Properties

 Label 2000.1 Level 2000 Weight 1 Dimension 40 Nonzero newspaces 5 Newform subspaces 5 Sturm bound 240000 Trace bound 9

Defining parameters

 Level: $$N$$ = $$2000\( 2000 = 2^{4} \cdot 5^{3}$$ \) Weight: $$k$$ = $$1$$ Nonzero newspaces: $$5$$ Newform subspaces: $$5$$ Sturm bound: $$240000$$ Trace bound: $$9$$

Dimensions

The following table gives the dimensions of various subspaces of $$M_{1}(\Gamma_1(2000))$$.

Total New Old
Modular forms 2749 616 2133
Cusp forms 229 40 189
Eisenstein series 2520 576 1944

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

$$D_n$$ $$A_4$$ $$S_4$$ $$A_5$$
Dimension 40 0 0 0

Trace form

 $$40q - q^{9} + O(q^{10})$$ $$40q - q^{9} + 2q^{29} + 10q^{37} + 10q^{41} + 9q^{49} + 10q^{53} + 10q^{61} + 5q^{65} + 5q^{81} - 5q^{85} - 8q^{89} + O(q^{100})$$

Decomposition of $$S_{1}^{\mathrm{new}}(\Gamma_1(2000))$$

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
2000.1.b $$\chi_{2000}(751, \cdot)$$ 2000.1.b.a 4 1
2000.1.e $$\chi_{2000}(999, \cdot)$$ None 0 1
2000.1.g $$\chi_{2000}(1751, \cdot)$$ None 0 1
2000.1.h $$\chi_{2000}(1999, \cdot)$$ 2000.1.h.a 4 1
2000.1.i $$\chi_{2000}(1557, \cdot)$$ None 0 2
2000.1.k $$\chi_{2000}(499, \cdot)$$ None 0 2
2000.1.m $$\chi_{2000}(57, \cdot)$$ None 0 2
2000.1.p $$\chi_{2000}(193, \cdot)$$ None 0 2
2000.1.r $$\chi_{2000}(251, \cdot)$$ None 0 2
2000.1.t $$\chi_{2000}(557, \cdot)$$ None 0 2
2000.1.v $$\chi_{2000}(151, \cdot)$$ None 0 4
2000.1.x $$\chi_{2000}(399, \cdot)$$ 2000.1.x.a 4 4
2000.1.z $$\chi_{2000}(351, \cdot)$$ 2000.1.z.a 8 4
2000.1.ba $$\chi_{2000}(199, \cdot)$$ None 0 4
2000.1.bc $$\chi_{2000}(157, \cdot)$$ None 0 8
2000.1.bf $$\chi_{2000}(99, \cdot)$$ None 0 8
2000.1.bg $$\chi_{2000}(257, \cdot)$$ None 0 8
2000.1.bj $$\chi_{2000}(393, \cdot)$$ None 0 8
2000.1.bk $$\chi_{2000}(51, \cdot)$$ None 0 8
2000.1.bn $$\chi_{2000}(93, \cdot)$$ None 0 8
2000.1.bp $$\chi_{2000}(79, \cdot)$$ 2000.1.bp.a 20 20
2000.1.bq $$\chi_{2000}(39, \cdot)$$ None 0 20
2000.1.bs $$\chi_{2000}(71, \cdot)$$ None 0 20
2000.1.bv $$\chi_{2000}(31, \cdot)$$ None 0 20
2000.1.bw $$\chi_{2000}(11, \cdot)$$ None 0 40
2000.1.bz $$\chi_{2000}(13, \cdot)$$ None 0 40
2000.1.ca $$\chi_{2000}(17, \cdot)$$ None 0 40
2000.1.cd $$\chi_{2000}(73, \cdot)$$ None 0 40
2000.1.ce $$\chi_{2000}(53, \cdot)$$ None 0 40
2000.1.ch $$\chi_{2000}(19, \cdot)$$ None 0 40

Decomposition of $$S_{1}^{\mathrm{old}}(\Gamma_1(2000))$$ into lower level spaces

$$S_{1}^{\mathrm{old}}(\Gamma_1(2000)) \cong$$ $$S_{1}^{\mathrm{new}}(\Gamma_1(80))$$$$^{\oplus 3}$$$$\oplus$$$$S_{1}^{\mathrm{new}}(\Gamma_1(100))$$$$^{\oplus 6}$$$$\oplus$$$$S_{1}^{\mathrm{new}}(\Gamma_1(200))$$$$^{\oplus 4}$$$$\oplus$$$$S_{1}^{\mathrm{new}}(\Gamma_1(400))$$$$^{\oplus 2}$$$$\oplus$$$$S_{1}^{\mathrm{new}}(\Gamma_1(500))$$$$^{\oplus 3}$$$$\oplus$$$$S_{1}^{\mathrm{new}}(\Gamma_1(1000))$$$$^{\oplus 2}$$

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ ($$( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} )$$)($$1 - T^{2} + T^{4} - T^{6} + T^{8}$$)($$1 - T^{2} + T^{4} - T^{6} + T^{8}$$)($$( 1 - T + T^{2} - T^{3} + T^{4} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} )^{2}$$)($$1 - T^{10} + T^{20} - T^{30} + T^{40}$$)
$5$ ($$1 - T^{5} + T^{10} - T^{15} + T^{20}$$)
$7$ ($$( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} )$$)($$1 - T^{2} + T^{4} - T^{6} + T^{8}$$)($$( 1 + T^{2} )^{4}$$)($$( 1 - T )^{8}( 1 + T )^{8}$$)($$( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{5}$$)
$11$ ($$( 1 - T )^{4}( 1 + T )^{4}$$)($$( 1 - T )^{4}( 1 + T )^{4}$$)($$( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} )$$)($$( 1 - T + T^{2} - T^{3} + T^{4} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} )^{2}$$)($$( 1 - T^{5} + T^{10} - T^{15} + T^{20} )( 1 + T^{5} + T^{10} + T^{15} + T^{20} )$$)
$13$ ($$( 1 + T^{2} )^{4}$$)($$( 1 - T )^{4}( 1 + T )^{4}$$)($$( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} )$$)($$( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2}$$)($$( 1 - T^{5} + T^{10} - T^{15} + T^{20} )( 1 + T^{5} + T^{10} + T^{15} + T^{20} )$$)
$17$ ($$( 1 + T^{2} )^{4}$$)($$( 1 - T )^{4}( 1 + T )^{4}$$)($$( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} )$$)($$( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2}$$)($$( 1 - T^{5} + T^{10} - T^{15} + T^{20} )( 1 + T^{5} + T^{10} + T^{15} + T^{20} )$$)
$19$ ($$( 1 - T )^{4}( 1 + T )^{4}$$)($$( 1 - T )^{4}( 1 + T )^{4}$$)($$( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} )$$)($$( 1 - T + T^{2} - T^{3} + T^{4} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} )^{2}$$)($$( 1 - T^{5} + T^{10} - T^{15} + T^{20} )( 1 + T^{5} + T^{10} + T^{15} + T^{20} )$$)
$23$ ($$( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} )$$)($$1 - T^{2} + T^{4} - T^{6} + T^{8}$$)($$1 - T^{2} + T^{4} - T^{6} + T^{8}$$)($$( 1 - T + T^{2} - T^{3} + T^{4} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} )^{2}$$)($$1 - T^{10} + T^{20} - T^{30} + T^{40}$$)
$29$ ($$( 1 - T + T^{2} - T^{3} + T^{4} )^{2}$$)($$( 1 + T + T^{2} + T^{3} + T^{4} )^{2}$$)($$( 1 + T + T^{2} + T^{3} + T^{4} )^{2}$$)($$( 1 - T + T^{2} - T^{3} + T^{4} )^{4}$$)($$( 1 + T^{5} + T^{10} + T^{15} + T^{20} )^{2}$$)
$31$ ($$( 1 - T )^{4}( 1 + T )^{4}$$)($$( 1 - T )^{4}( 1 + T )^{4}$$)($$( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} )$$)($$( 1 - T + T^{2} - T^{3} + T^{4} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} )^{2}$$)($$( 1 - T^{5} + T^{10} - T^{15} + T^{20} )( 1 + T^{5} + T^{10} + T^{15} + T^{20} )$$)
$37$ ($$( 1 + T^{2} )^{4}$$)($$( 1 - T )^{4}( 1 + T )^{4}$$)($$( 1 - T )^{4}( 1 - T + T^{2} - T^{3} + T^{4} )$$)($$( 1 + T^{2} )^{4}( 1 - T^{2} + T^{4} - T^{6} + T^{8} )$$)($$( 1 - T + T^{2} - T^{3} + T^{4} )^{5}( 1 + T^{5} + T^{10} + T^{15} + T^{20} )$$)
$41$ ($$( 1 - T + T^{2} - T^{3} + T^{4} )^{2}$$)($$( 1 - T + T^{2} - T^{3} + T^{4} )^{2}$$)($$( 1 - T + T^{2} - T^{3} + T^{4} )^{2}$$)($$( 1 - T + T^{2} - T^{3} + T^{4} )^{4}$$)($$( 1 - T^{5} + T^{10} - T^{15} + T^{20} )^{2}$$)
$43$ ($$( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} )$$)($$1 - T^{2} + T^{4} - T^{6} + T^{8}$$)($$( 1 + T^{2} )^{4}$$)($$( 1 - T )^{8}( 1 + T )^{8}$$)($$( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{5}$$)
$47$ ($$( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} )$$)($$1 - T^{2} + T^{4} - T^{6} + T^{8}$$)($$1 - T^{2} + T^{4} - T^{6} + T^{8}$$)($$( 1 - T + T^{2} - T^{3} + T^{4} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} )^{2}$$)($$1 - T^{10} + T^{20} - T^{30} + T^{40}$$)
$53$ ($$( 1 + T^{2} )^{4}$$)($$( 1 - T )^{4}( 1 + T )^{4}$$)($$( 1 - T )^{4}( 1 - T + T^{2} - T^{3} + T^{4} )$$)($$( 1 + T^{2} )^{4}( 1 - T^{2} + T^{4} - T^{6} + T^{8} )$$)($$( 1 - T + T^{2} - T^{3} + T^{4} )^{5}( 1 + T^{5} + T^{10} + T^{15} + T^{20} )$$)
$59$ ($$( 1 - T )^{4}( 1 + T )^{4}$$)($$( 1 - T )^{4}( 1 + T )^{4}$$)($$( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} )$$)($$( 1 - T + T^{2} - T^{3} + T^{4} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} )^{2}$$)($$( 1 - T^{5} + T^{10} - T^{15} + T^{20} )( 1 + T^{5} + T^{10} + T^{15} + T^{20} )$$)
$61$ ($$( 1 - T + T^{2} - T^{3} + T^{4} )^{2}$$)($$( 1 - T + T^{2} - T^{3} + T^{4} )^{2}$$)($$( 1 - T + T^{2} - T^{3} + T^{4} )^{2}$$)($$( 1 - T + T^{2} - T^{3} + T^{4} )^{4}$$)($$( 1 - T^{5} + T^{10} - T^{15} + T^{20} )^{2}$$)
$67$ ($$( 1 - T )^{4}( 1 + T )^{4}$$)($$( 1 + T^{2} )^{4}$$)($$1 - T^{2} + T^{4} - T^{6} + T^{8}$$)($$( 1 - T + T^{2} - T^{3} + T^{4} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} )^{2}$$)($$1 - T^{10} + T^{20} - T^{30} + T^{40}$$)
$71$ ($$( 1 - T )^{4}( 1 + T )^{4}$$)($$( 1 - T )^{4}( 1 + T )^{4}$$)($$( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} )$$)($$( 1 - T + T^{2} - T^{3} + T^{4} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} )^{2}$$)($$( 1 - T^{5} + T^{10} - T^{15} + T^{20} )( 1 + T^{5} + T^{10} + T^{15} + T^{20} )$$)
$73$ ($$( 1 + T^{2} )^{4}$$)($$( 1 - T )^{4}( 1 + T )^{4}$$)($$( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} )$$)($$( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2}$$)($$( 1 - T^{5} + T^{10} - T^{15} + T^{20} )( 1 + T^{5} + T^{10} + T^{15} + T^{20} )$$)
$79$ ($$( 1 - T )^{4}( 1 + T )^{4}$$)($$( 1 - T )^{4}( 1 + T )^{4}$$)($$( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} )$$)($$( 1 - T + T^{2} - T^{3} + T^{4} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} )^{2}$$)($$( 1 - T^{5} + T^{10} - T^{15} + T^{20} )( 1 + T^{5} + T^{10} + T^{15} + T^{20} )$$)
$83$ ($$( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} )$$)($$1 - T^{2} + T^{4} - T^{6} + T^{8}$$)($$1 - T^{2} + T^{4} - T^{6} + T^{8}$$)($$( 1 - T + T^{2} - T^{3} + T^{4} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} )^{2}$$)($$1 - T^{10} + T^{20} - T^{30} + T^{40}$$)
$89$ ($$( 1 - T + T^{2} - T^{3} + T^{4} )^{2}$$)($$( 1 + T + T^{2} + T^{3} + T^{4} )^{2}$$)($$( 1 - T )^{4}( 1 + T + T^{2} + T^{3} + T^{4} )$$)($$( 1 + T )^{8}( 1 - T + T^{2} - T^{3} + T^{4} )^{2}$$)($$( 1 + T + T^{2} + T^{3} + T^{4} )^{5}( 1 + T^{5} + T^{10} + T^{15} + T^{20} )$$)
$97$ ($$( 1 + T^{2} )^{4}$$)($$( 1 - T )^{4}( 1 + T )^{4}$$)($$( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} )$$)($$( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2}$$)($$( 1 - T^{5} + T^{10} - T^{15} + T^{20} )( 1 + T^{5} + T^{10} + T^{15} + T^{20} )$$)