Properties

Label 2880.1
Level 2880
Weight 1
Dimension 69
Nonzero newspaces 10
Newform subspaces 18
Sturm bound 442368
Trace bound 25

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Defining parameters

Level: \( N \) = \( 2880 = 2^{6} \cdot 3^{2} \cdot 5 \)
Weight: \( k \) = \( 1 \)
Nonzero newspaces: \( 10 \)
Newform subspaces: \( 18 \)
Sturm bound: \(442368\)
Trace bound: \(25\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{1}(\Gamma_1(2880))\).

Total New Old
Modular forms 5132 663 4469
Cusp forms 524 69 455
Eisenstein series 4608 594 4014

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

\(D_n\) \(A_4\) \(S_4\) \(A_5\)
Dimension 69 0 0 0

Trace form

\( 69q - q^{5} + O(q^{10}) \) \( 69q - q^{5} - 2q^{13} + 2q^{17} - 4q^{19} + 8q^{21} - 11q^{25} + 10q^{29} + 2q^{37} + 2q^{41} - 7q^{49} + 2q^{53} + 10q^{61} - 2q^{65} - 2q^{73} - 16q^{76} + 16q^{79} + 2q^{85} - 10q^{89} - 16q^{94} - 2q^{97} + O(q^{100}) \)

Decomposition of \(S_{1}^{\mathrm{new}}(\Gamma_1(2880))\)

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
2880.1.c \(\chi_{2880}(449, \cdot)\) 2880.1.c.a 4 1
2880.1.e \(\chi_{2880}(2431, \cdot)\) None 0 1
2880.1.g \(\chi_{2880}(991, \cdot)\) None 0 1
2880.1.i \(\chi_{2880}(1889, \cdot)\) 2880.1.i.a 4 1
2880.1.i.b 4
2880.1.j \(\chi_{2880}(1279, \cdot)\) 2880.1.j.a 1 1
2880.1.j.b 2
2880.1.l \(\chi_{2880}(1601, \cdot)\) None 0 1
2880.1.n \(\chi_{2880}(161, \cdot)\) None 0 1
2880.1.p \(\chi_{2880}(2719, \cdot)\) 2880.1.p.a 2 1
2880.1.p.b 2
2880.1.r \(\chi_{2880}(559, \cdot)\) 2880.1.r.a 4 2
2880.1.s \(\chi_{2880}(881, \cdot)\) None 0 2
2880.1.v \(\chi_{2880}(287, \cdot)\) None 0 2
2880.1.y \(\chi_{2880}(2017, \cdot)\) None 0 2
2880.1.ba \(\chi_{2880}(1583, \cdot)\) None 0 2
2880.1.bb \(\chi_{2880}(1873, \cdot)\) None 0 2
2880.1.be \(\chi_{2880}(143, \cdot)\) None 0 2
2880.1.bf \(\chi_{2880}(433, \cdot)\) None 0 2
2880.1.bh \(\chi_{2880}(577, \cdot)\) 2880.1.bh.a 2 2
2880.1.bh.b 2
2880.1.bh.c 2
2880.1.bk \(\chi_{2880}(1727, \cdot)\) 2880.1.bk.a 4 2
2880.1.bk.b 4
2880.1.bn \(\chi_{2880}(1169, \cdot)\) None 0 2
2880.1.bo \(\chi_{2880}(271, \cdot)\) None 0 2
2880.1.bp \(\chi_{2880}(799, \cdot)\) None 0 2
2880.1.bq \(\chi_{2880}(1121, \cdot)\) None 0 2
2880.1.bs \(\chi_{2880}(641, \cdot)\) None 0 2
2880.1.bu \(\chi_{2880}(319, \cdot)\) 2880.1.bu.a 2 2
2880.1.bu.b 2
2880.1.bu.c 4
2880.1.bx \(\chi_{2880}(929, \cdot)\) None 0 2
2880.1.bz \(\chi_{2880}(31, \cdot)\) None 0 2
2880.1.cb \(\chi_{2880}(511, \cdot)\) None 0 2
2880.1.cd \(\chi_{2880}(1409, \cdot)\) 2880.1.cd.a 8 2
2880.1.cf \(\chi_{2880}(73, \cdot)\) None 0 4
2880.1.cg \(\chi_{2880}(1223, \cdot)\) None 0 4
2880.1.cj \(\chi_{2880}(631, \cdot)\) None 0 4
2880.1.ck \(\chi_{2880}(89, \cdot)\) None 0 4
2880.1.cm \(\chi_{2880}(199, \cdot)\) None 0 4
2880.1.cp \(\chi_{2880}(521, \cdot)\) None 0 4
2880.1.cq \(\chi_{2880}(503, \cdot)\) None 0 4
2880.1.ct \(\chi_{2880}(793, \cdot)\) None 0 4
2880.1.cw \(\chi_{2880}(751, \cdot)\) None 0 4
2880.1.cx \(\chi_{2880}(209, \cdot)\) None 0 4
2880.1.cz \(\chi_{2880}(193, \cdot)\) None 0 4
2880.1.da \(\chi_{2880}(383, \cdot)\) None 0 4
2880.1.dd \(\chi_{2880}(337, \cdot)\) None 0 4
2880.1.de \(\chi_{2880}(47, \cdot)\) None 0 4
2880.1.dh \(\chi_{2880}(817, \cdot)\) None 0 4
2880.1.di \(\chi_{2880}(527, \cdot)\) None 0 4
2880.1.dl \(\chi_{2880}(1247, \cdot)\) None 0 4
2880.1.dm \(\chi_{2880}(97, \cdot)\) None 0 4
2880.1.do \(\chi_{2880}(401, \cdot)\) None 0 4
2880.1.dp \(\chi_{2880}(79, \cdot)\) None 0 4
2880.1.ds \(\chi_{2880}(397, \cdot)\) None 0 8
2880.1.dv \(\chi_{2880}(107, \cdot)\) None 0 8
2880.1.dx \(\chi_{2880}(341, \cdot)\) None 0 8
2880.1.dz \(\chi_{2880}(269, \cdot)\) None 0 8
2880.1.ea \(\chi_{2880}(91, \cdot)\) None 0 8
2880.1.ec \(\chi_{2880}(19, \cdot)\) 2880.1.ec.a 16 8
2880.1.ee \(\chi_{2880}(37, \cdot)\) None 0 8
2880.1.eh \(\chi_{2880}(467, \cdot)\) None 0 8
2880.1.ei \(\chi_{2880}(313, \cdot)\) None 0 8
2880.1.el \(\chi_{2880}(23, \cdot)\) None 0 8
2880.1.en \(\chi_{2880}(329, \cdot)\) None 0 8
2880.1.eo \(\chi_{2880}(151, \cdot)\) None 0 8
2880.1.eq \(\chi_{2880}(41, \cdot)\) None 0 8
2880.1.et \(\chi_{2880}(439, \cdot)\) None 0 8
2880.1.ev \(\chi_{2880}(263, \cdot)\) None 0 8
2880.1.ew \(\chi_{2880}(553, \cdot)\) None 0 8
2880.1.ez \(\chi_{2880}(83, \cdot)\) None 0 16
2880.1.fa \(\chi_{2880}(133, \cdot)\) None 0 16
2880.1.fc \(\chi_{2880}(139, \cdot)\) None 0 16
2880.1.fe \(\chi_{2880}(211, \cdot)\) None 0 16
2880.1.fh \(\chi_{2880}(29, \cdot)\) None 0 16
2880.1.fj \(\chi_{2880}(101, \cdot)\) None 0 16
2880.1.fl \(\chi_{2880}(203, \cdot)\) None 0 16
2880.1.fm \(\chi_{2880}(13, \cdot)\) None 0 16

Decomposition of \(S_{1}^{\mathrm{old}}(\Gamma_1(2880))\) into lower level spaces

\( S_{1}^{\mathrm{old}}(\Gamma_1(2880)) \cong \) \(S_{1}^{\mathrm{new}}(\Gamma_1(72))\)\(^{\oplus 8}\)\(\oplus\)\(S_{1}^{\mathrm{new}}(\Gamma_1(80))\)\(^{\oplus 9}\)\(\oplus\)\(S_{1}^{\mathrm{new}}(\Gamma_1(120))\)\(^{\oplus 8}\)\(\oplus\)\(S_{1}^{\mathrm{new}}(\Gamma_1(144))\)\(^{\oplus 6}\)\(\oplus\)\(S_{1}^{\mathrm{new}}(\Gamma_1(160))\)\(^{\oplus 6}\)\(\oplus\)\(S_{1}^{\mathrm{new}}(\Gamma_1(180))\)\(^{\oplus 5}\)\(\oplus\)\(S_{1}^{\mathrm{new}}(\Gamma_1(192))\)\(^{\oplus 4}\)\(\oplus\)\(S_{1}^{\mathrm{new}}(\Gamma_1(240))\)\(^{\oplus 6}\)\(\oplus\)\(S_{1}^{\mathrm{new}}(\Gamma_1(288))\)\(^{\oplus 4}\)\(\oplus\)\(S_{1}^{\mathrm{new}}(\Gamma_1(320))\)\(^{\oplus 3}\)\(\oplus\)\(S_{1}^{\mathrm{new}}(\Gamma_1(360))\)\(^{\oplus 4}\)\(\oplus\)\(S_{1}^{\mathrm{new}}(\Gamma_1(480))\)\(^{\oplus 4}\)\(\oplus\)\(S_{1}^{\mathrm{new}}(\Gamma_1(576))\)\(^{\oplus 2}\)\(\oplus\)\(S_{1}^{\mathrm{new}}(\Gamma_1(720))\)\(^{\oplus 3}\)\(\oplus\)\(S_{1}^{\mathrm{new}}(\Gamma_1(960))\)\(^{\oplus 2}\)\(\oplus\)\(S_{1}^{\mathrm{new}}(\Gamma_1(1440))\)\(^{\oplus 2}\)

Hecke Characteristic Polynomials

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