Properties

Label 3136.1
Level 3136
Weight 1
Dimension 115
Nonzero newspaces 14
Newform subspaces 19
Sturm bound 602112
Trace bound 57

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Defining parameters

Level: \( N \) = \( 3136\( 3136 = 2^{6} \cdot 7^{2} \) \)
Weight: \( k \) = \( 1 \)
Nonzero newspaces: \( 14 \)
Newform subspaces: \( 19 \)
Sturm bound: \(602112\)
Trace bound: \(57\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{1}(\Gamma_1(3136))\).

Total New Old
Modular forms 4728 1182 3546
Cusp forms 408 115 293
Eisenstein series 4320 1067 3253

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

\(D_n\) \(A_4\) \(S_4\) \(A_5\)
Dimension 107 8 0 0

Trace form

\( 115q - 2q^{5} + O(q^{10}) \) \( 115q - 2q^{5} - 2q^{11} + 2q^{17} + 8q^{22} + 8q^{29} + 2q^{33} + 4q^{37} + 2q^{43} + 8q^{44} + 16q^{57} + 2q^{61} + 10q^{67} + 4q^{69} - 2q^{73} - 8q^{74} + 4q^{85} - 2q^{89} - 2q^{93} + 10q^{99} + O(q^{100}) \)

Decomposition of \(S_{1}^{\mathrm{new}}(\Gamma_1(3136))\)

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
3136.1.c \(\chi_{3136}(1665, \cdot)\) 3136.1.c.a 4 1
3136.1.d \(\chi_{3136}(1471, \cdot)\) 3136.1.d.a 1 1
3136.1.d.b 2
3136.1.d.c 2
3136.1.d.d 2
3136.1.g \(\chi_{3136}(3039, \cdot)\) 3136.1.g.a 4 1
3136.1.h \(\chi_{3136}(97, \cdot)\) 3136.1.h.a 8 1
3136.1.k \(\chi_{3136}(687, \cdot)\) 3136.1.k.a 2 2
3136.1.l \(\chi_{3136}(881, \cdot)\) None 0 2
3136.1.n \(\chi_{3136}(1697, \cdot)\) 3136.1.n.a 16 2
3136.1.o \(\chi_{3136}(863, \cdot)\) 3136.1.o.a 8 2
3136.1.r \(\chi_{3136}(2431, \cdot)\) 3136.1.r.a 2 2
3136.1.r.b 4
3136.1.r.c 4
3136.1.s \(\chi_{3136}(129, \cdot)\) 3136.1.s.a 8 2
3136.1.w \(\chi_{3136}(489, \cdot)\) None 0 4
3136.1.x \(\chi_{3136}(295, \cdot)\) None 0 4
3136.1.z \(\chi_{3136}(79, \cdot)\) 3136.1.z.a 4 4
3136.1.bc \(\chi_{3136}(913, \cdot)\) 3136.1.bc.a 4 4
3136.1.bd \(\chi_{3136}(545, \cdot)\) None 0 6
3136.1.be \(\chi_{3136}(351, \cdot)\) None 0 6
3136.1.bh \(\chi_{3136}(127, \cdot)\) None 0 6
3136.1.bi \(\chi_{3136}(321, \cdot)\) None 0 6
3136.1.bm \(\chi_{3136}(99, \cdot)\) 3136.1.bm.a 8 8
3136.1.bn \(\chi_{3136}(293, \cdot)\) None 0 8
3136.1.bp \(\chi_{3136}(313, \cdot)\) None 0 8
3136.1.bs \(\chi_{3136}(263, \cdot)\) None 0 8
3136.1.bu \(\chi_{3136}(209, \cdot)\) None 0 12
3136.1.bv \(\chi_{3136}(15, \cdot)\) None 0 12
3136.1.by \(\chi_{3136}(257, \cdot)\) None 0 12
3136.1.bz \(\chi_{3136}(191, \cdot)\) None 0 12
3136.1.cc \(\chi_{3136}(95, \cdot)\) None 0 12
3136.1.cd \(\chi_{3136}(33, \cdot)\) None 0 12
3136.1.ce \(\chi_{3136}(117, \cdot)\) 3136.1.ce.a 16 16
3136.1.cf \(\chi_{3136}(67, \cdot)\) 3136.1.cf.a 16 16
3136.1.cj \(\chi_{3136}(71, \cdot)\) None 0 24
3136.1.ck \(\chi_{3136}(41, \cdot)\) None 0 24
3136.1.cm \(\chi_{3136}(17, \cdot)\) None 0 24
3136.1.cp \(\chi_{3136}(207, \cdot)\) None 0 24
3136.1.cs \(\chi_{3136}(13, \cdot)\) None 0 48
3136.1.ct \(\chi_{3136}(43, \cdot)\) None 0 48
3136.1.cu \(\chi_{3136}(23, \cdot)\) None 0 48
3136.1.cx \(\chi_{3136}(73, \cdot)\) None 0 48
3136.1.cy \(\chi_{3136}(11, \cdot)\) None 0 96
3136.1.cz \(\chi_{3136}(5, \cdot)\) None 0 96

Decomposition of \(S_{1}^{\mathrm{old}}(\Gamma_1(3136))\) into lower level spaces

\( S_{1}^{\mathrm{old}}(\Gamma_1(3136)) \cong \) \(S_{1}^{\mathrm{new}}(\Gamma_1(56))\)\(^{\oplus 8}\)\(\oplus\)\(S_{1}^{\mathrm{new}}(\Gamma_1(112))\)\(^{\oplus 6}\)\(\oplus\)\(S_{1}^{\mathrm{new}}(\Gamma_1(196))\)\(^{\oplus 5}\)\(\oplus\)\(S_{1}^{\mathrm{new}}(\Gamma_1(224))\)\(^{\oplus 4}\)\(\oplus\)\(S_{1}^{\mathrm{new}}(\Gamma_1(392))\)\(^{\oplus 4}\)\(\oplus\)\(S_{1}^{\mathrm{new}}(\Gamma_1(448))\)\(^{\oplus 2}\)\(\oplus\)\(S_{1}^{\mathrm{new}}(\Gamma_1(784))\)\(^{\oplus 3}\)\(\oplus\)\(S_{1}^{\mathrm{new}}(\Gamma_1(1568))\)\(^{\oplus 2}\)

Hecke characteristic polynomials

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