# Properties

 Label 2013.1 Level 2013 Weight 1 Dimension 32 Nonzero newspaces 1 Newform subspaces 4 Sturm bound 297600 Trace bound 0

## Defining parameters

 Level: $$N$$ = $$2013\( 2013 = 3 \cdot 11 \cdot 61$$ \) Weight: $$k$$ = $$1$$ Nonzero newspaces: $$1$$ Newform subspaces: $$4$$ Sturm bound: $$297600$$ Trace bound: $$0$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{1}(\Gamma_1(2013))$$.

Total New Old
Modular forms 2490 1096 1394
Cusp forms 90 32 58
Eisenstein series 2400 1064 1336

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

$$D_n$$ $$A_4$$ $$S_4$$ $$A_5$$
Dimension 32 0 0 0

## Trace form

 $$32q - 8q^{4} - 8q^{9} + O(q^{10})$$ $$32q - 8q^{4} - 8q^{9} - 8q^{16} - 8q^{25} - 8q^{36} - 16q^{46} - 16q^{48} - 8q^{49} - 16q^{52} + 24q^{58} - 8q^{64} - 8q^{66} + 64q^{76} - 8q^{81} - 8q^{88} + O(q^{100})$$

## Decomposition of $$S_{1}^{\mathrm{new}}(\Gamma_1(2013))$$

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
2013.1.c $$\chi_{2013}(1343, \cdot)$$ None 0 1
2013.1.d $$\chi_{2013}(1099, \cdot)$$ None 0 1
2013.1.g $$\chi_{2013}(914, \cdot)$$ None 0 1
2013.1.h $$\chi_{2013}(670, \cdot)$$ None 0 1
2013.1.k $$\chi_{2013}(133, \cdot)$$ None 0 2
2013.1.l $$\chi_{2013}(560, \cdot)$$ None 0 2
2013.1.t $$\chi_{2013}(109, \cdot)$$ None 0 2
2013.1.u $$\chi_{2013}(353, \cdot)$$ None 0 2
2013.1.x $$\chi_{2013}(901, \cdot)$$ None 0 2
2013.1.y $$\chi_{2013}(1145, \cdot)$$ None 0 2
2013.1.bb $$\chi_{2013}(424, \cdot)$$ None 0 4
2013.1.bc $$\chi_{2013}(20, \cdot)$$ None 0 4
2013.1.be $$\chi_{2013}(125, \cdot)$$ None 0 4
2013.1.bg $$\chi_{2013}(637, \cdot)$$ None 0 4
2013.1.bh $$\chi_{2013}(601, \cdot)$$ None 0 4
2013.1.bi $$\chi_{2013}(304, \cdot)$$ None 0 4
2013.1.bj $$\chi_{2013}(613, \cdot)$$ None 0 4
2013.1.bk $$\chi_{2013}(224, \cdot)$$ None 0 4
2013.1.bm $$\chi_{2013}(548, \cdot)$$ 2013.1.bm.a 4 4
2013.1.bm.b 4
2013.1.bm.c 8
2013.1.bm.d 16
2013.1.bp $$\chi_{2013}(1406, \cdot)$$ None 0 4
2013.1.br $$\chi_{2013}(881, \cdot)$$ None 0 4
2013.1.bs $$\chi_{2013}(712, \cdot)$$ None 0 4
2013.1.bt $$\chi_{2013}(851, \cdot)$$ None 0 4
2013.1.bw $$\chi_{2013}(1168, \cdot)$$ None 0 4
2013.1.by $$\chi_{2013}(184, \cdot)$$ None 0 4
2013.1.bz $$\chi_{2013}(142, \cdot)$$ None 0 4
2013.1.cb $$\chi_{2013}(217, \cdot)$$ None 0 4
2013.1.ce $$\chi_{2013}(119, \cdot)$$ None 0 4
2013.1.cg $$\chi_{2013}(386, \cdot)$$ None 0 4
2013.1.ch $$\chi_{2013}(245, \cdot)$$ None 0 4
2013.1.cj $$\chi_{2013}(680, \cdot)$$ None 0 4
2013.1.cm $$\chi_{2013}(985, \cdot)$$ None 0 4
2013.1.cn $$\chi_{2013}(52, \cdot)$$ None 0 4
2013.1.co $$\chi_{2013}(113, \cdot)$$ None 0 4
2013.1.cr $$\chi_{2013}(32, \cdot)$$ None 0 4
2013.1.cs $$\chi_{2013}(265, \cdot)$$ None 0 4
2013.1.db $$\chi_{2013}(206, \cdot)$$ None 0 8
2013.1.dc $$\chi_{2013}(577, \cdot)$$ None 0 8
2013.1.de $$\chi_{2013}(130, \cdot)$$ None 0 8
2013.1.dg $$\chi_{2013}(98, \cdot)$$ None 0 8
2013.1.dk $$\chi_{2013}(8, \cdot)$$ None 0 8
2013.1.dl $$\chi_{2013}(50, \cdot)$$ None 0 8
2013.1.dm $$\chi_{2013}(272, \cdot)$$ None 0 8
2013.1.dp $$\chi_{2013}(37, \cdot)$$ None 0 8
2013.1.dq $$\chi_{2013}(355, \cdot)$$ None 0 8
2013.1.dr $$\chi_{2013}(394, \cdot)$$ None 0 8
2013.1.dv $$\chi_{2013}(496, \cdot)$$ None 0 8
2013.1.dx $$\chi_{2013}(821, \cdot)$$ None 0 8
2013.1.dz $$\chi_{2013}(290, \cdot)$$ None 0 8
2013.1.ea $$\chi_{2013}(838, \cdot)$$ None 0 8
2013.1.eb $$\chi_{2013}(205, \cdot)$$ None 0 8
2013.1.ee $$\chi_{2013}(137, \cdot)$$ None 0 8
2013.1.eg $$\chi_{2013}(47, \cdot)$$ None 0 8
2013.1.eh $$\chi_{2013}(56, \cdot)$$ None 0 8
2013.1.ej $$\chi_{2013}(269, \cdot)$$ None 0 8
2013.1.em $$\chi_{2013}(391, \cdot)$$ None 0 8
2013.1.eo $$\chi_{2013}(76, \cdot)$$ None 0 8
2013.1.ep $$\chi_{2013}(13, \cdot)$$ None 0 8
2013.1.er $$\chi_{2013}(178, \cdot)$$ None 0 8
2013.1.eu $$\chi_{2013}(317, \cdot)$$ None 0 8
2013.1.ev $$\chi_{2013}(46, \cdot)$$ None 0 8
2013.1.ew $$\chi_{2013}(188, \cdot)$$ None 0 8
2013.1.ey $$\chi_{2013}(476, \cdot)$$ None 0 8
2013.1.fb $$\chi_{2013}(14, \cdot)$$ None 0 8
2013.1.fd $$\chi_{2013}(80, \cdot)$$ None 0 8
2013.1.fe $$\chi_{2013}(19, \cdot)$$ None 0 8
2013.1.ff $$\chi_{2013}(292, \cdot)$$ None 0 8
2013.1.fg $$\chi_{2013}(370, \cdot)$$ None 0 8
2013.1.fh $$\chi_{2013}(472, \cdot)$$ None 0 8
2013.1.fj $$\chi_{2013}(5, \cdot)$$ None 0 8
2013.1.fl $$\chi_{2013}(86, \cdot)$$ None 0 8
2013.1.fm $$\chi_{2013}(73, \cdot)$$ None 0 8
2013.1.fo $$\chi_{2013}(17, \cdot)$$ None 0 16
2013.1.fq $$\chi_{2013}(67, \cdot)$$ None 0 16
2013.1.fu $$\chi_{2013}(82, \cdot)$$ None 0 16
2013.1.fv $$\chi_{2013}(115, \cdot)$$ None 0 16
2013.1.fw $$\chi_{2013}(124, \cdot)$$ None 0 16
2013.1.fz $$\chi_{2013}(200, \cdot)$$ None 0 16
2013.1.ga $$\chi_{2013}(2, \cdot)$$ None 0 16
2013.1.gb $$\chi_{2013}(29, \cdot)$$ None 0 16
2013.1.gf $$\chi_{2013}(494, \cdot)$$ None 0 16
2013.1.gh $$\chi_{2013}(91, \cdot)$$ None 0 16
2013.1.gj $$\chi_{2013}(31, \cdot)$$ None 0 16
2013.1.gk $$\chi_{2013}(68, \cdot)$$ None 0 16

## Decomposition of $$S_{1}^{\mathrm{old}}(\Gamma_1(2013))$$ into lower level spaces

$$S_{1}^{\mathrm{old}}(\Gamma_1(2013)) \cong$$ $$S_{1}^{\mathrm{new}}(\Gamma_1(183))$$$$^{\oplus 2}$$$$\oplus$$$$S_{1}^{\mathrm{new}}(\Gamma_1(671))$$$$^{\oplus 2}$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ ($$( 1 + T )^{4}( 1 - T + T^{2} - T^{3} + T^{4} )$$)($$( 1 - T )^{4}( 1 + T + T^{2} + T^{3} + T^{4} )$$)($$( 1 + T^{2} )^{4}( 1 - T^{2} + T^{4} - T^{6} + T^{8} )$$)($$( 1 + T^{4} )^{4}( 1 - T^{4} + T^{8} - T^{12} + T^{16} )$$)
$3$ ($$1 + T + T^{2} + T^{3} + T^{4}$$)($$1 + T + T^{2} + T^{3} + T^{4}$$)($$( 1 + T + T^{2} + T^{3} + T^{4} )^{2}$$)($$( 1 - T + T^{2} - T^{3} + T^{4} )^{4}$$)
$5$ ($$( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} )$$)($$( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} )$$)($$( 1 - T + T^{2} - T^{3} + T^{4} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} )^{2}$$)($$( 1 - T + T^{2} - T^{3} + T^{4} )^{4}( 1 + T + T^{2} + T^{3} + T^{4} )^{4}$$)
$7$ ($$( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} )$$)($$( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} )$$)($$( 1 - T + T^{2} - T^{3} + T^{4} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} )^{2}$$)($$( 1 - T + T^{2} - T^{3} + T^{4} )^{4}( 1 + T + T^{2} + T^{3} + T^{4} )^{4}$$)
$11$ ($$1 - T + T^{2} - T^{3} + T^{4}$$)($$1 + T + T^{2} + T^{3} + T^{4}$$)($$1 - T^{2} + T^{4} - T^{6} + T^{8}$$)($$1 - T^{4} + T^{8} - T^{12} + T^{16}$$)
$13$ ($$( 1 - T )^{4}( 1 + T + T^{2} + T^{3} + T^{4} )$$)($$( 1 - T )^{4}( 1 + T + T^{2} + T^{3} + T^{4} )$$)($$( 1 + T )^{8}( 1 - T + T^{2} - T^{3} + T^{4} )^{2}$$)($$( 1 + T^{2} )^{8}( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2}$$)
$17$ ($$( 1 - T + T^{2} - T^{3} + T^{4} )^{2}$$)($$( 1 + T + T^{2} + T^{3} + T^{4} )^{2}$$)($$( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2}$$)($$( 1 - T^{4} + T^{8} - T^{12} + T^{16} )^{2}$$)
$19$ ($$( 1 - T )^{4}( 1 + T + T^{2} + T^{3} + T^{4} )$$)($$( 1 - T )^{4}( 1 + T + T^{2} + T^{3} + T^{4} )$$)($$( 1 + T )^{8}( 1 - T + T^{2} - T^{3} + T^{4} )^{2}$$)($$( 1 + T^{2} )^{8}( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2}$$)
$23$ ($$( 1 - T + T^{2} - T^{3} + T^{4} )^{2}$$)($$( 1 + T + T^{2} + T^{3} + T^{4} )^{2}$$)($$( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2}$$)($$( 1 - T^{4} + T^{8} - T^{12} + T^{16} )^{2}$$)
$29$ ($$( 1 - T + T^{2} - T^{3} + T^{4} )^{2}$$)($$( 1 + T + T^{2} + T^{3} + T^{4} )^{2}$$)($$( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2}$$)($$( 1 - T^{4} + T^{8} - T^{12} + T^{16} )^{2}$$)
$31$ ($$( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} )$$)($$( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} )$$)($$( 1 - T + T^{2} - T^{3} + T^{4} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} )^{2}$$)($$( 1 - T + T^{2} - T^{3} + T^{4} )^{4}( 1 + T + T^{2} + T^{3} + T^{4} )^{4}$$)
$37$ ($$( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} )$$)($$( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} )$$)($$( 1 - T + T^{2} - T^{3} + T^{4} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} )^{2}$$)($$( 1 - T + T^{2} - T^{3} + T^{4} )^{4}( 1 + T + T^{2} + T^{3} + T^{4} )^{4}$$)
$41$ ($$( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} )$$)($$( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} )$$)($$( 1 - T + T^{2} - T^{3} + T^{4} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} )^{2}$$)($$( 1 - T + T^{2} - T^{3} + T^{4} )^{4}( 1 + T + T^{2} + T^{3} + T^{4} )^{4}$$)
$43$ ($$( 1 - T )^{4}( 1 + T )^{4}$$)($$( 1 - T )^{4}( 1 + T )^{4}$$)($$( 1 - T )^{8}( 1 + T )^{8}$$)($$( 1 - T )^{16}( 1 + T )^{16}$$)
$47$ ($$( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} )$$)($$( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} )$$)($$( 1 - T + T^{2} - T^{3} + T^{4} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} )^{2}$$)($$( 1 - T + T^{2} - T^{3} + T^{4} )^{4}( 1 + T + T^{2} + T^{3} + T^{4} )^{4}$$)
$53$ ($$( 1 + T )^{4}( 1 - T + T^{2} - T^{3} + T^{4} )$$)($$( 1 - T )^{4}( 1 + T + T^{2} + T^{3} + T^{4} )$$)($$( 1 + T^{2} )^{4}( 1 - T^{2} + T^{4} - T^{6} + T^{8} )$$)($$( 1 + T^{4} )^{4}( 1 - T^{4} + T^{8} - T^{12} + T^{16} )$$)
$59$ ($$( 1 + T )^{4}( 1 - T + T^{2} - T^{3} + T^{4} )$$)($$( 1 - T )^{4}( 1 + T + T^{2} + T^{3} + T^{4} )$$)($$( 1 + T^{2} )^{4}( 1 - T^{2} + T^{4} - T^{6} + T^{8} )$$)($$( 1 + T^{4} )^{4}( 1 - T^{4} + T^{8} - T^{12} + T^{16} )$$)
$61$ ($$1 + T + T^{2} + T^{3} + T^{4}$$)($$1 + T + T^{2} + T^{3} + T^{4}$$)($$( 1 + T + T^{2} + T^{3} + T^{4} )^{2}$$)($$( 1 - T + T^{2} - T^{3} + T^{4} )^{4}$$)
$67$ ($$( 1 - T )^{4}( 1 + T )^{4}$$)($$( 1 - T )^{4}( 1 + T )^{4}$$)($$( 1 - T )^{8}( 1 + T )^{8}$$)($$( 1 - T )^{16}( 1 + T )^{16}$$)
$71$ ($$( 1 - T + T^{2} - T^{3} + T^{4} )^{2}$$)($$( 1 + T + T^{2} + T^{3} + T^{4} )^{2}$$)($$( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2}$$)($$( 1 - T^{4} + T^{8} - T^{12} + T^{16} )^{2}$$)
$73$ ($$( 1 + T + T^{2} + T^{3} + T^{4} )^{2}$$)($$( 1 + T + T^{2} + T^{3} + T^{4} )^{2}$$)($$( 1 + T + T^{2} + T^{3} + T^{4} )^{4}$$)($$( 1 - T + T^{2} - T^{3} + T^{4} )^{8}$$)
$79$ ($$( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} )$$)($$( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} )$$)($$( 1 - T + T^{2} - T^{3} + T^{4} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} )^{2}$$)($$( 1 - T + T^{2} - T^{3} + T^{4} )^{4}( 1 + T + T^{2} + T^{3} + T^{4} )^{4}$$)
$83$ ($$( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} )$$)($$( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} )$$)($$( 1 - T + T^{2} - T^{3} + T^{4} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} )^{2}$$)($$( 1 - T + T^{2} - T^{3} + T^{4} )^{4}( 1 + T + T^{2} + T^{3} + T^{4} )^{4}$$)
$89$ ($$( 1 - T + T^{2} - T^{3} + T^{4} )^{2}$$)($$( 1 + T + T^{2} + T^{3} + T^{4} )^{2}$$)($$( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2}$$)($$( 1 - T^{4} + T^{8} - T^{12} + T^{16} )^{2}$$)
$97$ ($$( 1 + T + T^{2} + T^{3} + T^{4} )^{2}$$)($$( 1 + T + T^{2} + T^{3} + T^{4} )^{2}$$)($$( 1 - T + T^{2} - T^{3} + T^{4} )^{4}$$)($$( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{4}$$)