# Properties

 Label 1800.4 Level 1800 Weight 4 Dimension 106015 Nonzero newspaces 36 Sturm bound 691200 Trace bound 16

## Defining parameters

 Level: $$N$$ = $$1800 = 2^{3} \cdot 3^{2} \cdot 5^{2}$$ Weight: $$k$$ = $$4$$ Nonzero newspaces: $$36$$ Sturm bound: $$691200$$ Trace bound: $$16$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{4}(\Gamma_1(1800))$$.

Total New Old
Modular forms 261888 106753 155135
Cusp forms 256512 106015 150497
Eisenstein series 5376 738 4638

## Trace form

 $$106015 q - 40 q^{2} - 51 q^{3} - 50 q^{4} + 11 q^{5} - 100 q^{6} - 102 q^{7} - 10 q^{8} - 75 q^{9} + O(q^{10})$$ $$106015 q - 40 q^{2} - 51 q^{3} - 50 q^{4} + 11 q^{5} - 100 q^{6} - 102 q^{7} - 10 q^{8} - 75 q^{9} - 144 q^{10} - 9 q^{11} + 58 q^{12} - 44 q^{13} - 154 q^{14} - 64 q^{15} + 298 q^{16} - 42 q^{17} - 168 q^{18} + 154 q^{19} + 332 q^{20} - 460 q^{21} - 138 q^{22} - 246 q^{23} - 312 q^{24} + 87 q^{25} - 824 q^{26} + 1344 q^{27} - 1020 q^{28} + 1044 q^{29} - 64 q^{30} + 148 q^{31} + 810 q^{32} + 251 q^{33} + 770 q^{34} + 372 q^{35} - 846 q^{36} - 795 q^{37} - 1522 q^{38} - 2276 q^{39} - 1608 q^{40} - 2401 q^{41} + 266 q^{42} + 353 q^{43} - 602 q^{44} + 3220 q^{46} + 5838 q^{47} + 7332 q^{48} + 4402 q^{49} + 5092 q^{50} + 1433 q^{51} + 1426 q^{52} + 857 q^{53} - 5456 q^{54} - 2280 q^{55} - 13228 q^{56} - 3647 q^{57} - 11654 q^{58} - 8735 q^{59} - 7144 q^{60} - 530 q^{61} - 14816 q^{62} - 5632 q^{63} - 5576 q^{64} - 4977 q^{65} - 1388 q^{66} + 399 q^{67} + 5128 q^{68} - 2222 q^{69} + 6468 q^{70} + 2900 q^{71} + 16426 q^{72} + 7490 q^{73} + 19522 q^{74} + 10088 q^{75} + 4574 q^{76} + 6612 q^{77} + 7226 q^{78} - 2888 q^{79} - 1708 q^{80} - 399 q^{81} + 12956 q^{82} + 4372 q^{83} + 1128 q^{84} + 11307 q^{85} - 290 q^{86} - 5846 q^{87} + 8530 q^{88} - 9119 q^{89} - 64 q^{90} + 4428 q^{91} - 2142 q^{92} - 11038 q^{93} - 11000 q^{94} - 5000 q^{95} - 4112 q^{96} - 5273 q^{97} - 28578 q^{98} + 4318 q^{99} + O(q^{100})$$

## Decomposition of $$S_{4}^{\mathrm{new}}(\Gamma_1(1800))$$

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 available newforms together with their dimension.

Label $$\chi$$ Newforms Dimension $$\chi$$ degree
1800.4.a $$\chi_{1800}(1, \cdot)$$ 1800.4.a.a 1 1
1800.4.a.b 1
1800.4.a.c 1
1800.4.a.d 1
1800.4.a.e 1
1800.4.a.f 1
1800.4.a.g 1
1800.4.a.h 1
1800.4.a.i 1
1800.4.a.j 1
1800.4.a.k 1
1800.4.a.l 1
1800.4.a.m 1
1800.4.a.n 1
1800.4.a.o 1
1800.4.a.p 1
1800.4.a.q 1
1800.4.a.r 1
1800.4.a.s 1
1800.4.a.t 1
1800.4.a.u 1
1800.4.a.v 1
1800.4.a.w 1
1800.4.a.x 1
1800.4.a.y 1
1800.4.a.z 1
1800.4.a.ba 1
1800.4.a.bb 1
1800.4.a.bc 1
1800.4.a.bd 1
1800.4.a.be 1
1800.4.a.bf 1
1800.4.a.bg 1
1800.4.a.bh 1
1800.4.a.bi 1
1800.4.a.bj 2
1800.4.a.bk 2
1800.4.a.bl 2
1800.4.a.bm 2
1800.4.a.bn 2
1800.4.a.bo 2
1800.4.a.bp 2
1800.4.a.bq 2
1800.4.a.br 3
1800.4.a.bs 3
1800.4.a.bt 3
1800.4.a.bu 3
1800.4.a.bv 4
1800.4.a.bw 4
1800.4.b $$\chi_{1800}(251, \cdot)$$ n/a 228 1
1800.4.d $$\chi_{1800}(1549, \cdot)$$ n/a 268 1
1800.4.f $$\chi_{1800}(649, \cdot)$$ 1800.4.f.a 2 1
1800.4.f.b 2
1800.4.f.c 2
1800.4.f.d 2
1800.4.f.e 2
1800.4.f.f 2
1800.4.f.g 2
1800.4.f.h 2
1800.4.f.i 2
1800.4.f.j 2
1800.4.f.k 2
1800.4.f.l 2
1800.4.f.m 2
1800.4.f.n 2
1800.4.f.o 2
1800.4.f.p 2
1800.4.f.q 2
1800.4.f.r 2
1800.4.f.s 2
1800.4.f.t 2
1800.4.f.u 2
1800.4.f.v 2
1800.4.f.w 2
1800.4.f.x 2
1800.4.f.y 4
1800.4.f.z 4
1800.4.f.ba 6
1800.4.f.bb 6
1800.4.h $$\chi_{1800}(1151, \cdot)$$ None 0 1
1800.4.k $$\chi_{1800}(901, \cdot)$$ n/a 282 1
1800.4.m $$\chi_{1800}(899, \cdot)$$ n/a 216 1
1800.4.o $$\chi_{1800}(1799, \cdot)$$ None 0 1
1800.4.q $$\chi_{1800}(601, \cdot)$$ n/a 342 2
1800.4.s $$\chi_{1800}(593, \cdot)$$ n/a 108 2
1800.4.t $$\chi_{1800}(343, \cdot)$$ None 0 2
1800.4.w $$\chi_{1800}(307, \cdot)$$ n/a 536 2
1800.4.x $$\chi_{1800}(557, \cdot)$$ n/a 432 2
1800.4.z $$\chi_{1800}(361, \cdot)$$ n/a 452 4
1800.4.bc $$\chi_{1800}(599, \cdot)$$ None 0 2
1800.4.be $$\chi_{1800}(299, \cdot)$$ n/a 1288 2
1800.4.bg $$\chi_{1800}(301, \cdot)$$ n/a 1356 2
1800.4.bh $$\chi_{1800}(551, \cdot)$$ None 0 2
1800.4.bj $$\chi_{1800}(49, \cdot)$$ n/a 324 2
1800.4.bl $$\chi_{1800}(349, \cdot)$$ n/a 1288 2
1800.4.bn $$\chi_{1800}(851, \cdot)$$ n/a 1356 2
1800.4.bq $$\chi_{1800}(71, \cdot)$$ None 0 4
1800.4.bs $$\chi_{1800}(289, \cdot)$$ n/a 448 4
1800.4.bu $$\chi_{1800}(109, \cdot)$$ n/a 1792 4
1800.4.bw $$\chi_{1800}(611, \cdot)$$ n/a 1440 4
1800.4.by $$\chi_{1800}(359, \cdot)$$ None 0 4
1800.4.ca $$\chi_{1800}(179, \cdot)$$ n/a 1440 4
1800.4.cc $$\chi_{1800}(181, \cdot)$$ n/a 1792 4
1800.4.ce $$\chi_{1800}(43, \cdot)$$ n/a 2576 4
1800.4.ch $$\chi_{1800}(293, \cdot)$$ n/a 2576 4
1800.4.ci $$\chi_{1800}(257, \cdot)$$ n/a 648 4
1800.4.cl $$\chi_{1800}(7, \cdot)$$ None 0 4
1800.4.cm $$\chi_{1800}(121, \cdot)$$ n/a 2160 8
1800.4.co $$\chi_{1800}(53, \cdot)$$ n/a 2880 8
1800.4.cp $$\chi_{1800}(163, \cdot)$$ n/a 3584 8
1800.4.cs $$\chi_{1800}(127, \cdot)$$ None 0 8
1800.4.ct $$\chi_{1800}(17, \cdot)$$ n/a 720 8
1800.4.cv $$\chi_{1800}(61, \cdot)$$ n/a 8608 8
1800.4.cx $$\chi_{1800}(59, \cdot)$$ n/a 8608 8
1800.4.cz $$\chi_{1800}(119, \cdot)$$ None 0 8
1800.4.dd $$\chi_{1800}(11, \cdot)$$ n/a 8608 8
1800.4.df $$\chi_{1800}(229, \cdot)$$ n/a 8608 8
1800.4.dh $$\chi_{1800}(169, \cdot)$$ n/a 2160 8
1800.4.dj $$\chi_{1800}(191, \cdot)$$ None 0 8
1800.4.dk $$\chi_{1800}(103, \cdot)$$ None 0 16
1800.4.dn $$\chi_{1800}(113, \cdot)$$ n/a 4320 16
1800.4.do $$\chi_{1800}(77, \cdot)$$ n/a 17216 16
1800.4.dr $$\chi_{1800}(67, \cdot)$$ n/a 17216 16

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

## Decomposition of $$S_{4}^{\mathrm{old}}(\Gamma_1(1800))$$ into lower level spaces

$$S_{4}^{\mathrm{old}}(\Gamma_1(1800)) \cong$$ $$S_{4}^{\mathrm{new}}(\Gamma_1(1))$$$$^{\oplus 36}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(2))$$$$^{\oplus 27}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(3))$$$$^{\oplus 24}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(4))$$$$^{\oplus 18}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(5))$$$$^{\oplus 24}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(6))$$$$^{\oplus 18}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(8))$$$$^{\oplus 9}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(9))$$$$^{\oplus 12}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(10))$$$$^{\oplus 18}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(12))$$$$^{\oplus 12}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(15))$$$$^{\oplus 16}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(18))$$$$^{\oplus 9}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(20))$$$$^{\oplus 12}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(24))$$$$^{\oplus 6}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(25))$$$$^{\oplus 12}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(30))$$$$^{\oplus 12}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(36))$$$$^{\oplus 6}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(40))$$$$^{\oplus 6}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(45))$$$$^{\oplus 8}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(50))$$$$^{\oplus 9}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(60))$$$$^{\oplus 8}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(72))$$$$^{\oplus 3}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(75))$$$$^{\oplus 8}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(90))$$$$^{\oplus 6}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(100))$$$$^{\oplus 6}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(120))$$$$^{\oplus 4}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(150))$$$$^{\oplus 6}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(180))$$$$^{\oplus 4}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(200))$$$$^{\oplus 3}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(225))$$$$^{\oplus 4}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(300))$$$$^{\oplus 4}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(360))$$$$^{\oplus 2}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(450))$$$$^{\oplus 3}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(600))$$$$^{\oplus 2}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(900))$$$$^{\oplus 2}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(1800))$$$$^{\oplus 1}$$