# Properties

 Label 1089.3 Level 1089 Weight 3 Dimension 67202 Nonzero newspaces 16 Sturm bound 261360 Trace bound 2

## 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

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

## 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

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