# Properties

 Label 2004.1 Level 2004 Weight 1 Dimension 26 Nonzero newspaces 1 Newform subspaces 6 Sturm bound 223104 Trace bound 0

## Defining parameters

 Level: $$N$$ = $$2004\( 2004 = 2^{2} \cdot 3 \cdot 167$$ \) Weight: $$k$$ = $$1$$ Nonzero newspaces: $$1$$ Newform subspaces: $$6$$ Sturm bound: $$223104$$ Trace bound: $$0$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{1}(\Gamma_1(2004))$$.

Total New Old
Modular forms 1716 358 1358
Cusp forms 56 26 30
Eisenstein series 1660 332 1328

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

$$D_n$$ $$A_4$$ $$S_4$$ $$A_5$$
Dimension 26 0 0 0

## Trace form

 $$26q + 4q^{4} - 4q^{6} + 4q^{9} + O(q^{10})$$ $$26q + 4q^{4} - 4q^{6} + 4q^{9} + 4q^{16} - 4q^{24} - 18q^{25} + 4q^{36} - 11q^{42} - 11q^{48} - 18q^{49} + 7q^{54} + 4q^{64} + 11q^{72} + 4q^{81} + 11q^{84} - 8q^{85} - 4q^{96} - 8q^{97} + O(q^{100})$$

## Decomposition of $$S_{1}^{\mathrm{new}}(\Gamma_1(2004))$$

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
2004.1.d $$\chi_{2004}(1337, \cdot)$$ None 0 1
2004.1.e $$\chi_{2004}(1669, \cdot)$$ None 0 1
2004.1.f $$\chi_{2004}(1003, \cdot)$$ None 0 1
2004.1.g $$\chi_{2004}(2003, \cdot)$$ 2004.1.g.a 1 1
2004.1.g.b 1
2004.1.g.c 2
2004.1.g.d 2
2004.1.g.e 10
2004.1.g.f 10
2004.1.k $$\chi_{2004}(23, \cdot)$$ None 0 82
2004.1.l $$\chi_{2004}(7, \cdot)$$ None 0 82
2004.1.m $$\chi_{2004}(13, \cdot)$$ None 0 82
2004.1.n $$\chi_{2004}(29, \cdot)$$ None 0 82

## Decomposition of $$S_{1}^{\mathrm{old}}(\Gamma_1(2004))$$ into lower level spaces

$$S_{1}^{\mathrm{old}}(\Gamma_1(2004)) \cong$$ $$S_{1}^{\mathrm{new}}(\Gamma_1(167))$$$$^{\oplus 6}$$

## Hecke characteristic polynomials

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