# Properties

 Label 2015.1 Level 2015 Weight 1 Dimension 52 Nonzero newspaces 3 Newform subspaces 11 Sturm bound 322560 Trace bound 1

## Defining parameters

 Level: $$N$$ = $$2015 = 5 \cdot 13 \cdot 31$$ Weight: $$k$$ = $$1$$ Nonzero newspaces: $$3$$ Newform subspaces: $$11$$ Sturm bound: $$322560$$ Trace bound: $$1$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{1}(\Gamma_1(2015))$$.

Total New Old
Modular forms 2950 1968 982
Cusp forms 70 52 18
Eisenstein series 2880 1916 964

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

$$D_n$$ $$A_4$$ $$S_4$$ $$A_5$$
Dimension 44 8 0 0

## Trace form

 $$52q + 16q^{4} + 16q^{9} + O(q^{10})$$ $$52q + 16q^{4} + 16q^{9} - 8q^{10} - 16q^{14} + 16q^{16} + 4q^{19} + 12q^{25} - 8q^{31} - 8q^{35} + 4q^{36} + 20q^{39} + 20q^{41} - 8q^{45} + 16q^{49} - 24q^{51} - 12q^{56} - 4q^{59} - 8q^{66} - 20q^{69} + 4q^{71} + 8q^{81} - 12q^{90} - 8q^{94} - 8q^{95} + O(q^{100})$$

## Decomposition of $$S_{1}^{\mathrm{new}}(\Gamma_1(2015))$$

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
2015.1.b $$\chi_{2015}(1611, \cdot)$$ None 0 1
2015.1.e $$\chi_{2015}(1704, \cdot)$$ None 0 1
2015.1.g $$\chi_{2015}(1301, \cdot)$$ None 0 1
2015.1.h $$\chi_{2015}(2014, \cdot)$$ 2015.1.h.a 4 1
2015.1.h.b 6
2015.1.h.c 6
2015.1.h.d 6
2015.1.h.e 6
2015.1.n $$\chi_{2015}(278, \cdot)$$ None 0 2
2015.1.p $$\chi_{2015}(1334, \cdot)$$ None 0 2
2015.1.r $$\chi_{2015}(1117, \cdot)$$ None 0 2
2015.1.s $$\chi_{2015}(807, \cdot)$$ None 0 2
2015.1.v $$\chi_{2015}(931, \cdot)$$ None 0 2
2015.1.x $$\chi_{2015}(1487, \cdot)$$ None 0 2
2015.1.z $$\chi_{2015}(274, \cdot)$$ None 0 2
2015.1.bc $$\chi_{2015}(181, \cdot)$$ None 0 2
2015.1.bd $$\chi_{2015}(1556, \cdot)$$ None 0 2
2015.1.bf $$\chi_{2015}(309, \cdot)$$ 2015.1.bf.a 8 2
2015.1.bg $$\chi_{2015}(1804, \cdot)$$ None 0 2
2015.1.bi $$\chi_{2015}(471, \cdot)$$ None 0 2
2015.1.bj $$\chi_{2015}(61, \cdot)$$ None 0 2
2015.1.bl $$\chi_{2015}(719, \cdot)$$ None 0 2
2015.1.bn $$\chi_{2015}(316, \cdot)$$ None 0 2
2015.1.bp $$\chi_{2015}(874, \cdot)$$ None 0 2
2015.1.bq $$\chi_{2015}(464, \cdot)$$ 2015.1.bq.a 2 2
2015.1.bq.b 2
2015.1.bq.c 4
2015.1.bq.d 4
2015.1.bq.e 4
2015.1.bt $$\chi_{2015}(836, \cdot)$$ None 0 2
2015.1.bu $$\chi_{2015}(1401, \cdot)$$ None 0 2
2015.1.bw $$\chi_{2015}(1959, \cdot)$$ None 0 2
2015.1.by $$\chi_{2015}(584, \cdot)$$ None 0 2
2015.1.bz $$\chi_{2015}(781, \cdot)$$ None 0 2
2015.1.cb $$\chi_{2015}(519, \cdot)$$ None 0 4
2015.1.cc $$\chi_{2015}(976, \cdot)$$ None 0 4
2015.1.ce $$\chi_{2015}(209, \cdot)$$ None 0 4
2015.1.ch $$\chi_{2015}(116, \cdot)$$ None 0 4
2015.1.cj $$\chi_{2015}(502, \cdot)$$ None 0 4
2015.1.ck $$\chi_{2015}(553, \cdot)$$ None 0 4
2015.1.cm $$\chi_{2015}(522, \cdot)$$ None 0 4
2015.1.co $$\chi_{2015}(1332, \cdot)$$ None 0 4
2015.1.cq $$\chi_{2015}(149, \cdot)$$ None 0 4
2015.1.ct $$\chi_{2015}(346, \cdot)$$ None 0 4
2015.1.cu $$\chi_{2015}(466, \cdot)$$ None 0 4
2015.1.cw $$\chi_{2015}(366, \cdot)$$ None 0 4
2015.1.cz $$\chi_{2015}(118, \cdot)$$ None 0 4
2015.1.da $$\chi_{2015}(428, \cdot)$$ None 0 4
2015.1.dd $$\chi_{2015}(563, \cdot)$$ None 0 4
2015.1.df $$\chi_{2015}(87, \cdot)$$ None 0 4
2015.1.dh $$\chi_{2015}(373, \cdot)$$ None 0 4
2015.1.di $$\chi_{2015}(842, \cdot)$$ None 0 4
2015.1.dk $$\chi_{2015}(218, \cdot)$$ None 0 4
2015.1.dm $$\chi_{2015}(997, \cdot)$$ None 0 4
2015.1.dp $$\chi_{2015}(304, \cdot)$$ None 0 4
2015.1.dq $$\chi_{2015}(769, \cdot)$$ None 0 4
2015.1.ds $$\chi_{2015}(249, \cdot)$$ None 0 4
2015.1.du $$\chi_{2015}(1276, \cdot)$$ None 0 4
2015.1.dw $$\chi_{2015}(657, \cdot)$$ None 0 4
2015.1.dy $$\chi_{2015}(123, \cdot)$$ None 0 4
2015.1.ea $$\chi_{2015}(37, \cdot)$$ None 0 4
2015.1.ed $$\chi_{2015}(57, \cdot)$$ None 0 4
2015.1.ei $$\chi_{2015}(122, \cdot)$$ None 0 8
2015.1.ek $$\chi_{2015}(281, \cdot)$$ None 0 8
2015.1.em $$\chi_{2015}(233, \cdot)$$ None 0 8
2015.1.ep $$\chi_{2015}(157, \cdot)$$ None 0 8
2015.1.eq $$\chi_{2015}(109, \cdot)$$ None 0 8
2015.1.es $$\chi_{2015}(213, \cdot)$$ None 0 8
2015.1.ev $$\chi_{2015}(261, \cdot)$$ None 0 8
2015.1.ew $$\chi_{2015}(259, \cdot)$$ None 0 8
2015.1.ey $$\chi_{2015}(74, \cdot)$$ None 0 8
2015.1.ez $$\chi_{2015}(426, \cdot)$$ None 0 8
2015.1.fc $$\chi_{2015}(296, \cdot)$$ None 0 8
2015.1.fd $$\chi_{2015}(269, \cdot)$$ None 0 8
2015.1.fg $$\chi_{2015}(29, \cdot)$$ None 0 8
2015.1.fh $$\chi_{2015}(166, \cdot)$$ None 0 8
2015.1.fj $$\chi_{2015}(114, \cdot)$$ None 0 8
2015.1.fk $$\chi_{2015}(321, \cdot)$$ None 0 8
2015.1.fn $$\chi_{2015}(581, \cdot)$$ None 0 8
2015.1.fo $$\chi_{2015}(244, \cdot)$$ None 0 8
2015.1.fp $$\chi_{2015}(179, \cdot)$$ None 0 8
2015.1.fr $$\chi_{2015}(146, \cdot)$$ None 0 8
2015.1.fs $$\chi_{2015}(571, \cdot)$$ None 0 8
2015.1.fv $$\chi_{2015}(79, \cdot)$$ None 0 8
2015.1.fw $$\chi_{2015}(73, \cdot)$$ None 0 16
2015.1.fz $$\chi_{2015}(137, \cdot)$$ None 0 16
2015.1.gb $$\chi_{2015}(58, \cdot)$$ None 0 16
2015.1.gd $$\chi_{2015}(362, \cdot)$$ None 0 16
2015.1.gf $$\chi_{2015}(41, \cdot)$$ None 0 16
2015.1.gh $$\chi_{2015}(219, \cdot)$$ None 0 16
2015.1.gj $$\chi_{2015}(319, \cdot)$$ None 0 16
2015.1.gk $$\chi_{2015}(164, \cdot)$$ None 0 16
2015.1.gm $$\chi_{2015}(82, \cdot)$$ None 0 16
2015.1.go $$\chi_{2015}(438, \cdot)$$ None 0 16
2015.1.gq $$\chi_{2015}(133, \cdot)$$ None 0 16
2015.1.gt $$\chi_{2015}(283, \cdot)$$ None 0 16
2015.1.gv $$\chi_{2015}(257, \cdot)$$ None 0 16
2015.1.gx $$\chi_{2015}(107, \cdot)$$ None 0 16
2015.1.gy $$\chi_{2015}(183, \cdot)$$ None 0 16
2015.1.hb $$\chi_{2015}(38, \cdot)$$ None 0 16
2015.1.hd $$\chi_{2015}(71, \cdot)$$ None 0 16
2015.1.hf $$\chi_{2015}(171, \cdot)$$ None 0 16
2015.1.hg $$\chi_{2015}(226, \cdot)$$ None 0 16
2015.1.hj $$\chi_{2015}(19, \cdot)$$ None 0 16
2015.1.hl $$\chi_{2015}(457, \cdot)$$ None 0 16
2015.1.hn $$\chi_{2015}(197, \cdot)$$ None 0 16
2015.1.hp $$\chi_{2015}(228, \cdot)$$ None 0 16
2015.1.hq $$\chi_{2015}(83, \cdot)$$ None 0 16

## Decomposition of $$S_{1}^{\mathrm{old}}(\Gamma_1(2015))$$ into lower level spaces

$$S_{1}^{\mathrm{old}}(\Gamma_1(2015)) \cong$$ $$S_{1}^{\mathrm{new}}(\Gamma_1(31))$$$$^{\oplus 4}$$$$\oplus$$$$S_{1}^{\mathrm{new}}(\Gamma_1(155))$$$$^{\oplus 2}$$$$\oplus$$$$S_{1}^{\mathrm{new}}(\Gamma_1(403))$$$$^{\oplus 2}$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ ($$( 1 + T^{2} )^{4}$$)($$1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} + T^{11} + T^{12}$$)($$1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} + T^{11} + T^{12}$$)($$1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} - T^{11} + T^{12}$$)($$1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} - T^{11} + T^{12}$$)($$( 1 - T^{2} + T^{4} )^{4}$$)($$( 1 - T + T^{2} )( 1 + T + T^{2} )$$)($$( 1 - T + T^{2} )( 1 + T + T^{2} )$$)($$( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} )$$)($$( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2}$$)($$( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} )$$)
$3$ ($$( 1 + T^{4} )^{2}$$)($$1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} + T^{11} + T^{12}$$)($$1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} - T^{11} + T^{12}$$)($$1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} + T^{11} + T^{12}$$)($$1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} - T^{11} + T^{12}$$)($$( 1 - T^{4} + T^{8} )^{2}$$)($$( 1 + T + T^{2} )^{2}$$)($$( 1 - T + T^{2} )^{2}$$)($$( 1 + T )^{4}( 1 - T + T^{2} )^{2}$$)($$( 1 - T^{2} + T^{4} )^{2}$$)($$( 1 - T )^{4}( 1 + T + T^{2} )^{2}$$)
$5$ ($$( 1 + T^{2} )^{2}$$)($$( 1 - T )^{6}$$)($$( 1 - T )^{6}$$)($$( 1 + T )^{6}$$)($$( 1 + T )^{6}$$)($$( 1 + T^{2} )^{4}$$)($$( 1 - T )^{2}$$)($$( 1 - T )^{2}$$)($$( 1 + T^{2} )^{2}$$)($$( 1 + T )^{4}$$)($$( 1 + T^{2} )^{2}$$)
$7$ ($$( 1 + T^{2} )^{4}$$)($$1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} + T^{11} + T^{12}$$)($$1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} + T^{11} + T^{12}$$)($$1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} - T^{11} + T^{12}$$)($$1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} - T^{11} + T^{12}$$)($$( 1 - T^{2} + T^{4} )^{4}$$)($$( 1 - T + T^{2} )( 1 + T + T^{2} )$$)($$( 1 - T + T^{2} )( 1 + T + T^{2} )$$)($$( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} )$$)($$( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2}$$)($$( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} )$$)
$11$ ($$( 1 + T^{2} )^{4}$$)($$1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} + T^{11} + T^{12}$$)($$1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} - T^{11} + T^{12}$$)($$1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} - T^{11} + T^{12}$$)($$1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} + T^{11} + T^{12}$$)($$( 1 - T^{2} + T^{4} )^{4}$$)($$( 1 - T + T^{2} )( 1 + T + T^{2} )$$)($$( 1 - T + T^{2} )( 1 + T + T^{2} )$$)($$( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} )$$)($$( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2}$$)($$( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} )$$)
$13$ ($$1 + T^{4}$$)($$( 1 - T )^{6}$$)($$( 1 + T )^{6}$$)($$( 1 - T )^{6}$$)($$( 1 + T )^{6}$$)($$1 - T^{4} + T^{8}$$)($$1 + T + T^{2}$$)($$1 - T + T^{2}$$)($$( 1 + T )^{4}$$)($$1 - T^{2} + T^{4}$$)($$( 1 - T )^{4}$$)
$17$ ($$( 1 + T^{4} )^{2}$$)($$1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} + T^{11} + T^{12}$$)($$1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} - T^{11} + T^{12}$$)($$1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} + T^{11} + T^{12}$$)($$1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} - T^{11} + T^{12}$$)($$( 1 + T^{4} )^{2}( 1 - T^{4} + T^{8} )$$)($$( 1 - T )^{2}( 1 + T + T^{2} )$$)($$( 1 + T )^{2}( 1 - T + T^{2} )$$)($$( 1 - T )^{4}( 1 + T + T^{2} )^{2}$$)($$( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} )$$)($$( 1 + T )^{4}( 1 - T + T^{2} )^{2}$$)
$19$ ($$( 1 + T^{2} )^{4}$$)($$( 1 - T )^{6}( 1 + T )^{6}$$)($$( 1 - T )^{6}( 1 + T )^{6}$$)($$( 1 - T )^{6}( 1 + T )^{6}$$)($$( 1 - T )^{6}( 1 + T )^{6}$$)($$( 1 + T^{2} )^{4}( 1 - T^{2} + T^{4} )^{2}$$)($$( 1 - T )^{2}( 1 + T + T^{2} )$$)($$( 1 - T )^{2}( 1 + T + T^{2} )$$)($$( 1 - T )^{4}( 1 + T + T^{2} )^{2}$$)($$( 1 + T )^{4}( 1 - T + T^{2} )^{2}$$)($$( 1 - T )^{4}( 1 + T + T^{2} )^{2}$$)
$23$ ($$( 1 + T^{4} )^{2}$$)($$1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} + T^{11} + T^{12}$$)($$1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} - T^{11} + T^{12}$$)($$1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} + T^{11} + T^{12}$$)($$1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} - T^{11} + T^{12}$$)($$( 1 + T^{4} )^{2}( 1 - T^{4} + T^{8} )$$)($$( 1 - T )^{2}( 1 + T + T^{2} )$$)($$( 1 + T )^{2}( 1 - T + T^{2} )$$)($$( 1 + T )^{4}( 1 - T + T^{2} )^{2}$$)($$( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} )$$)($$( 1 - T )^{4}( 1 + T + T^{2} )^{2}$$)
$29$ ($$( 1 - T )^{4}( 1 + T )^{4}$$)($$( 1 - T )^{6}( 1 + T )^{6}$$)($$( 1 - T )^{6}( 1 + T )^{6}$$)($$( 1 - T )^{6}( 1 + T )^{6}$$)($$( 1 - T )^{6}( 1 + T )^{6}$$)($$( 1 - T + T^{2} )^{4}( 1 + T + T^{2} )^{4}$$)($$( 1 - T + T^{2} )( 1 + T + T^{2} )$$)($$( 1 - T + T^{2} )( 1 + T + T^{2} )$$)($$( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} )$$)($$( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2}$$)($$( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} )$$)
$31$ ($$( 1 + T^{2} )^{2}$$)($$( 1 - T )^{6}$$)($$( 1 + T )^{6}$$)($$( 1 + T )^{6}$$)($$( 1 - T )^{6}$$)($$( 1 + T^{2} )^{4}$$)($$( 1 - T )^{2}$$)($$( 1 - T )^{2}$$)($$( 1 + T )^{4}$$)($$( 1 + T )^{4}$$)($$( 1 + T )^{4}$$)
$37$ ($$( 1 + T^{4} )^{2}$$)($$( 1 - T )^{6}( 1 + T )^{6}$$)($$( 1 - T )^{6}( 1 + T )^{6}$$)($$( 1 - T )^{6}( 1 + T )^{6}$$)($$( 1 - T )^{6}( 1 + T )^{6}$$)($$( 1 + T^{4} )^{2}( 1 - T^{4} + T^{8} )$$)($$( 1 - T )^{2}( 1 + T + T^{2} )$$)($$( 1 + T )^{2}( 1 - T + T^{2} )$$)($$( 1 - T )^{4}( 1 + T + T^{2} )^{2}$$)($$( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} )$$)($$( 1 + T )^{4}( 1 - T + T^{2} )^{2}$$)
$41$ ($$( 1 - T )^{4}( 1 + T )^{4}$$)($$( 1 - T )^{6}( 1 + T )^{6}$$)($$( 1 - T )^{6}( 1 + T )^{6}$$)($$( 1 - T )^{6}( 1 + T )^{6}$$)($$( 1 - T )^{6}( 1 + T )^{6}$$)($$( 1 - T )^{8}( 1 - T + T^{2} )^{4}$$)($$( 1 - T )^{2}( 1 + T + T^{2} )$$)($$( 1 - T )^{2}( 1 + T + T^{2} )$$)($$( 1 - T )^{4}( 1 + T + T^{2} )^{2}$$)($$( 1 - T )^{4}( 1 + T + T^{2} )^{2}$$)($$( 1 - T )^{4}( 1 + T + T^{2} )^{2}$$)
$43$ ($$( 1 + T^{4} )^{2}$$)($$1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} + T^{11} + T^{12}$$)($$1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} - T^{11} + T^{12}$$)($$1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} + T^{11} + T^{12}$$)($$1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} - T^{11} + T^{12}$$)($$( 1 + T^{4} )^{2}( 1 - T^{4} + T^{8} )$$)($$( 1 - T )^{2}( 1 + T + T^{2} )$$)($$( 1 + T )^{2}( 1 - T + T^{2} )$$)($$( 1 - T )^{4}( 1 + T + T^{2} )^{2}$$)($$( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} )$$)($$( 1 + T )^{4}( 1 - T + T^{2} )^{2}$$)
$47$ ($$( 1 + T^{2} )^{4}$$)($$1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} + T^{11} + T^{12}$$)($$1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} + T^{11} + T^{12}$$)($$1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} - T^{11} + T^{12}$$)($$1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} - T^{11} + T^{12}$$)($$( 1 + T^{2} )^{8}$$)($$( 1 - T )^{2}( 1 + T )^{2}$$)($$( 1 - T )^{2}( 1 + T )^{2}$$)($$( 1 - T )^{4}( 1 + T )^{4}$$)($$( 1 - T )^{4}( 1 + T )^{4}$$)($$( 1 - T )^{4}( 1 + T )^{4}$$)
$53$ ($$( 1 + T^{4} )^{2}$$)($$1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} + T^{11} + T^{12}$$)($$1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} - T^{11} + T^{12}$$)($$1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} + T^{11} + T^{12}$$)($$1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} - T^{11} + T^{12}$$)($$( 1 - T^{4} + T^{8} )^{2}$$)($$( 1 + T + T^{2} )^{2}$$)($$( 1 - T + T^{2} )^{2}$$)($$( 1 + T^{2} )^{4}$$)($$( 1 - T^{2} + T^{4} )^{2}$$)($$( 1 + T^{2} )^{4}$$)
$59$ ($$( 1 - T )^{4}( 1 + T )^{4}$$)($$( 1 - T )^{6}( 1 + T )^{6}$$)($$( 1 - T )^{6}( 1 + T )^{6}$$)($$( 1 - T )^{6}( 1 + T )^{6}$$)($$( 1 - T )^{6}( 1 + T )^{6}$$)($$( 1 + T )^{8}( 1 + T + T^{2} )^{4}$$)($$( 1 - T )^{2}( 1 + T + T^{2} )$$)($$( 1 - T )^{2}( 1 + T + T^{2} )$$)($$( 1 - T )^{4}( 1 + T + T^{2} )^{2}$$)($$( 1 - T )^{4}( 1 + T + T^{2} )^{2}$$)($$( 1 - T )^{4}( 1 + T + T^{2} )^{2}$$)
$61$ ($$( 1 - T )^{4}( 1 + T )^{4}$$)($$( 1 - T )^{6}( 1 + T )^{6}$$)($$( 1 - T )^{6}( 1 + T )^{6}$$)($$( 1 - T )^{6}( 1 + T )^{6}$$)($$( 1 - T )^{6}( 1 + T )^{6}$$)($$( 1 - T + T^{2} )^{4}( 1 + T + T^{2} )^{4}$$)($$( 1 - T + T^{2} )( 1 + T + T^{2} )$$)($$( 1 - T + T^{2} )( 1 + T + T^{2} )$$)($$( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} )$$)($$( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2}$$)($$( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} )$$)
$67$ ($$( 1 + T^{2} )^{4}$$)($$1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} + T^{11} + T^{12}$$)($$1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} + T^{11} + T^{12}$$)($$1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} - T^{11} + T^{12}$$)($$1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} - T^{11} + T^{12}$$)($$( 1 - T^{2} + T^{4} )^{4}$$)($$( 1 - T + T^{2} )( 1 + T + T^{2} )$$)($$( 1 - T + T^{2} )( 1 + T + T^{2} )$$)($$( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} )$$)($$( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2}$$)($$( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} )$$)
$71$ ($$( 1 + T^{2} )^{4}$$)($$( 1 - T )^{6}( 1 + T )^{6}$$)($$( 1 - T )^{6}( 1 + T )^{6}$$)($$( 1 - T )^{6}( 1 + T )^{6}$$)($$( 1 - T )^{6}( 1 + T )^{6}$$)($$( 1 + T^{2} )^{4}( 1 - T^{2} + T^{4} )^{2}$$)($$( 1 - T )^{2}( 1 + T + T^{2} )$$)($$( 1 - T )^{2}( 1 + T + T^{2} )$$)($$( 1 - T )^{4}( 1 + T + T^{2} )^{2}$$)($$( 1 + T )^{4}( 1 - T + T^{2} )^{2}$$)($$( 1 - T )^{4}( 1 + T + T^{2} )^{2}$$)
$73$ ($$( 1 + T^{4} )^{2}$$)($$( 1 - T )^{6}( 1 + T )^{6}$$)($$( 1 - T )^{6}( 1 + T )^{6}$$)($$( 1 - T )^{6}( 1 + T )^{6}$$)($$( 1 - T )^{6}( 1 + T )^{6}$$)($$( 1 + T^{4} )^{4}$$)($$( 1 - T )^{4}$$)($$( 1 + T )^{4}$$)($$( 1 + T^{2} )^{4}$$)($$( 1 + T^{2} )^{4}$$)($$( 1 + T^{2} )^{4}$$)
$79$ ($$( 1 - T )^{4}( 1 + T )^{4}$$)($$( 1 - T )^{6}( 1 + T )^{6}$$)($$( 1 - T )^{6}( 1 + T )^{6}$$)($$( 1 - T )^{6}( 1 + T )^{6}$$)($$( 1 - T )^{6}( 1 + T )^{6}$$)($$( 1 - T )^{8}( 1 + T )^{8}$$)($$( 1 - T )^{2}( 1 + T )^{2}$$)($$( 1 - T )^{2}( 1 + T )^{2}$$)($$( 1 + T^{2} )^{4}$$)($$( 1 - T )^{4}( 1 + T )^{4}$$)($$( 1 + T^{2} )^{4}$$)
$83$ ($$( 1 + T^{4} )^{2}$$)($$( 1 - T )^{6}( 1 + T )^{6}$$)($$( 1 - T )^{6}( 1 + T )^{6}$$)($$( 1 - T )^{6}( 1 + T )^{6}$$)($$( 1 - T )^{6}( 1 + T )^{6}$$)($$( 1 - T^{4} + T^{8} )^{2}$$)($$( 1 + T + T^{2} )^{2}$$)($$( 1 - T + T^{2} )^{2}$$)($$( 1 - T )^{8}$$)($$( 1 - T^{2} + T^{4} )^{2}$$)($$( 1 + T )^{8}$$)
$89$ ($$( 1 + T^{2} )^{4}$$)($$1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} + T^{11} + T^{12}$$)($$1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} - T^{11} + T^{12}$$)($$1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} - T^{11} + T^{12}$$)($$1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} + T^{11} + T^{12}$$)($$( 1 - T^{2} + T^{4} )^{4}$$)($$( 1 - T + T^{2} )( 1 + T + T^{2} )$$)($$( 1 - T + T^{2} )( 1 + T + T^{2} )$$)($$( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} )$$)($$( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2}$$)($$( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} )$$)
$97$ ($$( 1 + T^{2} )^{4}$$)($$1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} + T^{11} + T^{12}$$)($$1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} + T^{11} + T^{12}$$)($$1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} - T^{11} + T^{12}$$)($$1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} - T^{11} + T^{12}$$)($$( 1 - T^{2} + T^{4} )^{4}$$)($$( 1 - T + T^{2} )( 1 + T + T^{2} )$$)($$( 1 - T + T^{2} )( 1 + T + T^{2} )$$)($$( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} )$$)($$( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2}$$)($$( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} )$$)