# Properties

 Label 1089.6 Level 1089 Weight 6 Dimension 168954 Nonzero newspaces 16 Sturm bound 522720 Trace bound 2

## Defining parameters

 Level: $$N$$ = $$1089 = 3^{2} \cdot 11^{2}$$ Weight: $$k$$ = $$6$$ Nonzero newspaces: $$16$$ Sturm bound: $$522720$$ Trace bound: $$2$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{6}(\Gamma_1(1089))$$.

Total New Old
Modular forms 219080 170219 48861
Cusp forms 216520 168954 47566
Eisenstein series 2560 1265 1295

## Trace form

 $$168954q - 126q^{2} - 192q^{3} - 180q^{4} - 63q^{5} - 9q^{6} - 737q^{7} - 1373q^{8} - 594q^{9} + O(q^{10})$$ $$168954q - 126q^{2} - 192q^{3} - 180q^{4} - 63q^{5} - 9q^{6} - 737q^{7} - 1373q^{8} - 594q^{9} + 1629q^{10} + 295q^{11} + 2384q^{12} - 689q^{13} + 2007q^{14} - 2232q^{15} - 17840q^{16} - 8883q^{17} - 8280q^{18} + 4461q^{19} + 22615q^{20} + 8490q^{21} + 12420q^{22} - 30221q^{23} - 3471q^{24} + 21302q^{25} + 46675q^{26} + 29928q^{27} + 6001q^{28} - 17885q^{29} - 44796q^{30} - 13523q^{31} - 126962q^{32} - 49750q^{33} - 108902q^{34} - 126229q^{35} + 9871q^{36} - 24371q^{37} + 191854q^{38} + 116700q^{39} + 401567q^{40} + 235173q^{41} + 76158q^{42} + 42147q^{43} - 127135q^{44} - 112722q^{45} + 12965q^{46} - 42133q^{47} - 78065q^{48} - 134402q^{49} - 326666q^{50} - 230156q^{51} - 619083q^{52} - 134923q^{53} + 335201q^{54} + 265290q^{55} + 1449195q^{56} + 471362q^{57} + 858275q^{58} + 345405q^{59} + 165128q^{60} - 20205q^{61} - 787401q^{62} - 383296q^{63} - 1921339q^{64} - 1314879q^{65} - 540870q^{66} - 1050901q^{67} - 1384582q^{68} - 398962q^{69} + 321401q^{70} + 586317q^{71} + 934971q^{72} + 784287q^{73} + 2332095q^{74} + 943588q^{75} + 1673040q^{76} + 662085q^{77} + 1161282q^{78} - 179761q^{79} + 344343q^{80} + 991662q^{81} - 3240003q^{82} - 537781q^{83} - 1482366q^{84} + 725023q^{85} - 1786466q^{86} - 2017336q^{87} + 571930q^{88} - 1190327q^{89} - 2573672q^{90} + 1491377q^{91} + 963109q^{92} + 573518q^{93} - 133509q^{94} + 1383879q^{95} + 5208928q^{96} - 1768017q^{97} + 67007q^{98} + 1520600q^{99} + O(q^{100})$$

## Decomposition of $$S_{6}^{\mathrm{new}}(\Gamma_1(1089))$$

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
1089.6.a $$\chi_{1089}(1, \cdot)$$ 1089.6.a.a 1 1
1089.6.a.b 1
1089.6.a.c 1
1089.6.a.d 1
1089.6.a.e 1
1089.6.a.f 1
1089.6.a.g 1
1089.6.a.h 1
1089.6.a.i 1
1089.6.a.j 2
1089.6.a.k 2
1089.6.a.l 2
1089.6.a.m 2
1089.6.a.n 2
1089.6.a.o 2
1089.6.a.p 2
1089.6.a.q 3
1089.6.a.r 3
1089.6.a.s 3
1089.6.a.t 4
1089.6.a.u 4
1089.6.a.v 5
1089.6.a.w 5
1089.6.a.x 5
1089.6.a.y 5
1089.6.a.z 6
1089.6.a.ba 6
1089.6.a.bb 8
1089.6.a.bc 8
1089.6.a.bd 8
1089.6.a.be 8
1089.6.a.bf 8
1089.6.a.bg 8
1089.6.a.bh 10
1089.6.a.bi 10
1089.6.a.bj 10
1089.6.a.bk 10
1089.6.a.bl 10
1089.6.a.bm 12
1089.6.a.bn 20
1089.6.a.bo 20
1089.6.d $$\chi_{1089}(1088, \cdot)$$ n/a 180 1
1089.6.e $$\chi_{1089}(364, \cdot)$$ n/a 1072 2
1089.6.f $$\chi_{1089}(487, \cdot)$$ n/a 884 4
1089.6.g $$\chi_{1089}(362, \cdot)$$ n/a 1064 2
1089.6.j $$\chi_{1089}(161, \cdot)$$ n/a 720 4
1089.6.m $$\chi_{1089}(100, \cdot)$$ n/a 2740 10
1089.6.n $$\chi_{1089}(124, \cdot)$$ n/a 4256 8
1089.6.o $$\chi_{1089}(98, \cdot)$$ n/a 2200 10
1089.6.t $$\chi_{1089}(239, \cdot)$$ n/a 4256 8
1089.6.u $$\chi_{1089}(34, \cdot)$$ n/a 13160 20
1089.6.v $$\chi_{1089}(37, \cdot)$$ n/a 10960 40
1089.6.y $$\chi_{1089}(32, \cdot)$$ n/a 13160 20
1089.6.bb $$\chi_{1089}(8, \cdot)$$ n/a 8800 40
1089.6.bc $$\chi_{1089}(4, \cdot)$$ n/a 52640 80
1089.6.bd $$\chi_{1089}(2, \cdot)$$ n/a 52640 80

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

## Decomposition of $$S_{6}^{\mathrm{old}}(\Gamma_1(1089))$$ into lower level spaces

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