# Properties

 Label 2009.2 Level 2009 Weight 2 Dimension 155912 Nonzero newspaces 32 Sturm bound 658560 Trace bound 3

## Defining parameters

 Level: $$N$$ = $$2009 = 7^{2} \cdot 41$$ Weight: $$k$$ = $$2$$ Nonzero newspaces: $$32$$ Sturm bound: $$658560$$ Trace bound: $$3$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(\Gamma_1(2009))$$.

Total New Old
Modular forms 167040 159694 7346
Cusp forms 162241 155912 6329
Eisenstein series 4799 3782 1017

## Trace form

 $$155912q - 584q^{2} - 586q^{3} - 592q^{4} - 590q^{5} - 602q^{6} - 692q^{7} - 1052q^{8} - 604q^{9} + O(q^{10})$$ $$155912q - 584q^{2} - 586q^{3} - 592q^{4} - 590q^{5} - 602q^{6} - 692q^{7} - 1052q^{8} - 604q^{9} - 614q^{10} - 602q^{11} - 634q^{12} - 606q^{13} - 720q^{14} - 1070q^{15} - 640q^{16} - 614q^{17} - 656q^{18} - 618q^{19} - 662q^{20} - 734q^{21} - 1094q^{22} - 626q^{23} - 698q^{24} - 640q^{25} - 662q^{26} - 658q^{27} - 776q^{28} - 1082q^{29} - 762q^{30} - 662q^{31} - 774q^{32} - 734q^{33} - 736q^{34} - 762q^{35} - 1236q^{36} - 658q^{37} - 654q^{38} - 672q^{39} - 646q^{40} - 619q^{41} - 1398q^{42} - 1046q^{43} - 614q^{44} - 604q^{45} - 510q^{46} - 650q^{47} - 616q^{48} - 608q^{49} - 1888q^{50} - 614q^{51} - 536q^{52} - 622q^{53} - 606q^{54} - 512q^{55} - 636q^{56} - 1098q^{57} - 590q^{58} - 614q^{59} - 662q^{60} - 604q^{61} - 686q^{62} - 804q^{63} - 1132q^{64} - 756q^{65} - 946q^{66} - 774q^{67} - 930q^{68} - 850q^{69} - 930q^{70} - 1246q^{71} - 1148q^{72} - 806q^{73} - 926q^{74} - 986q^{75} - 1158q^{76} - 846q^{77} - 1342q^{78} - 818q^{79} - 1064q^{80} - 842q^{81} - 860q^{82} - 1278q^{83} - 608q^{84} - 1240q^{85} - 748q^{86} - 562q^{87} - 676q^{88} - 670q^{89} - 548q^{90} - 706q^{91} - 1184q^{92} - 410q^{93} - 668q^{94} - 562q^{95} - 484q^{96} - 644q^{97} - 384q^{98} - 1856q^{99} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(\Gamma_1(2009))$$

We only show spaces with even 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
2009.2.a $$\chi_{2009}(1, \cdot)$$ 2009.2.a.a 2 1
2009.2.a.b 2
2009.2.a.c 2
2009.2.a.d 2
2009.2.a.e 2
2009.2.a.f 2
2009.2.a.g 3
2009.2.a.h 3
2009.2.a.i 3
2009.2.a.j 3
2009.2.a.k 3
2009.2.a.l 5
2009.2.a.m 5
2009.2.a.n 5
2009.2.a.o 6
2009.2.a.p 7
2009.2.a.q 7
2009.2.a.r 17
2009.2.a.s 17
2009.2.a.t 20
2009.2.a.u 20
2009.2.c $$\chi_{2009}(491, \cdot)$$ n/a 138 1
2009.2.e $$\chi_{2009}(165, \cdot)$$ n/a 268 2
2009.2.f $$\chi_{2009}(50, \cdot)$$ n/a 278 2
2009.2.h $$\chi_{2009}(344, \cdot)$$ n/a 552 4
2009.2.j $$\chi_{2009}(655, \cdot)$$ n/a 272 2
2009.2.l $$\chi_{2009}(288, \cdot)$$ n/a 1128 6
2009.2.m $$\chi_{2009}(342, \cdot)$$ n/a 544 4
2009.2.o $$\chi_{2009}(148, \cdot)$$ n/a 552 4
2009.2.s $$\chi_{2009}(214, \cdot)$$ n/a 544 4
2009.2.u $$\chi_{2009}(204, \cdot)$$ n/a 1164 6
2009.2.w $$\chi_{2009}(18, \cdot)$$ n/a 1088 8
2009.2.y $$\chi_{2009}(197, \cdot)$$ n/a 1112 8
2009.2.z $$\chi_{2009}(247, \cdot)$$ n/a 2232 12
2009.2.bb $$\chi_{2009}(68, \cdot)$$ n/a 1088 8
2009.2.bd $$\chi_{2009}(155, \cdot)$$ n/a 2328 12
2009.2.bg $$\chi_{2009}(312, \cdot)$$ n/a 1088 8
2009.2.bh $$\chi_{2009}(57, \cdot)$$ n/a 4656 24
2009.2.bj $$\chi_{2009}(48, \cdot)$$ n/a 2176 16
2009.2.bl $$\chi_{2009}(81, \cdot)$$ n/a 2328 12
2009.2.bo $$\chi_{2009}(27, \cdot)$$ n/a 4656 24
2009.2.bp $$\chi_{2009}(128, \cdot)$$ n/a 2176 16
2009.2.bt $$\chi_{2009}(64, \cdot)$$ n/a 4656 24
2009.2.bu $$\chi_{2009}(9, \cdot)$$ n/a 4656 24
2009.2.bw $$\chi_{2009}(16, \cdot)$$ n/a 9312 48
2009.2.bx $$\chi_{2009}(19, \cdot)$$ n/a 4352 32
2009.2.bz $$\chi_{2009}(8, \cdot)$$ n/a 9312 48
2009.2.cb $$\chi_{2009}(3, \cdot)$$ n/a 9312 48
2009.2.cd $$\chi_{2009}(4, \cdot)$$ n/a 9312 48
2009.2.cg $$\chi_{2009}(6, \cdot)$$ n/a 18624 96
2009.2.cj $$\chi_{2009}(2, \cdot)$$ n/a 18624 96
2009.2.cl $$\chi_{2009}(12, \cdot)$$ n/a 37248 192

"n/a" means that newforms for that character have not been added to the database yet

## Decomposition of $$S_{2}^{\mathrm{old}}(\Gamma_1(2009))$$ into lower level spaces

$$S_{2}^{\mathrm{old}}(\Gamma_1(2009)) \cong$$ $$S_{2}^{\mathrm{new}}(\Gamma_1(41))$$$$^{\oplus 3}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(49))$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(287))$$$$^{\oplus 2}$$