# Properties

 Label 810.2.e Level $810$ Weight $2$ Character orbit 810.e Rep. character $\chi_{810}(271,\cdot)$ Character field $\Q(\zeta_{3})$ Dimension $32$ Newform subspaces $14$ 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.e (of order $$3$$ and degree $$2$$) Character conductor: $$\operatorname{cond}(\chi)$$ $$=$$ $$9$$ Character field: $$\Q(\zeta_{3})$$ Newform subspaces: $$14$$ Sturm bound: $$324$$ Trace bound: $$11$$ Distinguishing $$T_p$$: $$7$$, $$11$$, $$13$$, $$17$$, $$23$$

## Dimensions

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

Total New Old
Modular forms 372 32 340
Cusp forms 276 32 244
Eisenstein series 96 0 96

## Trace form

 $$32q - 16q^{4} - 20q^{7} + O(q^{10})$$ $$32q - 16q^{4} - 20q^{7} - 20q^{13} - 16q^{16} + 16q^{19} + 12q^{22} - 16q^{25} + 40q^{28} - 20q^{31} + 12q^{34} + 40q^{37} - 8q^{43} - 24q^{46} - 24q^{49} - 20q^{52} + 16q^{61} + 32q^{64} + 16q^{67} + 12q^{70} - 32q^{73} - 8q^{76} + 4q^{79} - 24q^{82} + 24q^{85} + 12q^{88} - 16q^{91} + 12q^{94} - 8q^{97} + O(q^{100})$$

## 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.e.a $$2$$ $$6.468$$ $$\Q(\sqrt{-3})$$ None $$-1$$ $$0$$ $$-1$$ $$-2$$ $$q+(-1+\zeta_{6})q^{2}-\zeta_{6}q^{4}-\zeta_{6}q^{5}+(-2+\cdots)q^{7}+\cdots$$
810.2.e.b $$2$$ $$6.468$$ $$\Q(\sqrt{-3})$$ None $$-1$$ $$0$$ $$-1$$ $$4$$ $$q+(-1+\zeta_{6})q^{2}-\zeta_{6}q^{4}-\zeta_{6}q^{5}+(4+\cdots)q^{7}+\cdots$$
810.2.e.c $$2$$ $$6.468$$ $$\Q(\sqrt{-3})$$ None $$-1$$ $$0$$ $$1$$ $$-5$$ $$q+(-1+\zeta_{6})q^{2}-\zeta_{6}q^{4}+\zeta_{6}q^{5}+(-5+\cdots)q^{7}+\cdots$$
810.2.e.d $$2$$ $$6.468$$ $$\Q(\sqrt{-3})$$ None $$-1$$ $$0$$ $$1$$ $$-2$$ $$q+(-1+\zeta_{6})q^{2}-\zeta_{6}q^{4}+\zeta_{6}q^{5}+(-2+\cdots)q^{7}+\cdots$$
810.2.e.e $$2$$ $$6.468$$ $$\Q(\sqrt{-3})$$ None $$-1$$ $$0$$ $$1$$ $$-2$$ $$q+(-1+\zeta_{6})q^{2}-\zeta_{6}q^{4}+\zeta_{6}q^{5}+(-2+\cdots)q^{7}+\cdots$$
810.2.e.f $$2$$ $$6.468$$ $$\Q(\sqrt{-3})$$ None $$-1$$ $$0$$ $$1$$ $$1$$ $$q+(-1+\zeta_{6})q^{2}-\zeta_{6}q^{4}+\zeta_{6}q^{5}+(1+\cdots)q^{7}+\cdots$$
810.2.e.g $$2$$ $$6.468$$ $$\Q(\sqrt{-3})$$ None $$1$$ $$0$$ $$-1$$ $$-5$$ $$q+(1-\zeta_{6})q^{2}-\zeta_{6}q^{4}-\zeta_{6}q^{5}+(-5+\cdots)q^{7}+\cdots$$
810.2.e.h $$2$$ $$6.468$$ $$\Q(\sqrt{-3})$$ None $$1$$ $$0$$ $$-1$$ $$-2$$ $$q+(1-\zeta_{6})q^{2}-\zeta_{6}q^{4}-\zeta_{6}q^{5}+(-2+\cdots)q^{7}+\cdots$$
810.2.e.i $$2$$ $$6.468$$ $$\Q(\sqrt{-3})$$ None $$1$$ $$0$$ $$-1$$ $$-2$$ $$q+(1-\zeta_{6})q^{2}-\zeta_{6}q^{4}-\zeta_{6}q^{5}+(-2+\cdots)q^{7}+\cdots$$
810.2.e.j $$2$$ $$6.468$$ $$\Q(\sqrt{-3})$$ None $$1$$ $$0$$ $$-1$$ $$1$$ $$q+(1-\zeta_{6})q^{2}-\zeta_{6}q^{4}-\zeta_{6}q^{5}+(1-\zeta_{6})q^{7}+\cdots$$
810.2.e.k $$2$$ $$6.468$$ $$\Q(\sqrt{-3})$$ None $$1$$ $$0$$ $$1$$ $$-2$$ $$q+(1-\zeta_{6})q^{2}-\zeta_{6}q^{4}+\zeta_{6}q^{5}+(-2+\cdots)q^{7}+\cdots$$
810.2.e.l $$2$$ $$6.468$$ $$\Q(\sqrt{-3})$$ None $$1$$ $$0$$ $$1$$ $$4$$ $$q+(1-\zeta_{6})q^{2}-\zeta_{6}q^{4}+\zeta_{6}q^{5}+(4-4\zeta_{6})q^{7}+\cdots$$
810.2.e.m $$4$$ $$6.468$$ $$\Q(\zeta_{12})$$ None $$-2$$ $$0$$ $$-2$$ $$-4$$ $$q-\zeta_{12}q^{2}+(-1+\zeta_{12})q^{4}+(-1+\zeta_{12}+\cdots)q^{5}+\cdots$$
810.2.e.n $$4$$ $$6.468$$ $$\Q(\zeta_{12})$$ None $$2$$ $$0$$ $$2$$ $$-4$$ $$q+\zeta_{12}q^{2}+(-1+\zeta_{12})q^{4}+(1-\zeta_{12}+\cdots)q^{5}+\cdots$$

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

$$S_{2}^{\mathrm{old}}(810, [\chi]) \cong$$ $$S_{2}^{\mathrm{new}}(18, [\chi])$$$$^{\oplus 6}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(45, [\chi])$$$$^{\oplus 6}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(54, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(81, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(90, [\chi])$$$$^{\oplus 3}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(135, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(162, [\chi])$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(270, [\chi])$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(405, [\chi])$$$$^{\oplus 2}$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ ($$1 + T + T^{2}$$)($$1 + T + T^{2}$$)($$1 + T + T^{2}$$)($$1 + T + T^{2}$$)($$1 + T + T^{2}$$)($$1 + T + T^{2}$$)($$1 - T + T^{2}$$)($$1 - T + T^{2}$$)($$1 - T + T^{2}$$)($$1 - T + T^{2}$$)($$1 - T + T^{2}$$)($$1 - T + T^{2}$$)($$( 1 + T + T^{2} )^{2}$$)($$( 1 - T + T^{2} )^{2}$$)
$3$ 1
$5$ ($$1 + T + T^{2}$$)($$1 + T + T^{2}$$)($$1 - T + T^{2}$$)($$1 - T + T^{2}$$)($$1 - T + T^{2}$$)($$1 - T + T^{2}$$)($$1 + T + T^{2}$$)($$1 + T + T^{2}$$)($$1 + T + T^{2}$$)($$1 + T + T^{2}$$)($$1 - T + T^{2}$$)($$1 - T + T^{2}$$)($$( 1 + T + T^{2} )^{2}$$)($$( 1 - T + T^{2} )^{2}$$)
$7$ ($$1 + 2 T - 3 T^{2} + 14 T^{3} + 49 T^{4}$$)($$( 1 - 5 T + 7 T^{2} )( 1 + T + 7 T^{2} )$$)($$( 1 + T + 7 T^{2} )( 1 + 4 T + 7 T^{2} )$$)($$1 + 2 T - 3 T^{2} + 14 T^{3} + 49 T^{4}$$)($$1 + 2 T - 3 T^{2} + 14 T^{3} + 49 T^{4}$$)($$( 1 - 5 T + 7 T^{2} )( 1 + 4 T + 7 T^{2} )$$)($$( 1 + T + 7 T^{2} )( 1 + 4 T + 7 T^{2} )$$)($$1 + 2 T - 3 T^{2} + 14 T^{3} + 49 T^{4}$$)($$1 + 2 T - 3 T^{2} + 14 T^{3} + 49 T^{4}$$)($$( 1 - 5 T + 7 T^{2} )( 1 + 4 T + 7 T^{2} )$$)($$1 + 2 T - 3 T^{2} + 14 T^{3} + 49 T^{4}$$)($$( 1 - 5 T + 7 T^{2} )( 1 + T + 7 T^{2} )$$)($$1 + 4 T + T^{2} + 4 T^{3} + 64 T^{4} + 28 T^{5} + 49 T^{6} + 1372 T^{7} + 2401 T^{8}$$)($$1 + 4 T + T^{2} + 4 T^{3} + 64 T^{4} + 28 T^{5} + 49 T^{6} + 1372 T^{7} + 2401 T^{8}$$)
$11$ ($$1 - 3 T - 2 T^{2} - 33 T^{3} + 121 T^{4}$$)($$1 - 11 T^{2} + 121 T^{4}$$)($$1 - 11 T^{2} + 121 T^{4}$$)($$1 + 3 T - 2 T^{2} + 33 T^{3} + 121 T^{4}$$)($$1 - 6 T + 25 T^{2} - 66 T^{3} + 121 T^{4}$$)($$1 - 11 T^{2} + 121 T^{4}$$)($$1 - 11 T^{2} + 121 T^{4}$$)($$1 + 6 T + 25 T^{2} + 66 T^{3} + 121 T^{4}$$)($$1 - 3 T - 2 T^{2} - 33 T^{3} + 121 T^{4}$$)($$1 - 11 T^{2} + 121 T^{4}$$)($$1 + 3 T - 2 T^{2} + 33 T^{3} + 121 T^{4}$$)($$1 - 11 T^{2} + 121 T^{4}$$)($$1 - 10 T^{2} - 21 T^{4} - 1210 T^{6} + 14641 T^{8}$$)($$1 - 10 T^{2} - 21 T^{4} - 1210 T^{6} + 14641 T^{8}$$)
$13$ ($$1 - T - 12 T^{2} - 13 T^{3} + 169 T^{4}$$)($$( 1 - 5 T + 13 T^{2} )( 1 + 7 T + 13 T^{2} )$$)($$1 - T - 12 T^{2} - 13 T^{3} + 169 T^{4}$$)($$( 1 - 2 T + 13 T^{2} )( 1 + 7 T + 13 T^{2} )$$)($$1 - 4 T + 3 T^{2} - 52 T^{3} + 169 T^{4}$$)($$( 1 - 2 T + 13 T^{2} )( 1 + 7 T + 13 T^{2} )$$)($$1 - T - 12 T^{2} - 13 T^{3} + 169 T^{4}$$)($$1 - 4 T + 3 T^{2} - 52 T^{3} + 169 T^{4}$$)($$( 1 - 2 T + 13 T^{2} )( 1 + 7 T + 13 T^{2} )$$)($$( 1 - 2 T + 13 T^{2} )( 1 + 7 T + 13 T^{2} )$$)($$1 - T - 12 T^{2} - 13 T^{3} + 169 T^{4}$$)($$( 1 - 5 T + 13 T^{2} )( 1 + 7 T + 13 T^{2} )$$)($$1 + 4 T - 11 T^{2} + 4 T^{3} + 352 T^{4} + 52 T^{5} - 1859 T^{6} + 8788 T^{7} + 28561 T^{8}$$)($$1 + 4 T - 11 T^{2} + 4 T^{3} + 352 T^{4} + 52 T^{5} - 1859 T^{6} + 8788 T^{7} + 28561 T^{8}$$)
$17$ ($$( 1 + 3 T + 17 T^{2} )^{2}$$)($$( 1 + 6 T + 17 T^{2} )^{2}$$)($$( 1 + 6 T + 17 T^{2} )^{2}$$)($$( 1 + 3 T + 17 T^{2} )^{2}$$)($$( 1 - 6 T + 17 T^{2} )^{2}$$)($$( 1 - 6 T + 17 T^{2} )^{2}$$)($$( 1 - 6 T + 17 T^{2} )^{2}$$)($$( 1 + 6 T + 17 T^{2} )^{2}$$)($$( 1 - 3 T + 17 T^{2} )^{2}$$)($$( 1 + 6 T + 17 T^{2} )^{2}$$)($$( 1 - 3 T + 17 T^{2} )^{2}$$)($$( 1 - 6 T + 17 T^{2} )^{2}$$)($$( 1 + 22 T^{2} + 289 T^{4} )^{2}$$)($$( 1 + 22 T^{2} + 289 T^{4} )^{2}$$)
$19$ ($$( 1 - 8 T + 19 T^{2} )^{2}$$)($$( 1 + 4 T + 19 T^{2} )^{2}$$)($$( 1 - 5 T + 19 T^{2} )^{2}$$)($$( 1 + 4 T + 19 T^{2} )^{2}$$)($$( 1 + 4 T + 19 T^{2} )^{2}$$)($$( 1 - 5 T + 19 T^{2} )^{2}$$)($$( 1 - 5 T + 19 T^{2} )^{2}$$)($$( 1 + 4 T + 19 T^{2} )^{2}$$)($$( 1 + 4 T + 19 T^{2} )^{2}$$)($$( 1 - 5 T + 19 T^{2} )^{2}$$)($$( 1 - 8 T + 19 T^{2} )^{2}$$)($$( 1 + 4 T + 19 T^{2} )^{2}$$)($$( 1 + 2 T - 9 T^{2} + 38 T^{3} + 361 T^{4} )^{2}$$)($$( 1 + 2 T - 9 T^{2} + 38 T^{3} + 361 T^{4} )^{2}$$)
$23$ ($$1 + 3 T - 14 T^{2} + 69 T^{3} + 529 T^{4}$$)($$1 - 23 T^{2} + 529 T^{4}$$)($$1 + 9 T + 58 T^{2} + 207 T^{3} + 529 T^{4}$$)($$1 - 9 T + 58 T^{2} - 207 T^{3} + 529 T^{4}$$)($$1 - 23 T^{2} + 529 T^{4}$$)($$1 - 3 T - 14 T^{2} - 69 T^{3} + 529 T^{4}$$)($$1 - 9 T + 58 T^{2} - 207 T^{3} + 529 T^{4}$$)($$1 - 23 T^{2} + 529 T^{4}$$)($$1 + 9 T + 58 T^{2} + 207 T^{3} + 529 T^{4}$$)($$1 + 3 T - 14 T^{2} + 69 T^{3} + 529 T^{4}$$)($$1 - 3 T - 14 T^{2} - 69 T^{3} + 529 T^{4}$$)($$1 - 23 T^{2} + 529 T^{4}$$)($$1 - 6 T - 7 T^{2} + 18 T^{3} + 492 T^{4} + 414 T^{5} - 3703 T^{6} - 73002 T^{7} + 279841 T^{8}$$)($$1 + 6 T - 7 T^{2} - 18 T^{3} + 492 T^{4} - 414 T^{5} - 3703 T^{6} + 73002 T^{7} + 279841 T^{8}$$)
$29$ ($$1 - 9 T + 52 T^{2} - 261 T^{3} + 841 T^{4}$$)($$1 + 6 T + 7 T^{2} + 174 T^{3} + 841 T^{4}$$)($$1 - 29 T^{2} + 841 T^{4}$$)($$1 - 3 T - 20 T^{2} - 87 T^{3} + 841 T^{4}$$)($$1 + 6 T + 7 T^{2} + 174 T^{3} + 841 T^{4}$$)($$1 - 29 T^{2} + 841 T^{4}$$)($$1 - 29 T^{2} + 841 T^{4}$$)($$1 - 6 T + 7 T^{2} - 174 T^{3} + 841 T^{4}$$)($$1 + 3 T - 20 T^{2} + 87 T^{3} + 841 T^{4}$$)($$1 - 29 T^{2} + 841 T^{4}$$)($$1 + 9 T + 52 T^{2} + 261 T^{3} + 841 T^{4}$$)($$1 - 6 T + 7 T^{2} - 174 T^{3} + 841 T^{4}$$)($$1 - 10 T^{2} - 741 T^{4} - 8410 T^{6} + 707281 T^{8}$$)($$1 - 10 T^{2} - 741 T^{4} - 8410 T^{6} + 707281 T^{8}$$)
$31$ ($$( 1 - 11 T + 31 T^{2} )( 1 + 4 T + 31 T^{2} )$$)($$1 + 8 T + 33 T^{2} + 248 T^{3} + 961 T^{4}$$)($$( 1 - 11 T + 31 T^{2} )( 1 + 7 T + 31 T^{2} )$$)($$1 + 5 T - 6 T^{2} + 155 T^{3} + 961 T^{4}$$)($$( 1 - 11 T + 31 T^{2} )( 1 + 7 T + 31 T^{2} )$$)($$1 + 8 T + 33 T^{2} + 248 T^{3} + 961 T^{4}$$)($$( 1 - 11 T + 31 T^{2} )( 1 + 7 T + 31 T^{2} )$$)($$( 1 - 11 T + 31 T^{2} )( 1 + 7 T + 31 T^{2} )$$)($$1 + 5 T - 6 T^{2} + 155 T^{3} + 961 T^{4}$$)($$1 + 8 T + 33 T^{2} + 248 T^{3} + 961 T^{4}$$)($$( 1 - 11 T + 31 T^{2} )( 1 + 4 T + 31 T^{2} )$$)($$1 + 8 T + 33 T^{2} + 248 T^{3} + 961 T^{4}$$)($$1 + 4 T - 38 T^{2} - 32 T^{3} + 1459 T^{4} - 992 T^{5} - 36518 T^{6} + 119164 T^{7} + 923521 T^{8}$$)($$1 + 4 T - 38 T^{2} - 32 T^{3} + 1459 T^{4} - 992 T^{5} - 36518 T^{6} + 119164 T^{7} + 923521 T^{8}$$)
$37$ ($$( 1 - 2 T + 37 T^{2} )^{2}$$)($$( 1 - 2 T + 37 T^{2} )^{2}$$)($$( 1 + 10 T + 37 T^{2} )^{2}$$)($$( 1 + 10 T + 37 T^{2} )^{2}$$)($$( 1 - 8 T + 37 T^{2} )^{2}$$)($$( 1 - 2 T + 37 T^{2} )^{2}$$)($$( 1 + 10 T + 37 T^{2} )^{2}$$)($$( 1 - 8 T + 37 T^{2} )^{2}$$)($$( 1 + 10 T + 37 T^{2} )^{2}$$)($$( 1 - 2 T + 37 T^{2} )^{2}$$)($$( 1 - 2 T + 37 T^{2} )^{2}$$)($$( 1 - 2 T + 37 T^{2} )^{2}$$)($$( 1 - 8 T + 37 T^{2} )^{4}$$)($$( 1 - 8 T + 37 T^{2} )^{4}$$)
$41$ ($$1 - 12 T + 103 T^{2} - 492 T^{3} + 1681 T^{4}$$)($$1 + 6 T - 5 T^{2} + 246 T^{3} + 1681 T^{4}$$)($$1 + 3 T - 32 T^{2} + 123 T^{3} + 1681 T^{4}$$)($$1 - 41 T^{2} + 1681 T^{4}$$)($$1 - 41 T^{2} + 1681 T^{4}$$)($$1 - 3 T - 32 T^{2} - 123 T^{3} + 1681 T^{4}$$)($$1 - 3 T - 32 T^{2} - 123 T^{3} + 1681 T^{4}$$)($$1 - 41 T^{2} + 1681 T^{4}$$)($$1 - 41 T^{2} + 1681 T^{4}$$)($$1 + 3 T - 32 T^{2} + 123 T^{3} + 1681 T^{4}$$)($$1 + 12 T + 103 T^{2} + 492 T^{3} + 1681 T^{4}$$)($$1 - 6 T - 5 T^{2} - 246 T^{3} + 1681 T^{4}$$)($$1 - 55 T^{2} + 1344 T^{4} - 92455 T^{6} + 2825761 T^{8}$$)($$1 - 55 T^{2} + 1344 T^{4} - 92455 T^{6} + 2825761 T^{8}$$)
$43$ ($$1 - 7 T + 6 T^{2} - 301 T^{3} + 1849 T^{4}$$)($$1 - 4 T - 27 T^{2} - 172 T^{3} + 1849 T^{4}$$)($$( 1 - 5 T + 43 T^{2} )( 1 + 13 T + 43 T^{2} )$$)($$1 - T - 42 T^{2} - 43 T^{3} + 1849 T^{4}$$)($$( 1 - 5 T + 43 T^{2} )( 1 + 13 T + 43 T^{2} )$$)($$1 - 4 T - 27 T^{2} - 172 T^{3} + 1849 T^{4}$$)($$( 1 - 5 T + 43 T^{2} )( 1 + 13 T + 43 T^{2} )$$)($$( 1 - 5 T + 43 T^{2} )( 1 + 13 T + 43 T^{2} )$$)($$1 - T - 42 T^{2} - 43 T^{3} + 1849 T^{4}$$)($$1 - 4 T - 27 T^{2} - 172 T^{3} + 1849 T^{4}$$)($$1 - 7 T + 6 T^{2} - 301 T^{3} + 1849 T^{4}$$)($$1 - 4 T - 27 T^{2} - 172 T^{3} + 1849 T^{4}$$)($$1 + 4 T - 62 T^{2} - 32 T^{3} + 3547 T^{4} - 1376 T^{5} - 114638 T^{6} + 318028 T^{7} + 3418801 T^{8}$$)($$1 + 4 T - 62 T^{2} - 32 T^{3} + 3547 T^{4} - 1376 T^{5} - 114638 T^{6} + 318028 T^{7} + 3418801 T^{8}$$)
$47$ ($$1 - 3 T - 38 T^{2} - 141 T^{3} + 2209 T^{4}$$)($$1 - 47 T^{2} + 2209 T^{4}$$)($$1 + 3 T - 38 T^{2} + 141 T^{3} + 2209 T^{4}$$)($$1 + 9 T + 34 T^{2} + 423 T^{3} + 2209 T^{4}$$)($$1 - 47 T^{2} + 2209 T^{4}$$)($$1 - 9 T + 34 T^{2} - 423 T^{3} + 2209 T^{4}$$)($$1 - 3 T - 38 T^{2} - 141 T^{3} + 2209 T^{4}$$)($$1 - 47 T^{2} + 2209 T^{4}$$)($$1 - 9 T + 34 T^{2} - 423 T^{3} + 2209 T^{4}$$)($$1 + 9 T + 34 T^{2} + 423 T^{3} + 2209 T^{4}$$)($$1 + 3 T - 38 T^{2} + 141 T^{3} + 2209 T^{4}$$)($$1 - 47 T^{2} + 2209 T^{4}$$)($$1 - 6 T - 55 T^{2} + 18 T^{3} + 4380 T^{4} + 846 T^{5} - 121495 T^{6} - 622938 T^{7} + 4879681 T^{8}$$)($$1 + 6 T - 55 T^{2} - 18 T^{3} + 4380 T^{4} - 846 T^{5} - 121495 T^{6} + 622938 T^{7} + 4879681 T^{8}$$)
$53$ ($$( 1 - 12 T + 53 T^{2} )^{2}$$)($$( 1 - 6 T + 53 T^{2} )^{2}$$)($$( 1 + 3 T + 53 T^{2} )^{2}$$)($$( 1 + 12 T + 53 T^{2} )^{2}$$)($$( 1 - 6 T + 53 T^{2} )^{2}$$)($$( 1 - 9 T + 53 T^{2} )^{2}$$)($$( 1 - 3 T + 53 T^{2} )^{2}$$)($$( 1 + 6 T + 53 T^{2} )^{2}$$)($$( 1 - 12 T + 53 T^{2} )^{2}$$)($$( 1 + 9 T + 53 T^{2} )^{2}$$)($$( 1 + 12 T + 53 T^{2} )^{2}$$)($$( 1 + 6 T + 53 T^{2} )^{2}$$)($$( 1 + 18 T + 175 T^{2} + 954 T^{3} + 2809 T^{4} )^{2}$$)($$( 1 - 18 T + 175 T^{2} - 954 T^{3} + 2809 T^{4} )^{2}$$)
$59$ ($$1 + 12 T + 85 T^{2} + 708 T^{3} + 3481 T^{4}$$)($$1 - 59 T^{2} + 3481 T^{4}$$)($$1 - 9 T + 22 T^{2} - 531 T^{3} + 3481 T^{4}$$)($$1 + 12 T + 85 T^{2} + 708 T^{3} + 3481 T^{4}$$)($$1 - 6 T - 23 T^{2} - 354 T^{3} + 3481 T^{4}$$)($$1 - 15 T + 166 T^{2} - 885 T^{3} + 3481 T^{4}$$)($$1 + 9 T + 22 T^{2} + 531 T^{3} + 3481 T^{4}$$)($$1 + 6 T - 23 T^{2} + 354 T^{3} + 3481 T^{4}$$)($$1 - 12 T + 85 T^{2} - 708 T^{3} + 3481 T^{4}$$)($$1 + 15 T + 166 T^{2} + 885 T^{3} + 3481 T^{4}$$)($$1 - 12 T + 85 T^{2} - 708 T^{3} + 3481 T^{4}$$)($$1 - 59 T^{2} + 3481 T^{4}$$)($$1 - 43 T^{2} - 1632 T^{4} - 149683 T^{6} + 12117361 T^{8}$$)($$1 - 43 T^{2} - 1632 T^{4} - 149683 T^{6} + 12117361 T^{8}$$)
$61$ ($$1 - 10 T + 39 T^{2} - 610 T^{3} + 3721 T^{4}$$)($$1 - 10 T + 39 T^{2} - 610 T^{3} + 3721 T^{4}$$)($$1 + 8 T + 3 T^{2} + 488 T^{3} + 3721 T^{4}$$)($$1 + 2 T - 57 T^{2} + 122 T^{3} + 3721 T^{4}$$)($$1 + 2 T - 57 T^{2} + 122 T^{3} + 3721 T^{4}$$)($$1 - 4 T - 45 T^{2} - 244 T^{3} + 3721 T^{4}$$)($$1 + 8 T + 3 T^{2} + 488 T^{3} + 3721 T^{4}$$)($$1 + 2 T - 57 T^{2} + 122 T^{3} + 3721 T^{4}$$)($$1 + 2 T - 57 T^{2} + 122 T^{3} + 3721 T^{4}$$)($$1 - 4 T - 45 T^{2} - 244 T^{3} + 3721 T^{4}$$)($$1 - 10 T + 39 T^{2} - 610 T^{3} + 3721 T^{4}$$)($$1 - 10 T + 39 T^{2} - 610 T^{3} + 3721 T^{4}$$)($$1 + 4 T - 2 T^{2} - 416 T^{3} - 4373 T^{4} - 25376 T^{5} - 7442 T^{6} + 907924 T^{7} + 13845841 T^{8}$$)($$1 + 4 T - 2 T^{2} - 416 T^{3} - 4373 T^{4} - 25376 T^{5} - 7442 T^{6} + 907924 T^{7} + 13845841 T^{8}$$)
$67$ ($$1 - 4 T - 51 T^{2} - 268 T^{3} + 4489 T^{4}$$)($$1 - 4 T - 51 T^{2} - 268 T^{3} + 4489 T^{4}$$)($$1 - 4 T - 51 T^{2} - 268 T^{3} + 4489 T^{4}$$)($$1 - 4 T - 51 T^{2} - 268 T^{3} + 4489 T^{4}$$)($$1 - 4 T - 51 T^{2} - 268 T^{3} + 4489 T^{4}$$)($$1 - 4 T - 51 T^{2} - 268 T^{3} + 4489 T^{4}$$)($$1 - 4 T - 51 T^{2} - 268 T^{3} + 4489 T^{4}$$)($$1 - 4 T - 51 T^{2} - 268 T^{3} + 4489 T^{4}$$)($$1 - 4 T - 51 T^{2} - 268 T^{3} + 4489 T^{4}$$)($$1 - 4 T - 51 T^{2} - 268 T^{3} + 4489 T^{4}$$)($$1 - 4 T - 51 T^{2} - 268 T^{3} + 4489 T^{4}$$)($$1 - 4 T - 51 T^{2} - 268 T^{3} + 4489 T^{4}$$)($$( 1 + 8 T - 3 T^{2} + 536 T^{3} + 4489 T^{4} )^{2}$$)($$( 1 + 8 T - 3 T^{2} + 536 T^{3} + 4489 T^{4} )^{2}$$)
$71$ ($$( 1 + 71 T^{2} )^{2}$$)($$( 1 + 71 T^{2} )^{2}$$)($$( 1 + 6 T + 71 T^{2} )^{2}$$)($$( 1 - 12 T + 71 T^{2} )^{2}$$)($$( 1 - 12 T + 71 T^{2} )^{2}$$)($$( 1 + 6 T + 71 T^{2} )^{2}$$)($$( 1 - 6 T + 71 T^{2} )^{2}$$)($$( 1 + 12 T + 71 T^{2} )^{2}$$)($$( 1 + 12 T + 71 T^{2} )^{2}$$)($$( 1 - 6 T + 71 T^{2} )^{2}$$)($$( 1 + 71 T^{2} )^{2}$$)($$( 1 + 71 T^{2} )^{2}$$)($$( 1 + 6 T + 71 T^{2} )^{4}$$)($$( 1 - 6 T + 71 T^{2} )^{4}$$)
$73$ ($$( 1 - 2 T + 73 T^{2} )^{2}$$)($$( 1 - 2 T + 73 T^{2} )^{2}$$)($$( 1 - 2 T + 73 T^{2} )^{2}$$)($$( 1 + 10 T + 73 T^{2} )^{2}$$)($$( 1 + 10 T + 73 T^{2} )^{2}$$)($$( 1 - 14 T + 73 T^{2} )^{2}$$)($$( 1 - 2 T + 73 T^{2} )^{2}$$)($$( 1 + 10 T + 73 T^{2} )^{2}$$)($$( 1 + 10 T + 73 T^{2} )^{2}$$)($$( 1 - 14 T + 73 T^{2} )^{2}$$)($$( 1 - 2 T + 73 T^{2} )^{2}$$)($$( 1 - 2 T + 73 T^{2} )^{2}$$)($$( 1 + 8 T + 54 T^{2} + 584 T^{3} + 5329 T^{4} )^{2}$$)($$( 1 + 8 T + 54 T^{2} + 584 T^{3} + 5329 T^{4} )^{2}$$)
$79$ ($$1 - T - 78 T^{2} - 79 T^{3} + 6241 T^{4}$$)($$1 + 8 T - 15 T^{2} + 632 T^{3} + 6241 T^{4}$$)($$1 + 2 T - 75 T^{2} + 158 T^{3} + 6241 T^{4}$$)($$( 1 - 17 T + 79 T^{2} )( 1 + 4 T + 79 T^{2} )$$)($$( 1 - 17 T + 79 T^{2} )( 1 + 13 T + 79 T^{2} )$$)($$1 + 14 T + 117 T^{2} + 1106 T^{3} + 6241 T^{4}$$)($$1 + 2 T - 75 T^{2} + 158 T^{3} + 6241 T^{4}$$)($$( 1 - 17 T + 79 T^{2} )( 1 + 13 T + 79 T^{2} )$$)($$( 1 - 17 T + 79 T^{2} )( 1 + 4 T + 79 T^{2} )$$)($$1 + 14 T + 117 T^{2} + 1106 T^{3} + 6241 T^{4}$$)($$1 - T - 78 T^{2} - 79 T^{3} + 6241 T^{4}$$)($$1 + 8 T - 15 T^{2} + 632 T^{3} + 6241 T^{4}$$)($$1 - 8 T - 2 T^{2} + 736 T^{3} - 6941 T^{4} + 58144 T^{5} - 12482 T^{6} - 3944312 T^{7} + 38950081 T^{8}$$)($$1 - 8 T - 2 T^{2} + 736 T^{3} - 6941 T^{4} + 58144 T^{5} - 12482 T^{6} - 3944312 T^{7} + 38950081 T^{8}$$)
$83$ ($$1 + 18 T + 241 T^{2} + 1494 T^{3} + 6889 T^{4}$$)($$1 - 12 T + 61 T^{2} - 996 T^{3} + 6889 T^{4}$$)($$1 - 6 T - 47 T^{2} - 498 T^{3} + 6889 T^{4}$$)($$1 + 6 T - 47 T^{2} + 498 T^{3} + 6889 T^{4}$$)($$1 - 12 T + 61 T^{2} - 996 T^{3} + 6889 T^{4}$$)($$1 + 6 T - 47 T^{2} + 498 T^{3} + 6889 T^{4}$$)($$1 + 6 T - 47 T^{2} + 498 T^{3} + 6889 T^{4}$$)($$1 + 12 T + 61 T^{2} + 996 T^{3} + 6889 T^{4}$$)($$1 - 6 T - 47 T^{2} - 498 T^{3} + 6889 T^{4}$$)($$1 - 6 T - 47 T^{2} - 498 T^{3} + 6889 T^{4}$$)($$1 - 18 T + 241 T^{2} - 1494 T^{3} + 6889 T^{4}$$)($$1 + 12 T + 61 T^{2} + 996 T^{3} + 6889 T^{4}$$)($$1 - 24 T + 278 T^{2} - 3168 T^{3} + 34107 T^{4} - 262944 T^{5} + 1915142 T^{6} - 13722888 T^{7} + 47458321 T^{8}$$)($$1 + 24 T + 278 T^{2} + 3168 T^{3} + 34107 T^{4} + 262944 T^{5} + 1915142 T^{6} + 13722888 T^{7} + 47458321 T^{8}$$)
$89$ ($$( 1 + 89 T^{2} )^{2}$$)($$( 1 + 18 T + 89 T^{2} )^{2}$$)($$( 1 + 6 T + 89 T^{2} )^{2}$$)($$( 1 + 12 T + 89 T^{2} )^{2}$$)($$( 1 + 12 T + 89 T^{2} )^{2}$$)($$( 1 + 18 T + 89 T^{2} )^{2}$$)($$( 1 - 6 T + 89 T^{2} )^{2}$$)($$( 1 - 12 T + 89 T^{2} )^{2}$$)($$( 1 - 12 T + 89 T^{2} )^{2}$$)($$( 1 - 18 T + 89 T^{2} )^{2}$$)($$( 1 + 89 T^{2} )^{2}$$)($$( 1 - 18 T + 89 T^{2} )^{2}$$)($$( 1 - 12 T + 89 T^{2} )^{4}$$)($$( 1 + 12 T + 89 T^{2} )^{4}$$)
$97$ ($$( 1 - 5 T + 97 T^{2} )( 1 + 19 T + 97 T^{2} )$$)($$1 + 2 T - 93 T^{2} + 194 T^{3} + 9409 T^{4}$$)($$1 - 16 T + 159 T^{2} - 1552 T^{3} + 9409 T^{4}$$)($$1 + 2 T - 93 T^{2} + 194 T^{3} + 9409 T^{4}$$)($$1 + 2 T - 93 T^{2} + 194 T^{3} + 9409 T^{4}$$)($$1 - 16 T + 159 T^{2} - 1552 T^{3} + 9409 T^{4}$$)($$1 - 16 T + 159 T^{2} - 1552 T^{3} + 9409 T^{4}$$)($$1 + 2 T - 93 T^{2} + 194 T^{3} + 9409 T^{4}$$)($$1 + 2 T - 93 T^{2} + 194 T^{3} + 9409 T^{4}$$)($$1 - 16 T + 159 T^{2} - 1552 T^{3} + 9409 T^{4}$$)($$( 1 - 5 T + 97 T^{2} )( 1 + 19 T + 97 T^{2} )$$)($$1 + 2 T - 93 T^{2} + 194 T^{3} + 9409 T^{4}$$)($$1 + 16 T + 46 T^{2} + 256 T^{3} + 9859 T^{4} + 24832 T^{5} + 432814 T^{6} + 14602768 T^{7} + 88529281 T^{8}$$)($$1 + 16 T + 46 T^{2} + 256 T^{3} + 9859 T^{4} + 24832 T^{5} + 432814 T^{6} + 14602768 T^{7} + 88529281 T^{8}$$)