# Properties

 Label 400.3 Level 400 Weight 3 Dimension 4885 Nonzero newspaces 14 Newform subspaces 60 Sturm bound 28800 Trace bound 5

## Defining parameters

 Level: $$N$$ = $$400 = 2^{4} \cdot 5^{2}$$ Weight: $$k$$ = $$3$$ Nonzero newspaces: $$14$$ Newform subspaces: $$60$$ Sturm bound: $$28800$$ Trace bound: $$5$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{3}(\Gamma_1(400))$$.

Total New Old
Modular forms 9992 5069 4923
Cusp forms 9208 4885 4323
Eisenstein series 784 184 600

## Trace form

 $$4885 q - 26 q^{2} - 20 q^{3} - 32 q^{4} - 40 q^{5} - 48 q^{6} - 22 q^{7} - 20 q^{8} - q^{9} + O(q^{10})$$ $$4885 q - 26 q^{2} - 20 q^{3} - 32 q^{4} - 40 q^{5} - 48 q^{6} - 22 q^{7} - 20 q^{8} - q^{9} - 32 q^{10} - 48 q^{11} + 28 q^{12} - 70 q^{13} - 12 q^{14} - 72 q^{15} - 80 q^{16} - 120 q^{17} - 98 q^{18} + 16 q^{19} - 32 q^{20} - 6 q^{21} - 76 q^{22} + 138 q^{23} + 24 q^{24} + 40 q^{25} + 16 q^{26} + 274 q^{27} + 288 q^{28} + 230 q^{29} + 256 q^{30} + 98 q^{31} + 544 q^{32} + 362 q^{33} + 540 q^{34} + 24 q^{35} + 292 q^{36} + 106 q^{37} + 128 q^{38} - 194 q^{39} - 112 q^{40} - 120 q^{41} - 528 q^{42} - 260 q^{43} - 564 q^{44} - 330 q^{45} - 908 q^{46} - 498 q^{47} - 1200 q^{48} - 1005 q^{49} - 472 q^{50} - 480 q^{51} - 756 q^{52} - 374 q^{53} - 568 q^{54} - 170 q^{55} - 208 q^{56} - 170 q^{57} - 200 q^{58} + 192 q^{59} - 160 q^{60} + 22 q^{61} - 168 q^{62} + 1214 q^{63} - 680 q^{64} + 176 q^{65} - 1508 q^{66} + 1580 q^{67} - 1144 q^{68} + 810 q^{69} - 680 q^{70} + 1566 q^{71} - 1652 q^{72} + 284 q^{73} - 996 q^{74} + 728 q^{75} - 652 q^{76} + 50 q^{77} - 348 q^{78} - 30 q^{79} + 48 q^{80} + 13 q^{81} + 696 q^{82} + 76 q^{83} + 1728 q^{84} + 936 q^{85} + 1124 q^{86} - 914 q^{87} + 1552 q^{88} + 472 q^{89} + 1288 q^{90} - 1570 q^{91} + 1936 q^{92} - 122 q^{93} + 1704 q^{94} - 550 q^{95} + 2056 q^{96} + 120 q^{97} + 1682 q^{98} - 1546 q^{99} + O(q^{100})$$

## Decomposition of $$S_{3}^{\mathrm{new}}(\Gamma_1(400))$$

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
400.3.b $$\chi_{400}(351, \cdot)$$ 400.3.b.a 1 1
400.3.b.b 2
400.3.b.c 2
400.3.b.d 2
400.3.b.e 2
400.3.b.f 2
400.3.b.g 4
400.3.b.h 4
400.3.e $$\chi_{400}(199, \cdot)$$ None 0 1
400.3.g $$\chi_{400}(151, \cdot)$$ None 0 1
400.3.h $$\chi_{400}(399, \cdot)$$ 400.3.h.a 2 1
400.3.h.b 4
400.3.h.c 4
400.3.h.d 8
400.3.i $$\chi_{400}(93, \cdot)$$ 400.3.i.a 32 2
400.3.i.b 44
400.3.i.c 64
400.3.k $$\chi_{400}(99, \cdot)$$ 400.3.k.a 4 2
400.3.k.b 4
400.3.k.c 6
400.3.k.d 6
400.3.k.e 28
400.3.k.f 28
400.3.k.g 32
400.3.k.h 32
400.3.m $$\chi_{400}(57, \cdot)$$ None 0 2
400.3.p $$\chi_{400}(193, \cdot)$$ 400.3.p.a 2 2
400.3.p.b 2
400.3.p.c 2
400.3.p.d 2
400.3.p.e 2
400.3.p.f 2
400.3.p.g 2
400.3.p.h 4
400.3.p.i 4
400.3.p.j 4
400.3.p.k 4
400.3.p.l 4
400.3.r $$\chi_{400}(51, \cdot)$$ 400.3.r.a 4 2
400.3.r.b 4
400.3.r.c 6
400.3.r.d 28
400.3.r.e 28
400.3.r.f 32
400.3.r.g 44
400.3.t $$\chi_{400}(157, \cdot)$$ 400.3.t.a 32 2
400.3.t.b 44
400.3.t.c 64
400.3.v $$\chi_{400}(71, \cdot)$$ None 0 4
400.3.x $$\chi_{400}(79, \cdot)$$ 400.3.x.a 40 4
400.3.x.b 80
400.3.z $$\chi_{400}(31, \cdot)$$ 400.3.z.a 8 4
400.3.z.b 32
400.3.z.c 80
400.3.ba $$\chi_{400}(39, \cdot)$$ None 0 4
400.3.bc $$\chi_{400}(53, \cdot)$$ 400.3.bc.a 944 8
400.3.bf $$\chi_{400}(19, \cdot)$$ 400.3.bf.a 944 8
400.3.bg $$\chi_{400}(17, \cdot)$$ 400.3.bg.a 16 8
400.3.bg.b 24
400.3.bg.c 32
400.3.bg.d 40
400.3.bg.e 56
400.3.bg.f 64
400.3.bj $$\chi_{400}(73, \cdot)$$ None 0 8
400.3.bk $$\chi_{400}(11, \cdot)$$ 400.3.bk.a 944 8
400.3.bn $$\chi_{400}(13, \cdot)$$ 400.3.bn.a 944 8

## Decomposition of $$S_{3}^{\mathrm{old}}(\Gamma_1(400))$$ into lower level spaces

$$S_{3}^{\mathrm{old}}(\Gamma_1(400)) \cong$$ $$S_{3}^{\mathrm{new}}(\Gamma_1(8))$$$$^{\oplus 6}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(\Gamma_1(10))$$$$^{\oplus 8}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(\Gamma_1(16))$$$$^{\oplus 3}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(\Gamma_1(20))$$$$^{\oplus 6}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(\Gamma_1(25))$$$$^{\oplus 5}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(\Gamma_1(40))$$$$^{\oplus 4}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(\Gamma_1(50))$$$$^{\oplus 4}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(\Gamma_1(80))$$$$^{\oplus 2}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(\Gamma_1(100))$$$$^{\oplus 3}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(\Gamma_1(200))$$$$^{\oplus 2}$$