# Properties

 Label 1800.2 Level 1800 Weight 2 Dimension 35092 Nonzero newspaces 36 Sturm bound 345600 Trace bound 16

## Defining parameters

 Level: $$N$$ = $$1800 = 2^{3} \cdot 3^{2} \cdot 5^{2}$$ Weight: $$k$$ = $$2$$ Nonzero newspaces: $$36$$ Sturm bound: $$345600$$ Trace bound: $$16$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(\Gamma_1(1800))$$.

Total New Old
Modular forms 89088 35830 53258
Cusp forms 83713 35092 48621
Eisenstein series 5375 738 4637

## Trace form

 $$35092 q - 38 q^{2} - 51 q^{3} - 38 q^{4} + q^{5} - 88 q^{6} - 30 q^{7} - 50 q^{8} - 105 q^{9} + O(q^{10})$$ $$35092 q - 38 q^{2} - 51 q^{3} - 38 q^{4} + q^{5} - 88 q^{6} - 30 q^{7} - 50 q^{8} - 105 q^{9} - 144 q^{10} - 69 q^{11} - 62 q^{12} + 2 q^{13} - 74 q^{14} - 64 q^{15} - 102 q^{16} - 96 q^{17} - 48 q^{18} - 162 q^{19} - 68 q^{20} - 28 q^{21} - 62 q^{22} - 126 q^{23} - 24 q^{24} - 123 q^{25} - 124 q^{26} - 96 q^{27} - 92 q^{28} - 78 q^{29} - 64 q^{30} - 156 q^{31} + 42 q^{32} - 151 q^{33} + 14 q^{34} - 108 q^{35} - 30 q^{36} - 75 q^{37} + 106 q^{38} - 44 q^{39} - 8 q^{40} - 173 q^{41} + 50 q^{42} - 99 q^{43} + 158 q^{44} - 52 q^{46} + 6 q^{47} + 84 q^{48} - 103 q^{49} + 92 q^{50} - 55 q^{51} + 210 q^{52} + 25 q^{53} + 148 q^{54} - 120 q^{55} + 244 q^{56} - 29 q^{57} + 242 q^{58} + 197 q^{59} + 56 q^{60} + 16 q^{61} + 260 q^{62} + 128 q^{63} + 184 q^{64} - 27 q^{65} + 40 q^{66} + 163 q^{67} + 248 q^{68} + 70 q^{69} + 108 q^{70} + 172 q^{71} + 10 q^{72} - 228 q^{73} + 110 q^{74} + 8 q^{75} - 42 q^{76} + 108 q^{77} - 58 q^{78} + 32 q^{79} + 52 q^{80} - 21 q^{81} + 56 q^{82} + 248 q^{83} - 48 q^{84} + 17 q^{85} - 22 q^{86} + 178 q^{87} + 210 q^{88} + 17 q^{89} - 64 q^{90} - 84 q^{91} + 114 q^{92} + 182 q^{93} + 236 q^{94} + 200 q^{95} - 32 q^{96} - 61 q^{97} + 84 q^{98} + 286 q^{99} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(\Gamma_1(1800))$$

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
1800.2.a $$\chi_{1800}(1, \cdot)$$ 1800.2.a.a 1 1
1800.2.a.b 1
1800.2.a.c 1
1800.2.a.d 1
1800.2.a.e 1
1800.2.a.f 1
1800.2.a.g 1
1800.2.a.h 1
1800.2.a.i 1
1800.2.a.j 1
1800.2.a.k 1
1800.2.a.l 1
1800.2.a.m 1
1800.2.a.n 1
1800.2.a.o 1
1800.2.a.p 1
1800.2.a.q 1
1800.2.a.r 1
1800.2.a.s 1
1800.2.a.t 1
1800.2.a.u 1
1800.2.a.v 1
1800.2.a.w 1
1800.2.a.x 1
1800.2.b $$\chi_{1800}(251, \cdot)$$ 1800.2.b.a 2 1
1800.2.b.b 2
1800.2.b.c 4
1800.2.b.d 6
1800.2.b.e 6
1800.2.b.f 8
1800.2.b.g 16
1800.2.b.h 16
1800.2.b.i 16
1800.2.d $$\chi_{1800}(1549, \cdot)$$ 1800.2.d.a 2 1
1800.2.d.b 2
1800.2.d.c 2
1800.2.d.d 2
1800.2.d.e 2
1800.2.d.f 2
1800.2.d.g 2
1800.2.d.h 2
1800.2.d.i 2
1800.2.d.j 2
1800.2.d.k 4
1800.2.d.l 4
1800.2.d.m 4
1800.2.d.n 4
1800.2.d.o 4
1800.2.d.p 4
1800.2.d.q 6
1800.2.d.r 6
1800.2.d.s 8
1800.2.d.t 8
1800.2.d.u 16
1800.2.f $$\chi_{1800}(649, \cdot)$$ 1800.2.f.a 2 1
1800.2.f.b 2
1800.2.f.c 2
1800.2.f.d 2
1800.2.f.e 2
1800.2.f.f 2
1800.2.f.g 2
1800.2.f.h 2
1800.2.f.i 2
1800.2.f.j 2
1800.2.f.k 2
1800.2.h $$\chi_{1800}(1151, \cdot)$$ None 0 1
1800.2.k $$\chi_{1800}(901, \cdot)$$ 1800.2.k.a 2 1
1800.2.k.b 2
1800.2.k.c 2
1800.2.k.d 2
1800.2.k.e 2
1800.2.k.f 2
1800.2.k.g 2
1800.2.k.h 2
1800.2.k.i 2
1800.2.k.j 4
1800.2.k.k 4
1800.2.k.l 4
1800.2.k.m 4
1800.2.k.n 4
1800.2.k.o 4
1800.2.k.p 6
1800.2.k.q 8
1800.2.k.r 8
1800.2.k.s 8
1800.2.k.t 8
1800.2.k.u 12
1800.2.m $$\chi_{1800}(899, \cdot)$$ 1800.2.m.a 4 1
1800.2.m.b 4
1800.2.m.c 8
1800.2.m.d 12
1800.2.m.e 12
1800.2.m.f 32
1800.2.o $$\chi_{1800}(1799, \cdot)$$ None 0 1
1800.2.q $$\chi_{1800}(601, \cdot)$$ n/a 114 2
1800.2.s $$\chi_{1800}(593, \cdot)$$ 1800.2.s.a 4 2
1800.2.s.b 4
1800.2.s.c 4
1800.2.s.d 8
1800.2.s.e 8
1800.2.s.f 8
1800.2.t $$\chi_{1800}(343, \cdot)$$ None 0 2
1800.2.w $$\chi_{1800}(307, \cdot)$$ n/a 176 2
1800.2.x $$\chi_{1800}(557, \cdot)$$ n/a 144 2
1800.2.z $$\chi_{1800}(361, \cdot)$$ n/a 148 4
1800.2.bc $$\chi_{1800}(599, \cdot)$$ None 0 2
1800.2.be $$\chi_{1800}(299, \cdot)$$ n/a 424 2
1800.2.bg $$\chi_{1800}(301, \cdot)$$ n/a 444 2
1800.2.bh $$\chi_{1800}(551, \cdot)$$ None 0 2
1800.2.bj $$\chi_{1800}(49, \cdot)$$ n/a 108 2
1800.2.bl $$\chi_{1800}(349, \cdot)$$ n/a 424 2
1800.2.bn $$\chi_{1800}(851, \cdot)$$ n/a 444 2
1800.2.bq $$\chi_{1800}(71, \cdot)$$ None 0 4
1800.2.bs $$\chi_{1800}(289, \cdot)$$ n/a 152 4
1800.2.bu $$\chi_{1800}(109, \cdot)$$ n/a 592 4
1800.2.bw $$\chi_{1800}(611, \cdot)$$ n/a 480 4
1800.2.by $$\chi_{1800}(359, \cdot)$$ None 0 4
1800.2.ca $$\chi_{1800}(179, \cdot)$$ n/a 480 4
1800.2.cc $$\chi_{1800}(181, \cdot)$$ n/a 592 4
1800.2.ce $$\chi_{1800}(43, \cdot)$$ n/a 848 4
1800.2.ch $$\chi_{1800}(293, \cdot)$$ n/a 848 4
1800.2.ci $$\chi_{1800}(257, \cdot)$$ n/a 216 4
1800.2.cl $$\chi_{1800}(7, \cdot)$$ None 0 4
1800.2.cm $$\chi_{1800}(121, \cdot)$$ n/a 720 8
1800.2.co $$\chi_{1800}(53, \cdot)$$ n/a 960 8
1800.2.cp $$\chi_{1800}(163, \cdot)$$ n/a 1184 8
1800.2.cs $$\chi_{1800}(127, \cdot)$$ None 0 8
1800.2.ct $$\chi_{1800}(17, \cdot)$$ n/a 240 8
1800.2.cv $$\chi_{1800}(61, \cdot)$$ n/a 2848 8
1800.2.cx $$\chi_{1800}(59, \cdot)$$ n/a 2848 8
1800.2.cz $$\chi_{1800}(119, \cdot)$$ None 0 8
1800.2.dd $$\chi_{1800}(11, \cdot)$$ n/a 2848 8
1800.2.df $$\chi_{1800}(229, \cdot)$$ n/a 2848 8
1800.2.dh $$\chi_{1800}(169, \cdot)$$ n/a 720 8
1800.2.dj $$\chi_{1800}(191, \cdot)$$ None 0 8
1800.2.dk $$\chi_{1800}(103, \cdot)$$ None 0 16
1800.2.dn $$\chi_{1800}(113, \cdot)$$ n/a 1440 16
1800.2.do $$\chi_{1800}(77, \cdot)$$ n/a 5696 16
1800.2.dr $$\chi_{1800}(67, \cdot)$$ n/a 5696 16

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

## Decomposition of $$S_{2}^{\mathrm{old}}(\Gamma_1(1800))$$ into lower level spaces

$$S_{2}^{\mathrm{old}}(\Gamma_1(1800)) \cong$$ $$S_{2}^{\mathrm{new}}(\Gamma_1(15))$$$$^{\oplus 16}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(18))$$$$^{\oplus 9}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(20))$$$$^{\oplus 12}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(24))$$$$^{\oplus 6}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(25))$$$$^{\oplus 12}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(30))$$$$^{\oplus 12}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(36))$$$$^{\oplus 6}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(40))$$$$^{\oplus 6}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(45))$$$$^{\oplus 8}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(50))$$$$^{\oplus 9}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(60))$$$$^{\oplus 8}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(72))$$$$^{\oplus 3}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(75))$$$$^{\oplus 8}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(90))$$$$^{\oplus 6}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(100))$$$$^{\oplus 6}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(120))$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(150))$$$$^{\oplus 6}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(180))$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(200))$$$$^{\oplus 3}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(225))$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(300))$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(360))$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(450))$$$$^{\oplus 3}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(600))$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(900))$$$$^{\oplus 2}$$