# Properties

 Label 2004.1.g Level 2004 Weight 1 Character orbit g Rep. character $$\chi_{2004}(2003,\cdot)$$ Character field $$\Q$$ Dimension 26 Newform subspaces 6 Sturm bound 336 Trace bound 2

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$2004 = 2^{2} \cdot 3 \cdot 167$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 2004.g (of order $$2$$ and degree $$1$$) Character conductor: $$\operatorname{cond}(\chi)$$ $$=$$ $$2004$$ Character field: $$\Q$$ Newform subspaces: $$6$$ Sturm bound: $$336$$ Trace bound: $$2$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{1}(2004, [\chi])$$.

Total New Old
Modular forms 30 30 0
Cusp forms 26 26 0
Eisenstein series 4 4 0

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 $$S_{1}^{\mathrm{new}}(2004, [\chi])$$ into newform subspaces

Label Dim. $$A$$ Field Image CM RM Traces $q$-expansion
$$a_2$$ $$a_3$$ $$a_5$$ $$a_7$$
2004.1.g.a $$1$$ $$1.000$$ $$\Q$$ $$D_{2}$$ $$\Q(\sqrt{-167})$$, $$\Q(\sqrt{-501})$$ $$\Q(\sqrt{3})$$ $$-1$$ $$-1$$ $$0$$ $$0$$ $$q-q^{2}-q^{3}+q^{4}+q^{6}-q^{8}+q^{9}+\cdots$$
2004.1.g.b $$1$$ $$1.000$$ $$\Q$$ $$D_{2}$$ $$\Q(\sqrt{-167})$$, $$\Q(\sqrt{-501})$$ $$\Q(\sqrt{3})$$ $$1$$ $$1$$ $$0$$ $$0$$ $$q+q^{2}+q^{3}+q^{4}+q^{6}+q^{8}+q^{9}+\cdots$$
2004.1.g.c $$2$$ $$1.000$$ $$\Q(\sqrt{2})$$ $$D_{4}$$ $$\Q(\sqrt{-501})$$ None $$-2$$ $$2$$ $$0$$ $$0$$ $$q-q^{2}+q^{3}+q^{4}-\beta q^{5}-q^{6}-q^{8}+\cdots$$
2004.1.g.d $$2$$ $$1.000$$ $$\Q(\sqrt{2})$$ $$D_{4}$$ $$\Q(\sqrt{-501})$$ None $$2$$ $$-2$$ $$0$$ $$0$$ $$q+q^{2}-q^{3}+q^{4}-\beta q^{5}-q^{6}+q^{8}+\cdots$$
2004.1.g.e $$10$$ $$1.000$$ $$\Q(\zeta_{22})$$ $$D_{22}$$ $$\Q(\sqrt{-167})$$ None $$-1$$ $$-1$$ $$0$$ $$0$$ $$q+\zeta_{22}^{6}q^{2}+\zeta_{22}^{2}q^{3}-\zeta_{22}q^{4}+\zeta_{22}^{8}q^{6}+\cdots$$
2004.1.g.f $$10$$ $$1.000$$ $$\Q(\zeta_{22})$$ $$D_{22}$$ $$\Q(\sqrt{-167})$$ None $$1$$ $$1$$ $$0$$ $$0$$ $$q-\zeta_{22}^{6}q^{2}+\zeta_{22}^{9}q^{3}-\zeta_{22}q^{4}+\zeta_{22}^{4}q^{6}+\cdots$$

## Hecke characteristic polynomials

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