Properties

Label 3600.1
Level 3600
Weight 1
Dimension 82
Nonzero newspaces 14
Newform subspaces 23
Sturm bound 691200
Trace bound 49

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Defining parameters

Level: \( N \) = \( 3600 = 2^{4} \cdot 3^{2} \cdot 5^{2} \)
Weight: \( k \) = \( 1 \)
Nonzero newspaces: \( 14 \)
Newform subspaces: \( 23 \)
Sturm bound: \(691200\)
Trace bound: \(49\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{1}(\Gamma_1(3600))\).

Total New Old
Modular forms 7192 912 6280
Cusp forms 920 82 838
Eisenstein series 6272 830 5442

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

\(D_n\) \(A_4\) \(S_4\) \(A_5\)
Dimension 74 0 8 0

Trace form

\( 82q - q^{5} - 2q^{9} + O(q^{10}) \) \( 82q - q^{5} - 2q^{9} - 2q^{13} + 4q^{16} - 4q^{19} + 8q^{21} + q^{25} + 2q^{29} - 8q^{31} + 4q^{34} + 7q^{37} + 10q^{41} + 12q^{46} + 4q^{49} + 5q^{53} + 8q^{61} + 5q^{65} + 6q^{69} + 14q^{73} + 4q^{76} + 2q^{81} + 3q^{85} + 3q^{89} + 20q^{91} - 4q^{94} + 14q^{97} + O(q^{100}) \)

Decomposition of \(S_{1}^{\mathrm{new}}(\Gamma_1(3600))\)

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
3600.1.c \(\chi_{3600}(449, \cdot)\) 3600.1.c.a 4 1
3600.1.e \(\chi_{3600}(3151, \cdot)\) 3600.1.e.a 1 1
3600.1.e.b 1
3600.1.e.c 2
3600.1.e.d 2
3600.1.g \(\chi_{3600}(1351, \cdot)\) None 0 1
3600.1.i \(\chi_{3600}(2249, \cdot)\) None 0 1
3600.1.j \(\chi_{3600}(1999, \cdot)\) 3600.1.j.a 2 1
3600.1.j.b 4
3600.1.l \(\chi_{3600}(1601, \cdot)\) 3600.1.l.a 2 1
3600.1.l.b 2
3600.1.n \(\chi_{3600}(3401, \cdot)\) None 0 1
3600.1.p \(\chi_{3600}(199, \cdot)\) None 0 1
3600.1.r \(\chi_{3600}(1099, \cdot)\) None 0 2
3600.1.s \(\chi_{3600}(701, \cdot)\) None 0 2
3600.1.v \(\chi_{3600}(1943, \cdot)\) None 0 2
3600.1.y \(\chi_{3600}(793, \cdot)\) None 0 2
3600.1.ba \(\chi_{3600}(107, \cdot)\) None 0 2
3600.1.bb \(\chi_{3600}(757, \cdot)\) 3600.1.bb.a 2 2
3600.1.bb.b 2
3600.1.be \(\chi_{3600}(1907, \cdot)\) None 0 2
3600.1.bf \(\chi_{3600}(2557, \cdot)\) 3600.1.bf.a 2 2
3600.1.bf.b 2
3600.1.bh \(\chi_{3600}(2593, \cdot)\) 3600.1.bh.a 2 2
3600.1.bh.b 4
3600.1.bk \(\chi_{3600}(143, \cdot)\) 3600.1.bk.a 4 2
3600.1.bn \(\chi_{3600}(1349, \cdot)\) None 0 2
3600.1.bo \(\chi_{3600}(451, \cdot)\) 3600.1.bo.a 4 2
3600.1.bq \(\chi_{3600}(1399, \cdot)\) None 0 2
3600.1.br \(\chi_{3600}(1001, \cdot)\) None 0 2
3600.1.bt \(\chi_{3600}(401, \cdot)\) None 0 2
3600.1.bv \(\chi_{3600}(799, \cdot)\) None 0 2
3600.1.by \(\chi_{3600}(1049, \cdot)\) None 0 2
3600.1.ca \(\chi_{3600}(151, \cdot)\) None 0 2
3600.1.cc \(\chi_{3600}(751, \cdot)\) 3600.1.cc.a 2 2
3600.1.cc.b 2
3600.1.ce \(\chi_{3600}(1649, \cdot)\) None 0 2
3600.1.cf \(\chi_{3600}(89, \cdot)\) None 0 4
3600.1.ch \(\chi_{3600}(631, \cdot)\) None 0 4
3600.1.cj \(\chi_{3600}(271, \cdot)\) 3600.1.cj.a 8 4
3600.1.cl \(\chi_{3600}(1169, \cdot)\) None 0 4
3600.1.cn \(\chi_{3600}(919, \cdot)\) None 0 4
3600.1.cp \(\chi_{3600}(521, \cdot)\) None 0 4
3600.1.cr \(\chi_{3600}(161, \cdot)\) None 0 4
3600.1.ct \(\chi_{3600}(559, \cdot)\) 3600.1.ct.a 4 4
3600.1.cw \(\chi_{3600}(1051, \cdot)\) None 0 4
3600.1.cx \(\chi_{3600}(149, \cdot)\) None 0 4
3600.1.cz \(\chi_{3600}(193, \cdot)\) None 0 4
3600.1.da \(\chi_{3600}(1343, \cdot)\) 3600.1.da.a 8 4
3600.1.dd \(\chi_{3600}(157, \cdot)\) None 0 4
3600.1.de \(\chi_{3600}(443, \cdot)\) None 0 4
3600.1.dh \(\chi_{3600}(493, \cdot)\) None 0 4
3600.1.di \(\chi_{3600}(1307, \cdot)\) None 0 4
3600.1.dl \(\chi_{3600}(407, \cdot)\) None 0 4
3600.1.dm \(\chi_{3600}(457, \cdot)\) None 0 4
3600.1.do \(\chi_{3600}(101, \cdot)\) None 0 4
3600.1.dp \(\chi_{3600}(499, \cdot)\) None 0 4
3600.1.dv \(\chi_{3600}(341, \cdot)\) None 0 8
3600.1.dw \(\chi_{3600}(19, \cdot)\) None 0 8
3600.1.dx \(\chi_{3600}(287, \cdot)\) 3600.1.dx.a 16 8
3600.1.ea \(\chi_{3600}(433, \cdot)\) None 0 8
3600.1.ec \(\chi_{3600}(397, \cdot)\) None 0 8
3600.1.ed \(\chi_{3600}(467, \cdot)\) None 0 8
3600.1.eg \(\chi_{3600}(37, \cdot)\) None 0 8
3600.1.eh \(\chi_{3600}(323, \cdot)\) None 0 8
3600.1.ej \(\chi_{3600}(73, \cdot)\) None 0 8
3600.1.em \(\chi_{3600}(503, \cdot)\) None 0 8
3600.1.en \(\chi_{3600}(91, \cdot)\) None 0 8
3600.1.eo \(\chi_{3600}(269, \cdot)\) None 0 8
3600.1.es \(\chi_{3600}(79, \cdot)\) None 0 8
3600.1.eu \(\chi_{3600}(641, \cdot)\) None 0 8
3600.1.ew \(\chi_{3600}(41, \cdot)\) None 0 8
3600.1.ex \(\chi_{3600}(439, \cdot)\) None 0 8
3600.1.ey \(\chi_{3600}(209, \cdot)\) None 0 8
3600.1.fa \(\chi_{3600}(31, \cdot)\) None 0 8
3600.1.fc \(\chi_{3600}(391, \cdot)\) None 0 8
3600.1.fe \(\chi_{3600}(329, \cdot)\) None 0 8
3600.1.fg \(\chi_{3600}(29, \cdot)\) None 0 16
3600.1.fh \(\chi_{3600}(211, \cdot)\) None 0 16
3600.1.fl \(\chi_{3600}(313, \cdot)\) None 0 16
3600.1.fm \(\chi_{3600}(23, \cdot)\) None 0 16
3600.1.fp \(\chi_{3600}(83, \cdot)\) None 0 16
3600.1.fq \(\chi_{3600}(13, \cdot)\) None 0 16
3600.1.ft \(\chi_{3600}(203, \cdot)\) None 0 16
3600.1.fu \(\chi_{3600}(133, \cdot)\) None 0 16
3600.1.fx \(\chi_{3600}(47, \cdot)\) None 0 16
3600.1.fy \(\chi_{3600}(97, \cdot)\) None 0 16
3600.1.gc \(\chi_{3600}(139, \cdot)\) None 0 16
3600.1.gd \(\chi_{3600}(221, \cdot)\) None 0 16

Decomposition of \(S_{1}^{\mathrm{old}}(\Gamma_1(3600))\) into lower level spaces

\( S_{1}^{\mathrm{old}}(\Gamma_1(3600)) \cong \) \(S_{1}^{\mathrm{new}}(\Gamma_1(72))\)\(^{\oplus 6}\)\(\oplus\)\(S_{1}^{\mathrm{new}}(\Gamma_1(80))\)\(^{\oplus 6}\)\(\oplus\)\(S_{1}^{\mathrm{new}}(\Gamma_1(100))\)\(^{\oplus 9}\)\(\oplus\)\(S_{1}^{\mathrm{new}}(\Gamma_1(120))\)\(^{\oplus 8}\)\(\oplus\)\(S_{1}^{\mathrm{new}}(\Gamma_1(144))\)\(^{\oplus 3}\)\(\oplus\)\(S_{1}^{\mathrm{new}}(\Gamma_1(180))\)\(^{\oplus 6}\)\(\oplus\)\(S_{1}^{\mathrm{new}}(\Gamma_1(200))\)\(^{\oplus 6}\)\(\oplus\)\(S_{1}^{\mathrm{new}}(\Gamma_1(225))\)\(^{\oplus 5}\)\(\oplus\)\(S_{1}^{\mathrm{new}}(\Gamma_1(240))\)\(^{\oplus 4}\)\(\oplus\)\(S_{1}^{\mathrm{new}}(\Gamma_1(300))\)\(^{\oplus 6}\)\(\oplus\)\(S_{1}^{\mathrm{new}}(\Gamma_1(360))\)\(^{\oplus 4}\)\(\oplus\)\(S_{1}^{\mathrm{new}}(\Gamma_1(400))\)\(^{\oplus 3}\)\(\oplus\)\(S_{1}^{\mathrm{new}}(\Gamma_1(600))\)\(^{\oplus 4}\)\(\oplus\)\(S_{1}^{\mathrm{new}}(\Gamma_1(720))\)\(^{\oplus 2}\)\(\oplus\)\(S_{1}^{\mathrm{new}}(\Gamma_1(900))\)\(^{\oplus 3}\)\(\oplus\)\(S_{1}^{\mathrm{new}}(\Gamma_1(1200))\)\(^{\oplus 2}\)\(\oplus\)\(S_{1}^{\mathrm{new}}(\Gamma_1(1800))\)\(^{\oplus 2}\)

Hecke characteristic polynomials

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