# Properties

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

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