Properties

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

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