# Properties

 Label 2009.1 Level 2009 Weight 1 Dimension 12 Nonzero newspaces 1 Newform subspaces 2 Sturm bound 329280 Trace bound 0

## Defining parameters

 Level: $$N$$ = $$2009\( 2009 = 7^{2} \cdot 41$$ \) Weight: $$k$$ = $$1$$ Nonzero newspaces: $$1$$ Newform subspaces: $$2$$ Sturm bound: $$329280$$ Trace bound: $$0$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{1}(\Gamma_1(2009))$$.

Total New Old
Modular forms 2424 1904 520
Cusp forms 24 12 12
Eisenstein series 2400 1892 508

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

$$D_n$$ $$A_4$$ $$S_4$$ $$A_5$$
Dimension 12 0 0 0

## Trace form

 $$12q + 2q^{2} - 4q^{4} - 8q^{8} - 4q^{9} + O(q^{10})$$ $$12q + 2q^{2} - 4q^{4} - 8q^{8} - 4q^{9} - 2q^{16} + 6q^{18} + 2q^{23} - 6q^{25} + 6q^{32} - 4q^{36} + 2q^{37} + 4q^{39} - 4q^{43} + 4q^{46} - 4q^{50} + 4q^{51} - 8q^{57} - 2q^{72} - 10q^{74} + 12q^{78} - 2q^{81} + 4q^{86} + 16q^{92} + O(q^{100})$$

## Decomposition of $$S_{1}^{\mathrm{new}}(\Gamma_1(2009))$$

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
2009.1.b $$\chi_{2009}(1518, \cdot)$$ None 0 1
2009.1.d $$\chi_{2009}(2008, \cdot)$$ None 0 1
2009.1.g $$\chi_{2009}(1567, \cdot)$$ None 0 2
2009.1.i $$\chi_{2009}(901, \cdot)$$ 2009.1.i.a 6 2
2009.1.i.b 6
2009.1.k $$\chi_{2009}(411, \cdot)$$ None 0 2
2009.1.n $$\chi_{2009}(834, \cdot)$$ None 0 4
2009.1.p $$\chi_{2009}(734, \cdot)$$ None 0 4
2009.1.q $$\chi_{2009}(146, \cdot)$$ None 0 4
2009.1.r $$\chi_{2009}(460, \cdot)$$ None 0 4
2009.1.t $$\chi_{2009}(286, \cdot)$$ None 0 6
2009.1.v $$\chi_{2009}(83, \cdot)$$ None 0 6
2009.1.x $$\chi_{2009}(244, \cdot)$$ None 0 8
2009.1.ba $$\chi_{2009}(79, \cdot)$$ None 0 8
2009.1.bc $$\chi_{2009}(132, \cdot)$$ None 0 12
2009.1.be $$\chi_{2009}(31, \cdot)$$ None 0 8
2009.1.bf $$\chi_{2009}(215, \cdot)$$ None 0 8
2009.1.bi $$\chi_{2009}(99, \cdot)$$ None 0 16
2009.1.bk $$\chi_{2009}(124, \cdot)$$ None 0 12
2009.1.bm $$\chi_{2009}(40, \cdot)$$ None 0 12
2009.1.bn $$\chi_{2009}(85, \cdot)$$ None 0 24
2009.1.bq $$\chi_{2009}(80, \cdot)$$ None 0 16
2009.1.br $$\chi_{2009}(209, \cdot)$$ None 0 24
2009.1.bs $$\chi_{2009}(139, \cdot)$$ None 0 24
2009.1.bv $$\chi_{2009}(73, \cdot)$$ None 0 24
2009.1.by $$\chi_{2009}(30, \cdot)$$ None 0 32
2009.1.ca $$\chi_{2009}(20, \cdot)$$ None 0 48
2009.1.cc $$\chi_{2009}(44, \cdot)$$ None 0 48
2009.1.ce $$\chi_{2009}(10, \cdot)$$ None 0 48
2009.1.cf $$\chi_{2009}(45, \cdot)$$ None 0 48
2009.1.ch $$\chi_{2009}(15, \cdot)$$ None 0 96
2009.1.ci $$\chi_{2009}(5, \cdot)$$ None 0 96
2009.1.ck $$\chi_{2009}(11, \cdot)$$ None 0 192

## Decomposition of $$S_{1}^{\mathrm{old}}(\Gamma_1(2009))$$ into lower level spaces

$$S_{1}^{\mathrm{old}}(\Gamma_1(2009)) \cong$$ $$S_{1}^{\mathrm{new}}(\Gamma_1(287))$$$$^{\oplus 2}$$

## Hecke characteristic polynomials

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