$$ None $$0$$ $$0$$ $$4$$ $$0$$ $$q-\zeta_{12}q^{2}+\zeta_{12}^{2}q^{4}+(\zeta_{12}+2\zeta_{12}^{2}+\cdots)q^{5}+\cdots$$
810.2.i.f $$4$$ $$6.468$$ $$\Q(\zeta_{12})$$ None $$0$$ $$0$$ $$4$$ $$0$$ $$q+\zeta_{12}q^{2}+\zeta_{12}^{2}q^{4}+(-\zeta_{12}+2\zeta_{12}^{2}+\cdots)q^{5}+\cdots$$
810.2.i.g $$8$$ $$6.468$$ 8.0.2702336256.1 None $$0$$ $$0$$ $$0$$ $$-6$$ $$q+(\beta _{4}+\beta _{6})q^{2}+(1+\beta _{2})q^{4}+(-\beta _{1}+\cdots)q^{5}+\cdots$$
810.2.i.h $$8$$ $$6.468$$ 8.0.2702336256.1 None $$0$$ $$0$$ $$0$$ $$0$$ $$q+\beta _{4}q^{2}-\beta _{2}q^{4}-\beta _{1}q^{5}+(1+\beta _{2}+\cdots)q^{7}+\cdots$$
810.2.i.i $$8$$ $$6.468$$ 8.0.2702336256.1 None $$0$$ $$0$$ $$0$$ $$6$$ $$q+(-\beta _{4}-\beta _{6})q^{2}+(1+\beta _{2})q^{4}+(\beta _{1}+\cdots)q^{5}+\cdots$$

## Decomposition of $$S_{2}^{\mathrm{old}}(810, [\chi])$$ into lower level spaces

$$S_{2}^{\mathrm{old}}(810, [\chi]) \cong$$ $$S_{2}^{\mathrm{new}}(45, [\chi])$$$$^{\oplus 6}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(90, [\chi])$$$$^{\oplus 3}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(135, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(270, [\chi])$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(405, [\chi])$$$$^{\oplus 2}$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ ($$1 - T^{2} + T^{4}$$)($$1 - T^{2} + T^{4}$$)($$1 - T^{2} + T^{4}$$)($$1 - T^{2} + T^{4}$$)($$1 - T^{2} + T^{4}$$)($$1 - T^{2} + T^{4}$$)($$( 1 - T^{2} + T^{4} )^{2}$$)($$( 1 - T^{2} + T^{4} )^{2}$$)($$( 1 - T^{2} + T^{4} )^{2}$$)
$3$ 1
$5$ ($$1 + 4 T + 11 T^{2} + 20 T^{3} + 25 T^{4}$$)($$1 + 4 T + 11 T^{2} + 20 T^{3} + 25 T^{4}$$)($$1 + 2 T - T^{2} + 10 T^{3} + 25 T^{4}$$)($$1 - 2 T - T^{2} - 10 T^{3} + 25 T^{4}$$)($$1 - 4 T + 11 T^{2} - 20 T^{3} + 25 T^{4}$$)($$1 - 4 T + 11 T^{2} - 20 T^{3} + 25 T^{4}$$)($$( 1 - 9 T^{2} + 25 T^{4} )^{2}$$)($$1 + 9 T^{2} + 56 T^{4} + 225 T^{6} + 625 T^{8}$$)($$1 + 9 T^{2} + 56 T^{4} + 225 T^{6} + 625 T^{8}$$)
$7$ ($$( 1 + 2 T^{2} + 49 T^{4} )( 1 + 11 T^{2} + 49 T^{4} )$$)($$1 + 10 T^{2} + 51 T^{4} + 490 T^{6} + 2401 T^{8}$$)($$( 1 - 13 T^{2} + 49 T^{4} )( 1 + 11 T^{2} + 49 T^{4} )$$)($$( 1 - 13 T^{2} + 49 T^{4} )( 1 + 11 T^{2} + 49 T^{4} )$$)($$1 + 10 T^{2} + 51 T^{4} + 490 T^{6} + 2401 T^{8}$$)($$( 1 + 2 T^{2} + 49 T^{4} )( 1 + 11 T^{2} + 49 T^{4} )$$)($$( 1 + 3 T + 13 T^{2} + 30 T^{3} + 72 T^{4} + 210 T^{5} + 637 T^{6} + 1029 T^{7} + 2401 T^{8} )^{2}$$)($$( 1 - 3 T + 2 T^{2} - 21 T^{3} + 49 T^{4} )^{2}( 1 + 3 T + 2 T^{2} + 21 T^{3} + 49 T^{4} )^{2}$$)($$( 1 - 3 T + 13 T^{2} - 30 T^{3} + 72 T^{4} - 210 T^{5} + 637 T^{6} - 1029 T^{7} + 2401 T^{8} )^{2}$$)
$11$ ($$( 1 + 4 T + 5 T^{2} + 44 T^{3} + 121 T^{4} )^{2}$$)($$( 1 - 2 T - 7 T^{2} - 22 T^{3} + 121 T^{4} )^{2}$$)($$( 1 + 5 T + 14 T^{2} + 55 T^{3} + 121 T^{4} )^{2}$$)($$( 1 - 5 T + 14 T^{2} - 55 T^{3} + 121 T^{4} )^{2}$$)($$( 1 + 2 T - 7 T^{2} + 22 T^{3} + 121 T^{4} )^{2}$$)($$( 1 - 4 T + 5 T^{2} - 44 T^{3} + 121 T^{4} )^{2}$$)($$( 1 - 6 T + 18 T^{2} - 36 T^{3} + 47 T^{4} - 396 T^{5} + 2178 T^{6} - 7986 T^{7} + 14641 T^{8} )( 1 + 6 T + 18 T^{2} + 36 T^{3} + 47 T^{4} + 396 T^{5} + 2178 T^{6} + 7986 T^{7} + 14641 T^{8} )$$)($$( 1 - 3 T^{2} - 112 T^{4} - 363 T^{6} + 14641 T^{8} )^{2}$$)($$( 1 - 6 T + 18 T^{2} - 36 T^{3} + 47 T^{4} - 396 T^{5} + 2178 T^{6} - 7986 T^{7} + 14641 T^{8} )( 1 + 6 T + 18 T^{2} + 36 T^{3} + 47 T^{4} + 396 T^{5} + 2178 T^{6} + 7986 T^{7} + 14641 T^{8} )$$)
$13$ ($$1 + 17 T^{2} + 120 T^{4} + 2873 T^{6} + 28561 T^{8}$$)($$( 1 - 4 T + 3 T^{2} - 52 T^{3} + 169 T^{4} )( 1 + 4 T + 3 T^{2} + 52 T^{3} + 169 T^{4} )$$)($$1 + 17 T^{2} + 120 T^{4} + 2873 T^{6} + 28561 T^{8}$$)($$1 + 17 T^{2} + 120 T^{4} + 2873 T^{6} + 28561 T^{8}$$)($$( 1 - 4 T + 3 T^{2} - 52 T^{3} + 169 T^{4} )( 1 + 4 T + 3 T^{2} + 52 T^{3} + 169 T^{4} )$$)($$1 + 17 T^{2} + 120 T^{4} + 2873 T^{6} + 28561 T^{8}$$)($$( 1 - 2 T + 13 T^{2} )^{4}( 1 + 5 T + 13 T^{2} )^{4}$$)($$( 1 + 13 T^{2} + 169 T^{4} )^{4}$$)($$( 1 - 5 T + 13 T^{2} )^{4}( 1 + 2 T + 13 T^{2} )^{4}$$)
$17$ ($$( 1 - 2 T + 17 T^{2} )^{2}( 1 + 2 T + 17 T^{2} )^{2}$$)($$( 1 - 8 T + 17 T^{2} )^{2}( 1 + 8 T + 17 T^{2} )^{2}$$)($$( 1 - 33 T^{2} + 289 T^{4} )^{2}$$)($$( 1 - 33 T^{2} + 289 T^{4} )^{2}$$)($$( 1 - 8 T + 17 T^{2} )^{2}( 1 + 8 T + 17 T^{2} )^{2}$$)($$( 1 - 2 T + 17 T^{2} )^{2}( 1 + 2 T + 17 T^{2} )^{2}$$)($$( 1 - 27 T^{2} + 404 T^{4} - 7803 T^{6} + 83521 T^{8} )^{2}$$)($$( 1 - 18 T^{2} + 289 T^{4} )^{4}$$)($$( 1 - 27 T^{2} + 404 T^{4} - 7803 T^{6} + 83521 T^{8} )^{2}$$)
$19$ ($$( 1 - 3 T + 19 T^{2} )^{4}$$)($$( 1 + 19 T^{2} )^{4}$$)($$( 1 - 6 T + 19 T^{2} )^{4}$$)($$( 1 - 6 T + 19 T^{2} )^{4}$$)($$( 1 + 19 T^{2} )^{4}$$)($$( 1 - 3 T + 19 T^{2} )^{4}$$)($$( 1 + 3 T + 26 T^{2} + 57 T^{3} + 361 T^{4} )^{4}$$)($$( 1 + 6 T + 19 T^{2} )^{8}$$)($$( 1 + 3 T + 26 T^{2} + 57 T^{3} + 361 T^{4} )^{4}$$)
$23$ ($$1 + 21 T^{2} - 88 T^{4} + 11109 T^{6} + 279841 T^{8}$$)($$1 + 30 T^{2} + 371 T^{4} + 15870 T^{6} + 279841 T^{8}$$)($$1 + 45 T^{2} + 1496 T^{4} + 23805 T^{6} + 279841 T^{8}$$)($$1 + 45 T^{2} + 1496 T^{4} + 23805 T^{6} + 279841 T^{8}$$)($$1 + 30 T^{2} + 371 T^{4} + 15870 T^{6} + 279841 T^{8}$$)($$1 + 21 T^{2} - 88 T^{4} + 11109 T^{6} + 279841 T^{8}$$)($$1 + 51 T^{2} + 1249 T^{4} + 14994 T^{6} + 172134 T^{8} + 7931826 T^{10} + 349521409 T^{12} + 7549830339 T^{14} + 78310985281 T^{16}$$)($$( 1 + 42 T^{2} + 1235 T^{4} + 22218 T^{6} + 279841 T^{8} )^{2}$$)($$1 + 51 T^{2} + 1249 T^{4} + 14994 T^{6} + 172134 T^{8} + 7931826 T^{10} + 349521409 T^{12} + 7549830339 T^{14} + 78310985281 T^{16}$$)
$29$ ($$( 1 - 29 T^{2} + 841 T^{4} )^{2}$$)($$( 1 - 29 T^{2} + 841 T^{4} )^{2}$$)($$( 1 - 9 T + 52 T^{2} - 261 T^{3} + 841 T^{4} )^{2}$$)($$( 1 + 9 T + 52 T^{2} + 261 T^{3} + 841 T^{4} )^{2}$$)($$( 1 - 29 T^{2} + 841 T^{4} )^{2}$$)($$( 1 - 29 T^{2} + 841 T^{4} )^{2}$$)($$1 - 17 T^{2} - 311 T^{4} + 18394 T^{6} - 590330 T^{8} + 15469354 T^{10} - 219964391 T^{12} - 10111996457 T^{14} + 500246412961 T^{16}$$)($$( 1 - 29 T^{2} + 841 T^{4} )^{4}$$)($$1 - 17 T^{2} - 311 T^{4} + 18394 T^{6} - 590330 T^{8} + 15469354 T^{10} - 219964391 T^{12} - 10111996457 T^{14} + 500246412961 T^{16}$$)
$31$ ($$( 1 - 2 T - 27 T^{2} - 62 T^{3} + 961 T^{4} )^{2}$$)($$( 1 - 8 T + 33 T^{2} - 248 T^{3} + 961 T^{4} )^{2}$$)($$( 1 - 5 T - 6 T^{2} - 155 T^{3} + 961 T^{4} )^{2}$$)($$( 1 - 5 T - 6 T^{2} - 155 T^{3} + 961 T^{4} )^{2}$$)($$( 1 - 8 T + 33 T^{2} - 248 T^{3} + 961 T^{4} )^{2}$$)($$( 1 - 2 T - 27 T^{2} - 62 T^{3} + 961 T^{4} )^{2}$$)($$( 1 + 2 T - 2 T^{2} - 112 T^{3} - 1049 T^{4} - 3472 T^{5} - 1922 T^{6} + 59582 T^{7} + 923521 T^{8} )^{2}$$)($$( 1 - 4 T + 31 T^{2} )^{4}( 1 + 11 T + 31 T^{2} )^{4}$$)($$( 1 + 2 T - 2 T^{2} - 112 T^{3} - 1049 T^{4} - 3472 T^{5} - 1922 T^{6} + 59582 T^{7} + 923521 T^{8} )^{2}$$)
$37$ ($$( 1 - 12 T + 37 T^{2} )^{2}( 1 + 12 T + 37 T^{2} )^{2}$$)($$( 1 - 12 T + 37 T^{2} )^{2}( 1 + 12 T + 37 T^{2} )^{2}$$)($$( 1 - 12 T + 37 T^{2} )^{2}( 1 + 12 T + 37 T^{2} )^{2}$$)($$( 1 - 12 T + 37 T^{2} )^{2}( 1 + 12 T + 37 T^{2} )^{2}$$)($$( 1 - 12 T + 37 T^{2} )^{2}( 1 + 12 T + 37 T^{2} )^{2}$$)($$( 1 - 12 T + 37 T^{2} )^{2}( 1 + 12 T + 37 T^{2} )^{2}$$)($$( 1 - 17 T^{2} + 1656 T^{4} - 23273 T^{6} + 1874161 T^{8} )^{2}$$)($$( 1 + 2 T^{2} + 1369 T^{4} )^{4}$$)($$( 1 - 17 T^{2} + 1656 T^{4} - 23273 T^{6} + 1874161 T^{8} )^{2}$$)
$41$ ($$( 1 + 7 T + 8 T^{2} + 287 T^{3} + 1681 T^{4} )^{2}$$)($$( 1 - 2 T - 37 T^{2} - 82 T^{3} + 1681 T^{4} )^{2}$$)($$( 1 + 2 T - 37 T^{2} + 82 T^{3} + 1681 T^{4} )^{2}$$)($$( 1 - 2 T - 37 T^{2} - 82 T^{3} + 1681 T^{4} )^{2}$$)($$( 1 + 2 T - 37 T^{2} + 82 T^{3} + 1681 T^{4} )^{2}$$)($$( 1 - 7 T + 8 T^{2} - 287 T^{3} + 1681 T^{4} )^{2}$$)($$1 - 33 T^{2} - 1391 T^{4} + 29106 T^{6} + 1494030 T^{8} + 48927186 T^{10} - 3930633551 T^{12} - 156753439953 T^{14} + 7984925229121 T^{16}$$)($$( 1 - 6 T^{2} - 1645 T^{4} - 10086 T^{6} + 2825761 T^{8} )^{2}$$)($$1 - 33 T^{2} - 1391 T^{4} + 29106 T^{6} + 1494030 T^{8} + 48927186 T^{10} - 3930633551 T^{12} - 156753439953 T^{14} + 7984925229121 T^{16}$$)
$43$ ($$1 - 14 T^{2} - 1653 T^{4} - 25886 T^{6} + 3418801 T^{8}$$)($$1 + 70 T^{2} + 3051 T^{4} + 129430 T^{6} + 3418801 T^{8}$$)($$1 + 85 T^{2} + 5376 T^{4} + 157165 T^{6} + 3418801 T^{8}$$)($$1 + 85 T^{2} + 5376 T^{4} + 157165 T^{6} + 3418801 T^{8}$$)($$1 + 70 T^{2} + 3051 T^{4} + 129430 T^{6} + 3418801 T^{8}$$)($$1 - 14 T^{2} - 1653 T^{4} - 25886 T^{6} + 3418801 T^{8}$$)($$( 1 + 6 T + 82 T^{2} + 420 T^{3} + 3567 T^{4} + 18060 T^{5} + 151618 T^{6} + 477042 T^{7} + 3418801 T^{8} )^{2}$$)($$( 1 + 10 T^{2} - 1749 T^{4} + 18490 T^{6} + 3418801 T^{8} )^{2}$$)($$( 1 - 6 T + 82 T^{2} - 420 T^{3} + 3567 T^{4} - 18060 T^{5} + 151618 T^{6} - 477042 T^{7} + 3418801 T^{8} )^{2}$$)
$47$ ($$1 + 45 T^{2} - 184 T^{4} + 99405 T^{6} + 4879681 T^{8}$$)($$1 + 30 T^{2} - 1309 T^{4} + 66270 T^{6} + 4879681 T^{8}$$)($$1 - 75 T^{2} + 3416 T^{4} - 165675 T^{6} + 4879681 T^{8}$$)($$1 - 75 T^{2} + 3416 T^{4} - 165675 T^{6} + 4879681 T^{8}$$)($$1 + 30 T^{2} - 1309 T^{4} + 66270 T^{6} + 4879681 T^{8}$$)($$1 + 45 T^{2} - 184 T^{4} + 99405 T^{6} + 4879681 T^{8}$$)($$( 1 + 75 T^{2} + 2209 T^{4} )^{2}( 1 - 75 T^{2} + 3416 T^{4} - 165675 T^{6} + 4879681 T^{8} )$$)($$( 1 + 90 T^{2} + 5891 T^{4} + 198810 T^{6} + 4879681 T^{8} )^{2}$$)($$( 1 + 75 T^{2} + 2209 T^{4} )^{2}( 1 - 75 T^{2} + 3416 T^{4} - 165675 T^{6} + 4879681 T^{8} )$$)
$53$ ($$( 1 - 97 T^{2} + 2809 T^{4} )^{2}$$)($$( 1 - 70 T^{2} + 2809 T^{4} )^{2}$$)($$( 1 - 53 T^{2} )^{4}$$)($$( 1 - 53 T^{2} )^{4}$$)($$( 1 - 70 T^{2} + 2809 T^{4} )^{2}$$)($$( 1 - 97 T^{2} + 2809 T^{4} )^{2}$$)($$( 1 - 143 T^{2} + 9576 T^{4} - 401687 T^{6} + 7890481 T^{8} )^{2}$$)($$( 1 - 97 T^{2} + 2809 T^{4} )^{4}$$)($$( 1 - 143 T^{2} + 9576 T^{4} - 401687 T^{6} + 7890481 T^{8} )^{2}$$)
$59$ ($$( 1 + T - 58 T^{2} + 59 T^{3} + 3481 T^{4} )^{2}$$)($$( 1 + 10 T + 41 T^{2} + 590 T^{3} + 3481 T^{4} )^{2}$$)($$( 1 - 4 T - 43 T^{2} - 236 T^{3} + 3481 T^{4} )^{2}$$)($$( 1 + 4 T - 43 T^{2} + 236 T^{3} + 3481 T^{4} )^{2}$$)($$( 1 - 10 T + 41 T^{2} - 590 T^{3} + 3481 T^{4} )^{2}$$)($$( 1 - T - 58 T^{2} - 59 T^{3} + 3481 T^{4} )^{2}$$)($$1 - 153 T^{2} + 11293 T^{4} - 788562 T^{6} + 53193366 T^{8} - 2744984322 T^{10} + 136841357773 T^{12} - 6453621647073 T^{14} + 146830437604321 T^{16}$$)($$( 1 - 42 T^{2} - 1717 T^{4} - 146202 T^{6} + 12117361 T^{8} )^{2}$$)($$1 - 153 T^{2} + 11293 T^{4} - 788562 T^{6} + 53193366 T^{8} - 2744984322 T^{10} + 136841357773 T^{12} - 6453621647073 T^{14} + 146830437604321 T^{16}$$)
$61$ ($$( 1 - 10 T + 39 T^{2} - 610 T^{3} + 3721 T^{4} )^{2}$$)($$( 1 + 2 T - 57 T^{2} + 122 T^{3} + 3721 T^{4} )^{2}$$)($$( 1 + 8 T + 3 T^{2} + 488 T^{3} + 3721 T^{4} )^{2}$$)($$( 1 + 8 T + 3 T^{2} + 488 T^{3} + 3721 T^{4} )^{2}$$)($$( 1 + 2 T - 57 T^{2} + 122 T^{3} + 3721 T^{4} )^{2}$$)($$( 1 - 10 T + 39 T^{2} - 610 T^{3} + 3721 T^{4} )^{2}$$)($$( 1 + 7 T - 71 T^{2} - 14 T^{3} + 7690 T^{4} - 854 T^{5} - 264191 T^{6} + 1588867 T^{7} + 13845841 T^{8} )^{2}$$)($$( 1 - 4 T - 45 T^{2} - 244 T^{3} + 3721 T^{4} )^{4}$$)($$( 1 + 7 T - 71 T^{2} - 14 T^{3} + 7690 T^{4} - 854 T^{5} - 264191 T^{6} + 1588867 T^{7} + 13845841 T^{8} )^{2}$$)
$67$ ($$( 1 - 109 T^{2} + 4489 T^{4} )( 1 - 13 T^{2} + 4489 T^{4} )$$)($$1 + 70 T^{2} + 411 T^{4} + 314230 T^{6} + 20151121 T^{8}$$)($$1 + 118 T^{2} + 9435 T^{4} + 529702 T^{6} + 20151121 T^{8}$$)($$1 + 118 T^{2} + 9435 T^{4} + 529702 T^{6} + 20151121 T^{8}$$)($$1 + 70 T^{2} + 411 T^{4} + 314230 T^{6} + 20151121 T^{8}$$)($$( 1 - 109 T^{2} + 4489 T^{4} )( 1 - 13 T^{2} + 4489 T^{4} )$$)($$( 1 - 12 T + 118 T^{2} - 840 T^{3} + 3627 T^{4} - 56280 T^{5} + 529702 T^{6} - 3609156 T^{7} + 20151121 T^{8} )^{2}$$)($$( 1 + 58 T^{2} - 1125 T^{4} + 260362 T^{6} + 20151121 T^{8} )^{2}$$)($$( 1 + 12 T + 118 T^{2} + 840 T^{3} + 3627 T^{4} + 56280 T^{5} + 529702 T^{6} + 3609156 T^{7} + 20151121 T^{8} )^{2}$$)
$71$ ($$( 1 + 12 T + 71 T^{2} )^{4}$$)($$( 1 + 12 T + 71 T^{2} )^{4}$$)($$( 1 - 6 T + 71 T^{2} )^{4}$$)($$( 1 + 6 T + 71 T^{2} )^{4}$$)($$( 1 - 12 T + 71 T^{2} )^{4}$$)($$( 1 - 12 T + 71 T^{2} )^{4}$$)($$( 1 + 94 T^{2} + 5041 T^{4} )^{4}$$)($$( 1 + 71 T^{2} )^{8}$$)($$( 1 + 94 T^{2} + 5041 T^{4} )^{4}$$)
$73$ ($$( 1 - 142 T^{2} + 5329 T^{4} )^{2}$$)($$( 1 - 130 T^{2} + 5329 T^{4} )^{2}$$)($$( 1 - 142 T^{2} + 5329 T^{4} )^{2}$$)($$( 1 - 142 T^{2} + 5329 T^{4} )^{2}$$)($$( 1 - 130 T^{2} + 5329 T^{4} )^{2}$$)($$( 1 - 142 T^{2} + 5329 T^{4} )^{2}$$)($$( 1 + 19 T^{2} - 6708 T^{4} + 101251 T^{6} + 28398241 T^{8} )^{2}$$)($$( 1 - 127 T^{2} + 5329 T^{4} )^{4}$$)($$( 1 + 19 T^{2} - 6708 T^{4} + 101251 T^{6} + 28398241 T^{8} )^{2}$$)
$79$ ($$( 1 - 12 T + 65 T^{2} - 948 T^{3} + 6241 T^{4} )^{2}$$)($$( 1 - 79 T^{2} + 6241 T^{4} )^{2}$$)($$( 1 - 9 T + 2 T^{2} - 711 T^{3} + 6241 T^{4} )^{2}$$)($$( 1 - 9 T + 2 T^{2} - 711 T^{3} + 6241 T^{4} )^{2}$$)($$( 1 - 79 T^{2} + 6241 T^{4} )^{2}$$)($$( 1 - 12 T + 65 T^{2} - 948 T^{3} + 6241 T^{4} )^{2}$$)($$( 1 - 79 T^{2} + 6241 T^{4} )^{4}$$)($$( 1 - 79 T^{2} + 6241 T^{4} )^{4}$$)($$( 1 - 79 T^{2} + 6241 T^{4} )^{4}$$)
$83$ ($$1 + 102 T^{2} + 3515 T^{4} + 702678 T^{6} + 47458321 T^{8}$$)($$1 + 150 T^{2} + 15611 T^{4} + 1033350 T^{6} + 47458321 T^{8}$$)($$1 + 150 T^{2} + 15611 T^{4} + 1033350 T^{6} + 47458321 T^{8}$$)($$1 + 150 T^{2} + 15611 T^{4} + 1033350 T^{6} + 47458321 T^{8}$$)($$1 + 150 T^{2} + 15611 T^{4} + 1033350 T^{6} + 47458321 T^{8}$$)($$1 + 102 T^{2} + 3515 T^{4} + 702678 T^{6} + 47458321 T^{8}$$)($$( 1 + 150 T^{2} + 15611 T^{4} + 1033350 T^{6} + 47458321 T^{8} )^{2}$$)($$( 1 + 141 T^{2} + 12992 T^{4} + 971349 T^{6} + 47458321 T^{8} )^{2}$$)($$( 1 + 150 T^{2} + 15611 T^{4} + 1033350 T^{6} + 47458321 T^{8} )^{2}$$)
$89$ ($$( 1 - 14 T + 89 T^{2} )^{4}$$)($$( 1 + 10 T + 89 T^{2} )^{4}$$)($$( 1 + 14 T + 89 T^{2} )^{4}$$)($$( 1 - 14 T + 89 T^{2} )^{4}$$)($$( 1 - 10 T + 89 T^{2} )^{4}$$)($$( 1 + 14 T + 89 T^{2} )^{4}$$)($$( 1 + 345 T^{2} + 45584 T^{4} + 2732745 T^{6} + 62742241 T^{8} )^{2}$$)($$( 1 + 102 T^{2} + 7921 T^{4} )^{4}$$)($$( 1 + 345 T^{2} + 45584 T^{4} + 2732745 T^{6} + 62742241 T^{8} )^{2}$$)
$97$ ($$( 1 - 18 T + 227 T^{2} - 1746 T^{3} + 9409 T^{4} )( 1 + 18 T + 227 T^{2} + 1746 T^{3} + 9409 T^{4} )$$)($$( 1 - 18 T + 227 T^{2} - 1746 T^{3} + 9409 T^{4} )( 1 + 18 T + 227 T^{2} + 1746 T^{3} + 9409 T^{4} )$$)($$1 + 94 T^{2} - 573 T^{4} + 884446 T^{6} + 88529281 T^{8}$$)($$1 + 94 T^{2} - 573 T^{4} + 884446 T^{6} + 88529281 T^{8}$$)($$( 1 - 18 T + 227 T^{2} - 1746 T^{3} + 9409 T^{4} )( 1 + 18 T + 227 T^{2} + 1746 T^{3} + 9409 T^{4} )$$)($$( 1 - 18 T + 227 T^{2} - 1746 T^{3} + 9409 T^{4} )( 1 + 18 T + 227 T^{2} + 1746 T^{3} + 9409 T^{4} )$$)($$( 1 + 36 T + 658 T^{2} + 8136 T^{3} + 83571 T^{4} + 789192 T^{5} + 6191122 T^{6} + 32856228 T^{7} + 88529281 T^{8} )^{2}$$)($$( 1 + 175 T^{2} + 21216 T^{4} + 1646575 T^{6} + 88529281 T^{8} )^{2}$$)($$( 1 - 36 T + 658 T^{2} - 8136 T^{3} + 83571 T^{4} - 789192 T^{5} + 6191122 T^{6} - 32856228 T^{7} + 88529281 T^{8} )^{2}$$)