Properties

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

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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

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