## Defining parameters

 Level: $$N$$ = $$1089 = 3^{2} \cdot 11^{2}$$ Weight: $$k$$ = $$3$$ Nonzero newspaces: $$16$$ Sturm bound: $$261360$$ Trace bound: $$2$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{3}(\Gamma_1(1089))$$.

Total New Old
Modular forms 88400 68466 19934
Cusp forms 85840 67202 18638
Eisenstein series 2560 1264 1296

## Trace form

 $$67202q - 138q^{2} - 183q^{3} - 136q^{4} - 129q^{5} - 171q^{6} - 167q^{7} - 175q^{8} - 189q^{9} + O(q^{10})$$ $$67202q - 138q^{2} - 183q^{3} - 136q^{4} - 129q^{5} - 171q^{6} - 167q^{7} - 175q^{8} - 189q^{9} - 427q^{10} - 145q^{11} - 334q^{12} - 101q^{13} - 119q^{14} - 180q^{15} - 204q^{16} - 145q^{17} - 180q^{18} - 443q^{19} - 71q^{20} - 186q^{21} - 120q^{22} + 7q^{23} + 255q^{24} + 432q^{25} + 835q^{26} + 174q^{27} + 669q^{28} + 373q^{29} + 78q^{30} + 93q^{31} - 78q^{32} - 250q^{33} - 882q^{34} - 905q^{35} - 749q^{36} - 1013q^{37} - 1498q^{38} - 624q^{39} - 1915q^{40} - 1086q^{41} - 900q^{42} - 964q^{43} - 855q^{44} - 654q^{45} - 379q^{46} - 269q^{47} - 467q^{48} + 20q^{49} + 984q^{50} + 181q^{51} + 1719q^{52} + 1675q^{53} + 1499q^{54} + 410q^{55} + 3095q^{56} + 1067q^{57} + 1937q^{58} + 1362q^{59} + 1238q^{60} + 419q^{61} + 1255q^{62} + 176q^{63} + 253q^{64} - 81q^{65} - 330q^{66} - 494q^{67} - 1372q^{68} - 820q^{69} - 2453q^{70} - 2095q^{71} - 2145q^{72} - 2165q^{73} - 4543q^{74} - 1619q^{75} - 3676q^{76} - 1875q^{77} - 2904q^{78} - 2563q^{79} - 6025q^{80} - 2901q^{81} - 1223q^{82} - 2299q^{83} - 5646q^{84} + 321q^{85} - 2198q^{86} - 2614q^{87} + 590q^{88} - 1225q^{89} - 3326q^{90} + 1409q^{91} - 2537q^{92} - 988q^{93} + 339q^{94} - 359q^{95} - 1280q^{96} + 190q^{97} - 165q^{98} - 40q^{99} + O(q^{100})$$

## Decomposition of $$S_{3}^{\mathrm{new}}(\Gamma_1(1089))$$

We only show spaces with odd 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
1089.3.b $$\chi_{1089}(485, \cdot)$$ 1089.3.b.a 2 1
1089.3.b.b 2
1089.3.b.c 4
1089.3.b.d 4
1089.3.b.e 4
1089.3.b.f 8
1089.3.b.g 8
1089.3.b.h 8
1089.3.b.i 16
1089.3.b.j 16
1089.3.c $$\chi_{1089}(604, \cdot)$$ 1089.3.c.a 2 1
1089.3.c.b 4
1089.3.c.c 4
1089.3.c.d 4
1089.3.c.e 4
1089.3.c.f 4
1089.3.c.g 4
1089.3.c.h 4
1089.3.c.i 8
1089.3.c.j 8
1089.3.c.k 8
1089.3.c.l 16
1089.3.c.m 16
1089.3.h $$\chi_{1089}(241, \cdot)$$ n/a 416 2
1089.3.i $$\chi_{1089}(122, \cdot)$$ n/a 418 2
1089.3.k $$\chi_{1089}(118, \cdot)$$ n/a 344 4
1089.3.l $$\chi_{1089}(251, \cdot)$$ n/a 288 4
1089.3.p $$\chi_{1089}(10, \cdot)$$ n/a 1090 10
1089.3.q $$\chi_{1089}(89, \cdot)$$ n/a 880 10
1089.3.r $$\chi_{1089}(245, \cdot)$$ n/a 1664 8
1089.3.s $$\chi_{1089}(40, \cdot)$$ n/a 1664 8
1089.3.w $$\chi_{1089}(23, \cdot)$$ n/a 5240 20
1089.3.x $$\chi_{1089}(43, \cdot)$$ n/a 5240 20
1089.3.z $$\chi_{1089}(26, \cdot)$$ n/a 3520 40
1089.3.ba $$\chi_{1089}(19, \cdot)$$ n/a 4360 40
1089.3.be $$\chi_{1089}(7, \cdot)$$ n/a 20960 80
1089.3.bf $$\chi_{1089}(5, \cdot)$$ n/a 20960 80

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

## Decomposition of $$S_{3}^{\mathrm{old}}(\Gamma_1(1089))$$ into lower level spaces

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