Properties

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

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Defining parameters

Level: \( N \) = \( 2015\( 2015 = 5 \cdot 13 \cdot 31 \) \)
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

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