Properties

Label 1224.1
Level 1224
Weight 1
Dimension 83
Nonzero newspaces 9
Newform subspaces 17
Sturm bound 82944
Trace bound 11

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Defining parameters

Level: \( N \) = \( 1224\( 1224 = 2^{3} \cdot 3^{2} \cdot 17 \) \)
Weight: \( k \) = \( 1 \)
Nonzero newspaces: \( 9 \)
Newform subspaces: \( 17 \)
Sturm bound: \(82944\)
Trace bound: \(11\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{1}(\Gamma_1(1224))\).

Total New Old
Modular forms 1788 353 1435
Cusp forms 252 83 169
Eisenstein series 1536 270 1266

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

\(D_n\) \(A_4\) \(S_4\) \(A_5\)
Dimension 79 0 4 0

Trace form

\( 83q + 3q^{2} + 2q^{3} + 5q^{4} + 2q^{6} + 8q^{7} - 3q^{8} + 2q^{9} + O(q^{10}) \) \( 83q + 3q^{2} + 2q^{3} + 5q^{4} + 2q^{6} + 8q^{7} - 3q^{8} + 2q^{9} - 4q^{12} - 4q^{13} - 3q^{16} + 3q^{17} - 4q^{18} - 2q^{19} - 4q^{22} + 2q^{24} + 5q^{25} - 4q^{27} + 3q^{32} - 2q^{33} - 6q^{34} + 2q^{36} - 24q^{38} + 6q^{44} + 2q^{48} + 9q^{49} - 21q^{50} - q^{51} - 14q^{54} + 4q^{55} - 2q^{57} - q^{64} - 12q^{66} - 4q^{67} - 8q^{70} + 2q^{72} + 2q^{73} + 2q^{75} - 4q^{76} - 8q^{79} + 2q^{81} + 2q^{82} - 18q^{83} + 4q^{85} - 4q^{88} - 6q^{89} - 8q^{94} - 4q^{96} - 4q^{97} - 3q^{98} - 12q^{99} + O(q^{100}) \)

Decomposition of \(S_{1}^{\mathrm{new}}(\Gamma_1(1224))\)

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
1224.1.b \(\chi_{1224}(917, \cdot)\) None 0 1
1224.1.d \(\chi_{1224}(307, \cdot)\) None 0 1
1224.1.g \(\chi_{1224}(953, \cdot)\) None 0 1
1224.1.i \(\chi_{1224}(271, \cdot)\) None 0 1
1224.1.k \(\chi_{1224}(919, \cdot)\) None 0 1
1224.1.m \(\chi_{1224}(305, \cdot)\) 1224.1.m.a 2 1
1224.1.m.b 2
1224.1.n \(\chi_{1224}(883, \cdot)\) 1224.1.n.a 1 1
1224.1.n.b 2
1224.1.n.c 2
1224.1.p \(\chi_{1224}(341, \cdot)\) None 0 1
1224.1.s \(\chi_{1224}(523, \cdot)\) 1224.1.s.a 2 2
1224.1.u \(\chi_{1224}(557, \cdot)\) None 0 2
1224.1.v \(\chi_{1224}(89, \cdot)\) None 0 2
1224.1.x \(\chi_{1224}(55, \cdot)\) None 0 2
1224.1.z \(\chi_{1224}(749, \cdot)\) None 0 2
1224.1.bb \(\chi_{1224}(67, \cdot)\) 1224.1.bb.a 2 2
1224.1.bb.b 2
1224.1.bc \(\chi_{1224}(713, \cdot)\) None 0 2
1224.1.be \(\chi_{1224}(103, \cdot)\) None 0 2
1224.1.bg \(\chi_{1224}(679, \cdot)\) None 0 2
1224.1.bi \(\chi_{1224}(137, \cdot)\) None 0 2
1224.1.bl \(\chi_{1224}(715, \cdot)\) None 0 2
1224.1.bn \(\chi_{1224}(101, \cdot)\) None 0 2
1224.1.bo \(\chi_{1224}(161, \cdot)\) None 0 4
1224.1.bp \(\chi_{1224}(127, \cdot)\) None 0 4
1224.1.bu \(\chi_{1224}(53, \cdot)\) 1224.1.bu.a 4 4
1224.1.bu.b 4
1224.1.bv \(\chi_{1224}(19, \cdot)\) 1224.1.bv.a 4 4
1224.1.bw \(\chi_{1224}(353, \cdot)\) None 0 4
1224.1.by \(\chi_{1224}(319, \cdot)\) None 0 4
1224.1.cb \(\chi_{1224}(115, \cdot)\) 1224.1.cb.a 4 4
1224.1.cb.b 4
1224.1.cd \(\chi_{1224}(149, \cdot)\) None 0 4
1224.1.ce \(\chi_{1224}(107, \cdot)\) None 0 8
1224.1.ch \(\chi_{1224}(37, \cdot)\) None 0 8
1224.1.ci \(\chi_{1224}(73, \cdot)\) None 0 8
1224.1.cl \(\chi_{1224}(71, \cdot)\) None 0 8
1224.1.co \(\chi_{1224}(77, \cdot)\) None 0 8
1224.1.cp \(\chi_{1224}(43, \cdot)\) 1224.1.cp.a 8 8
1224.1.cp.b 8
1224.1.cq \(\chi_{1224}(185, \cdot)\) None 0 8
1224.1.cr \(\chi_{1224}(151, \cdot)\) None 0 8
1224.1.cu \(\chi_{1224}(61, \cdot)\) None 0 16
1224.1.cx \(\chi_{1224}(11, \cdot)\) 1224.1.cx.a 16 16
1224.1.cx.b 16
1224.1.cy \(\chi_{1224}(23, \cdot)\) None 0 16
1224.1.db \(\chi_{1224}(97, \cdot)\) None 0 16

Decomposition of \(S_{1}^{\mathrm{old}}(\Gamma_1(1224))\) into lower level spaces

\( S_{1}^{\mathrm{old}}(\Gamma_1(1224)) \cong \) \(S_{1}^{\mathrm{new}}(\Gamma_1(68))\)\(^{\oplus 6}\)\(\oplus\)\(S_{1}^{\mathrm{new}}(\Gamma_1(72))\)\(^{\oplus 2}\)\(\oplus\)\(S_{1}^{\mathrm{new}}(\Gamma_1(136))\)\(^{\oplus 3}\)\(\oplus\)\(S_{1}^{\mathrm{new}}(\Gamma_1(204))\)\(^{\oplus 4}\)\(\oplus\)\(S_{1}^{\mathrm{new}}(\Gamma_1(408))\)\(^{\oplus 2}\)\(\oplus\)\(S_{1}^{\mathrm{new}}(\Gamma_1(612))\)\(^{\oplus 2}\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ (\( 1 - T \))(\( ( 1 + T )^{2} \))(\( ( 1 - T )^{2} \))(\( 1 + T^{2} \))(\( 1 - T + T^{2} \))(\( 1 - T + T^{2} \))(\( 1 + T^{4} \))(\( 1 + T^{4} \))(\( 1 + T^{4} \))(\( 1 - T^{2} + T^{4} \))(\( 1 - T^{2} + T^{4} \))(\( 1 - T^{4} + T^{8} \))(\( 1 - T^{4} + T^{8} \))(\( 1 - T^{8} + T^{16} \))(\( 1 - T^{8} + T^{16} \))
$3$ (\( 1 + T + T^{2} \))(\( 1 - T + T^{2} \))(\( 1 - T^{2} + T^{4} \))(\( ( 1 - T + T^{2} )^{2} \))(\( 1 - T^{4} + T^{8} \))(\( ( 1 - T^{2} + T^{4} )^{2} \))(\( ( 1 - T^{4} + T^{8} )^{2} \))(\( 1 - T^{8} + T^{16} \))
$5$ (\( ( 1 + T + T^{2} )^{2} \))(\( ( 1 - T + T^{2} )^{2} \))(\( 1 + T^{2} \))(\( 1 + T^{4} \))(\( 1 + T^{4} \))(\( 1 + T^{4} \))(\( 1 - T^{2} + T^{4} \))(\( 1 - T^{2} + T^{4} \))(\( 1 + T^{8} \))(\( 1 + T^{8} \))(\( 1 + T^{8} \))(\( 1 - T^{4} + T^{8} \))(\( 1 - T^{4} + T^{8} \))(\( 1 - T^{8} + T^{16} \))(\( 1 - T^{8} + T^{16} \))(\( 1 - T^{16} + T^{32} \))(\( 1 - T^{16} + T^{32} \))
$7$ (\( 1 + T^{4} \))(\( 1 + T^{4} \))(\( 1 + T^{2} \))(\( 1 + T^{4} \))(\( 1 + T^{4} \))(\( 1 + T^{4} \))(\( 1 - T^{2} + T^{4} \))(\( 1 - T^{2} + T^{4} \))(\( ( 1 - T )^{4}( 1 + T^{4} ) \))(\( ( 1 - T )^{4}( 1 + T^{4} ) \))(\( 1 + T^{8} \))(\( 1 - T^{4} + T^{8} \))(\( 1 - T^{4} + T^{8} \))(\( 1 - T^{8} + T^{16} \))(\( 1 - T^{8} + T^{16} \))(\( 1 - T^{16} + T^{32} \))(\( 1 - T^{16} + T^{32} \))
$11$ (\( ( 1 + T + T^{2} )^{2} \))(\( ( 1 - T + T^{2} )^{2} \))(\( ( 1 - T )( 1 + T ) \))(\( ( 1 - T )^{2}( 1 + T )^{2} \))(\( ( 1 - T )^{2}( 1 + T )^{2} \))(\( ( 1 + T )^{2}( 1 + T^{2} ) \))(\( ( 1 - T )^{2}( 1 - T + T^{2} ) \))(\( ( 1 + T )^{2}( 1 + T + T^{2} ) \))(\( 1 + T^{8} \))(\( 1 + T^{8} \))(\( ( 1 - T )^{4}( 1 + T^{4} ) \))(\( ( 1 + T^{2} )^{2}( 1 + T + T^{2} )^{2} \))(\( ( 1 - T )^{4}( 1 - T^{2} + T^{4} ) \))(\( ( 1 - T + T^{2} )^{4}( 1 + T^{4} )^{2} \))(\( ( 1 + T )^{8}( 1 - T^{4} + T^{8} ) \))(\( ( 1 + T^{2} )^{8}( 1 - T^{8} + T^{16} ) \))(\( ( 1 - T^{2} + T^{4} )^{4}( 1 + T^{8} )^{2} \))
$13$ (\( ( 1 + T + T^{2} )^{2} \))(\( ( 1 + T + T^{2} )^{2} \))(\( ( 1 - T )( 1 + T ) \))(\( ( 1 - T )^{2}( 1 + T )^{2} \))(\( ( 1 - T )^{2}( 1 + T )^{2} \))(\( ( 1 - T )^{2}( 1 + T )^{2} \))(\( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \))(\( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \))(\( ( 1 + T^{2} )^{4} \))(\( ( 1 + T^{2} )^{4} \))(\( ( 1 + T^{2} )^{4} \))(\( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \))(\( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \))(\( ( 1 - T^{2} + T^{4} )^{4} \))(\( ( 1 - T^{2} + T^{4} )^{4} \))(\( ( 1 - T^{4} + T^{8} )^{4} \))(\( ( 1 - T^{4} + T^{8} )^{4} \))
$17$ (\( ( 1 + T )^{2} \))(\( ( 1 - T )^{2} \))(\( 1 - T \))(\( ( 1 - T )^{2} \))(\( ( 1 + T )^{2} \))(\( 1 + T^{2} \))(\( 1 - T + T^{2} \))(\( 1 - T + T^{2} \))(\( ( 1 + T^{2} )^{2} \))(\( ( 1 + T^{2} )^{2} \))(\( 1 + T^{4} \))(\( 1 - T^{2} + T^{4} \))(\( 1 - T^{2} + T^{4} \))(\( 1 - T^{4} + T^{8} \))(\( 1 - T^{4} + T^{8} \))(\( 1 - T^{8} + T^{16} \))(\( 1 - T^{8} + T^{16} \))
$19$ (\( ( 1 + T + T^{2} )^{2} \))(\( ( 1 + T + T^{2} )^{2} \))(\( ( 1 + T )^{2} \))(\( ( 1 + T^{2} )^{2} \))(\( ( 1 + T^{2} )^{2} \))(\( ( 1 + T^{2} )^{2} \))(\( ( 1 - T + T^{2} )^{2} \))(\( ( 1 - T + T^{2} )^{2} \))(\( ( 1 + T^{4} )^{2} \))(\( ( 1 + T^{4} )^{2} \))(\( ( 1 + T^{4} )^{2} \))(\( ( 1 - T^{2} + T^{4} )^{2} \))(\( ( 1 - T^{2} + T^{4} )^{2} \))(\( ( 1 - T^{4} + T^{8} )^{2} \))(\( ( 1 - T^{4} + T^{8} )^{2} \))(\( ( 1 - T^{8} + T^{16} )^{2} \))(\( ( 1 - T^{8} + T^{16} )^{2} \))
$23$ (\( ( 1 - T + T^{2} )^{2} \))(\( ( 1 + T + T^{2} )^{2} \))(\( 1 + T^{2} \))(\( 1 + T^{4} \))(\( 1 + T^{4} \))(\( 1 + T^{4} \))(\( 1 - T^{2} + T^{4} \))(\( 1 - T^{2} + T^{4} \))(\( ( 1 + T^{2} )^{2}( 1 + T^{4} ) \))(\( ( 1 + T^{2} )^{2}( 1 + T^{4} ) \))(\( 1 + T^{8} \))(\( 1 - T^{4} + T^{8} \))(\( 1 - T^{4} + T^{8} \))(\( 1 - T^{8} + T^{16} \))(\( 1 - T^{8} + T^{16} \))(\( 1 - T^{16} + T^{32} \))(\( 1 - T^{16} + T^{32} \))
$29$ (\( ( 1 + T^{2} )^{2} \))(\( ( 1 + T^{2} )^{2} \))(\( 1 + T^{2} \))(\( 1 + T^{4} \))(\( 1 + T^{4} \))(\( 1 + T^{4} \))(\( 1 - T^{2} + T^{4} \))(\( 1 - T^{2} + T^{4} \))(\( 1 + T^{8} \))(\( 1 + T^{8} \))(\( 1 + T^{8} \))(\( 1 - T^{4} + T^{8} \))(\( 1 - T^{4} + T^{8} \))(\( 1 - T^{8} + T^{16} \))(\( 1 - T^{8} + T^{16} \))(\( 1 - T^{16} + T^{32} \))(\( 1 - T^{16} + T^{32} \))
$31$ (\( 1 + T^{4} \))(\( 1 + T^{4} \))(\( 1 + T^{2} \))(\( 1 + T^{4} \))(\( 1 + T^{4} \))(\( 1 + T^{4} \))(\( 1 - T^{2} + T^{4} \))(\( 1 - T^{2} + T^{4} \))(\( ( 1 + T^{2} )^{2}( 1 + T^{4} ) \))(\( ( 1 + T^{2} )^{2}( 1 + T^{4} ) \))(\( 1 + T^{8} \))(\( 1 - T^{4} + T^{8} \))(\( 1 - T^{4} + T^{8} \))(\( 1 - T^{8} + T^{16} \))(\( 1 - T^{8} + T^{16} \))(\( 1 - T^{16} + T^{32} \))(\( 1 - T^{16} + T^{32} \))
$37$ (\( 1 + T^{4} \))(\( 1 + T^{4} \))(\( 1 + T^{2} \))(\( 1 + T^{4} \))(\( 1 + T^{4} \))(\( 1 + T^{4} \))(\( ( 1 + T^{2} )^{2} \))(\( ( 1 + T^{2} )^{2} \))(\( 1 + T^{8} \))(\( 1 + T^{8} \))(\( 1 + T^{8} \))(\( ( 1 + T^{4} )^{2} \))(\( ( 1 + T^{4} )^{2} \))(\( ( 1 + T^{8} )^{2} \))(\( ( 1 + T^{8} )^{2} \))(\( ( 1 + T^{16} )^{2} \))(\( ( 1 + T^{16} )^{2} \))
$41$ (\( ( 1 + T + T^{2} )^{2} \))(\( ( 1 - T + T^{2} )^{2} \))(\( ( 1 - T )( 1 + T ) \))(\( ( 1 - T )^{2}( 1 + T )^{2} \))(\( ( 1 - T )^{2}( 1 + T )^{2} \))(\( ( 1 - T )^{2}( 1 + T^{2} ) \))(\( ( 1 + T )^{2}( 1 + T + T^{2} ) \))(\( ( 1 - T )^{2}( 1 - T + T^{2} ) \))(\( ( 1 + T^{2} )^{2}( 1 + T^{4} ) \))(\( ( 1 + T^{2} )^{2}( 1 + T^{4} ) \))(\( ( 1 + T^{2} )^{2}( 1 + T^{4} ) \))(\( ( 1 + T )^{4}( 1 - T^{2} + T^{4} ) \))(\( ( 1 - T + T^{2} )^{2}( 1 + T^{2} )^{2} \))(\( ( 1 + T^{2} )^{4}( 1 - T^{4} + T^{8} ) \))(\( ( 1 - T^{2} + T^{4} )^{2}( 1 + T^{4} )^{2} \))(\( ( 1 - T^{4} + T^{8} )^{2}( 1 + T^{8} )^{2} \))(\( ( 1 + T^{4} )^{4}( 1 - T^{8} + T^{16} ) \))
$43$ (\( ( 1 - T + T^{2} )^{2} \))(\( ( 1 - T + T^{2} )^{2} \))(\( ( 1 - T )^{2} \))(\( ( 1 + T )^{4} \))(\( ( 1 + T )^{4} \))(\( ( 1 - T )^{2}( 1 + T )^{2} \))(\( ( 1 - T )^{2}( 1 + T + T^{2} ) \))(\( ( 1 - T )^{2}( 1 + T + T^{2} ) \))(\( ( 1 + T^{4} )^{2} \))(\( ( 1 + T^{4} )^{2} \))(\( ( 1 - T )^{4}( 1 + T^{2} )^{2} \))(\( ( 1 - T )^{4}( 1 - T + T^{2} )^{2} \))(\( ( 1 + T )^{4}( 1 + T + T^{2} )^{2} \))(\( ( 1 - T )^{8}( 1 - T^{2} + T^{4} )^{2} \))(\( ( 1 + T^{2} )^{4}( 1 + T + T^{2} )^{4} \))(\( ( 1 - T + T^{2} )^{8}( 1 + T^{4} )^{4} \))(\( ( 1 + T )^{16}( 1 - T^{4} + T^{8} )^{2} \))
$47$ (\( 1 + T^{4} \))(\( 1 + T^{4} \))(\( ( 1 - T )( 1 + T ) \))(\( ( 1 - T )^{2}( 1 + T )^{2} \))(\( ( 1 - T )^{2}( 1 + T )^{2} \))(\( ( 1 - T )^{2}( 1 + T )^{2} \))(\( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \))(\( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \))(\( ( 1 + T^{4} )^{2} \))(\( ( 1 + T^{4} )^{2} \))(\( ( 1 + T^{2} )^{4} \))(\( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \))(\( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \))(\( ( 1 - T^{2} + T^{4} )^{4} \))(\( ( 1 - T^{2} + T^{4} )^{4} \))(\( ( 1 - T^{4} + T^{8} )^{4} \))(\( ( 1 - T^{4} + T^{8} )^{4} \))
$53$ (\( 1 + T^{4} \))(\( 1 + T^{4} \))(\( ( 1 - T )( 1 + T ) \))(\( ( 1 - T )^{2}( 1 + T )^{2} \))(\( ( 1 - T )^{2}( 1 + T )^{2} \))(\( ( 1 + T^{2} )^{2} \))(\( ( 1 - T )^{2}( 1 + T )^{2} \))(\( ( 1 - T )^{2}( 1 + T )^{2} \))(\( ( 1 + T^{4} )^{2} \))(\( ( 1 + T^{4} )^{2} \))(\( ( 1 + T^{4} )^{2} \))(\( ( 1 + T^{2} )^{4} \))(\( ( 1 + T^{2} )^{4} \))(\( ( 1 + T^{4} )^{4} \))(\( ( 1 + T^{4} )^{4} \))(\( ( 1 + T^{8} )^{4} \))(\( ( 1 + T^{8} )^{4} \))
$59$ (\( ( 1 - T )^{2}( 1 + T )^{2} \))(\( ( 1 - T )^{2}( 1 + T )^{2} \))(\( ( 1 + T )^{2} \))(\( ( 1 + T^{2} )^{2} \))(\( ( 1 + T^{2} )^{2} \))(\( ( 1 - T )^{2}( 1 + T )^{2} \))(\( ( 1 - T )^{2}( 1 + T + T^{2} ) \))(\( ( 1 - T )^{2}( 1 + T + T^{2} ) \))(\( ( 1 + T^{4} )^{2} \))(\( ( 1 + T^{4} )^{2} \))(\( ( 1 - T )^{4}( 1 + T^{2} )^{2} \))(\( ( 1 - T )^{4}( 1 - T + T^{2} )^{2} \))(\( ( 1 + T )^{4}( 1 + T + T^{2} )^{2} \))(\( ( 1 + T )^{8}( 1 - T^{2} + T^{4} )^{2} \))(\( ( 1 - T + T^{2} )^{4}( 1 + T^{2} )^{4} \))(\( ( 1 - T^{2} + T^{4} )^{4}( 1 + T^{4} )^{4} \))(\( ( 1 + T^{2} )^{8}( 1 - T^{4} + T^{8} )^{2} \))
$61$ (\( 1 + T^{4} \))(\( 1 + T^{4} \))(\( 1 + T^{2} \))(\( 1 + T^{4} \))(\( 1 + T^{4} \))(\( 1 + T^{4} \))(\( 1 - T^{2} + T^{4} \))(\( 1 - T^{2} + T^{4} \))(\( 1 + T^{8} \))(\( 1 + T^{8} \))(\( 1 + T^{8} \))(\( 1 - T^{4} + T^{8} \))(\( 1 - T^{4} + T^{8} \))(\( 1 - T^{8} + T^{16} \))(\( 1 - T^{8} + T^{16} \))(\( 1 - T^{16} + T^{32} \))(\( 1 - T^{16} + T^{32} \))
$67$ (\( ( 1 + T^{2} )^{2} \))(\( ( 1 + T^{2} )^{2} \))(\( ( 1 + T )^{2} \))(\( ( 1 + T^{2} )^{2} \))(\( ( 1 + T^{2} )^{2} \))(\( ( 1 + T^{2} )^{2} \))(\( ( 1 + T )^{2}( 1 - T + T^{2} ) \))(\( ( 1 + T )^{2}( 1 - T + T^{2} ) \))(\( ( 1 - T )^{4}( 1 + T )^{4} \))(\( ( 1 - T )^{4}( 1 + T )^{4} \))(\( ( 1 + T^{4} )^{2} \))(\( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \))(\( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \))(\( ( 1 + T^{4} )^{2}( 1 - T^{4} + T^{8} ) \))(\( ( 1 + T^{4} )^{2}( 1 - T^{4} + T^{8} ) \))(\( ( 1 + T^{8} )^{2}( 1 - T^{8} + T^{16} ) \))(\( ( 1 + T^{8} )^{2}( 1 - T^{8} + T^{16} ) \))
$71$ (\( ( 1 + T^{2} )^{2} \))(\( ( 1 + T^{2} )^{2} \))(\( 1 + T^{2} \))(\( 1 + T^{4} \))(\( 1 + T^{4} \))(\( 1 + T^{4} \))(\( ( 1 + T^{2} )^{2} \))(\( ( 1 + T^{2} )^{2} \))(\( ( 1 - T )^{4}( 1 + T^{4} ) \))(\( ( 1 + T )^{4}( 1 + T^{4} ) \))(\( 1 + T^{8} \))(\( ( 1 + T^{4} )^{2} \))(\( ( 1 + T^{4} )^{2} \))(\( ( 1 + T^{8} )^{2} \))(\( ( 1 + T^{8} )^{2} \))(\( ( 1 + T^{16} )^{2} \))(\( ( 1 + T^{16} )^{2} \))
$73$ (\( 1 + T^{4} \))(\( 1 + T^{4} \))(\( ( 1 - T )( 1 + T ) \))(\( ( 1 - T )^{2}( 1 + T )^{2} \))(\( ( 1 - T )^{2}( 1 + T )^{2} \))(\( ( 1 + T )^{2}( 1 + T^{2} ) \))(\( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \))(\( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \))(\( ( 1 + T^{2} )^{2}( 1 + T^{4} ) \))(\( ( 1 + T^{2} )^{2}( 1 + T^{4} ) \))(\( ( 1 + T^{2} )^{2}( 1 + T^{4} ) \))(\( ( 1 - T + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \))(\( ( 1 - T + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \))(\( ( 1 - T^{2} + T^{4} )^{2}( 1 - T^{4} + T^{8} ) \))(\( ( 1 - T^{2} + T^{4} )^{2}( 1 - T^{4} + T^{8} ) \))(\( ( 1 - T^{4} + T^{8} )^{2}( 1 - T^{8} + T^{16} ) \))(\( ( 1 - T^{4} + T^{8} )^{2}( 1 - T^{8} + T^{16} ) \))
$79$ (\( ( 1 - T )^{2}( 1 + T )^{2} \))(\( ( 1 - T )^{2}( 1 + T )^{2} \))(\( 1 + T^{2} \))(\( 1 + T^{4} \))(\( 1 + T^{4} \))(\( 1 + T^{4} \))(\( 1 - T^{2} + T^{4} \))(\( 1 - T^{2} + T^{4} \))(\( ( 1 + T )^{4}( 1 + T^{4} ) \))(\( ( 1 + T )^{4}( 1 + T^{4} ) \))(\( 1 + T^{8} \))(\( 1 - T^{4} + T^{8} \))(\( 1 - T^{4} + T^{8} \))(\( 1 - T^{8} + T^{16} \))(\( 1 - T^{8} + T^{16} \))(\( 1 - T^{16} + T^{32} \))(\( 1 - T^{16} + T^{32} \))
$83$ (\( 1 + T^{4} \))(\( 1 + T^{4} \))(\( ( 1 + T )^{2} \))(\( ( 1 + T^{2} )^{2} \))(\( ( 1 + T^{2} )^{2} \))(\( ( 1 - T )^{2}( 1 + T )^{2} \))(\( ( 1 + T + T^{2} )^{2} \))(\( ( 1 + T + T^{2} )^{2} \))(\( ( 1 + T^{4} )^{2} \))(\( ( 1 + T^{4} )^{2} \))(\( ( 1 + T )^{4}( 1 + T^{2} )^{2} \))(\( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \))(\( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \))(\( ( 1 + T + T^{2} )^{4}( 1 - T^{2} + T^{4} )^{2} \))(\( ( 1 + T + T^{2} )^{4}( 1 - T^{2} + T^{4} )^{2} \))(\( ( 1 - T + T^{2} )^{8}( 1 - T^{4} + T^{8} )^{2} \))(\( ( 1 + T + T^{2} )^{8}( 1 - T^{4} + T^{8} )^{2} \))
$89$ (\( 1 + T^{4} \))(\( 1 + T^{4} \))(\( ( 1 - T )^{2} \))(\( ( 1 - T )^{4} \))(\( ( 1 + T )^{4} \))(\( ( 1 + T^{2} )^{2} \))(\( ( 1 + T )^{4} \))(\( ( 1 + T )^{4} \))(\( ( 1 + T^{4} )^{2} \))(\( ( 1 + T^{4} )^{2} \))(\( ( 1 + T^{4} )^{2} \))(\( ( 1 + T^{2} )^{4} \))(\( ( 1 + T^{2} )^{4} \))(\( ( 1 + T^{4} )^{4} \))(\( ( 1 + T^{4} )^{4} \))(\( ( 1 + T^{8} )^{4} \))(\( ( 1 + T^{8} )^{4} \))
$97$ (\( ( 1 - T )^{2}( 1 + T )^{2} \))(\( ( 1 - T )^{2}( 1 + T )^{2} \))(\( ( 1 - T )( 1 + T ) \))(\( ( 1 - T )^{2}( 1 + T )^{2} \))(\( ( 1 - T )^{2}( 1 + T )^{2} \))(\( ( 1 - T )^{2}( 1 + T^{2} ) \))(\( ( 1 + T )^{2}( 1 + T + T^{2} ) \))(\( ( 1 - T )^{2}( 1 - T + T^{2} ) \))(\( ( 1 + T^{2} )^{2}( 1 + T^{4} ) \))(\( ( 1 + T^{2} )^{2}( 1 + T^{4} ) \))(\( ( 1 + T )^{4}( 1 + T^{4} ) \))(\( ( 1 - T )^{4}( 1 - T^{2} + T^{4} ) \))(\( ( 1 + T^{2} )^{2}( 1 + T + T^{2} )^{2} \))(\( ( 1 + T )^{8}( 1 - T^{4} + T^{8} ) \))(\( ( 1 - T + T^{2} )^{4}( 1 + T^{4} )^{2} \))(\( ( 1 - T^{2} + T^{4} )^{4}( 1 + T^{8} )^{2} \))(\( ( 1 + T^{2} )^{8}( 1 - T^{8} + T^{16} ) \))
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