# Properties

 Label 400.6 Level 400 Weight 6 Dimension 12350 Nonzero newspaces 14 Sturm bound 57600 Trace bound 3

## Defining parameters

 Level: $$N$$ = $$400 = 2^{4} \cdot 5^{2}$$ Weight: $$k$$ = $$6$$ Nonzero newspaces: $$14$$ Sturm bound: $$57600$$ Trace bound: $$3$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{6}(\Gamma_1(400))$$.

Total New Old
Modular forms 24392 12535 11857
Cusp forms 23608 12350 11258
Eisenstein series 784 185 599

## Trace form

 $$12350 q - 26 q^{2} - 28 q^{3} - 48 q^{4} - 40 q^{5} + 72 q^{6} + 94 q^{7} + 220 q^{8} + 56 q^{9} + O(q^{10})$$ $$12350 q - 26 q^{2} - 28 q^{3} - 48 q^{4} - 40 q^{5} + 72 q^{6} + 94 q^{7} + 220 q^{8} + 56 q^{9} - 32 q^{10} - 1300 q^{11} - 20 q^{12} + 516 q^{13} - 124 q^{14} - 2892 q^{15} - 912 q^{16} + 954 q^{17} - 3162 q^{18} + 2400 q^{19} - 32 q^{20} + 7570 q^{21} + 4396 q^{22} - 3754 q^{23} + 8344 q^{24} - 6240 q^{25} + 7288 q^{26} - 7774 q^{27} - 23744 q^{28} - 16960 q^{29} - 5504 q^{30} + 4226 q^{31} + 78304 q^{32} + 72370 q^{33} + 96668 q^{34} + 2364 q^{35} - 44988 q^{36} - 75932 q^{37} - 95848 q^{38} - 63286 q^{39} - 118512 q^{40} - 68790 q^{41} - 80384 q^{42} + 4576 q^{43} + 42620 q^{44} + 67770 q^{45} + 194788 q^{46} + 150430 q^{47} + 210256 q^{48} + 33804 q^{49} + 40328 q^{50} - 141160 q^{51} - 222116 q^{52} - 182812 q^{53} + 84232 q^{54} + 9110 q^{55} + 191088 q^{56} + 216998 q^{57} + 106752 q^{58} + 211668 q^{59} + 8160 q^{60} - 5192 q^{61} - 273896 q^{62} - 267758 q^{63} + 269208 q^{64} - 65724 q^{65} - 223156 q^{66} + 159380 q^{67} - 473240 q^{68} + 75706 q^{69} - 399560 q^{70} + 563042 q^{71} - 249540 q^{72} + 78870 q^{73} + 244100 q^{74} - 156632 q^{75} + 399492 q^{76} - 183246 q^{77} + 1146388 q^{78} - 1268718 q^{79} + 492848 q^{80} - 290318 q^{81} + 640808 q^{82} - 851976 q^{83} + 610688 q^{84} + 362896 q^{85} + 106268 q^{86} + 2136530 q^{87} - 655600 q^{88} + 452906 q^{89} - 1191512 q^{90} + 1512974 q^{91} - 1623440 q^{92} + 391214 q^{93} - 907704 q^{94} - 555550 q^{95} + 619272 q^{96} - 1027454 q^{97} + 908386 q^{98} - 2738474 q^{99} + O(q^{100})$$

## Decomposition of $$S_{6}^{\mathrm{new}}(\Gamma_1(400))$$

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
400.6.a $$\chi_{400}(1, \cdot)$$ 400.6.a.a 1 1
400.6.a.b 1
400.6.a.c 1
400.6.a.d 1
400.6.a.e 1
400.6.a.f 1
400.6.a.g 1
400.6.a.h 1
400.6.a.i 1
400.6.a.j 1
400.6.a.k 1
400.6.a.l 1
400.6.a.m 1
400.6.a.n 1
400.6.a.o 2
400.6.a.p 2
400.6.a.q 2
400.6.a.r 2
400.6.a.s 2
400.6.a.t 2
400.6.a.u 2
400.6.a.v 2
400.6.a.w 2
400.6.a.x 3
400.6.a.y 3
400.6.a.z 4
400.6.a.ba 4
400.6.c $$\chi_{400}(49, \cdot)$$ 400.6.c.a 2 1
400.6.c.b 2
400.6.c.c 2
400.6.c.d 2
400.6.c.e 2
400.6.c.f 2
400.6.c.g 2
400.6.c.h 2
400.6.c.i 2
400.6.c.j 2
400.6.c.k 2
400.6.c.l 4
400.6.c.m 4
400.6.c.n 4
400.6.c.o 4
400.6.c.p 6
400.6.d $$\chi_{400}(201, \cdot)$$ None 0 1
400.6.f $$\chi_{400}(249, \cdot)$$ None 0 1
400.6.j $$\chi_{400}(43, \cdot)$$ n/a 356 2
400.6.l $$\chi_{400}(101, \cdot)$$ n/a 374 2
400.6.n $$\chi_{400}(143, \cdot)$$ 400.6.n.a 2 2
400.6.n.b 4
400.6.n.c 4
400.6.n.d 4
400.6.n.e 16
400.6.n.f 16
400.6.n.g 20
400.6.n.h 24
400.6.o $$\chi_{400}(7, \cdot)$$ None 0 2
400.6.q $$\chi_{400}(149, \cdot)$$ n/a 356 2
400.6.s $$\chi_{400}(107, \cdot)$$ n/a 356 2
400.6.u $$\chi_{400}(81, \cdot)$$ n/a 296 4
400.6.w $$\chi_{400}(9, \cdot)$$ None 0 4
400.6.y $$\chi_{400}(129, \cdot)$$ n/a 296 4
400.6.bb $$\chi_{400}(41, \cdot)$$ None 0 4
400.6.bd $$\chi_{400}(3, \cdot)$$ n/a 2384 8
400.6.be $$\chi_{400}(21, \cdot)$$ n/a 2384 8
400.6.bh $$\chi_{400}(23, \cdot)$$ None 0 8
400.6.bi $$\chi_{400}(47, \cdot)$$ n/a 600 8
400.6.bl $$\chi_{400}(29, \cdot)$$ n/a 2384 8
400.6.bm $$\chi_{400}(67, \cdot)$$ n/a 2384 8

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

## Decomposition of $$S_{6}^{\mathrm{old}}(\Gamma_1(400))$$ into lower level spaces

$$S_{6}^{\mathrm{old}}(\Gamma_1(400)) \cong$$ $$S_{6}^{\mathrm{new}}(\Gamma_1(4))$$$$^{\oplus 9}$$$$\oplus$$$$S_{6}^{\mathrm{new}}(\Gamma_1(5))$$$$^{\oplus 10}$$$$\oplus$$$$S_{6}^{\mathrm{new}}(\Gamma_1(8))$$$$^{\oplus 6}$$$$\oplus$$$$S_{6}^{\mathrm{new}}(\Gamma_1(10))$$$$^{\oplus 8}$$$$\oplus$$$$S_{6}^{\mathrm{new}}(\Gamma_1(16))$$$$^{\oplus 3}$$$$\oplus$$$$S_{6}^{\mathrm{new}}(\Gamma_1(20))$$$$^{\oplus 6}$$$$\oplus$$$$S_{6}^{\mathrm{new}}(\Gamma_1(25))$$$$^{\oplus 5}$$$$\oplus$$$$S_{6}^{\mathrm{new}}(\Gamma_1(40))$$$$^{\oplus 4}$$$$\oplus$$$$S_{6}^{\mathrm{new}}(\Gamma_1(50))$$$$^{\oplus 4}$$$$\oplus$$$$S_{6}^{\mathrm{new}}(\Gamma_1(80))$$$$^{\oplus 2}$$$$\oplus$$$$S_{6}^{\mathrm{new}}(\Gamma_1(100))$$$$^{\oplus 3}$$$$\oplus$$$$S_{6}^{\mathrm{new}}(\Gamma_1(200))$$$$^{\oplus 2}$$