# Properties

 Label 2880.1 Level 2880 Weight 1 Dimension 69 Nonzero newspaces 10 Newform subspaces 18 Sturm bound 442368 Trace bound 25

## Defining parameters

 Level: $$N$$ = $$2880 = 2^{6} \cdot 3^{2} \cdot 5$$ Weight: $$k$$ = $$1$$ Nonzero newspaces: $$10$$ Newform subspaces: $$18$$ Sturm bound: $$442368$$ Trace bound: $$25$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{1}(\Gamma_1(2880))$$.

Total New Old
Modular forms 5132 663 4469
Cusp forms 524 69 455
Eisenstein series 4608 594 4014

The following table gives the dimensions of subspaces with specified projective image type.

$$D_n$$ $$A_4$$ $$S_4$$ $$A_5$$
Dimension 69 0 0 0

## Trace form

 $$69q - q^{5} + O(q^{10})$$ $$69q - q^{5} - 2q^{13} + 2q^{17} - 4q^{19} + 8q^{21} - 11q^{25} + 10q^{29} + 2q^{37} + 2q^{41} - 7q^{49} + 2q^{53} + 10q^{61} - 2q^{65} - 2q^{73} - 16q^{76} + 16q^{79} + 2q^{85} - 10q^{89} - 16q^{94} - 2q^{97} + O(q^{100})$$

## Decomposition of $$S_{1}^{\mathrm{new}}(\Gamma_1(2880))$$

We only show spaces with odd parity, since no modular forms exist when this condition is not satisfied. Within each space $$S_k^{\mathrm{new}}(N, \chi)$$ we list the newforms together with their dimension.

Label $$\chi$$ Newforms Dimension $$\chi$$ degree
2880.1.c $$\chi_{2880}(449, \cdot)$$ 2880.1.c.a 4 1
2880.1.e $$\chi_{2880}(2431, \cdot)$$ None 0 1
2880.1.g $$\chi_{2880}(991, \cdot)$$ None 0 1
2880.1.i $$\chi_{2880}(1889, \cdot)$$ 2880.1.i.a 4 1
2880.1.i.b 4
2880.1.j $$\chi_{2880}(1279, \cdot)$$ 2880.1.j.a 1 1
2880.1.j.b 2
2880.1.l $$\chi_{2880}(1601, \cdot)$$ None 0 1
2880.1.n $$\chi_{2880}(161, \cdot)$$ None 0 1
2880.1.p $$\chi_{2880}(2719, \cdot)$$ 2880.1.p.a 2 1
2880.1.p.b 2
2880.1.r $$\chi_{2880}(559, \cdot)$$ 2880.1.r.a 4 2
2880.1.s $$\chi_{2880}(881, \cdot)$$ None 0 2
2880.1.v $$\chi_{2880}(287, \cdot)$$ None 0 2
2880.1.y $$\chi_{2880}(2017, \cdot)$$ None 0 2
2880.1.ba $$\chi_{2880}(1583, \cdot)$$ None 0 2
2880.1.bb $$\chi_{2880}(1873, \cdot)$$ None 0 2
2880.1.be $$\chi_{2880}(143, \cdot)$$ None 0 2
2880.1.bf $$\chi_{2880}(433, \cdot)$$ None 0 2
2880.1.bh $$\chi_{2880}(577, \cdot)$$ 2880.1.bh.a 2 2
2880.1.bh.b 2
2880.1.bh.c 2
2880.1.bk $$\chi_{2880}(1727, \cdot)$$ 2880.1.bk.a 4 2
2880.1.bk.b 4
2880.1.bn $$\chi_{2880}(1169, \cdot)$$ None 0 2
2880.1.bo $$\chi_{2880}(271, \cdot)$$ None 0 2
2880.1.bp $$\chi_{2880}(799, \cdot)$$ None 0 2
2880.1.bq $$\chi_{2880}(1121, \cdot)$$ None 0 2
2880.1.bs $$\chi_{2880}(641, \cdot)$$ None 0 2
2880.1.bu $$\chi_{2880}(319, \cdot)$$ 2880.1.bu.a 2 2
2880.1.bu.b 2
2880.1.bu.c 4
2880.1.bx $$\chi_{2880}(929, \cdot)$$ None 0 2
2880.1.bz $$\chi_{2880}(31, \cdot)$$ None 0 2
2880.1.cb $$\chi_{2880}(511, \cdot)$$ None 0 2
2880.1.cd $$\chi_{2880}(1409, \cdot)$$ 2880.1.cd.a 8 2
2880.1.cf $$\chi_{2880}(73, \cdot)$$ None 0 4
2880.1.cg $$\chi_{2880}(1223, \cdot)$$ None 0 4
2880.1.cj $$\chi_{2880}(631, \cdot)$$ None 0 4
2880.1.ck $$\chi_{2880}(89, \cdot)$$ None 0 4
2880.1.cm $$\chi_{2880}(199, \cdot)$$ None 0 4
2880.1.cp $$\chi_{2880}(521, \cdot)$$ None 0 4
2880.1.cq $$\chi_{2880}(503, \cdot)$$ None 0 4
2880.1.ct $$\chi_{2880}(793, \cdot)$$ None 0 4
2880.1.cw $$\chi_{2880}(751, \cdot)$$ None 0 4
2880.1.cx $$\chi_{2880}(209, \cdot)$$ None 0 4
2880.1.cz $$\chi_{2880}(193, \cdot)$$ None 0 4
2880.1.da $$\chi_{2880}(383, \cdot)$$ None 0 4
2880.1.dd $$\chi_{2880}(337, \cdot)$$ None 0 4
2880.1.de $$\chi_{2880}(47, \cdot)$$ None 0 4
2880.1.dh $$\chi_{2880}(817, \cdot)$$ None 0 4
2880.1.di $$\chi_{2880}(527, \cdot)$$ None 0 4
2880.1.dl $$\chi_{2880}(1247, \cdot)$$ None 0 4
2880.1.dm $$\chi_{2880}(97, \cdot)$$ None 0 4
2880.1.do $$\chi_{2880}(401, \cdot)$$ None 0 4
2880.1.dp $$\chi_{2880}(79, \cdot)$$ None 0 4
2880.1.ds $$\chi_{2880}(397, \cdot)$$ None 0 8
2880.1.dv $$\chi_{2880}(107, \cdot)$$ None 0 8
2880.1.dx $$\chi_{2880}(341, \cdot)$$ None 0 8
2880.1.dz $$\chi_{2880}(269, \cdot)$$ None 0 8
2880.1.ea $$\chi_{2880}(91, \cdot)$$ None 0 8
2880.1.ec $$\chi_{2880}(19, \cdot)$$ 2880.1.ec.a 16 8
2880.1.ee $$\chi_{2880}(37, \cdot)$$ None 0 8
2880.1.eh $$\chi_{2880}(467, \cdot)$$ None 0 8
2880.1.ei $$\chi_{2880}(313, \cdot)$$ None 0 8
2880.1.el $$\chi_{2880}(23, \cdot)$$ None 0 8
2880.1.en $$\chi_{2880}(329, \cdot)$$ None 0 8
2880.1.eo $$\chi_{2880}(151, \cdot)$$ None 0 8
2880.1.eq $$\chi_{2880}(41, \cdot)$$ None 0 8
2880.1.et $$\chi_{2880}(439, \cdot)$$ None 0 8
2880.1.ev $$\chi_{2880}(263, \cdot)$$ None 0 8
2880.1.ew $$\chi_{2880}(553, \cdot)$$ None 0 8
2880.1.ez $$\chi_{2880}(83, \cdot)$$ None 0 16
2880.1.fa $$\chi_{2880}(133, \cdot)$$ None 0 16
2880.1.fc $$\chi_{2880}(139, \cdot)$$ None 0 16
2880.1.fe $$\chi_{2880}(211, \cdot)$$ None 0 16
2880.1.fh $$\chi_{2880}(29, \cdot)$$ None 0 16
2880.1.fj $$\chi_{2880}(101, \cdot)$$ None 0 16
2880.1.fl $$\chi_{2880}(203, \cdot)$$ None 0 16
2880.1.fm $$\chi_{2880}(13, \cdot)$$ None 0 16

## Decomposition of $$S_{1}^{\mathrm{old}}(\Gamma_1(2880))$$ into lower level spaces

$$S_{1}^{\mathrm{old}}(\Gamma_1(2880)) \cong$$ $$S_{1}^{\mathrm{new}}(\Gamma_1(72))$$$$^{\oplus 8}$$$$\oplus$$$$S_{1}^{\mathrm{new}}(\Gamma_1(80))$$$$^{\oplus 9}$$$$\oplus$$$$S_{1}^{\mathrm{new}}(\Gamma_1(120))$$$$^{\oplus 8}$$$$\oplus$$$$S_{1}^{\mathrm{new}}(\Gamma_1(144))$$$$^{\oplus 6}$$$$\oplus$$$$S_{1}^{\mathrm{new}}(\Gamma_1(160))$$$$^{\oplus 6}$$$$\oplus$$$$S_{1}^{\mathrm{new}}(\Gamma_1(180))$$$$^{\oplus 5}$$$$\oplus$$$$S_{1}^{\mathrm{new}}(\Gamma_1(192))$$$$^{\oplus 4}$$$$\oplus$$$$S_{1}^{\mathrm{new}}(\Gamma_1(240))$$$$^{\oplus 6}$$$$\oplus$$$$S_{1}^{\mathrm{new}}(\Gamma_1(288))$$$$^{\oplus 4}$$$$\oplus$$$$S_{1}^{\mathrm{new}}(\Gamma_1(320))$$$$^{\oplus 3}$$$$\oplus$$$$S_{1}^{\mathrm{new}}(\Gamma_1(360))$$$$^{\oplus 4}$$$$\oplus$$$$S_{1}^{\mathrm{new}}(\Gamma_1(480))$$$$^{\oplus 4}$$$$\oplus$$$$S_{1}^{\mathrm{new}}(\Gamma_1(576))$$$$^{\oplus 2}$$$$\oplus$$$$S_{1}^{\mathrm{new}}(\Gamma_1(720))$$$$^{\oplus 3}$$$$\oplus$$$$S_{1}^{\mathrm{new}}(\Gamma_1(960))$$$$^{\oplus 2}$$$$\oplus$$$$S_{1}^{\mathrm{new}}(\Gamma_1(1440))$$$$^{\oplus 2}$$

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ ($$1 + T^{16}$$)
$3$ ($$1 + T + T^{2}$$)($$1 - T + T^{2}$$)($$1 - T^{2} + T^{4}$$)($$1 - T^{4} + T^{8}$$)
$5$ ($$1 + T^{4}$$)($$( 1 + T^{2} )^{2}$$)($$( 1 + T^{2} )^{2}$$)($$1 + T$$)($$1 + T^{2}$$)($$1 + T^{2}$$)($$1 + T^{2}$$)($$1 + T^{4}$$)($$( 1 + T )^{2}$$)($$1 + T^{2}$$)($$( 1 - T )^{2}$$)($$1 + T^{4}$$)($$1 + T^{4}$$)($$1 - T + T^{2}$$)($$1 - T + T^{2}$$)($$( 1 + T + T^{2} )^{2}$$)($$( 1 - T^{2} + T^{4} )^{2}$$)($$1 + T^{16}$$)
$7$ ($$( 1 - T )^{4}( 1 + T )^{4}$$)($$( 1 + T^{4} )^{2}$$)($$( 1 + T^{4} )^{2}$$)($$1 + T^{2}$$)($$( 1 + T^{2} )^{2}$$)($$( 1 + T^{2} )^{2}$$)($$( 1 + T^{2} )^{2}$$)($$( 1 - T )^{4}( 1 + T )^{4}$$)($$1 + T^{4}$$)($$1 + T^{4}$$)($$1 + T^{4}$$)($$( 1 + T^{4} )^{2}$$)($$( 1 + T^{4} )^{2}$$)($$( 1 + T )^{2}( 1 - T + T^{2} )$$)($$( 1 - T )^{2}( 1 + T + T^{2} )$$)($$( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} )$$)($$( 1 + T^{4} )^{2}( 1 - T^{4} + T^{8} )$$)($$( 1 + T^{8} )^{4}$$)
$11$ ($$( 1 - T )^{4}( 1 + T )^{4}$$)($$( 1 + T^{4} )^{2}$$)($$( 1 + T^{4} )^{2}$$)($$( 1 - T )( 1 + T )$$)($$( 1 - T )^{2}( 1 + T )^{2}$$)($$( 1 + T )^{4}$$)($$( 1 - T )^{4}$$)($$( 1 + T^{4} )^{2}$$)($$( 1 + T^{2} )^{2}$$)($$( 1 + T^{2} )^{2}$$)($$( 1 + T^{2} )^{2}$$)($$( 1 + T^{2} )^{4}$$)($$( 1 + T^{2} )^{4}$$)($$( 1 - T + T^{2} )( 1 + T + T^{2} )$$)($$( 1 - T + T^{2} )( 1 + T + T^{2} )$$)($$( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2}$$)($$( 1 - T + T^{2} )^{4}( 1 + T + T^{2} )^{4}$$)($$( 1 + T^{16} )^{2}$$)
$13$ ($$( 1 + T^{2} )^{4}$$)($$( 1 + T^{4} )^{2}$$)($$( 1 + T^{4} )^{2}$$)($$( 1 - T )( 1 + T )$$)($$( 1 - T )^{2}( 1 + T )^{2}$$)($$( 1 + T^{2} )^{2}$$)($$( 1 + T^{2} )^{2}$$)($$( 1 + T^{4} )^{2}$$)($$( 1 + T )^{2}( 1 + T^{2} )$$)($$( 1 - T )^{2}( 1 + T^{2} )$$)($$( 1 + T )^{2}( 1 + T^{2} )$$)($$( 1 + T )^{4}( 1 + T^{2} )^{2}$$)($$( 1 - T )^{4}( 1 + T^{2} )^{2}$$)($$( 1 - T + T^{2} )( 1 + T + T^{2} )$$)($$( 1 - T + T^{2} )( 1 + T + T^{2} )$$)($$( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2}$$)($$( 1 - T + T^{2} )^{4}( 1 + T + T^{2} )^{4}$$)($$( 1 + T^{16} )^{2}$$)
$17$ ($$( 1 + T^{4} )^{2}$$)($$( 1 + T^{2} )^{4}$$)($$( 1 + T^{2} )^{4}$$)($$( 1 - T )( 1 + T )$$)($$( 1 + T^{2} )^{2}$$)($$( 1 - T )^{2}( 1 + T )^{2}$$)($$( 1 - T )^{2}( 1 + T )^{2}$$)($$( 1 + T^{4} )^{2}$$)($$( 1 + T )^{2}( 1 + T^{2} )$$)($$( 1 - T )^{2}( 1 + T^{2} )$$)($$( 1 - T )^{2}( 1 + T^{2} )$$)($$( 1 + T^{4} )^{2}$$)($$( 1 + T^{4} )^{2}$$)($$( 1 - T )^{2}( 1 + T )^{2}$$)($$( 1 - T )^{2}( 1 + T )^{2}$$)($$( 1 - T )^{4}( 1 + T )^{4}$$)($$( 1 + T^{2} )^{8}$$)($$( 1 + T^{16} )^{2}$$)
$19$ ($$( 1 + T^{2} )^{4}$$)($$( 1 - T )^{4}( 1 + T )^{4}$$)($$( 1 - T )^{4}( 1 + T )^{4}$$)($$( 1 - T )( 1 + T )$$)($$( 1 - T )^{2}( 1 + T )^{2}$$)($$( 1 + T^{2} )^{2}$$)($$( 1 + T^{2} )^{2}$$)($$( 1 + T )^{4}( 1 + T^{2} )^{2}$$)($$( 1 - T )^{2}( 1 + T )^{2}$$)($$( 1 - T )^{2}( 1 + T )^{2}$$)($$( 1 - T )^{2}( 1 + T )^{2}$$)($$( 1 + T^{2} )^{4}$$)($$( 1 + T^{2} )^{4}$$)($$( 1 - T )^{2}( 1 + T )^{2}$$)($$( 1 - T )^{2}( 1 + T )^{2}$$)($$( 1 - T )^{4}( 1 + T )^{4}$$)($$( 1 + T^{2} )^{8}$$)($$( 1 + T^{4} )^{4}( 1 + T^{8} )^{2}$$)
$23$ ($$( 1 + T^{2} )^{4}$$)($$( 1 + T )^{8}$$)($$( 1 - T )^{8}$$)($$1 + T^{2}$$)($$( 1 + T^{2} )^{2}$$)($$( 1 + T^{2} )^{2}$$)($$( 1 + T^{2} )^{2}$$)($$( 1 + T^{4} )^{2}$$)($$1 + T^{4}$$)($$1 + T^{4}$$)($$1 + T^{4}$$)($$( 1 + T^{4} )^{2}$$)($$( 1 + T^{4} )^{2}$$)($$( 1 + T )^{2}( 1 - T + T^{2} )$$)($$( 1 - T )^{2}( 1 + T + T^{2} )$$)($$( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} )$$)($$( 1 + T^{4} )^{2}( 1 - T^{4} + T^{8} )$$)($$( 1 + T^{16} )^{2}$$)
$29$ ($$( 1 + T^{4} )^{2}$$)($$( 1 + T^{2} )^{4}$$)($$( 1 + T^{2} )^{4}$$)($$( 1 - T )^{2}$$)($$( 1 + T^{2} )^{2}$$)($$( 1 + T^{2} )^{2}$$)($$( 1 + T^{2} )^{2}$$)($$( 1 + T^{4} )^{2}$$)($$( 1 + T^{2} )^{2}$$)($$( 1 - T )^{2}( 1 + T )^{2}$$)($$( 1 + T^{2} )^{2}$$)($$( 1 + T^{4} )^{2}$$)($$( 1 + T^{4} )^{2}$$)($$( 1 + T )^{2}( 1 - T + T^{2} )$$)($$( 1 + T )^{2}( 1 - T + T^{2} )$$)($$( 1 + T )^{4}( 1 - T + T^{2} )^{2}$$)($$( 1 - T )^{8}( 1 - T + T^{2} )^{4}$$)($$( 1 + T^{16} )^{2}$$)
$31$ ($$( 1 + T^{2} )^{4}$$)($$( 1 + T^{2} )^{4}$$)($$( 1 + T^{2} )^{4}$$)($$( 1 - T )( 1 + T )$$)($$( 1 - T )^{2}( 1 + T )^{2}$$)($$( 1 + T^{2} )^{2}$$)($$( 1 + T^{2} )^{2}$$)($$( 1 + T^{2} )^{4}$$)($$( 1 + T^{2} )^{2}$$)($$( 1 + T^{2} )^{2}$$)($$( 1 + T^{2} )^{2}$$)($$( 1 - T )^{4}( 1 + T )^{4}$$)($$( 1 - T )^{4}( 1 + T )^{4}$$)($$( 1 - T + T^{2} )( 1 + T + T^{2} )$$)($$( 1 - T + T^{2} )( 1 + T + T^{2} )$$)($$( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2}$$)($$( 1 - T^{2} + T^{4} )^{4}$$)($$( 1 + T^{8} )^{4}$$)
$37$ ($$( 1 - T )^{4}( 1 + T )^{4}$$)($$( 1 + T^{4} )^{2}$$)($$( 1 + T^{4} )^{2}$$)($$( 1 - T )( 1 + T )$$)($$( 1 - T )^{2}( 1 + T )^{2}$$)($$( 1 + T^{2} )^{2}$$)($$( 1 + T^{2} )^{2}$$)($$( 1 + T^{4} )^{2}$$)($$( 1 + T )^{2}( 1 + T^{2} )$$)($$( 1 + T )^{2}( 1 + T^{2} )$$)($$( 1 + T )^{2}( 1 + T^{2} )$$)($$( 1 - T )^{4}( 1 + T^{2} )^{2}$$)($$( 1 - T )^{4}( 1 + T^{2} )^{2}$$)($$( 1 - T )^{2}( 1 + T )^{2}$$)($$( 1 - T )^{2}( 1 + T )^{2}$$)($$( 1 - T )^{4}( 1 + T )^{4}$$)($$( 1 - T )^{8}( 1 + T )^{8}$$)($$( 1 + T^{16} )^{2}$$)
$41$ ($$( 1 + T^{4} )^{2}$$)($$( 1 + T^{4} )^{2}$$)($$( 1 + T^{4} )^{2}$$)($$( 1 - T )^{2}$$)($$( 1 + T^{2} )^{2}$$)($$( 1 + T^{2} )^{2}$$)($$( 1 + T^{2} )^{2}$$)($$( 1 - T )^{4}( 1 + T )^{4}$$)($$( 1 + T )^{4}$$)($$( 1 + T^{2} )^{2}$$)($$( 1 - T )^{4}$$)($$( 1 + T^{4} )^{2}$$)($$( 1 + T^{4} )^{2}$$)($$( 1 - T )^{2}( 1 + T + T^{2} )$$)($$( 1 - T )^{2}( 1 + T + T^{2} )$$)($$( 1 + T )^{4}( 1 - T + T^{2} )^{2}$$)($$( 1 + T^{2} )^{4}( 1 - T^{2} + T^{4} )^{2}$$)($$( 1 + T^{8} )^{4}$$)
$43$ ($$( 1 - T )^{4}( 1 + T )^{4}$$)($$( 1 + T^{2} )^{4}$$)($$( 1 + T^{2} )^{4}$$)($$1 + T^{2}$$)($$( 1 + T^{2} )^{2}$$)($$( 1 - T )^{2}( 1 + T )^{2}$$)($$( 1 - T )^{2}( 1 + T )^{2}$$)($$( 1 + T^{4} )^{2}$$)($$1 + T^{4}$$)($$1 + T^{4}$$)($$1 + T^{4}$$)($$( 1 + T^{4} )^{2}$$)($$( 1 + T^{4} )^{2}$$)($$( 1 + T + T^{2} )^{2}$$)($$( 1 - T + T^{2} )^{2}$$)($$( 1 - T^{2} + T^{4} )^{2}$$)($$( 1 - T^{4} + T^{8} )^{2}$$)($$( 1 + T^{16} )^{2}$$)
$47$ ($$( 1 + T^{2} )^{4}$$)($$( 1 + T^{2} )^{4}$$)($$( 1 + T^{2} )^{4}$$)($$1 + T^{2}$$)($$( 1 + T^{2} )^{2}$$)($$( 1 + T^{2} )^{2}$$)($$( 1 + T^{2} )^{2}$$)($$( 1 + T^{4} )^{2}$$)($$1 + T^{4}$$)($$1 + T^{4}$$)($$1 + T^{4}$$)($$( 1 + T^{4} )^{2}$$)($$( 1 + T^{4} )^{2}$$)($$( 1 + T )^{2}( 1 - T + T^{2} )$$)($$( 1 - T )^{2}( 1 + T + T^{2} )$$)($$( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} )$$)($$( 1 + T^{4} )^{2}( 1 - T^{4} + T^{8} )$$)($$( 1 + T^{16} )^{2}$$)
$53$ ($$( 1 + T^{4} )^{2}$$)($$( 1 - T )^{4}( 1 + T )^{4}$$)($$( 1 - T )^{4}( 1 + T )^{4}$$)($$( 1 - T )( 1 + T )$$)($$( 1 + T^{2} )^{2}$$)($$( 1 + T^{2} )^{2}$$)($$( 1 + T^{2} )^{2}$$)($$( 1 + T^{4} )^{2}$$)($$( 1 + T )^{2}( 1 + T^{2} )$$)($$( 1 - T )^{2}( 1 + T^{2} )$$)($$( 1 - T )^{2}( 1 + T^{2} )$$)($$( 1 + T^{4} )^{2}$$)($$( 1 + T^{4} )^{2}$$)($$( 1 - T )^{2}( 1 + T )^{2}$$)($$( 1 - T )^{2}( 1 + T )^{2}$$)($$( 1 - T )^{4}( 1 + T )^{4}$$)($$( 1 + T^{2} )^{8}$$)($$( 1 + T^{16} )^{2}$$)
$59$ ($$( 1 - T )^{4}( 1 + T )^{4}$$)($$( 1 + T^{4} )^{2}$$)($$( 1 + T^{4} )^{2}$$)($$( 1 - T )( 1 + T )$$)($$( 1 - T )^{2}( 1 + T )^{2}$$)($$( 1 + T )^{4}$$)($$( 1 - T )^{4}$$)($$( 1 + T^{4} )^{2}$$)($$( 1 - T )^{2}( 1 + T )^{2}$$)($$( 1 - T )^{2}( 1 + T )^{2}$$)($$( 1 - T )^{2}( 1 + T )^{2}$$)($$( 1 - T )^{4}( 1 + T )^{4}$$)($$( 1 - T )^{4}( 1 + T )^{4}$$)($$( 1 - T + T^{2} )( 1 + T + T^{2} )$$)($$( 1 - T + T^{2} )( 1 + T + T^{2} )$$)($$( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2}$$)($$( 1 - T + T^{2} )^{4}( 1 + T + T^{2} )^{4}$$)($$( 1 + T^{16} )^{2}$$)
$61$ ($$( 1 - T )^{8}$$)($$( 1 - T )^{4}( 1 + T )^{4}$$)($$( 1 - T )^{4}( 1 + T )^{4}$$)($$( 1 - T )^{2}$$)($$( 1 - T )^{4}$$)($$( 1 - T )^{2}( 1 + T )^{2}$$)($$( 1 - T )^{2}( 1 + T )^{2}$$)($$( 1 + T )^{4}( 1 + T^{2} )^{2}$$)($$( 1 + T^{2} )^{2}$$)($$( 1 + T^{2} )^{2}$$)($$( 1 + T^{2} )^{2}$$)($$( 1 + T^{2} )^{4}$$)($$( 1 + T^{2} )^{4}$$)($$( 1 + T )^{2}( 1 - T + T^{2} )$$)($$( 1 + T )^{2}( 1 - T + T^{2} )$$)($$( 1 - T )^{4}( 1 + T + T^{2} )^{2}$$)($$( 1 + T^{2} )^{4}( 1 - T^{2} + T^{4} )^{2}$$)($$( 1 + T^{2} )^{8}( 1 + T^{8} )^{2}$$)
$67$ ($$( 1 - T )^{4}( 1 + T )^{4}$$)($$( 1 + T^{2} )^{4}$$)($$( 1 + T^{2} )^{4}$$)($$1 + T^{2}$$)($$( 1 + T^{2} )^{2}$$)($$( 1 - T )^{2}( 1 + T )^{2}$$)($$( 1 - T )^{2}( 1 + T )^{2}$$)($$( 1 + T^{4} )^{2}$$)($$1 + T^{4}$$)($$1 + T^{4}$$)($$1 + T^{4}$$)($$( 1 + T^{4} )^{2}$$)($$( 1 + T^{4} )^{2}$$)($$( 1 - T )^{2}( 1 + T + T^{2} )$$)($$( 1 + T )^{2}( 1 - T + T^{2} )$$)($$( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} )$$)($$( 1 + T^{4} )^{2}( 1 - T^{4} + T^{8} )$$)($$( 1 + T^{16} )^{2}$$)
$71$ ($$( 1 - T )^{4}( 1 + T )^{4}$$)($$( 1 - T )^{4}( 1 + T )^{4}$$)($$( 1 - T )^{4}( 1 + T )^{4}$$)($$( 1 - T )( 1 + T )$$)($$( 1 - T )^{2}( 1 + T )^{2}$$)($$( 1 - T )^{2}( 1 + T )^{2}$$)($$( 1 - T )^{2}( 1 + T )^{2}$$)($$( 1 + T^{2} )^{4}$$)($$( 1 + T^{2} )^{2}$$)($$( 1 + T^{2} )^{2}$$)($$( 1 + T^{2} )^{2}$$)($$( 1 + T^{2} )^{4}$$)($$( 1 + T^{2} )^{4}$$)($$( 1 - T )^{2}( 1 + T )^{2}$$)($$( 1 - T )^{2}( 1 + T )^{2}$$)($$( 1 - T )^{4}( 1 + T )^{4}$$)($$( 1 - T )^{8}( 1 + T )^{8}$$)($$( 1 + T^{8} )^{4}$$)
$73$ ($$( 1 + T^{2} )^{4}$$)($$( 1 - T )^{4}( 1 + T )^{4}$$)($$( 1 - T )^{4}( 1 + T )^{4}$$)($$( 1 - T )( 1 + T )$$)($$( 1 - T )^{2}( 1 + T )^{2}$$)($$( 1 - T )^{2}( 1 + T )^{2}$$)($$( 1 - T )^{2}( 1 + T )^{2}$$)($$( 1 + T^{2} )^{4}$$)($$( 1 + T )^{2}( 1 + T^{2} )$$)($$( 1 - T )^{2}( 1 + T^{2} )$$)($$( 1 + T )^{2}( 1 + T^{2} )$$)($$( 1 + T )^{4}( 1 + T^{2} )^{2}$$)($$( 1 - T )^{4}( 1 + T^{2} )^{2}$$)($$( 1 - T )^{2}( 1 + T )^{2}$$)($$( 1 - T )^{2}( 1 + T )^{2}$$)($$( 1 - T )^{4}( 1 + T )^{4}$$)($$( 1 - T )^{8}( 1 + T )^{8}$$)($$( 1 + T^{8} )^{4}$$)
$79$ ($$( 1 + T^{2} )^{4}$$)($$( 1 + T^{2} )^{4}$$)($$( 1 + T^{2} )^{4}$$)($$( 1 - T )( 1 + T )$$)($$( 1 - T )^{2}( 1 + T )^{2}$$)($$( 1 + T^{2} )^{2}$$)($$( 1 + T^{2} )^{2}$$)($$( 1 - T )^{4}( 1 + T )^{4}$$)($$( 1 - T )^{2}( 1 + T )^{2}$$)($$( 1 - T )^{2}( 1 + T )^{2}$$)($$( 1 - T )^{2}( 1 + T )^{2}$$)($$( 1 + T^{2} )^{4}$$)($$( 1 + T^{2} )^{4}$$)($$( 1 - T + T^{2} )( 1 + T + T^{2} )$$)($$( 1 - T + T^{2} )( 1 + T + T^{2} )$$)($$( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2}$$)($$( 1 - T^{2} + T^{4} )^{4}$$)($$( 1 - T )^{16}( 1 + T^{2} )^{8}$$)
$83$ ($$( 1 + T^{2} )^{4}$$)($$( 1 - T )^{4}( 1 + T )^{4}$$)($$( 1 - T )^{4}( 1 + T )^{4}$$)($$1 + T^{2}$$)($$( 1 + T^{2} )^{2}$$)($$( 1 - T )^{2}( 1 + T )^{2}$$)($$( 1 - T )^{2}( 1 + T )^{2}$$)($$( 1 + T^{4} )^{2}$$)($$1 + T^{4}$$)($$1 + T^{4}$$)($$1 + T^{4}$$)($$( 1 + T^{4} )^{2}$$)($$( 1 + T^{4} )^{2}$$)($$( 1 - T )^{2}( 1 + T + T^{2} )$$)($$( 1 + T )^{2}( 1 - T + T^{2} )$$)($$( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} )$$)($$( 1 + T^{4} )^{2}( 1 - T^{4} + T^{8} )$$)($$( 1 + T^{16} )^{2}$$)
$89$ ($$( 1 + T^{4} )^{2}$$)($$( 1 + T^{4} )^{2}$$)($$( 1 + T^{4} )^{2}$$)($$( 1 + T )^{2}$$)($$( 1 + T^{2} )^{2}$$)($$( 1 + T^{2} )^{2}$$)($$( 1 + T^{2} )^{2}$$)($$( 1 - T )^{4}( 1 + T )^{4}$$)($$( 1 + T^{2} )^{2}$$)($$( 1 - T )^{2}( 1 + T )^{2}$$)($$( 1 + T^{2} )^{2}$$)($$( 1 + T^{4} )^{2}$$)($$( 1 + T^{4} )^{2}$$)($$( 1 + T + T^{2} )^{2}$$)($$( 1 + T + T^{2} )^{2}$$)($$( 1 + T + T^{2} )^{4}$$)($$( 1 - T + T^{2} )^{4}( 1 + T + T^{2} )^{4}$$)($$( 1 + T^{8} )^{4}$$)
$97$ ($$( 1 + T^{2} )^{4}$$)($$( 1 - T )^{4}( 1 + T )^{4}$$)($$( 1 - T )^{4}( 1 + T )^{4}$$)($$( 1 - T )( 1 + T )$$)($$( 1 - T )^{2}( 1 + T )^{2}$$)($$( 1 - T )^{2}( 1 + T )^{2}$$)($$( 1 - T )^{2}( 1 + T )^{2}$$)($$( 1 - T )^{4}( 1 + T )^{4}$$)($$( 1 + T )^{2}( 1 + T^{2} )$$)($$( 1 - T )^{2}( 1 + T^{2} )$$)($$( 1 + T )^{2}( 1 + T^{2} )$$)($$( 1 + T )^{4}( 1 + T^{2} )^{2}$$)($$( 1 - T )^{4}( 1 + T^{2} )^{2}$$)($$( 1 - T + T^{2} )( 1 + T + T^{2} )$$)($$( 1 - T + T^{2} )( 1 + T + T^{2} )$$)($$( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2}$$)($$( 1 - T + T^{2} )^{4}( 1 + T + T^{2} )^{4}$$)($$( 1 + T^{2} )^{16}$$)