## Defining parameters

 Level: $$N$$ = $$224 = 2^{5} \cdot 7$$ Weight: $$k$$ = $$2$$ Nonzero newspaces: $$12$$ Newform subspaces: $$23$$ Sturm bound: $$6144$$ Trace bound: $$13$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(\Gamma_1(224))$$.

Total New Old
Modular forms 1728 878 850
Cusp forms 1345 778 567
Eisenstein series 383 100 283

## Trace form

 $$778q - 16q^{2} - 10q^{3} - 16q^{4} - 12q^{5} - 16q^{6} - 14q^{7} - 40q^{8} - 22q^{9} + O(q^{10})$$ $$778q - 16q^{2} - 10q^{3} - 16q^{4} - 12q^{5} - 16q^{6} - 14q^{7} - 40q^{8} - 22q^{9} - 32q^{10} - 10q^{11} - 48q^{12} - 28q^{13} - 36q^{14} - 36q^{15} - 56q^{16} - 16q^{17} - 56q^{18} - 10q^{19} - 48q^{20} - 20q^{21} - 64q^{22} - 26q^{23} + 8q^{24} - 26q^{25} + 24q^{26} - 64q^{27} - 20q^{29} + 48q^{30} - 50q^{31} + 24q^{32} - 44q^{33} + 8q^{34} - 38q^{35} + 16q^{36} - 12q^{37} - 32q^{38} - 76q^{39} - 40q^{40} - 48q^{41} - 40q^{42} - 56q^{43} - 96q^{44} - 100q^{45} - 80q^{46} - 54q^{47} - 120q^{48} - 46q^{49} - 120q^{50} - 62q^{51} - 32q^{52} - 124q^{53} - 40q^{54} + 12q^{55} - 32q^{56} - 160q^{57} - 8q^{58} + 6q^{59} - 8q^{60} - 108q^{61} + 24q^{62} - 30q^{63} + 32q^{64} - 68q^{65} + 32q^{66} + 34q^{67} - 56q^{68} - 128q^{69} - 8q^{70} + 24q^{71} - 64q^{72} - 16q^{73} - 48q^{74} - 16q^{75} - 16q^{76} - 36q^{77} - 120q^{78} - 18q^{79} - 8q^{80} + 6q^{81} - 16q^{82} - 96q^{83} - 40q^{84} - 72q^{85} + 24q^{86} - 140q^{87} - 24q^{88} - 48q^{89} + 104q^{90} - 20q^{91} + 56q^{92} + 56q^{93} + 104q^{94} - 94q^{95} + 216q^{96} + 16q^{97} + 168q^{98} - 72q^{99} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(\Gamma_1(224))$$

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
224.2.a $$\chi_{224}(1, \cdot)$$ 224.2.a.a 1 1
224.2.a.b 1
224.2.a.c 2
224.2.a.d 2
224.2.b $$\chi_{224}(113, \cdot)$$ 224.2.b.a 2 1
224.2.b.b 4
224.2.e $$\chi_{224}(111, \cdot)$$ 224.2.e.a 2 1
224.2.e.b 4
224.2.f $$\chi_{224}(223, \cdot)$$ 224.2.f.a 8 1
224.2.i $$\chi_{224}(65, \cdot)$$ 224.2.i.a 4 2
224.2.i.b 4
224.2.i.c 4
224.2.i.d 4
224.2.j $$\chi_{224}(55, \cdot)$$ None 0 2
224.2.m $$\chi_{224}(57, \cdot)$$ None 0 2
224.2.p $$\chi_{224}(31, \cdot)$$ 224.2.p.a 16 2
224.2.q $$\chi_{224}(47, \cdot)$$ 224.2.q.a 12 2
224.2.t $$\chi_{224}(81, \cdot)$$ 224.2.t.a 12 2
224.2.u $$\chi_{224}(29, \cdot)$$ 224.2.u.a 4 4
224.2.u.b 40
224.2.u.c 52
224.2.x $$\chi_{224}(27, \cdot)$$ 224.2.x.a 8 4
224.2.x.b 112
224.2.z $$\chi_{224}(87, \cdot)$$ None 0 4
224.2.ba $$\chi_{224}(9, \cdot)$$ None 0 4
224.2.bd $$\chi_{224}(37, \cdot)$$ 224.2.bd.a 240 8
224.2.be $$\chi_{224}(3, \cdot)$$ 224.2.be.a 240 8

## Decomposition of $$S_{2}^{\mathrm{old}}(\Gamma_1(224))$$ into lower level spaces

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