## 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

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

## 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 the 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

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