# Properties

 Label 1089.4 Level 1089 Weight 4 Dimension 101120 Nonzero newspaces 16 Sturm bound 348480 Trace bound 2

## Defining parameters

 Level: $$N$$ = $$1089 = 3^{2} \cdot 11^{2}$$ Weight: $$k$$ = $$4$$ Nonzero newspaces: $$16$$ Sturm bound: $$348480$$ Trace bound: $$2$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{4}(\Gamma_1(1089))$$.

Total New Old
Modular forms 131960 102385 29575
Cusp forms 129400 101120 28280
Eisenstein series 2560 1265 1295

## Trace form

 $$101120 q - 138 q^{2} - 183 q^{3} - 148 q^{4} - 150 q^{5} - 171 q^{6} - 102 q^{7} - 149 q^{8} - 135 q^{9} + O(q^{10})$$ $$101120 q - 138 q^{2} - 183 q^{3} - 148 q^{4} - 150 q^{5} - 171 q^{6} - 102 q^{7} - 149 q^{8} - 135 q^{9} - 583 q^{10} - 200 q^{11} - 496 q^{12} - 234 q^{13} + 195 q^{14} - 153 q^{15} + 696 q^{16} + 363 q^{17} + 36 q^{18} - 953 q^{19} - 1313 q^{20} - 159 q^{21} - 780 q^{22} - 2408 q^{23} - 2679 q^{24} - 1939 q^{25} - 1913 q^{26} - 492 q^{27} + 2977 q^{28} + 2416 q^{29} + 3228 q^{30} + 3310 q^{31} + 8758 q^{32} + 1400 q^{33} + 4928 q^{34} + 4931 q^{35} + 2395 q^{36} + 285 q^{37} + 136 q^{38} - 21 q^{39} - 6249 q^{40} - 6087 q^{41} - 5466 q^{42} - 5573 q^{43} - 5035 q^{44} - 3495 q^{45} - 2723 q^{46} - 1714 q^{47} - 1481 q^{48} - 1545 q^{49} - 4886 q^{50} - 2783 q^{51} - 15235 q^{52} - 10627 q^{53} - 9685 q^{54} - 2450 q^{55} - 7461 q^{56} - 2521 q^{57} + 395 q^{58} + 2877 q^{59} + 9584 q^{60} + 6168 q^{61} + 26703 q^{62} + 9863 q^{63} + 26149 q^{64} + 18630 q^{65} + 8250 q^{66} + 16337 q^{67} + 26822 q^{68} + 9191 q^{69} + 20857 q^{70} + 12861 q^{71} + 7611 q^{72} + 6415 q^{73} - 1941 q^{74} - 4043 q^{75} - 13844 q^{76} - 8640 q^{77} - 19530 q^{78} - 16088 q^{79} - 29193 q^{80} + 1893 q^{81} - 44815 q^{82} - 3118 q^{83} + 29442 q^{84} - 13129 q^{85} + 5812 q^{86} + 6467 q^{87} - 5790 q^{88} - 5747 q^{89} - 11336 q^{90} + 3873 q^{91} - 10307 q^{92} - 13993 q^{93} + 10815 q^{94} - 17043 q^{95} - 58520 q^{96} + 3769 q^{97} - 22351 q^{98} - 15880 q^{99} + O(q^{100})$$

## Decomposition of $$S_{4}^{\mathrm{new}}(\Gamma_1(1089))$$

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 available newforms together with their dimension.

Label $$\chi$$ Newforms Dimension $$\chi$$ degree
1089.4.a $$\chi_{1089}(1, \cdot)$$ 1089.4.a.a 1 1
1089.4.a.b 1
1089.4.a.c 1
1089.4.a.d 1
1089.4.a.e 1
1089.4.a.f 1
1089.4.a.g 1
1089.4.a.h 1
1089.4.a.i 1
1089.4.a.j 1
1089.4.a.k 2
1089.4.a.l 2
1089.4.a.m 2
1089.4.a.n 2
1089.4.a.o 2
1089.4.a.p 2
1089.4.a.q 2
1089.4.a.r 2
1089.4.a.s 2
1089.4.a.t 2
1089.4.a.u 2
1089.4.a.v 2
1089.4.a.w 2
1089.4.a.x 2
1089.4.a.y 4
1089.4.a.z 4
1089.4.a.ba 4
1089.4.a.bb 4
1089.4.a.bc 4
1089.4.a.bd 4
1089.4.a.be 4
1089.4.a.bf 4
1089.4.a.bg 4
1089.4.a.bh 4
1089.4.a.bi 6
1089.4.a.bj 6
1089.4.a.bk 6
1089.4.a.bl 12
1089.4.a.bm 12
1089.4.a.bn 12
1089.4.d $$\chi_{1089}(1088, \cdot)$$ n/a 108 1
1089.4.e $$\chi_{1089}(364, \cdot)$$ n/a 636 2
1089.4.f $$\chi_{1089}(487, \cdot)$$ n/a 524 4
1089.4.g $$\chi_{1089}(362, \cdot)$$ n/a 632 2
1089.4.j $$\chi_{1089}(161, \cdot)$$ n/a 432 4
1089.4.m $$\chi_{1089}(100, \cdot)$$ n/a 1640 10
1089.4.n $$\chi_{1089}(124, \cdot)$$ n/a 2528 8
1089.4.o $$\chi_{1089}(98, \cdot)$$ n/a 1320 10
1089.4.t $$\chi_{1089}(239, \cdot)$$ n/a 2528 8
1089.4.u $$\chi_{1089}(34, \cdot)$$ n/a 7880 20
1089.4.v $$\chi_{1089}(37, \cdot)$$ n/a 6560 40
1089.4.y $$\chi_{1089}(32, \cdot)$$ n/a 7880 20
1089.4.bb $$\chi_{1089}(8, \cdot)$$ n/a 5280 40
1089.4.bc $$\chi_{1089}(4, \cdot)$$ n/a 31520 80
1089.4.bd $$\chi_{1089}(2, \cdot)$$ n/a 31520 80

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

## Decomposition of $$S_{4}^{\mathrm{old}}(\Gamma_1(1089))$$ into lower level spaces

$$S_{4}^{\mathrm{old}}(\Gamma_1(1089)) \cong$$ $$S_{4}^{\mathrm{new}}(\Gamma_1(1))$$$$^{\oplus 9}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(3))$$$$^{\oplus 6}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(9))$$$$^{\oplus 3}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(11))$$$$^{\oplus 6}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(33))$$$$^{\oplus 4}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(99))$$$$^{\oplus 2}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(121))$$$$^{\oplus 3}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(363))$$$$^{\oplus 2}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(1089))$$$$^{\oplus 1}$$