# Properties

 Label 1575.1 Level 1575 Weight 1 Dimension 83 Nonzero newspaces 10 Newform subspaces 20 Sturm bound 172800 Trace bound 16

## Defining parameters

 Level: $$N$$ = $$1575 = 3^{2} \cdot 5^{2} \cdot 7$$ Weight: $$k$$ = $$1$$ Nonzero newspaces: $$10$$ Newform subspaces: $$20$$ Sturm bound: $$172800$$ Trace bound: $$16$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{1}(\Gamma_1(1575))$$.

Total New Old
Modular forms 2838 1006 1832
Cusp forms 150 83 67
Eisenstein series 2688 923 1765

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

$$D_n$$ $$A_4$$ $$S_4$$ $$A_5$$
Dimension 67 16 0 0

## Trace form

 $$83q + 3q^{4} + 8q^{6} + q^{7} - 4q^{9} + O(q^{10})$$ $$83q + 3q^{4} + 8q^{6} + q^{7} - 4q^{9} - 6q^{11} - 11q^{16} - 2q^{21} - 8q^{26} - q^{28} + 4q^{29} + 4q^{31} + 2q^{36} + 2q^{37} - 2q^{39} - 8q^{41} + 2q^{43} + 4q^{44} - 20q^{46} + q^{49} - 8q^{51} - 8q^{56} - 4q^{61} - 3q^{64} - 2q^{67} - 32q^{71} - 8q^{76} - 4q^{79} - 8q^{81} - 4q^{84} - 8q^{86} + 12q^{91} - 2q^{99} + O(q^{100})$$

## Decomposition of $$S_{1}^{\mathrm{new}}(\Gamma_1(1575))$$

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
1575.1.c $$\chi_{1575}(701, \cdot)$$ None 0 1
1575.1.e $$\chi_{1575}(874, \cdot)$$ 1575.1.e.a 2 1
1575.1.e.b 2
1575.1.e.c 4
1575.1.f $$\chi_{1575}(449, \cdot)$$ None 0 1
1575.1.h $$\chi_{1575}(1126, \cdot)$$ 1575.1.h.a 1 1
1575.1.h.b 1
1575.1.h.c 1
1575.1.h.d 2
1575.1.h.e 2
1575.1.n $$\chi_{1575}(818, \cdot)$$ 1575.1.n.a 4 2
1575.1.n.b 4
1575.1.n.c 8
1575.1.n.d 8
1575.1.o $$\chi_{1575}(568, \cdot)$$ None 0 2
1575.1.r $$\chi_{1575}(1174, \cdot)$$ None 0 2
1575.1.t $$\chi_{1575}(326, \cdot)$$ None 0 2
1575.1.w $$\chi_{1575}(599, \cdot)$$ None 0 2
1575.1.x $$\chi_{1575}(451, \cdot)$$ 1575.1.x.a 2 2
1575.1.x.b 2
1575.1.y $$\chi_{1575}(76, \cdot)$$ 1575.1.y.a 4 2
1575.1.z $$\chi_{1575}(674, \cdot)$$ None 0 2
1575.1.bb $$\chi_{1575}(974, \cdot)$$ None 0 2
1575.1.bd $$\chi_{1575}(376, \cdot)$$ None 0 2
1575.1.be $$\chi_{1575}(851, \cdot)$$ None 0 2
1575.1.bh $$\chi_{1575}(349, \cdot)$$ None 0 2
1575.1.bj $$\chi_{1575}(199, \cdot)$$ 1575.1.bj.a 4 2
1575.1.bl $$\chi_{1575}(176, \cdot)$$ None 0 2
1575.1.bn $$\chi_{1575}(926, \cdot)$$ None 0 2
1575.1.bo $$\chi_{1575}(124, \cdot)$$ None 0 2
1575.1.bq $$\chi_{1575}(1426, \cdot)$$ None 0 2
1575.1.bs $$\chi_{1575}(74, \cdot)$$ None 0 2
1575.1.bt $$\chi_{1575}(181, \cdot)$$ None 0 4
1575.1.bv $$\chi_{1575}(134, \cdot)$$ None 0 4
1575.1.bw $$\chi_{1575}(244, \cdot)$$ None 0 4
1575.1.by $$\chi_{1575}(71, \cdot)$$ None 0 4
1575.1.cb $$\chi_{1575}(718, \cdot)$$ 1575.1.cb.a 8 4
1575.1.cc $$\chi_{1575}(68, \cdot)$$ None 0 4
1575.1.ce $$\chi_{1575}(1118, \cdot)$$ None 0 4
1575.1.cg $$\chi_{1575}(43, \cdot)$$ None 0 4
1575.1.ci $$\chi_{1575}(793, \cdot)$$ 1575.1.ci.a 8 4
1575.1.cl $$\chi_{1575}(143, \cdot)$$ None 0 4
1575.1.cn $$\chi_{1575}(293, \cdot)$$ 1575.1.cn.a 8 4
1575.1.cp $$\chi_{1575}(193, \cdot)$$ 1575.1.cp.a 8 4
1575.1.cv $$\chi_{1575}(127, \cdot)$$ None 0 8
1575.1.cw $$\chi_{1575}(62, \cdot)$$ None 0 8
1575.1.cy $$\chi_{1575}(389, \cdot)$$ None 0 8
1575.1.da $$\chi_{1575}(166, \cdot)$$ None 0 8
1575.1.dc $$\chi_{1575}(94, \cdot)$$ None 0 8
1575.1.dd $$\chi_{1575}(116, \cdot)$$ None 0 8
1575.1.df $$\chi_{1575}(281, \cdot)$$ None 0 8
1575.1.dh $$\chi_{1575}(19, \cdot)$$ None 0 8
1575.1.dj $$\chi_{1575}(34, \cdot)$$ None 0 8
1575.1.dm $$\chi_{1575}(191, \cdot)$$ None 0 8
1575.1.dn $$\chi_{1575}(31, \cdot)$$ None 0 8
1575.1.dp $$\chi_{1575}(29, \cdot)$$ None 0 8
1575.1.dr $$\chi_{1575}(44, \cdot)$$ None 0 8
1575.1.ds $$\chi_{1575}(286, \cdot)$$ None 0 8
1575.1.dt $$\chi_{1575}(136, \cdot)$$ None 0 8
1575.1.du $$\chi_{1575}(254, \cdot)$$ None 0 8
1575.1.dx $$\chi_{1575}(11, \cdot)$$ None 0 8
1575.1.dz $$\chi_{1575}(229, \cdot)$$ None 0 8
1575.1.ea $$\chi_{1575}(67, \cdot)$$ None 0 16
1575.1.ec $$\chi_{1575}(83, \cdot)$$ None 0 16
1575.1.ee $$\chi_{1575}(17, \cdot)$$ None 0 16
1575.1.eh $$\chi_{1575}(37, \cdot)$$ None 0 16
1575.1.ej $$\chi_{1575}(22, \cdot)$$ None 0 16
1575.1.el $$\chi_{1575}(47, \cdot)$$ None 0 16
1575.1.en $$\chi_{1575}(38, \cdot)$$ None 0 16
1575.1.eo $$\chi_{1575}(58, \cdot)$$ None 0 16

## Decomposition of $$S_{1}^{\mathrm{old}}(\Gamma_1(1575))$$ into lower level spaces

$$S_{1}^{\mathrm{old}}(\Gamma_1(1575)) \cong$$ $$S_{1}^{\mathrm{new}}(\Gamma_1(63))$$$$^{\oplus 3}$$$$\oplus$$$$S_{1}^{\mathrm{new}}(\Gamma_1(175))$$$$^{\oplus 3}$$$$\oplus$$$$S_{1}^{\mathrm{new}}(\Gamma_1(225))$$$$^{\oplus 2}$$$$\oplus$$$$S_{1}^{\mathrm{new}}(\Gamma_1(315))$$$$^{\oplus 2}$$$$\oplus$$$$S_{1}^{\mathrm{new}}(\Gamma_1(525))$$$$^{\oplus 2}$$

## Hecke characteristic polynomials

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