LMFDB - Space of Modular Forms of Level 2880 and Weight 1

Defining parameters

Level:	$ N $	=	$ 2880 = 2^{6} \cdot 3^{2} \cdot 5 $
Weight:	$ k $	=	$ 1 $
Nonzero newspaces:			$ 10 $
Newform subspaces:			$ 18 $
Sturm bound:			$442368$
Trace bound:			$25$

Dimensions

The following table gives the dimensions of various subspaces of $M_{1}(\Gamma_1(2880))$.

	Total	New	Old
Modular forms	5132	663	4469
Cusp forms	524	69	455
Eisenstein series	4608	594	4014

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

	$D_n$	$A_4$	$S_4$	$A_5$
Dimension	69	0	0	0

Trace form

Decomposition of $S_{1}^{\mathrm{new}}(\Gamma_1(2880))$

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
2880.1.c	$\chi_{2880}(449, \cdot)$	2880.1.c.a	4	1
2880.1.e	$\chi_{2880}(2431, \cdot)$	None	0	1
2880.1.g	$\chi_{2880}(991, \cdot)$	None	0	1
2880.1.i	$\chi_{2880}(1889, \cdot)$	2880.1.i.a	4	1
2880.1.i	$\chi_{2880}(1889, \cdot)$	2880.1.i.b	4	1
2880.1.j	$\chi_{2880}(1279, \cdot)$	2880.1.j.a	1	1
2880.1.j	$\chi_{2880}(1279, \cdot)$	2880.1.j.b	2	1
2880.1.l	$\chi_{2880}(1601, \cdot)$	None	0	1
2880.1.n	$\chi_{2880}(161, \cdot)$	None	0	1
2880.1.p	$\chi_{2880}(2719, \cdot)$	2880.1.p.a	2	1
2880.1.p	$\chi_{2880}(2719, \cdot)$	2880.1.p.b	2	1
2880.1.r	$\chi_{2880}(559, \cdot)$	2880.1.r.a	4	2
2880.1.s	$\chi_{2880}(881, \cdot)$	None	0	2
2880.1.v	$\chi_{2880}(287, \cdot)$	None	0	2
2880.1.y	$\chi_{2880}(2017, \cdot)$	None	0	2
2880.1.ba	$\chi_{2880}(1583, \cdot)$	None	0	2
2880.1.bb	$\chi_{2880}(1873, \cdot)$	None	0	2
2880.1.be	$\chi_{2880}(143, \cdot)$	None	0	2
2880.1.bf	$\chi_{2880}(433, \cdot)$	None	0	2
2880.1.bh	$\chi_{2880}(577, \cdot)$	2880.1.bh.a	2	2
		2880.1.bh.b	2
		2880.1.bh.c	2
2880.1.bk	$\chi_{2880}(1727, \cdot)$	2880.1.bk.a	4	2
2880.1.bk	$\chi_{2880}(1727, \cdot)$	2880.1.bk.b	4	2
2880.1.bn	$\chi_{2880}(1169, \cdot)$	None	0	2
2880.1.bo	$\chi_{2880}(271, \cdot)$	None	0	2
2880.1.bp	$\chi_{2880}(799, \cdot)$	None	0	2
2880.1.bq	$\chi_{2880}(1121, \cdot)$	None	0	2
2880.1.bs	$\chi_{2880}(641, \cdot)$	None	0	2
2880.1.bu	$\chi_{2880}(319, \cdot)$	2880.1.bu.a	2	2
		2880.1.bu.b	2
		2880.1.bu.c	4
2880.1.bx	$\chi_{2880}(929, \cdot)$	None	0	2
2880.1.bz	$\chi_{2880}(31, \cdot)$	None	0	2
2880.1.cb	$\chi_{2880}(511, \cdot)$	None	0	2
2880.1.cd	$\chi_{2880}(1409, \cdot)$	2880.1.cd.a	8	2
2880.1.cf	$\chi_{2880}(73, \cdot)$	None	0	4
2880.1.cg	$\chi_{2880}(1223, \cdot)$	None	0	4
2880.1.cj	$\chi_{2880}(631, \cdot)$	None	0	4
2880.1.ck	$\chi_{2880}(89, \cdot)$	None	0	4
2880.1.cm	$\chi_{2880}(199, \cdot)$	None	0	4
2880.1.cp	$\chi_{2880}(521, \cdot)$	None	0	4
2880.1.cq	$\chi_{2880}(503, \cdot)$	None	0	4
2880.1.ct	$\chi_{2880}(793, \cdot)$	None	0	4
2880.1.cw	$\chi_{2880}(751, \cdot)$	None	0	4
2880.1.cx	$\chi_{2880}(209, \cdot)$	None	0	4
2880.1.cz	$\chi_{2880}(193, \cdot)$	None	0	4
2880.1.da	$\chi_{2880}(383, \cdot)$	None	0	4
2880.1.dd	$\chi_{2880}(337, \cdot)$	None	0	4
2880.1.de	$\chi_{2880}(47, \cdot)$	None	0	4
2880.1.dh	$\chi_{2880}(817, \cdot)$	None	0	4
2880.1.di	$\chi_{2880}(527, \cdot)$	None	0	4
2880.1.dl	$\chi_{2880}(1247, \cdot)$	None	0	4
2880.1.dm	$\chi_{2880}(97, \cdot)$	None	0	4
2880.1.do	$\chi_{2880}(401, \cdot)$	None	0	4
2880.1.dp	$\chi_{2880}(79, \cdot)$	None	0	4
2880.1.ds	$\chi_{2880}(397, \cdot)$	None	0	8
2880.1.dv	$\chi_{2880}(107, \cdot)$	None	0	8
2880.1.dx	$\chi_{2880}(341, \cdot)$	None	0	8
2880.1.dz	$\chi_{2880}(269, \cdot)$	None	0	8
2880.1.ea	$\chi_{2880}(91, \cdot)$	None	0	8
2880.1.ec	$\chi_{2880}(19, \cdot)$	2880.1.ec.a	16	8
2880.1.ee	$\chi_{2880}(37, \cdot)$	None	0	8
2880.1.eh	$\chi_{2880}(467, \cdot)$	None	0	8
2880.1.ei	$\chi_{2880}(313, \cdot)$	None	0	8
2880.1.el	$\chi_{2880}(23, \cdot)$	None	0	8
2880.1.en	$\chi_{2880}(329, \cdot)$	None	0	8
2880.1.eo	$\chi_{2880}(151, \cdot)$	None	0	8
2880.1.eq	$\chi_{2880}(41, \cdot)$	None	0	8
2880.1.et	$\chi_{2880}(439, \cdot)$	None	0	8
2880.1.ev	$\chi_{2880}(263, \cdot)$	None	0	8
2880.1.ew	$\chi_{2880}(553, \cdot)$	None	0	8
2880.1.ez	$\chi_{2880}(83, \cdot)$	None	0	16
2880.1.fa	$\chi_{2880}(133, \cdot)$	None	0	16
2880.1.fc	$\chi_{2880}(139, \cdot)$	None	0	16
2880.1.fe	$\chi_{2880}(211, \cdot)$	None	0	16
2880.1.fh	$\chi_{2880}(29, \cdot)$	None	0	16
2880.1.fj	$\chi_{2880}(101, \cdot)$	None	0	16
2880.1.fl	$\chi_{2880}(203, \cdot)$	None	0	16
2880.1.fm	$\chi_{2880}(13, \cdot)$	None	0	16

Decomposition of $S_{1}^{\mathrm{old}}(\Gamma_1(2880))$ into lower level spaces

Hecke Characteristic Polynomials

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

Level:	\( N \)	=	\( 2880 = 2^{6} \cdot 3^{2} \cdot 5 \)
Weight:	\( k \)	=	\( 1 \)
Nonzero newspaces:			\( 10 \)
Newform subspaces:			\( 18 \)
Sturm bound:			\(442368\)
Trace bound:			\(25\)

Introduction and more

L-functions

Modular Forms

Varieties

Fields

Representations

Groups

Knowledge

Properties

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Learn more about

Defining parameters

Dimensions

Trace form

Decomposition of \(S_{1}^{\mathrm{new}}(\Gamma_1(2880))\)

Decomposition of \(S_{1}^{\mathrm{old}}(\Gamma_1(2880))\) into lower level spaces

Hecke Characteristic Polynomials

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	2880.1

Level	2880
Weight	1
Dimension	69
Nonzero newspaces	10
Newform subspaces	18
Sturm bound	442368
Trace bound	25

Label	\(\chi\)	Newforms	Dimension	\(\chi\) degree
2880.1.c	\(\chi_{2880}(449, \cdot)\)	2880.1.c.a	4	1
2880.1.e	\(\chi_{2880}(2431, \cdot)\)	None	0	1
2880.1.g	\(\chi_{2880}(991, \cdot)\)	None	0	1
2880.1.i	\(\chi_{2880}(1889, \cdot)\)	2880.1.i.a	4	1
2880.1.i	\(\chi_{2880}(1889, \cdot)\)	2880.1.i.b	4	1
2880.1.j	\(\chi_{2880}(1279, \cdot)\)	2880.1.j.a	1	1
2880.1.j	\(\chi_{2880}(1279, \cdot)\)	2880.1.j.b	2	1
2880.1.l	\(\chi_{2880}(1601, \cdot)\)	None	0	1
2880.1.n	\(\chi_{2880}(161, \cdot)\)	None	0	1
2880.1.p	\(\chi_{2880}(2719, \cdot)\)	2880.1.p.a	2	1
2880.1.p	\(\chi_{2880}(2719, \cdot)\)	2880.1.p.b	2	1
2880.1.r	\(\chi_{2880}(559, \cdot)\)	2880.1.r.a	4	2
2880.1.s	\(\chi_{2880}(881, \cdot)\)	None	0	2
2880.1.v	\(\chi_{2880}(287, \cdot)\)	None	0	2
2880.1.y	\(\chi_{2880}(2017, \cdot)\)	None	0	2
2880.1.ba	\(\chi_{2880}(1583, \cdot)\)	None	0	2
2880.1.bb	\(\chi_{2880}(1873, \cdot)\)	None	0	2
2880.1.be	\(\chi_{2880}(143, \cdot)\)	None	0	2
2880.1.bf	\(\chi_{2880}(433, \cdot)\)	None	0	2
2880.1.bh	\(\chi_{2880}(577, \cdot)\)	2880.1.bh.a	2	2
		2880.1.bh.b	2
		2880.1.bh.c	2
2880.1.bk	\(\chi_{2880}(1727, \cdot)\)	2880.1.bk.a	4	2
2880.1.bk	\(\chi_{2880}(1727, \cdot)\)	2880.1.bk.b	4	2
2880.1.bn	\(\chi_{2880}(1169, \cdot)\)	None	0	2
2880.1.bo	\(\chi_{2880}(271, \cdot)\)	None	0	2
2880.1.bp	\(\chi_{2880}(799, \cdot)\)	None	0	2
2880.1.bq	\(\chi_{2880}(1121, \cdot)\)	None	0	2
2880.1.bs	\(\chi_{2880}(641, \cdot)\)	None	0	2
2880.1.bu	\(\chi_{2880}(319, \cdot)\)	2880.1.bu.a	2	2
		2880.1.bu.b	2
		2880.1.bu.c	4
2880.1.bx	\(\chi_{2880}(929, \cdot)\)	None	0	2
2880.1.bz	\(\chi_{2880}(31, \cdot)\)	None	0	2
2880.1.cb	\(\chi_{2880}(511, \cdot)\)	None	0	2
2880.1.cd	\(\chi_{2880}(1409, \cdot)\)	2880.1.cd.a	8	2
2880.1.cf	\(\chi_{2880}(73, \cdot)\)	None	0	4
2880.1.cg	\(\chi_{2880}(1223, \cdot)\)	None	0	4
2880.1.cj	\(\chi_{2880}(631, \cdot)\)	None	0	4
2880.1.ck	\(\chi_{2880}(89, \cdot)\)	None	0	4
2880.1.cm	\(\chi_{2880}(199, \cdot)\)	None	0	4
2880.1.cp	\(\chi_{2880}(521, \cdot)\)	None	0	4
2880.1.cq	\(\chi_{2880}(503, \cdot)\)	None	0	4
2880.1.ct	\(\chi_{2880}(793, \cdot)\)	None	0	4
2880.1.cw	\(\chi_{2880}(751, \cdot)\)	None	0	4
2880.1.cx	\(\chi_{2880}(209, \cdot)\)	None	0	4
2880.1.cz	\(\chi_{2880}(193, \cdot)\)	None	0	4
2880.1.da	\(\chi_{2880}(383, \cdot)\)	None	0	4
2880.1.dd	\(\chi_{2880}(337, \cdot)\)	None	0	4
2880.1.de	\(\chi_{2880}(47, \cdot)\)	None	0	4
2880.1.dh	\(\chi_{2880}(817, \cdot)\)	None	0	4
2880.1.di	\(\chi_{2880}(527, \cdot)\)	None	0	4
2880.1.dl	\(\chi_{2880}(1247, \cdot)\)	None	0	4
2880.1.dm	\(\chi_{2880}(97, \cdot)\)	None	0	4
2880.1.do	\(\chi_{2880}(401, \cdot)\)	None	0	4
2880.1.dp	\(\chi_{2880}(79, \cdot)\)	None	0	4
2880.1.ds	\(\chi_{2880}(397, \cdot)\)	None	0	8
2880.1.dv	\(\chi_{2880}(107, \cdot)\)	None	0	8
2880.1.dx	\(\chi_{2880}(341, \cdot)\)	None	0	8
2880.1.dz	\(\chi_{2880}(269, \cdot)\)	None	0	8
2880.1.ea	\(\chi_{2880}(91, \cdot)\)	None	0	8
2880.1.ec	\(\chi_{2880}(19, \cdot)\)	2880.1.ec.a	16	8
2880.1.ee	\(\chi_{2880}(37, \cdot)\)	None	0	8
2880.1.eh	\(\chi_{2880}(467, \cdot)\)	None	0	8
2880.1.ei	\(\chi_{2880}(313, \cdot)\)	None	0	8
2880.1.el	\(\chi_{2880}(23, \cdot)\)	None	0	8
2880.1.en	\(\chi_{2880}(329, \cdot)\)	None	0	8
2880.1.eo	\(\chi_{2880}(151, \cdot)\)	None	0	8
2880.1.eq	\(\chi_{2880}(41, \cdot)\)	None	0	8
2880.1.et	\(\chi_{2880}(439, \cdot)\)	None	0	8
2880.1.ev	\(\chi_{2880}(263, \cdot)\)	None	0	8
2880.1.ew	\(\chi_{2880}(553, \cdot)\)	None	0	8
2880.1.ez	\(\chi_{2880}(83, \cdot)\)	None	0	16
2880.1.fa	\(\chi_{2880}(133, \cdot)\)	None	0	16
2880.1.fc	\(\chi_{2880}(139, \cdot)\)	None	0	16
2880.1.fe	\(\chi_{2880}(211, \cdot)\)	None	0	16
2880.1.fh	\(\chi_{2880}(29, \cdot)\)	None	0	16
2880.1.fj	\(\chi_{2880}(101, \cdot)\)	None	0	16
2880.1.fl	\(\chi_{2880}(203, \cdot)\)	None	0	16
2880.1.fm	\(\chi_{2880}(13, \cdot)\)	None	0	16

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