# Properties

 Label 224.4 Level 224 Weight 4 Dimension 2434 Nonzero newspaces 12 Newform subspaces 28 Sturm bound 12288 Trace bound 13

## Defining parameters

 Level: $$N$$ = $$224 = 2^{5} \cdot 7$$ Weight: $$k$$ = $$4$$ Nonzero newspaces: $$12$$ Newform subspaces: $$28$$ Sturm bound: $$12288$$ Trace bound: $$13$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{4}(\Gamma_1(224))$$.

Total New Old
Modular forms 4800 2534 2266
Cusp forms 4416 2434 1982
Eisenstein series 384 100 284

## Trace form

 $$2434q - 16q^{2} - 10q^{3} - 16q^{4} - 20q^{5} - 16q^{6} - 30q^{7} - 40q^{8} - 118q^{9} + O(q^{10})$$ $$2434q - 16q^{2} - 10q^{3} - 16q^{4} - 20q^{5} - 16q^{6} - 30q^{7} - 40q^{8} - 118q^{9} - 256q^{10} - 10q^{11} + 80q^{12} + 220q^{13} + 188q^{14} + 188q^{15} + 584q^{16} + 352q^{17} + 344q^{18} - 10q^{19} - 176q^{20} - 276q^{21} - 432q^{22} - 1274q^{23} + 72q^{24} - 698q^{25} - 56q^{26} + 512q^{27} - 400q^{28} + 356q^{29} - 2400q^{30} + 2382q^{31} - 1256q^{32} + 900q^{33} - 1080q^{34} + 442q^{35} - 2960q^{36} - 1684q^{37} - 1984q^{38} - 2492q^{39} + 1048q^{40} - 928q^{41} + 2240q^{42} - 2856q^{43} + 4064q^{44} - 204q^{45} + 2864q^{46} + 714q^{47} + 4872q^{48} + 3402q^{49} + 7080q^{50} + 4850q^{51} + 5024q^{52} + 1436q^{53} + 2152q^{54} + 2540q^{55} - 1216q^{56} + 2960q^{57} - 6424q^{58} - 698q^{59} - 11528q^{60} + 3052q^{61} - 6792q^{62} - 2574q^{63} - 9952q^{64} - 5172q^{65} - 12016q^{66} - 6862q^{67} - 5560q^{68} - 5536q^{69} - 1424q^{70} - 2984q^{71} + 3392q^{72} + 1472q^{73} + 6928q^{74} + 2656q^{75} + 10224q^{76} - 644q^{77} + 23784q^{78} - 1874q^{79} + 20600q^{80} + 3310q^{81} + 12784q^{82} - 4896q^{83} + 6536q^{84} + 1896q^{85} + 1832q^{86} + 5764q^{87} - 2520q^{88} + 3552q^{89} - 9640q^{90} + 5644q^{91} - 13896q^{92} + 1048q^{93} + 2376q^{94} + 16386q^{95} - 4648q^{96} + 4736q^{97} + 680q^{98} + 13512q^{99} + O(q^{100})$$

## Decomposition of $$S_{4}^{\mathrm{new}}(\Gamma_1(224))$$

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
224.4.a $$\chi_{224}(1, \cdot)$$ 224.4.a.a 1 1
224.4.a.b 1
224.4.a.c 2
224.4.a.d 2
224.4.a.e 3
224.4.a.f 3
224.4.a.g 3
224.4.a.h 3
224.4.b $$\chi_{224}(113, \cdot)$$ 224.4.b.a 8 1
224.4.b.b 10
224.4.e $$\chi_{224}(111, \cdot)$$ 224.4.e.a 2 1
224.4.e.b 4
224.4.e.c 16
224.4.f $$\chi_{224}(223, \cdot)$$ 224.4.f.a 24 1
224.4.i $$\chi_{224}(65, \cdot)$$ 224.4.i.a 4 2
224.4.i.b 8
224.4.i.c 12
224.4.i.d 12
224.4.i.e 12
224.4.j $$\chi_{224}(55, \cdot)$$ None 0 2
224.4.m $$\chi_{224}(57, \cdot)$$ None 0 2
224.4.p $$\chi_{224}(31, \cdot)$$ 224.4.p.a 48 2
224.4.q $$\chi_{224}(47, \cdot)$$ 224.4.q.a 44 2
224.4.t $$\chi_{224}(81, \cdot)$$ 224.4.t.a 44 2
224.4.u $$\chi_{224}(29, \cdot)$$ 224.4.u.a 140 4
224.4.u.b 148
224.4.x $$\chi_{224}(27, \cdot)$$ 224.4.x.a 8 4
224.4.x.b 368
224.4.z $$\chi_{224}(87, \cdot)$$ None 0 4
224.4.ba $$\chi_{224}(9, \cdot)$$ None 0 4
224.4.bd $$\chi_{224}(37, \cdot)$$ 224.4.bd.a 752 8
224.4.be $$\chi_{224}(3, \cdot)$$ 224.4.be.a 752 8

## Decomposition of $$S_{4}^{\mathrm{old}}(\Gamma_1(224))$$ into lower level spaces

$$S_{4}^{\mathrm{old}}(\Gamma_1(224)) \cong$$ $$S_{4}^{\mathrm{new}}(\Gamma_1(7))$$$$^{\oplus 6}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(8))$$$$^{\oplus 6}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(14))$$$$^{\oplus 5}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(16))$$$$^{\oplus 4}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(28))$$$$^{\oplus 4}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(32))$$$$^{\oplus 2}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(56))$$$$^{\oplus 3}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(112))$$$$^{\oplus 2}$$