# Properties

 Label 200.4 Level 200 Weight 4 Dimension 1826 Nonzero newspaces 10 Newform subspaces 44 Sturm bound 9600 Trace bound 3

## Defining parameters

 Level: $$N$$ = $$200 = 2^{3} \cdot 5^{2}$$ Weight: $$k$$ = $$4$$ Nonzero newspaces: $$10$$ Newform subspaces: $$44$$ Sturm bound: $$9600$$ Trace bound: $$3$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{4}(\Gamma_1(200))$$.

Total New Old
Modular forms 3768 1908 1860
Cusp forms 3432 1826 1606
Eisenstein series 336 82 254

## Trace form

 $$1826 q - 14 q^{2} - 16 q^{3} - 24 q^{4} + 11 q^{5} + 8 q^{6} - 28 q^{7} + 28 q^{8} - 127 q^{9} + O(q^{10})$$ $$1826 q - 14 q^{2} - 16 q^{3} - 24 q^{4} + 11 q^{5} + 8 q^{6} - 28 q^{7} + 28 q^{8} - 127 q^{9} - 16 q^{10} - 120 q^{11} - 284 q^{12} - 86 q^{13} - 260 q^{14} + 56 q^{15} + 220 q^{16} + 170 q^{17} + 942 q^{18} + 480 q^{19} + 364 q^{20} + 656 q^{21} + 424 q^{22} + 4 q^{23} - 180 q^{24} - 137 q^{25} - 496 q^{26} - 1516 q^{27} - 676 q^{28} - 1350 q^{29} + 300 q^{30} - 1220 q^{31} + 316 q^{32} - 160 q^{33} - 560 q^{34} + 452 q^{35} - 1736 q^{36} + 939 q^{37} - 2344 q^{38} + 1356 q^{39} - 1576 q^{40} + 1334 q^{41} - 3156 q^{42} + 1936 q^{43} - 1420 q^{44} + 1991 q^{45} - 1492 q^{46} - 924 q^{47} - 1316 q^{48} - 1474 q^{49} + 164 q^{50} - 3276 q^{51} + 3412 q^{52} + 307 q^{53} + 5460 q^{54} + 1592 q^{55} + 4924 q^{56} + 3360 q^{57} + 4620 q^{58} + 3232 q^{59} + 3780 q^{60} + 622 q^{61} + 5900 q^{62} + 1596 q^{63} + 8172 q^{64} - 2129 q^{65} + 8308 q^{66} + 2000 q^{67} + 1772 q^{68} - 3472 q^{69} - 1660 q^{70} - 4460 q^{71} - 10248 q^{72} - 222 q^{73} - 11492 q^{74} - 4664 q^{75} - 7344 q^{76} + 2528 q^{77} - 12820 q^{78} - 6564 q^{79} - 3676 q^{80} + 4425 q^{81} + 4288 q^{82} + 1020 q^{83} + 4492 q^{84} + 7443 q^{85} - 2720 q^{86} + 764 q^{87} + 4460 q^{88} - 3897 q^{89} - 316 q^{90} + 1636 q^{91} + 2708 q^{92} - 5720 q^{93} - 6596 q^{94} - 4608 q^{95} + 9796 q^{96} - 6850 q^{97} - 6694 q^{98} - 10128 q^{99} + O(q^{100})$$

## Decomposition of $$S_{4}^{\mathrm{new}}(\Gamma_1(200))$$

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
200.4.a $$\chi_{200}(1, \cdot)$$ 200.4.a.a 1 1
200.4.a.b 1
200.4.a.c 1
200.4.a.d 1
200.4.a.e 1
200.4.a.f 1
200.4.a.g 1
200.4.a.h 1
200.4.a.i 1
200.4.a.j 1
200.4.a.k 2
200.4.a.l 2
200.4.c $$\chi_{200}(49, \cdot)$$ 200.4.c.a 2 1
200.4.c.b 2
200.4.c.c 2
200.4.c.d 2
200.4.c.e 2
200.4.c.f 2
200.4.c.g 2
200.4.d $$\chi_{200}(101, \cdot)$$ 200.4.d.a 2 1
200.4.d.b 12
200.4.d.c 12
200.4.d.d 12
200.4.d.e 16
200.4.f $$\chi_{200}(149, \cdot)$$ 200.4.f.a 4 1
200.4.f.b 12
200.4.f.c 12
200.4.f.d 24
200.4.j $$\chi_{200}(7, \cdot)$$ None 0 2
200.4.k $$\chi_{200}(43, \cdot)$$ 200.4.k.a 2 2
200.4.k.b 2
200.4.k.c 2
200.4.k.d 2
200.4.k.e 4
200.4.k.f 4
200.4.k.g 8
200.4.k.h 16
200.4.k.i 32
200.4.k.j 32
200.4.m $$\chi_{200}(41, \cdot)$$ 200.4.m.a 44 4
200.4.m.b 48
200.4.o $$\chi_{200}(29, \cdot)$$ 200.4.o.a 352 4
200.4.q $$\chi_{200}(9, \cdot)$$ 200.4.q.a 88 4
200.4.t $$\chi_{200}(21, \cdot)$$ 200.4.t.a 352 4
200.4.v $$\chi_{200}(3, \cdot)$$ 200.4.v.a 704 8
200.4.w $$\chi_{200}(23, \cdot)$$ None 0 8

## Decomposition of $$S_{4}^{\mathrm{old}}(\Gamma_1(200))$$ into lower level spaces

$$S_{4}^{\mathrm{old}}(\Gamma_1(200)) \cong$$ $$S_{4}^{\mathrm{new}}(\Gamma_1(1))$$$$^{\oplus 12}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(2))$$$$^{\oplus 9}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(4))$$$$^{\oplus 6}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(5))$$$$^{\oplus 8}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(8))$$$$^{\oplus 3}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(10))$$$$^{\oplus 6}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(20))$$$$^{\oplus 4}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(25))$$$$^{\oplus 4}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(40))$$$$^{\oplus 2}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(50))$$$$^{\oplus 3}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(100))$$$$^{\oplus 2}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(200))$$$$^{\oplus 1}$$