# Properties

 Label 2001.1 Level 2001 Weight 1 Dimension 84 Nonzero newspaces 2 Newform subspaces 8 Sturm bound 295680 Trace bound 1

## Defining parameters

 Level: $$N$$ = $$2001 = 3 \cdot 23 \cdot 29$$ Weight: $$k$$ = $$1$$ Nonzero newspaces: $$2$$ Newform subspaces: $$8$$ Sturm bound: $$295680$$ Trace bound: $$1$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{1}(\Gamma_1(2001))$$.

Total New Old
Modular forms 2634 1220 1414
Cusp forms 170 84 86
Eisenstein series 2464 1136 1328

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

$$D_n$$ $$A_4$$ $$S_4$$ $$A_5$$
Dimension 84 0 0 0

## Trace form

 $$84q + O(q^{10})$$ $$84q - 84q^{64} + 42q^{72} - 42q^{96} + O(q^{100})$$

## Decomposition of $$S_{1}^{\mathrm{new}}(\Gamma_1(2001))$$

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
2001.1.b $$\chi_{2001}(1333, \cdot)$$ None 0 1
2001.1.c $$\chi_{2001}(668, \cdot)$$ None 0 1
2001.1.f $$\chi_{2001}(436, \cdot)$$ None 0 1
2001.1.g $$\chi_{2001}(1565, \cdot)$$ None 0 1
2001.1.i $$\chi_{2001}(1172, \cdot)$$ 2001.1.i.a 2 2
2001.1.i.b 2
2001.1.i.c 4
2001.1.i.d 4
2001.1.j $$\chi_{2001}(70, \cdot)$$ None 0 2
2001.1.p $$\chi_{2001}(254, \cdot)$$ None 0 6
2001.1.q $$\chi_{2001}(574, \cdot)$$ None 0 6
2001.1.t $$\chi_{2001}(806, \cdot)$$ None 0 6
2001.1.u $$\chi_{2001}(22, \cdot)$$ None 0 6
2001.1.w $$\chi_{2001}(173, \cdot)$$ None 0 10
2001.1.x $$\chi_{2001}(88, \cdot)$$ None 0 10
2001.1.ba $$\chi_{2001}(59, \cdot)$$ None 0 10
2001.1.bb $$\chi_{2001}(28, \cdot)$$ None 0 10
2001.1.be $$\chi_{2001}(346, \cdot)$$ None 0 12
2001.1.bf $$\chi_{2001}(68, \cdot)$$ 2001.1.bf.a 12 12
2001.1.bf.b 12
2001.1.bf.c 24
2001.1.bf.d 24
2001.1.bi $$\chi_{2001}(133, \cdot)$$ None 0 20
2001.1.bj $$\chi_{2001}(17, \cdot)$$ None 0 20
2001.1.bl $$\chi_{2001}(34, \cdot)$$ None 0 60
2001.1.bm $$\chi_{2001}(110, \cdot)$$ None 0 60
2001.1.bp $$\chi_{2001}(7, \cdot)$$ None 0 60
2001.1.bq $$\chi_{2001}(35, \cdot)$$ None 0 60
2001.1.bs $$\chi_{2001}(11, \cdot)$$ None 0 120
2001.1.bt $$\chi_{2001}(31, \cdot)$$ None 0 120

## Decomposition of $$S_{1}^{\mathrm{old}}(\Gamma_1(2001))$$ into lower level spaces

$$S_{1}^{\mathrm{old}}(\Gamma_1(2001)) \cong$$ $$S_{1}^{\mathrm{new}}(\Gamma_1(23))$$$$^{\oplus 4}$$$$\oplus$$$$S_{1}^{\mathrm{new}}(\Gamma_1(87))$$$$^{\oplus 2}$$$$\oplus$$$$S_{1}^{\mathrm{new}}(\Gamma_1(667))$$$$^{\oplus 2}$$

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ ($$( 1 + T )^{2}( 1 + T^{2} )$$)($$( 1 - T )^{2}( 1 + T^{2} )$$)($$( 1 + T + T^{2} )^{2}( 1 - T^{2} + T^{4} )$$)($$( 1 - T + T^{2} )^{2}( 1 - T^{2} + T^{4} )$$)($$( 1 + T^{2} )^{6}( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} )^{2}$$)($$( 1 + T^{2} )^{6}( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )^{2}$$)($$( 1 - T^{2} + T^{4} )^{6}( 1 + T - T^{3} - T^{4} + T^{6} - T^{8} - T^{9} + T^{11} + T^{12} )^{2}$$)($$( 1 - T^{2} + T^{4} )^{6}( 1 - T + T^{3} - T^{4} + T^{6} - T^{8} + T^{9} - T^{11} + T^{12} )^{2}$$)
$3$ ($$( 1 - T )^{2}$$)($$1 + T^{2}$$)($$1 - T^{2} + T^{4}$$)($$( 1 + T + T^{2} )^{2}$$)($$1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12}$$)($$( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} )^{2}$$)($$( 1 - T + T^{3} - T^{4} + T^{6} - T^{8} + T^{9} - T^{11} + T^{12} )^{2}$$)($$1 + T^{2} - T^{6} - T^{8} + T^{12} - T^{16} - T^{18} + T^{22} + T^{24}$$)
$5$ ($$( 1 - T )^{2}( 1 + T )^{2}$$)($$( 1 - T )^{2}( 1 + T )^{2}$$)($$( 1 - T )^{4}( 1 + T )^{4}$$)($$( 1 - T )^{4}( 1 + T )^{4}$$)($$( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} )^{2}$$)($$( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} )^{2}$$)($$( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )^{4}( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} )^{4}$$)($$( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )^{4}( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} )^{4}$$)
$7$ ($$( 1 - T )^{2}( 1 + T )^{2}$$)($$( 1 - T )^{2}( 1 + T )^{2}$$)($$( 1 - T )^{4}( 1 + T )^{4}$$)($$( 1 - T )^{4}( 1 + T )^{4}$$)($$( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} )^{2}$$)($$( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} )^{2}$$)($$( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )^{4}( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} )^{4}$$)($$( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )^{4}( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} )^{4}$$)
$11$ ($$1 + T^{4}$$)($$1 + T^{4}$$)($$( 1 + T^{4} )^{2}$$)($$( 1 + T^{4} )^{2}$$)($$1 - T^{4} + T^{8} - T^{12} + T^{16} - T^{20} + T^{24}$$)($$1 - T^{4} + T^{8} - T^{12} + T^{16} - T^{20} + T^{24}$$)($$( 1 - T^{4} + T^{8} - T^{12} + T^{16} - T^{20} + T^{24} )^{2}$$)($$( 1 - T^{4} + T^{8} - T^{12} + T^{16} - T^{20} + T^{24} )^{2}$$)
$13$ ($$( 1 + T^{2} )^{2}$$)($$( 1 + T^{2} )^{2}$$)($$( 1 - T^{2} + T^{4} )^{2}$$)($$( 1 - T^{2} + T^{4} )^{2}$$)($$( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} )^{2}$$)($$( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} )^{2}$$)($$( 1 + T^{2} - T^{6} - T^{8} + T^{12} - T^{16} - T^{18} + T^{22} + T^{24} )^{2}$$)($$( 1 + T^{2} - T^{6} - T^{8} + T^{12} - T^{16} - T^{18} + T^{22} + T^{24} )^{2}$$)
$17$ ($$1 + T^{4}$$)($$1 + T^{4}$$)($$( 1 + T^{4} )^{2}$$)($$( 1 + T^{4} )^{2}$$)($$( 1 + T^{4} )^{6}$$)($$( 1 + T^{4} )^{6}$$)($$( 1 + T^{4} )^{12}$$)($$( 1 + T^{4} )^{12}$$)
$19$ ($$1 + T^{4}$$)($$1 + T^{4}$$)($$( 1 + T^{4} )^{2}$$)($$( 1 + T^{4} )^{2}$$)($$1 - T^{4} + T^{8} - T^{12} + T^{16} - T^{20} + T^{24}$$)($$1 - T^{4} + T^{8} - T^{12} + T^{16} - T^{20} + T^{24}$$)($$( 1 - T^{4} + T^{8} - T^{12} + T^{16} - T^{20} + T^{24} )^{2}$$)($$( 1 - T^{4} + T^{8} - T^{12} + T^{16} - T^{20} + T^{24} )^{2}$$)
$23$ ($$1 + T^{2}$$)($$1 + T^{2}$$)($$( 1 + T^{2} )^{2}$$)($$( 1 + T^{2} )^{2}$$)($$1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12}$$)($$1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12}$$)($$( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} )^{2}$$)($$( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} )^{2}$$)
$29$ ($$1 + T^{2}$$)($$1 + T^{2}$$)($$1 - T^{2} + T^{4}$$)($$1 - T^{2} + T^{4}$$)($$1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12}$$)($$1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12}$$)($$1 + T^{2} - T^{6} - T^{8} + T^{12} - T^{16} - T^{18} + T^{22} + T^{24}$$)($$1 + T^{2} - T^{6} - T^{8} + T^{12} - T^{16} - T^{18} + T^{22} + T^{24}$$)
$31$ ($$( 1 + T )^{2}( 1 + T^{2} )$$)($$( 1 + T )^{2}( 1 + T^{2} )$$)($$( 1 - T + T^{2} )^{2}( 1 - T^{2} + T^{4} )$$)($$( 1 - T + T^{2} )^{2}( 1 - T^{2} + T^{4} )$$)($$( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )^{2}( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} )$$)($$( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )^{2}( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} )$$)($$( 1 + T - T^{3} - T^{4} + T^{6} - T^{8} - T^{9} + T^{11} + T^{12} )^{2}( 1 + T^{2} - T^{6} - T^{8} + T^{12} - T^{16} - T^{18} + T^{22} + T^{24} )$$)($$( 1 + T - T^{3} - T^{4} + T^{6} - T^{8} - T^{9} + T^{11} + T^{12} )^{2}( 1 + T^{2} - T^{6} - T^{8} + T^{12} - T^{16} - T^{18} + T^{22} + T^{24} )$$)
$37$ ($$1 + T^{4}$$)($$1 + T^{4}$$)($$( 1 + T^{4} )^{2}$$)($$( 1 + T^{4} )^{2}$$)($$1 - T^{4} + T^{8} - T^{12} + T^{16} - T^{20} + T^{24}$$)($$1 - T^{4} + T^{8} - T^{12} + T^{16} - T^{20} + T^{24}$$)($$( 1 - T^{4} + T^{8} - T^{12} + T^{16} - T^{20} + T^{24} )^{2}$$)($$( 1 - T^{4} + T^{8} - T^{12} + T^{16} - T^{20} + T^{24} )^{2}$$)
$41$ ($$( 1 - T )^{2}( 1 + T^{2} )$$)($$( 1 + T )^{2}( 1 + T^{2} )$$)($$( 1 - T + T^{2} )^{2}( 1 - T^{2} + T^{4} )$$)($$( 1 + T + T^{2} )^{2}( 1 - T^{2} + T^{4} )$$)($$( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )^{2}( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} )$$)($$( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} )^{2}( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} )$$)($$( 1 - T + T^{3} - T^{4} + T^{6} - T^{8} + T^{9} - T^{11} + T^{12} )^{2}( 1 + T^{2} - T^{6} - T^{8} + T^{12} - T^{16} - T^{18} + T^{22} + T^{24} )$$)($$( 1 + T - T^{3} - T^{4} + T^{6} - T^{8} - T^{9} + T^{11} + T^{12} )^{2}( 1 + T^{2} - T^{6} - T^{8} + T^{12} - T^{16} - T^{18} + T^{22} + T^{24} )$$)
$43$ ($$1 + T^{4}$$)($$1 + T^{4}$$)($$( 1 + T^{4} )^{2}$$)($$( 1 + T^{4} )^{2}$$)($$1 - T^{4} + T^{8} - T^{12} + T^{16} - T^{20} + T^{24}$$)($$1 - T^{4} + T^{8} - T^{12} + T^{16} - T^{20} + T^{24}$$)($$( 1 - T^{4} + T^{8} - T^{12} + T^{16} - T^{20} + T^{24} )^{2}$$)($$( 1 - T^{4} + T^{8} - T^{12} + T^{16} - T^{20} + T^{24} )^{2}$$)
$47$ ($$( 1 + T )^{2}( 1 + T^{2} )$$)($$( 1 - T )^{2}( 1 + T^{2} )$$)($$( 1 + T + T^{2} )^{2}( 1 - T^{2} + T^{4} )$$)($$( 1 - T + T^{2} )^{2}( 1 - T^{2} + T^{4} )$$)($$( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} )^{2}( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} )$$)($$( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )^{2}( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} )$$)($$( 1 + T - T^{3} - T^{4} + T^{6} - T^{8} - T^{9} + T^{11} + T^{12} )^{2}( 1 + T^{2} - T^{6} - T^{8} + T^{12} - T^{16} - T^{18} + T^{22} + T^{24} )$$)($$( 1 - T + T^{3} - T^{4} + T^{6} - T^{8} + T^{9} - T^{11} + T^{12} )^{2}( 1 + T^{2} - T^{6} - T^{8} + T^{12} - T^{16} - T^{18} + T^{22} + T^{24} )$$)
$53$ ($$( 1 + T^{2} )^{2}$$)($$( 1 + T^{2} )^{2}$$)($$( 1 + T^{2} )^{4}$$)($$( 1 + T^{2} )^{4}$$)($$( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} )^{2}$$)($$( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} )^{2}$$)($$( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} )^{4}$$)($$( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} )^{4}$$)
$59$ ($$( 1 + T^{2} )^{2}$$)($$( 1 + T^{2} )^{2}$$)($$( 1 + T^{2} )^{4}$$)($$( 1 + T^{2} )^{4}$$)($$( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} )^{2}$$)($$( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} )^{2}$$)($$( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} )^{4}$$)($$( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} )^{4}$$)
$61$ ($$1 + T^{4}$$)($$1 + T^{4}$$)($$( 1 + T^{4} )^{2}$$)($$( 1 + T^{4} )^{2}$$)($$1 - T^{4} + T^{8} - T^{12} + T^{16} - T^{20} + T^{24}$$)($$1 - T^{4} + T^{8} - T^{12} + T^{16} - T^{20} + T^{24}$$)($$( 1 - T^{4} + T^{8} - T^{12} + T^{16} - T^{20} + T^{24} )^{2}$$)($$( 1 - T^{4} + T^{8} - T^{12} + T^{16} - T^{20} + T^{24} )^{2}$$)
$67$ ($$( 1 + T^{2} )^{2}$$)($$( 1 + T^{2} )^{2}$$)($$( 1 + T^{2} )^{4}$$)($$( 1 + T^{2} )^{4}$$)($$( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} )^{2}$$)($$( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} )^{2}$$)($$( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} )^{4}$$)($$( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} )^{4}$$)
$71$ ($$( 1 + T^{2} )^{2}$$)($$( 1 + T^{2} )^{2}$$)($$( 1 - T^{2} + T^{4} )^{2}$$)($$( 1 - T^{2} + T^{4} )^{2}$$)($$( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} )^{2}$$)($$( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} )^{2}$$)($$( 1 + T^{2} - T^{6} - T^{8} + T^{12} - T^{16} - T^{18} + T^{22} + T^{24} )^{2}$$)($$( 1 + T^{2} - T^{6} - T^{8} + T^{12} - T^{16} - T^{18} + T^{22} + T^{24} )^{2}$$)
$73$ ($$( 1 + T )^{2}( 1 + T^{2} )$$)($$( 1 + T )^{2}( 1 + T^{2} )$$)($$( 1 - T + T^{2} )^{2}( 1 - T^{2} + T^{4} )$$)($$( 1 - T + T^{2} )^{2}( 1 - T^{2} + T^{4} )$$)($$( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )^{2}( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} )$$)($$( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )^{2}( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} )$$)($$( 1 + T - T^{3} - T^{4} + T^{6} - T^{8} - T^{9} + T^{11} + T^{12} )^{2}( 1 + T^{2} - T^{6} - T^{8} + T^{12} - T^{16} - T^{18} + T^{22} + T^{24} )$$)($$( 1 + T - T^{3} - T^{4} + T^{6} - T^{8} - T^{9} + T^{11} + T^{12} )^{2}( 1 + T^{2} - T^{6} - T^{8} + T^{12} - T^{16} - T^{18} + T^{22} + T^{24} )$$)
$79$ ($$1 + T^{4}$$)($$1 + T^{4}$$)($$( 1 + T^{4} )^{2}$$)($$( 1 + T^{4} )^{2}$$)($$1 - T^{4} + T^{8} - T^{12} + T^{16} - T^{20} + T^{24}$$)($$1 - T^{4} + T^{8} - T^{12} + T^{16} - T^{20} + T^{24}$$)($$( 1 - T^{4} + T^{8} - T^{12} + T^{16} - T^{20} + T^{24} )^{2}$$)($$( 1 - T^{4} + T^{8} - T^{12} + T^{16} - T^{20} + T^{24} )^{2}$$)
$83$ ($$( 1 + T^{2} )^{2}$$)($$( 1 + T^{2} )^{2}$$)($$( 1 + T^{2} )^{4}$$)($$( 1 + T^{2} )^{4}$$)($$( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} )^{2}$$)($$( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} )^{2}$$)($$( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} )^{4}$$)($$( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} )^{4}$$)
$89$ ($$1 + T^{4}$$)($$1 + T^{4}$$)($$( 1 + T^{4} )^{2}$$)($$( 1 + T^{4} )^{2}$$)($$1 - T^{4} + T^{8} - T^{12} + T^{16} - T^{20} + T^{24}$$)($$1 - T^{4} + T^{8} - T^{12} + T^{16} - T^{20} + T^{24}$$)($$( 1 - T^{4} + T^{8} - T^{12} + T^{16} - T^{20} + T^{24} )^{2}$$)($$( 1 - T^{4} + T^{8} - T^{12} + T^{16} - T^{20} + T^{24} )^{2}$$)
$97$ ($$1 + T^{4}$$)($$1 + T^{4}$$)($$( 1 + T^{4} )^{2}$$)($$( 1 + T^{4} )^{2}$$)($$1 - T^{4} + T^{8} - T^{12} + T^{16} - T^{20} + T^{24}$$)($$1 - T^{4} + T^{8} - T^{12} + T^{16} - T^{20} + T^{24}$$)($$( 1 - T^{4} + T^{8} - T^{12} + T^{16} - T^{20} + T^{24} )^{2}$$)($$( 1 - T^{4} + T^{8} - T^{12} + T^{16} - T^{20} + T^{24} )^{2}$$)