## Defining parameters

 Level: $$N$$ = $$1080 = 2^{3} \cdot 3^{3} \cdot 5$$ Weight: $$k$$ = $$4$$ Nonzero newspaces: $$27$$ Sturm bound: $$248832$$ Trace bound: $$22$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{4}(\Gamma_1(1080))$$.

Total New Old
Modular forms 94752 38208 56544
Cusp forms 91872 37824 54048
Eisenstein series 2880 384 2496

## Trace form

 $$37824 q - 16 q^{2} - 24 q^{3} - 28 q^{4} - 10 q^{5} - 72 q^{6} - 76 q^{7} - 28 q^{8} - 48 q^{9} + O(q^{10})$$ $$37824 q - 16 q^{2} - 24 q^{3} - 28 q^{4} - 10 q^{5} - 72 q^{6} - 76 q^{7} - 28 q^{8} - 48 q^{9} + 30 q^{10} - 120 q^{11} - 24 q^{12} - 12 q^{13} + 116 q^{14} + 96 q^{15} + 36 q^{16} + 420 q^{17} - 24 q^{18} + 20 q^{19} - 306 q^{20} + 120 q^{21} - 468 q^{22} - 84 q^{23} + 300 q^{24} + 138 q^{25} + 2676 q^{26} + 870 q^{27} + 1208 q^{28} + 684 q^{29} - 276 q^{30} - 780 q^{31} - 3636 q^{32} - 570 q^{33} - 2236 q^{34} - 2126 q^{35} - 4272 q^{36} - 2220 q^{37} - 4580 q^{38} - 1680 q^{39} - 558 q^{40} + 444 q^{41} + 816 q^{42} - 964 q^{43} + 4620 q^{44} + 1674 q^{45} + 4196 q^{46} + 4432 q^{47} + 4332 q^{48} + 2794 q^{49} + 786 q^{50} - 3696 q^{51} - 500 q^{52} - 1088 q^{53} - 24 q^{54} - 190 q^{55} - 8540 q^{56} - 498 q^{57} - 7852 q^{58} + 8494 q^{59} - 3696 q^{60} + 900 q^{61} - 5388 q^{62} + 2760 q^{63} - 820 q^{64} - 1482 q^{65} + 7308 q^{66} - 9676 q^{67} + 13116 q^{68} - 192 q^{69} + 8090 q^{70} - 21068 q^{71} + 11904 q^{72} + 772 q^{73} + 22180 q^{74} + 1542 q^{75} + 12708 q^{76} + 1264 q^{77} + 3852 q^{78} + 3908 q^{79} + 5594 q^{80} + 408 q^{81} + 4496 q^{82} + 12932 q^{83} - 11244 q^{84} + 12 q^{85} - 22596 q^{86} + 7164 q^{87} - 7332 q^{88} - 8406 q^{89} - 21126 q^{90} - 3448 q^{91} - 27244 q^{92} - 12960 q^{93} - 452 q^{94} - 13080 q^{95} - 7872 q^{96} - 2900 q^{97} - 7828 q^{98} + 7596 q^{99} + O(q^{100})$$

## Decomposition of $$S_{4}^{\mathrm{new}}(\Gamma_1(1080))$$

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
1080.4.a $$\chi_{1080}(1, \cdot)$$ 1080.4.a.a 2 1
1080.4.a.b 2
1080.4.a.c 3
1080.4.a.d 3
1080.4.a.e 3
1080.4.a.f 3
1080.4.a.g 3
1080.4.a.h 3
1080.4.a.i 3
1080.4.a.j 3
1080.4.a.k 3
1080.4.a.l 3
1080.4.a.m 3
1080.4.a.n 3
1080.4.a.o 4
1080.4.a.p 4
1080.4.b $$\chi_{1080}(971, \cdot)$$ n/a 192 1
1080.4.d $$\chi_{1080}(109, \cdot)$$ n/a 288 1
1080.4.f $$\chi_{1080}(649, \cdot)$$ 1080.4.f.a 16 1
1080.4.f.b 18
1080.4.f.c 18
1080.4.f.d 20
1080.4.h $$\chi_{1080}(431, \cdot)$$ None 0 1
1080.4.k $$\chi_{1080}(541, \cdot)$$ n/a 192 1
1080.4.m $$\chi_{1080}(539, \cdot)$$ n/a 288 1
1080.4.o $$\chi_{1080}(1079, \cdot)$$ None 0 1
1080.4.q $$\chi_{1080}(361, \cdot)$$ 1080.4.q.a 2 2
1080.4.q.b 16
1080.4.q.c 16
1080.4.q.d 18
1080.4.q.e 20
1080.4.s $$\chi_{1080}(377, \cdot)$$ n/a 144 2
1080.4.t $$\chi_{1080}(487, \cdot)$$ None 0 2
1080.4.w $$\chi_{1080}(163, \cdot)$$ n/a 576 2
1080.4.x $$\chi_{1080}(53, \cdot)$$ n/a 576 2
1080.4.bb $$\chi_{1080}(359, \cdot)$$ None 0 2
1080.4.bd $$\chi_{1080}(179, \cdot)$$ n/a 424 2
1080.4.bf $$\chi_{1080}(181, \cdot)$$ n/a 288 2
1080.4.bg $$\chi_{1080}(71, \cdot)$$ None 0 2
1080.4.bi $$\chi_{1080}(289, \cdot)$$ n/a 108 2
1080.4.bk $$\chi_{1080}(469, \cdot)$$ n/a 424 2
1080.4.bm $$\chi_{1080}(251, \cdot)$$ n/a 288 2
1080.4.bo $$\chi_{1080}(121, \cdot)$$ n/a 648 6
1080.4.bp $$\chi_{1080}(307, \cdot)$$ n/a 848 4
1080.4.bs $$\chi_{1080}(197, \cdot)$$ n/a 848 4
1080.4.bt $$\chi_{1080}(17, \cdot)$$ n/a 216 4
1080.4.bw $$\chi_{1080}(127, \cdot)$$ None 0 4
1080.4.bx $$\chi_{1080}(59, \cdot)$$ n/a 3864 6
1080.4.cc $$\chi_{1080}(61, \cdot)$$ n/a 2592 6
1080.4.cd $$\chi_{1080}(119, \cdot)$$ None 0 6
1080.4.cg $$\chi_{1080}(49, \cdot)$$ n/a 972 6
1080.4.ch $$\chi_{1080}(11, \cdot)$$ n/a 2592 6
1080.4.ci $$\chi_{1080}(191, \cdot)$$ None 0 6
1080.4.cj $$\chi_{1080}(229, \cdot)$$ n/a 3864 6
1080.4.co $$\chi_{1080}(77, \cdot)$$ n/a 7728 12
1080.4.cp $$\chi_{1080}(7, \cdot)$$ None 0 12
1080.4.cs $$\chi_{1080}(113, \cdot)$$ n/a 1944 12
1080.4.ct $$\chi_{1080}(43, \cdot)$$ n/a 7728 12

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

## Decomposition of $$S_{4}^{\mathrm{old}}(\Gamma_1(1080))$$ into lower level spaces

$$S_{4}^{\mathrm{old}}(\Gamma_1(1080)) \cong$$ $$S_{4}^{\mathrm{new}}(\Gamma_1(5))$$$$^{\oplus 16}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(6))$$$$^{\oplus 18}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(8))$$$$^{\oplus 8}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(9))$$$$^{\oplus 16}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(10))$$$$^{\oplus 12}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(12))$$$$^{\oplus 12}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(15))$$$$^{\oplus 12}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(18))$$$$^{\oplus 12}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(20))$$$$^{\oplus 8}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(24))$$$$^{\oplus 6}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(27))$$$$^{\oplus 8}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(30))$$$$^{\oplus 9}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(36))$$$$^{\oplus 8}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(40))$$$$^{\oplus 4}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(45))$$$$^{\oplus 8}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(54))$$$$^{\oplus 6}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(60))$$$$^{\oplus 6}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(72))$$$$^{\oplus 4}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(90))$$$$^{\oplus 6}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(108))$$$$^{\oplus 4}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(120))$$$$^{\oplus 3}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(135))$$$$^{\oplus 4}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(180))$$$$^{\oplus 4}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(216))$$$$^{\oplus 2}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(270))$$$$^{\oplus 3}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(360))$$$$^{\oplus 2}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(540))$$$$^{\oplus 2}$$