## Defining parameters

 Level: $$N$$ = $$1089 = 3^{2} \cdot 11^{2}$$ Weight: $$k$$ = $$2$$ Nonzero newspaces: $$16$$ Sturm bound: $$174240$$ Trace bound: $$3$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(\Gamma_1(1089))$$.

Total New Old
Modular forms 44840 34550 10290
Cusp forms 42281 33285 8996
Eisenstein series 2559 1265 1294

## Trace form

 $$33285q - 135q^{2} - 180q^{3} - 135q^{4} - 135q^{5} - 180q^{6} - 125q^{7} - 115q^{8} - 180q^{9} + O(q^{10})$$ $$33285q - 135q^{2} - 180q^{3} - 135q^{4} - 135q^{5} - 180q^{6} - 125q^{7} - 115q^{8} - 180q^{9} - 375q^{10} - 145q^{11} - 340q^{12} - 125q^{13} - 105q^{14} - 180q^{15} - 75q^{16} - 105q^{17} - 180q^{18} - 375q^{19} - 145q^{20} - 180q^{21} - 120q^{22} - 265q^{23} - 240q^{24} - 155q^{25} - 205q^{26} - 240q^{27} - 535q^{28} - 205q^{29} - 300q^{30} - 175q^{31} - 355q^{32} - 250q^{33} - 335q^{34} - 245q^{35} - 320q^{36} - 425q^{37} - 265q^{38} - 240q^{39} - 205q^{40} - 165q^{41} - 300q^{42} - 125q^{43} - 140q^{44} - 360q^{45} - 275q^{46} - 65q^{47} - 260q^{48} - 25q^{49} - 175q^{50} - 200q^{51} - 125q^{52} - 185q^{53} - 340q^{54} - 470q^{55} - 435q^{56} - 340q^{57} - 285q^{58} - 315q^{59} - 460q^{60} - 205q^{61} - 525q^{62} - 340q^{63} - 705q^{64} - 345q^{65} - 330q^{66} - 445q^{67} - 485q^{68} - 340q^{69} - 345q^{70} - 315q^{71} - 480q^{72} - 435q^{73} - 465q^{74} - 320q^{75} - 185q^{76} - 225q^{77} - 540q^{78} - 125q^{79} + 75q^{80} - 180q^{81} - 475q^{82} + 55q^{83} - 5q^{85} + 155q^{86} + 20q^{87} - 70q^{88} - 25q^{89} + 220q^{90} - 415q^{91} + 275q^{92} + 20q^{93} + 15q^{94} + 255q^{95} + 400q^{96} - 15q^{97} + 385q^{98} - 40q^{99} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(\Gamma_1(1089))$$

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
1089.2.a $$\chi_{1089}(1, \cdot)$$ 1089.2.a.a 1 1
1089.2.a.b 1
1089.2.a.c 1
1089.2.a.d 1
1089.2.a.e 1
1089.2.a.f 1
1089.2.a.g 1
1089.2.a.h 1
1089.2.a.i 1
1089.2.a.j 1
1089.2.a.k 1
1089.2.a.l 2
1089.2.a.m 2
1089.2.a.n 2
1089.2.a.o 2
1089.2.a.p 2
1089.2.a.q 2
1089.2.a.r 2
1089.2.a.s 2
1089.2.a.t 2
1089.2.a.u 4
1089.2.a.v 4
1089.2.a.w 4
1089.2.d $$\chi_{1089}(1088, \cdot)$$ 1089.2.d.a 2 1
1089.2.d.b 2
1089.2.d.c 4
1089.2.d.d 4
1089.2.d.e 4
1089.2.d.f 4
1089.2.d.g 16
1089.2.e $$\chi_{1089}(364, \cdot)$$ 1089.2.e.a 2 2
1089.2.e.b 2
1089.2.e.c 2
1089.2.e.d 4
1089.2.e.e 4
1089.2.e.f 4
1089.2.e.g 4
1089.2.e.h 6
1089.2.e.i 8
1089.2.e.j 12
1089.2.e.k 16
1089.2.e.l 20
1089.2.e.m 20
1089.2.e.n 24
1089.2.e.o 36
1089.2.e.p 36
1089.2.f $$\chi_{1089}(487, \cdot)$$ n/a 164 4
1089.2.g $$\chi_{1089}(362, \cdot)$$ n/a 200 2
1089.2.j $$\chi_{1089}(161, \cdot)$$ n/a 144 4
1089.2.m $$\chi_{1089}(100, \cdot)$$ n/a 540 10
1089.2.n $$\chi_{1089}(124, \cdot)$$ n/a 800 8
1089.2.o $$\chi_{1089}(98, \cdot)$$ n/a 440 10
1089.2.t $$\chi_{1089}(239, \cdot)$$ n/a 800 8
1089.2.u $$\chi_{1089}(34, \cdot)$$ n/a 2600 20
1089.2.v $$\chi_{1089}(37, \cdot)$$ n/a 2160 40
1089.2.y $$\chi_{1089}(32, \cdot)$$ n/a 2600 20
1089.2.bb $$\chi_{1089}(8, \cdot)$$ n/a 1760 40
1089.2.bc $$\chi_{1089}(4, \cdot)$$ n/a 10400 80
1089.2.bd $$\chi_{1089}(2, \cdot)$$ n/a 10400 80

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

## Decomposition of $$S_{2}^{\mathrm{old}}(\Gamma_1(1089))$$ into lower level spaces

$$S_{2}^{\mathrm{old}}(\Gamma_1(1089)) \cong$$ $$S_{2}^{\mathrm{new}}(\Gamma_1(11))$$$$^{\oplus 6}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(33))$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(99))$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(121))$$$$^{\oplus 3}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(363))$$$$^{\oplus 2}$$