# Properties

 Label 4000.1 Level 4000 Weight 1 Dimension 128 Nonzero newspaces 9 Newform subspaces 16 Sturm bound 960000 Trace bound 29

## Defining parameters

 Level: $$N$$ = $$4000 = 2^{5} \cdot 5^{3}$$ Weight: $$k$$ = $$1$$ Nonzero newspaces: $$9$$ Newform subspaces: $$16$$ Sturm bound: $$960000$$ Trace bound: $$29$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{1}(\Gamma_1(4000))$$.

Total New Old
Modular forms 6292 1408 4884
Cusp forms 532 128 404
Eisenstein series 5760 1280 4480

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

$$D_n$$ $$A_4$$ $$S_4$$ $$A_5$$
Dimension 88 0 0 40

## Trace form

 $$128q + O(q^{10})$$ $$128q + 4q^{11} - 8q^{13} - 2q^{17} + 12q^{21} + 6q^{29} + 12q^{41} + 4q^{49} + 4q^{57} - 14q^{61} - 2q^{69} + 4q^{73} - 4q^{81} + 10q^{85} + 20q^{89} + 8q^{91} + 8q^{93} - 2q^{97} + O(q^{100})$$

## Decomposition of $$S_{1}^{\mathrm{new}}(\Gamma_1(4000))$$

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
4000.1.b $$\chi_{4000}(2751, \cdot)$$ 4000.1.b.a 4 1
4000.1.b.b 4
4000.1.e $$\chi_{4000}(1999, \cdot)$$ 4000.1.e.a 2 1
4000.1.e.b 2
4000.1.g $$\chi_{4000}(751, \cdot)$$ 4000.1.g.a 4 1
4000.1.h $$\chi_{4000}(3999, \cdot)$$ 4000.1.h.a 4 1
4000.1.h.b 4
4000.1.i $$\chi_{4000}(57, \cdot)$$ None 0 2
4000.1.k $$\chi_{4000}(999, \cdot)$$ None 0 2
4000.1.m $$\chi_{4000}(2193, \cdot)$$ None 0 2
4000.1.p $$\chi_{4000}(193, \cdot)$$ 4000.1.p.a 8 2
4000.1.p.b 8
4000.1.r $$\chi_{4000}(1751, \cdot)$$ None 0 2
4000.1.t $$\chi_{4000}(2057, \cdot)$$ None 0 2
4000.1.w $$\chi_{4000}(1557, \cdot)$$ None 0 4
4000.1.x $$\chi_{4000}(251, \cdot)$$ None 0 4
4000.1.z $$\chi_{4000}(499, \cdot)$$ None 0 4
4000.1.bc $$\chi_{4000}(557, \cdot)$$ None 0 4
4000.1.bd $$\chi_{4000}(1551, \cdot)$$ None 0 4
4000.1.bf $$\chi_{4000}(799, \cdot)$$ 4000.1.bf.a 8 4
4000.1.bf.b 8
4000.1.bh $$\chi_{4000}(351, \cdot)$$ 4000.1.bh.a 8 4
4000.1.bi $$\chi_{4000}(399, \cdot)$$ None 0 4
4000.1.bk $$\chi_{4000}(457, \cdot)$$ None 0 8
4000.1.bn $$\chi_{4000}(199, \cdot)$$ None 0 8
4000.1.bo $$\chi_{4000}(257, \cdot)$$ 4000.1.bo.a 8 8
4000.1.bo.b 8
4000.1.bo.c 8
4000.1.br $$\chi_{4000}(593, \cdot)$$ None 0 8
4000.1.bs $$\chi_{4000}(151, \cdot)$$ None 0 8
4000.1.bv $$\chi_{4000}(393, \cdot)$$ None 0 8
4000.1.bx $$\chi_{4000}(93, \cdot)$$ None 0 16
4000.1.ca $$\chi_{4000}(99, \cdot)$$ None 0 16
4000.1.cc $$\chi_{4000}(51, \cdot)$$ None 0 16
4000.1.cd $$\chi_{4000}(157, \cdot)$$ None 0 16
4000.1.cf $$\chi_{4000}(159, \cdot)$$ None 0 20
4000.1.cg $$\chi_{4000}(79, \cdot)$$ None 0 20
4000.1.ci $$\chi_{4000}(111, \cdot)$$ None 0 20
4000.1.cl $$\chi_{4000}(31, \cdot)$$ None 0 20
4000.1.cm $$\chi_{4000}(71, \cdot)$$ None 0 40
4000.1.cp $$\chi_{4000}(73, \cdot)$$ None 0 40
4000.1.cq $$\chi_{4000}(33, \cdot)$$ 4000.1.cq.a 40 40
4000.1.ct $$\chi_{4000}(17, \cdot)$$ None 0 40
4000.1.cu $$\chi_{4000}(137, \cdot)$$ None 0 40
4000.1.cx $$\chi_{4000}(39, \cdot)$$ None 0 40
4000.1.cz $$\chi_{4000}(53, \cdot)$$ None 0 80
4000.1.db $$\chi_{4000}(11, \cdot)$$ None 0 80
4000.1.dc $$\chi_{4000}(19, \cdot)$$ None 0 80
4000.1.de $$\chi_{4000}(13, \cdot)$$ None 0 80

## Decomposition of $$S_{1}^{\mathrm{old}}(\Gamma_1(4000))$$ into lower level spaces

$$S_{1}^{\mathrm{old}}(\Gamma_1(4000)) \cong$$ $$S_{1}^{\mathrm{new}}(\Gamma_1(80))$$$$^{\oplus 6}$$$$\oplus$$$$S_{1}^{\mathrm{new}}(\Gamma_1(100))$$$$^{\oplus 8}$$$$\oplus$$$$S_{1}^{\mathrm{new}}(\Gamma_1(160))$$$$^{\oplus 3}$$$$\oplus$$$$S_{1}^{\mathrm{new}}(\Gamma_1(200))$$$$^{\oplus 6}$$$$\oplus$$$$S_{1}^{\mathrm{new}}(\Gamma_1(400))$$$$^{\oplus 4}$$$$\oplus$$$$S_{1}^{\mathrm{new}}(\Gamma_1(500))$$$$^{\oplus 4}$$$$\oplus$$$$S_{1}^{\mathrm{new}}(\Gamma_1(800))$$$$^{\oplus 2}$$$$\oplus$$$$S_{1}^{\mathrm{new}}(\Gamma_1(1000))$$$$^{\oplus 3}$$$$\oplus$$$$S_{1}^{\mathrm{new}}(\Gamma_1(2000))$$$$^{\oplus 2}$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ ($$1 - T^{2} + T^{4} - T^{6} + T^{8}$$)($$1 - T^{2} + T^{4} - T^{6} + T^{8}$$)($$( 1 - T )^{2}( 1 + T )^{2}$$)($$( 1 - T )^{2}( 1 + T )^{2}$$)($$( 1 + T^{2} )^{4}$$)($$( 1 + T + T^{2} + T^{3} + T^{4} )^{2}$$)($$( 1 - T + T^{2} - T^{3} + T^{4} )^{2}$$)($$( 1 + T + T^{2} + T^{3} + T^{4} )^{2}( 1 - T^{2} + T^{4} - T^{6} + T^{8} )$$)($$( 1 - T + T^{2} - T^{3} + T^{4} )^{2}( 1 - T^{2} + T^{4} - T^{6} + T^{8} )$$)($$( 1 + T - T^{3} - T^{4} - T^{5} + T^{7} + T^{8} )^{2}$$)($$( 1 - T + T^{3} - T^{4} + T^{5} - T^{7} + T^{8} )^{2}$$)($$1 + T^{2} - T^{6} - T^{8} - T^{10} + T^{14} + T^{16}$$)($$1 - T^{4} + T^{8} - T^{12} + T^{16}$$)($$1 - T^{4} + T^{8} - T^{12} + T^{16}$$)($$1 - T^{4} + T^{8} - T^{12} + T^{16}$$)($$1 - T^{20} + T^{40} - T^{60} + T^{80}$$)
$5$ ($$1 - T^{10} + T^{20} - T^{30} + T^{40}$$)
$7$ ($$( 1 - T^{2} + T^{4} )^{2}$$)($$( 1 - T^{2} + T^{4} )^{2}$$)($$1 + T + T^{2} + T^{3} + T^{4}$$)($$1 - T + T^{2} - T^{3} + T^{4}$$)($$1 - T^{2} + T^{4} - T^{6} + T^{8}$$)($$( 1 - T + T^{2} )^{4}$$)($$( 1 + T + T^{2} )^{4}$$)($$( 1 - T + T^{2} - T^{3} + T^{4} )^{2}( 1 - T^{2} + T^{4} - T^{6} + T^{8} )$$)($$( 1 + T + T^{2} + T^{3} + T^{4} )^{2}( 1 - T^{2} + T^{4} - T^{6} + T^{8} )$$)($$( 1 - T + T^{2} - T^{3} + T^{4} )^{4}$$)($$( 1 + T + T^{2} + T^{3} + T^{4} )^{4}$$)($$( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2}$$)($$( 1 + T^{4} )^{4}$$)($$( 1 + T^{4} )^{4}$$)($$( 1 + T^{4} )^{4}$$)($$( 1 - T^{4} + T^{8} - T^{12} + T^{16} )^{5}$$)
$11$ ($$( 1 - T^{2} + T^{4} )^{2}$$)($$( 1 - T^{2} + T^{4} )^{2}$$)($$1 - T + T^{2} - T^{3} + T^{4}$$)($$1 - T + T^{2} - T^{3} + T^{4}$$)($$( 1 - T + T^{2} - T^{3} + T^{4} )^{2}$$)($$( 1 - T^{2} + T^{4} )^{2}$$)($$( 1 - T^{2} + T^{4} )^{2}$$)($$( 1 + T^{2} )^{8}$$)($$( 1 + T^{2} )^{8}$$)($$( 1 - T + T^{2} - T^{3} + T^{4} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} )^{2}$$)($$( 1 - T + T^{2} - T^{3} + T^{4} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} )^{2}$$)($$( 1 - T + T^{2} - T^{3} + T^{4} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} )^{2}$$)($$( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2}$$)($$( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2}$$)($$( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2}$$)($$( 1 - T^{10} + T^{20} - T^{30} + T^{40} )^{2}$$)
$13$ ($$( 1 + T + T^{2} + T^{3} + T^{4} )^{2}$$)($$( 1 - T + T^{2} - T^{3} + T^{4} )^{2}$$)($$1 - T + T^{2} - T^{3} + T^{4}$$)($$1 + T + T^{2} + T^{3} + T^{4}$$)($$1 - T^{2} + T^{4} - T^{6} + T^{8}$$)($$1 - T^{2} + T^{4} - T^{6} + T^{8}$$)($$1 - T^{2} + T^{4} - T^{6} + T^{8}$$)($$( 1 + T^{4} )^{4}$$)($$( 1 + T^{4} )^{4}$$)($$( 1 + T^{2} )^{4}( 1 - T^{2} + T^{4} - T^{6} + T^{8} )$$)($$( 1 + T^{2} )^{4}( 1 - T^{2} + T^{4} - T^{6} + T^{8} )$$)($$( 1 + T )^{8}( 1 - T + T^{2} - T^{3} + T^{4} )^{2}$$)($$( 1 + T + T^{2} + T^{3} + T^{4} )^{2}( 1 - T^{2} + T^{4} - T^{6} + T^{8} )$$)($$( 1 + T + T^{2} + T^{3} + T^{4} )^{2}( 1 - T^{2} + T^{4} - T^{6} + T^{8} )$$)($$( 1 - T + T^{2} - T^{3} + T^{4} )^{2}( 1 - T^{2} + T^{4} - T^{6} + T^{8} )$$)($$( 1 - T^{5} + T^{10} - T^{15} + T^{20} )^{2}( 1 - T^{10} + T^{20} - T^{30} + T^{40} )$$)
$17$ ($$( 1 + T + T^{2} )^{4}$$)($$( 1 - T + T^{2} )^{4}$$)($$( 1 - T )^{2}( 1 + T )^{2}$$)($$( 1 - T )^{2}( 1 + T )^{2}$$)($$( 1 + T^{2} )^{4}$$)($$( 1 - T^{2} + T^{4} )^{2}$$)($$( 1 - T^{2} + T^{4} )^{2}$$)($$( 1 + T^{4} )^{4}$$)($$( 1 + T^{4} )^{4}$$)($$( 1 - T + T^{2} - T^{3} + T^{4} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} )^{2}$$)($$( 1 - T + T^{2} - T^{3} + T^{4} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} )^{2}$$)($$( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2}$$)($$( 1 + T + T^{2} + T^{3} + T^{4} )^{2}( 1 - T^{2} + T^{4} - T^{6} + T^{8} )$$)($$( 1 + T + T^{2} + T^{3} + T^{4} )^{2}( 1 - T^{2} + T^{4} - T^{6} + T^{8} )$$)($$( 1 - T + T^{2} - T^{3} + T^{4} )^{2}( 1 - T^{2} + T^{4} - T^{6} + T^{8} )$$)($$( 1 - T^{5} + T^{10} - T^{15} + T^{20} )^{2}( 1 - T^{10} + T^{20} - T^{30} + T^{40} )$$)
$19$ ($$1 - T^{2} + T^{4} - T^{6} + T^{8}$$)($$1 - T^{2} + T^{4} - T^{6} + T^{8}$$)($$1 - T + T^{2} - T^{3} + T^{4}$$)($$1 - T + T^{2} - T^{3} + T^{4}$$)($$( 1 + T + T^{2} + T^{3} + T^{4} )^{2}$$)($$1 - T^{2} + T^{4} - T^{6} + T^{8}$$)($$1 - T^{2} + T^{4} - T^{6} + T^{8}$$)($$( 1 - T )^{8}( 1 + T )^{8}$$)($$( 1 - T )^{8}( 1 + T )^{8}$$)($$( 1 + T^{2} )^{4}( 1 - T^{2} + T^{4} - T^{6} + T^{8} )$$)($$( 1 + T^{2} )^{4}( 1 - T^{2} + T^{4} - T^{6} + T^{8} )$$)($$( 1 + T^{2} )^{4}( 1 - T^{2} + T^{4} - T^{6} + T^{8} )$$)($$( 1 - T + T^{2} - T^{3} + T^{4} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} )^{2}$$)($$( 1 - T + T^{2} - T^{3} + T^{4} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} )^{2}$$)($$( 1 - T + T^{2} - T^{3} + T^{4} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} )^{2}$$)($$( 1 - T^{5} + T^{10} - T^{15} + T^{20} )^{2}( 1 + T^{5} + T^{10} + T^{15} + T^{20} )^{2}$$)
$23$ ($$( 1 - T )^{4}( 1 + T )^{4}$$)($$( 1 - T )^{4}( 1 + T )^{4}$$)($$1 + T + T^{2} + T^{3} + T^{4}$$)($$1 - T + T^{2} - T^{3} + T^{4}$$)($$1 - T^{2} + T^{4} - T^{6} + T^{8}$$)($$( 1 + T^{2} )^{4}$$)($$( 1 + T^{2} )^{4}$$)($$( 1 + T + T^{2} + T^{3} + T^{4} )^{2}( 1 - T^{2} + T^{4} - T^{6} + T^{8} )$$)($$( 1 - T + T^{2} - T^{3} + T^{4} )^{2}( 1 - T^{2} + T^{4} - T^{6} + T^{8} )$$)($$( 1 + T - T^{3} - T^{4} - T^{5} + T^{7} + T^{8} )^{2}$$)($$( 1 - T + T^{3} - T^{4} + T^{5} - T^{7} + T^{8} )^{2}$$)($$1 + T^{2} - T^{6} - T^{8} - T^{10} + T^{14} + T^{16}$$)($$1 - T^{4} + T^{8} - T^{12} + T^{16}$$)($$1 - T^{4} + T^{8} - T^{12} + T^{16}$$)($$1 - T^{4} + T^{8} - T^{12} + T^{16}$$)($$1 - T^{20} + T^{40} - T^{60} + T^{80}$$)
$29$ ($$( 1 + T + T^{2} )^{4}$$)($$( 1 + T + T^{2} )^{4}$$)($$( 1 - T )^{2}( 1 + T )^{2}$$)($$( 1 - T )^{2}( 1 + T )^{2}$$)($$( 1 - T )^{4}( 1 + T )^{4}$$)($$( 1 - T + T^{2} )^{4}$$)($$( 1 - T + T^{2} )^{4}$$)($$( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2}$$)($$( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2}$$)($$( 1 - T )^{8}( 1 + T + T^{2} + T^{3} + T^{4} )^{2}$$)($$( 1 - T )^{8}( 1 + T + T^{2} + T^{3} + T^{4} )^{2}$$)($$( 1 + T )^{8}( 1 - T + T^{2} - T^{3} + T^{4} )^{2}$$)($$( 1 - T + T^{2} - T^{3} + T^{4} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} )^{2}$$)($$( 1 - T + T^{2} - T^{3} + T^{4} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} )^{2}$$)($$( 1 - T + T^{2} - T^{3} + T^{4} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} )^{2}$$)($$( 1 - T^{5} + T^{10} - T^{15} + T^{20} )^{2}( 1 + T^{5} + T^{10} + T^{15} + T^{20} )^{2}$$)
$31$ ($$1 - T^{2} + T^{4} - T^{6} + T^{8}$$)($$1 - T^{2} + T^{4} - T^{6} + T^{8}$$)($$( 1 - T )^{2}( 1 + T )^{2}$$)($$( 1 - T )^{2}( 1 + T )^{2}$$)($$( 1 - T )^{4}( 1 + T )^{4}$$)($$1 - T^{2} + T^{4} - T^{6} + T^{8}$$)($$1 - T^{2} + T^{4} - T^{6} + T^{8}$$)($$( 1 + T^{2} )^{8}$$)($$( 1 + T^{2} )^{8}$$)($$1 + T^{2} - T^{6} - T^{8} - T^{10} + T^{14} + T^{16}$$)($$1 + T^{2} - T^{6} - T^{8} - T^{10} + T^{14} + T^{16}$$)($$1 + T^{2} - T^{6} - T^{8} - T^{10} + T^{14} + T^{16}$$)($$( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2}$$)($$( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2}$$)($$( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2}$$)($$( 1 - T^{10} + T^{20} - T^{30} + T^{40} )^{2}$$)
$37$ ($$( 1 + T^{2} )^{4}$$)($$( 1 + T^{2} )^{4}$$)($$1 - T + T^{2} - T^{3} + T^{4}$$)($$1 + T + T^{2} + T^{3} + T^{4}$$)($$1 - T^{2} + T^{4} - T^{6} + T^{8}$$)($$( 1 - T )^{4}( 1 + T )^{4}$$)($$( 1 - T )^{4}( 1 + T )^{4}$$)($$( 1 + T^{4} )^{4}$$)($$( 1 + T^{4} )^{4}$$)($$1 + T^{2} - T^{6} - T^{8} - T^{10} + T^{14} + T^{16}$$)($$1 + T^{2} - T^{6} - T^{8} - T^{10} + T^{14} + T^{16}$$)($$( 1 + T - T^{3} - T^{4} - T^{5} + T^{7} + T^{8} )^{2}$$)($$( 1 + T )^{8}( 1 - T^{2} + T^{4} - T^{6} + T^{8} )$$)($$( 1 + T^{2} )^{4}( 1 - T + T^{2} - T^{3} + T^{4} )^{2}$$)($$( 1 - T )^{8}( 1 - T^{2} + T^{4} - T^{6} + T^{8} )$$)($$( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{5}( 1 + T^{5} + T^{10} + T^{15} + T^{20} )^{2}$$)
$41$ ($$( 1 - T + T^{2} )^{4}$$)($$( 1 - T + T^{2} )^{4}$$)($$1 + T + T^{2} + T^{3} + T^{4}$$)($$1 + T + T^{2} + T^{3} + T^{4}$$)($$( 1 + T + T^{2} + T^{3} + T^{4} )^{2}$$)($$( 1 - T + T^{2} )^{4}$$)($$( 1 - T + T^{2} )^{4}$$)($$( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2}$$)($$( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2}$$)($$( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2}$$)($$( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2}$$)($$( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2}$$)($$( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2}$$)($$( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2}$$)($$( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2}$$)($$( 1 - T^{10} + T^{20} - T^{30} + T^{40} )^{2}$$)
$43$ ($$( 1 - T^{2} + T^{4} )^{2}$$)($$( 1 - T^{2} + T^{4} )^{2}$$)($$( 1 - T )^{2}( 1 + T )^{2}$$)($$( 1 - T )^{2}( 1 + T )^{2}$$)($$( 1 + T^{2} )^{4}$$)($$( 1 + T + T^{2} )^{4}$$)($$( 1 - T + T^{2} )^{4}$$)($$( 1 - T + T^{2} - T^{3} + T^{4} )^{2}( 1 - T^{2} + T^{4} - T^{6} + T^{8} )$$)($$( 1 + T + T^{2} + T^{3} + T^{4} )^{2}( 1 - T^{2} + T^{4} - T^{6} + T^{8} )$$)($$( 1 + T + T^{2} + T^{3} + T^{4} )^{4}$$)($$( 1 - T + T^{2} - T^{3} + T^{4} )^{4}$$)($$( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2}$$)($$( 1 + T^{4} )^{4}$$)($$( 1 + T^{4} )^{4}$$)($$( 1 + T^{4} )^{4}$$)($$( 1 - T^{4} + T^{8} - T^{12} + T^{16} )^{5}$$)
$47$ ($$1 - T^{2} + T^{4} - T^{6} + T^{8}$$)($$1 - T^{2} + T^{4} - T^{6} + T^{8}$$)($$1 + T + T^{2} + T^{3} + T^{4}$$)($$1 - T + T^{2} - T^{3} + T^{4}$$)($$1 - T^{2} + T^{4} - T^{6} + T^{8}$$)($$( 1 - T + T^{2} - T^{3} + T^{4} )^{2}$$)($$( 1 + T + T^{2} + T^{3} + T^{4} )^{2}$$)($$( 1 + T + T^{2} + T^{3} + T^{4} )^{2}( 1 - T^{2} + T^{4} - T^{6} + T^{8} )$$)($$( 1 - T + T^{2} - T^{3} + T^{4} )^{2}( 1 - T^{2} + T^{4} - T^{6} + T^{8} )$$)($$( 1 + T + T^{2} + T^{3} + T^{4} )^{4}$$)($$( 1 - T + T^{2} - T^{3} + T^{4} )^{4}$$)($$( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2}$$)($$1 - T^{4} + T^{8} - T^{12} + T^{16}$$)($$1 - T^{4} + T^{8} - T^{12} + T^{16}$$)($$1 - T^{4} + T^{8} - T^{12} + T^{16}$$)($$1 - T^{20} + T^{40} - T^{60} + T^{80}$$)
$53$ ($$( 1 + T + T^{2} + T^{3} + T^{4} )^{2}$$)($$( 1 - T + T^{2} - T^{3} + T^{4} )^{2}$$)($$1 - T + T^{2} - T^{3} + T^{4}$$)($$1 + T + T^{2} + T^{3} + T^{4}$$)($$1 - T^{2} + T^{4} - T^{6} + T^{8}$$)($$1 - T^{2} + T^{4} - T^{6} + T^{8}$$)($$1 - T^{2} + T^{4} - T^{6} + T^{8}$$)($$( 1 + T^{4} )^{4}$$)($$( 1 + T^{4} )^{4}$$)($$1 + T^{2} - T^{6} - T^{8} - T^{10} + T^{14} + T^{16}$$)($$1 + T^{2} - T^{6} - T^{8} - T^{10} + T^{14} + T^{16}$$)($$( 1 + T - T^{3} - T^{4} - T^{5} + T^{7} + T^{8} )^{2}$$)($$( 1 + T )^{8}( 1 - T^{2} + T^{4} - T^{6} + T^{8} )$$)($$( 1 + T^{2} )^{4}( 1 - T + T^{2} - T^{3} + T^{4} )^{2}$$)($$( 1 - T )^{8}( 1 - T^{2} + T^{4} - T^{6} + T^{8} )$$)($$( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{5}( 1 + T^{5} + T^{10} + T^{15} + T^{20} )^{2}$$)
$59$ ($$1 - T^{2} + T^{4} - T^{6} + T^{8}$$)($$1 - T^{2} + T^{4} - T^{6} + T^{8}$$)($$1 - T + T^{2} - T^{3} + T^{4}$$)($$1 - T + T^{2} - T^{3} + T^{4}$$)($$( 1 + T + T^{2} + T^{3} + T^{4} )^{2}$$)($$1 - T^{2} + T^{4} - T^{6} + T^{8}$$)($$1 - T^{2} + T^{4} - T^{6} + T^{8}$$)($$( 1 - T )^{8}( 1 + T )^{8}$$)($$( 1 - T )^{8}( 1 + T )^{8}$$)($$( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2}$$)($$( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2}$$)($$( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2}$$)($$( 1 - T + T^{2} - T^{3} + T^{4} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} )^{2}$$)($$( 1 - T + T^{2} - T^{3} + T^{4} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} )^{2}$$)($$( 1 - T + T^{2} - T^{3} + T^{4} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} )^{2}$$)($$( 1 - T^{5} + T^{10} - T^{15} + T^{20} )^{2}( 1 + T^{5} + T^{10} + T^{15} + T^{20} )^{2}$$)
$61$ ($$( 1 + T + T^{2} + T^{3} + T^{4} )^{2}$$)($$( 1 + T + T^{2} + T^{3} + T^{4} )^{2}$$)($$( 1 - T )^{2}( 1 + T )^{2}$$)($$( 1 - T )^{2}( 1 + T )^{2}$$)($$( 1 - T )^{4}( 1 + T )^{4}$$)($$( 1 + T + T^{2} + T^{3} + T^{4} )^{2}$$)($$( 1 + T + T^{2} + T^{3} + T^{4} )^{2}$$)($$( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2}$$)($$( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2}$$)($$( 1 + T - T^{3} - T^{4} - T^{5} + T^{7} + T^{8} )^{2}$$)($$( 1 + T - T^{3} - T^{4} - T^{5} + T^{7} + T^{8} )^{2}$$)($$( 1 + T - T^{3} - T^{4} - T^{5} + T^{7} + T^{8} )^{2}$$)($$( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2}$$)($$( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2}$$)($$( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2}$$)($$( 1 - T^{10} + T^{20} - T^{30} + T^{40} )^{2}$$)
$67$ ($$1 - T^{2} + T^{4} - T^{6} + T^{8}$$)($$1 - T^{2} + T^{4} - T^{6} + T^{8}$$)($$( 1 - T )^{2}( 1 + T )^{2}$$)($$( 1 - T )^{2}( 1 + T )^{2}$$)($$( 1 + T^{2} )^{4}$$)($$( 1 + T + T^{2} + T^{3} + T^{4} )^{2}$$)($$( 1 - T + T^{2} - T^{3} + T^{4} )^{2}$$)($$( 1 + T )^{8}( 1 + T^{2} )^{4}$$)($$( 1 - T )^{8}( 1 + T^{2} )^{4}$$)($$( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2}$$)($$( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2}$$)($$( 1 - T + T^{2} - T^{3} + T^{4} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} )^{2}$$)($$1 - T^{4} + T^{8} - T^{12} + T^{16}$$)($$1 - T^{4} + T^{8} - T^{12} + T^{16}$$)($$1 - T^{4} + T^{8} - T^{12} + T^{16}$$)($$1 - T^{20} + T^{40} - T^{60} + T^{80}$$)
$71$ ($$( 1 - T^{2} + T^{4} )^{2}$$)($$( 1 - T^{2} + T^{4} )^{2}$$)($$( 1 - T )^{2}( 1 + T )^{2}$$)($$( 1 - T )^{2}( 1 + T )^{2}$$)($$( 1 - T )^{4}( 1 + T )^{4}$$)($$( 1 - T^{2} + T^{4} )^{2}$$)($$( 1 - T^{2} + T^{4} )^{2}$$)($$( 1 + T^{2} )^{8}$$)($$( 1 + T^{2} )^{8}$$)($$( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2}$$)($$( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2}$$)($$( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2}$$)($$( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2}$$)($$( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2}$$)($$( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2}$$)($$( 1 - T^{10} + T^{20} - T^{30} + T^{40} )^{2}$$)
$73$ ($$( 1 + T + T^{2} + T^{3} + T^{4} )^{2}$$)($$( 1 - T + T^{2} - T^{3} + T^{4} )^{2}$$)($$( 1 - T )^{2}( 1 + T )^{2}$$)($$( 1 - T )^{2}( 1 + T )^{2}$$)($$( 1 + T^{2} )^{4}$$)($$1 - T^{2} + T^{4} - T^{6} + T^{8}$$)($$1 - T^{2} + T^{4} - T^{6} + T^{8}$$)($$( 1 + T^{4} )^{4}$$)($$( 1 + T^{4} )^{4}$$)($$1 + T^{2} - T^{6} - T^{8} - T^{10} + T^{14} + T^{16}$$)($$1 + T^{2} - T^{6} - T^{8} - T^{10} + T^{14} + T^{16}$$)($$( 1 - T + T^{3} - T^{4} + T^{5} - T^{7} + T^{8} )^{2}$$)($$( 1 - T + T^{2} - T^{3} + T^{4} )^{2}( 1 - T^{2} + T^{4} - T^{6} + T^{8} )$$)($$( 1 - T + T^{2} - T^{3} + T^{4} )^{2}( 1 - T^{2} + T^{4} - T^{6} + T^{8} )$$)($$( 1 + T + T^{2} + T^{3} + T^{4} )^{2}( 1 - T^{2} + T^{4} - T^{6} + T^{8} )$$)($$( 1 + T^{5} + T^{10} + T^{15} + T^{20} )^{2}( 1 - T^{10} + T^{20} - T^{30} + T^{40} )$$)
$79$ ($$( 1 - T^{2} + T^{4} )^{2}$$)($$( 1 - T^{2} + T^{4} )^{2}$$)($$( 1 - T )^{2}( 1 + T )^{2}$$)($$( 1 - T )^{2}( 1 + T )^{2}$$)($$( 1 - T )^{4}( 1 + T )^{4}$$)($$( 1 - T^{2} + T^{4} )^{2}$$)($$( 1 - T^{2} + T^{4} )^{2}$$)($$( 1 - T )^{8}( 1 + T )^{8}$$)($$( 1 - T )^{8}( 1 + T )^{8}$$)($$( 1 + T^{2} )^{4}( 1 - T^{2} + T^{4} - T^{6} + T^{8} )$$)($$( 1 + T^{2} )^{4}( 1 - T^{2} + T^{4} - T^{6} + T^{8} )$$)($$( 1 + T^{2} )^{4}( 1 - T^{2} + T^{4} - T^{6} + T^{8} )$$)($$( 1 - T + T^{2} - T^{3} + T^{4} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} )^{2}$$)($$( 1 - T + T^{2} - T^{3} + T^{4} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} )^{2}$$)($$( 1 - T + T^{2} - T^{3} + T^{4} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} )^{2}$$)($$( 1 - T^{5} + T^{10} - T^{15} + T^{20} )^{2}( 1 + T^{5} + T^{10} + T^{15} + T^{20} )^{2}$$)
$83$ ($$( 1 - T )^{4}( 1 + T )^{4}$$)($$( 1 - T )^{4}( 1 + T )^{4}$$)($$( 1 - T )^{2}( 1 + T )^{2}$$)($$( 1 - T )^{2}( 1 + T )^{2}$$)($$( 1 + T^{2} )^{4}$$)($$( 1 + T^{2} )^{4}$$)($$( 1 + T^{2} )^{4}$$)($$( 1 - T + T^{2} - T^{3} + T^{4} )^{2}( 1 - T^{2} + T^{4} - T^{6} + T^{8} )$$)($$( 1 + T + T^{2} + T^{3} + T^{4} )^{2}( 1 - T^{2} + T^{4} - T^{6} + T^{8} )$$)($$( 1 - T + T^{3} - T^{4} + T^{5} - T^{7} + T^{8} )^{2}$$)($$( 1 + T - T^{3} - T^{4} - T^{5} + T^{7} + T^{8} )^{2}$$)($$1 + T^{2} - T^{6} - T^{8} - T^{10} + T^{14} + T^{16}$$)($$1 - T^{4} + T^{8} - T^{12} + T^{16}$$)($$1 - T^{4} + T^{8} - T^{12} + T^{16}$$)($$1 - T^{4} + T^{8} - T^{12} + T^{16}$$)($$1 - T^{20} + T^{40} - T^{60} + T^{80}$$)
$89$ ($$( 1 + T^{2} )^{4}$$)($$( 1 + T^{2} )^{4}$$)($$1 + T + T^{2} + T^{3} + T^{4}$$)($$1 + T + T^{2} + T^{3} + T^{4}$$)($$( 1 - T + T^{2} - T^{3} + T^{4} )^{2}$$)($$( 1 + T^{2} )^{4}$$)($$( 1 + T^{2} )^{4}$$)($$( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2}$$)($$( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2}$$)($$( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2}$$)($$( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2}$$)($$( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2}$$)($$( 1 - T )^{8}( 1 - T + T^{2} - T^{3} + T^{4} )^{2}$$)($$( 1 + T )^{8}( 1 + T + T^{2} + T^{3} + T^{4} )^{2}$$)($$( 1 - T )^{8}( 1 - T + T^{2} - T^{3} + T^{4} )^{2}$$)($$( 1 - T + T^{2} - T^{3} + T^{4} )^{10}( 1 + T^{5} + T^{10} + T^{15} + T^{20} )^{2}$$)
$97$ ($$( 1 - T + T^{2} - T^{3} + T^{4} )^{2}$$)($$( 1 + T + T^{2} + T^{3} + T^{4} )^{2}$$)($$( 1 - T )^{2}( 1 + T )^{2}$$)($$( 1 - T )^{2}( 1 + T )^{2}$$)($$( 1 + T^{2} )^{4}$$)($$1 - T^{2} + T^{4} - T^{6} + T^{8}$$)($$1 - T^{2} + T^{4} - T^{6} + T^{8}$$)($$( 1 + T^{4} )^{4}$$)($$( 1 + T^{4} )^{4}$$)($$( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2}$$)($$( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2}$$)($$( 1 + T + T^{2} + T^{3} + T^{4} )^{4}$$)($$( 1 - T + T^{2} - T^{3} + T^{4} )^{2}( 1 - T^{2} + T^{4} - T^{6} + T^{8} )$$)($$( 1 - T + T^{2} - T^{3} + T^{4} )^{2}( 1 - T^{2} + T^{4} - T^{6} + T^{8} )$$)($$( 1 + T + T^{2} + T^{3} + T^{4} )^{2}( 1 - T^{2} + T^{4} - T^{6} + T^{8} )$$)($$( 1 + T^{5} + T^{10} + T^{15} + T^{20} )^{2}( 1 - T^{10} + T^{20} - T^{30} + T^{40} )$$)