# Properties

 Label 112.4 Level 112 Weight 4 Dimension 599 Nonzero newspaces 8 Newform subspaces 29 Sturm bound 3072 Trace bound 3

## Defining parameters

 Level: $$N$$ = $$112 = 2^{4} \cdot 7$$ Weight: $$k$$ = $$4$$ Nonzero newspaces: $$8$$ Newform subspaces: $$29$$ Sturm bound: $$3072$$ Trace bound: $$3$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{4}(\Gamma_1(112))$$.

Total New Old
Modular forms 1236 643 593
Cusp forms 1068 599 469
Eisenstein series 168 44 124

## Trace form

 $$599q - 8q^{2} - 13q^{3} - 28q^{4} - 7q^{5} + 52q^{6} + 15q^{7} + 64q^{8} + 19q^{9} + O(q^{10})$$ $$599q - 8q^{2} - 13q^{3} - 28q^{4} - 7q^{5} + 52q^{6} + 15q^{7} + 64q^{8} + 19q^{9} + 124q^{10} - 133q^{11} - 212q^{12} - 58q^{13} - 200q^{14} + 246q^{15} - 572q^{16} - 119q^{17} - 360q^{18} - 139q^{19} + 380q^{20} - 151q^{21} + 1152q^{22} + 5q^{23} + 1684q^{24} + 743q^{25} + 516q^{26} + 578q^{27} - 292q^{28} + 278q^{29} - 2484q^{30} - 705q^{31} - 1948q^{32} - 443q^{33} - 884q^{34} - 1055q^{35} + 1168q^{36} - 427q^{37} + 2452q^{38} - 734q^{39} + 2660q^{40} + 390q^{41} + 3260q^{42} + 2368q^{43} + 1948q^{44} + 2112q^{45} - 176q^{46} + 3241q^{47} - 3428q^{48} - 4065q^{49} - 3576q^{50} + 1541q^{51} - 4184q^{52} - 1127q^{53} - 6164q^{54} - 1138q^{55} - 4144q^{56} - 4742q^{57} - 4144q^{58} - 5789q^{59} - 5508q^{60} - 4135q^{61} - 1256q^{62} - 2871q^{63} + 380q^{64} + 2350q^{65} + 7444q^{66} - 3265q^{67} + 7808q^{68} + 1422q^{69} + 7268q^{70} - 80q^{71} + 3948q^{72} - 959q^{73} + 892q^{74} + 7836q^{75} + 2444q^{76} + 3277q^{77} + 1520q^{78} + 10157q^{79} + 5284q^{80} + 3246q^{81} + 1396q^{82} + 10072q^{83} - 1972q^{84} + 4082q^{85} - 7540q^{86} + 2322q^{87} - 3068q^{88} + 1113q^{89} - 6396q^{90} + 3466q^{91} - 4516q^{92} + 5525q^{93} - 7556q^{94} - 2141q^{95} - 6628q^{96} - 1442q^{97} - 5604q^{98} - 7868q^{99} + O(q^{100})$$

## Decomposition of $$S_{4}^{\mathrm{new}}(\Gamma_1(112))$$

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
112.4.a $$\chi_{112}(1, \cdot)$$ 112.4.a.a 1 1
112.4.a.b 1
112.4.a.c 1
112.4.a.d 1
112.4.a.e 1
112.4.a.f 1
112.4.a.g 1
112.4.a.h 2
112.4.b $$\chi_{112}(57, \cdot)$$ None 0 1
112.4.e $$\chi_{112}(55, \cdot)$$ None 0 1
112.4.f $$\chi_{112}(111, \cdot)$$ 112.4.f.a 4 1
112.4.f.b 8
112.4.i $$\chi_{112}(65, \cdot)$$ 112.4.i.a 2 2
112.4.i.b 2
112.4.i.c 2
112.4.i.d 4
112.4.i.e 6
112.4.i.f 6
112.4.j $$\chi_{112}(27, \cdot)$$ 112.4.j.a 4 2
112.4.j.b 88
112.4.m $$\chi_{112}(29, \cdot)$$ 112.4.m.a 34 2
112.4.m.b 38
112.4.p $$\chi_{112}(31, \cdot)$$ 112.4.p.a 2 2
112.4.p.b 2
112.4.p.c 2
112.4.p.d 2
112.4.p.e 4
112.4.p.f 6
112.4.p.g 6
112.4.q $$\chi_{112}(87, \cdot)$$ None 0 2
112.4.t $$\chi_{112}(9, \cdot)$$ None 0 2
112.4.v $$\chi_{112}(3, \cdot)$$ 112.4.v.a 184 4
112.4.w $$\chi_{112}(37, \cdot)$$ 112.4.w.a 184 4

## Decomposition of $$S_{4}^{\mathrm{old}}(\Gamma_1(112))$$ into lower level spaces

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