Properties

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

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