## Defining parameters

 Level: $$N$$ = $$112 = 2^{4} \cdot 7$$ Weight: $$k$$ = $$2$$ Nonzero newspaces: $$8$$ Newforms: $$23$$ Sturm bound: $$1536$$ Trace bound: $$3$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(\Gamma_1(112))$$.

Total New Old
Modular forms 468 229 239
Cusp forms 301 185 116
Eisenstein series 167 44 123

## Trace form

 $$185q - 8q^{2} - 5q^{3} - 12q^{4} - 11q^{5} - 20q^{6} - 9q^{7} - 32q^{8} - 3q^{9} + O(q^{10})$$ $$185q - 8q^{2} - 5q^{3} - 12q^{4} - 11q^{5} - 20q^{6} - 9q^{7} - 32q^{8} - 3q^{9} - 12q^{10} - 13q^{11} - 4q^{12} - 14q^{13} - 8q^{14} - 26q^{15} + 4q^{16} - 19q^{17} - 16q^{18} - 27q^{19} - 20q^{20} - 31q^{21} - 24q^{22} - 27q^{23} - 12q^{24} - 27q^{25} - 20q^{26} - 14q^{27} - 20q^{28} - 54q^{29} - 4q^{30} - q^{31} - 28q^{32} - 43q^{33} - 20q^{34} - 23q^{35} - 16q^{36} - 39q^{37} + 12q^{38} - 6q^{39} + 4q^{40} - 6q^{41} + 60q^{42} - 32q^{43} + 44q^{44} + 20q^{45} + 24q^{46} - 23q^{47} + 92q^{48} + q^{49} + 72q^{50} + 13q^{51} + 104q^{52} + 53q^{53} + 156q^{54} + 30q^{55} + 80q^{56} + 90q^{57} + 96q^{58} + 27q^{59} + 156q^{60} + 69q^{61} + 64q^{62} + 17q^{63} + 60q^{64} - 2q^{65} + 100q^{66} + 23q^{67} + 48q^{68} + 30q^{69} + 28q^{70} - 48q^{71} + 76q^{72} - 27q^{73} - 12q^{74} - 60q^{75} - 36q^{76} - 67q^{77} - 16q^{78} - 51q^{79} - 28q^{80} - 104q^{81} - 12q^{82} - 56q^{83} - 4q^{84} - 86q^{85} - 12q^{86} - 54q^{87} - 28q^{88} - 51q^{89} - 76q^{90} + 50q^{91} + 12q^{92} - 83q^{93} - 52q^{94} + 51q^{95} - 100q^{96} - 46q^{97} - 76q^{98} + 100q^{99} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(\Gamma_1(112))$$

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
112.2.a $$\chi_{112}(1, \cdot)$$ 112.2.a.a 1 1
112.2.a.b 1
112.2.a.c 1
112.2.b $$\chi_{112}(57, \cdot)$$ None 0 1
112.2.e $$\chi_{112}(55, \cdot)$$ None 0 1
112.2.f $$\chi_{112}(111, \cdot)$$ 112.2.f.a 2 1
112.2.f.b 2
112.2.i $$\chi_{112}(65, \cdot)$$ 112.2.i.a 2 2
112.2.i.b 2
112.2.i.c 2
112.2.j $$\chi_{112}(27, \cdot)$$ 112.2.j.a 4 2
112.2.j.b 4
112.2.j.c 4
112.2.j.d 16
112.2.m $$\chi_{112}(29, \cdot)$$ 112.2.m.a 2 2
112.2.m.b 2
112.2.m.c 8
112.2.m.d 12
112.2.p $$\chi_{112}(31, \cdot)$$ 112.2.p.a 2 2
112.2.p.b 2
112.2.p.c 4
112.2.q $$\chi_{112}(87, \cdot)$$ None 0 2
112.2.t $$\chi_{112}(9, \cdot)$$ None 0 2
112.2.v $$\chi_{112}(3, \cdot)$$ 112.2.v.a 56 4
112.2.w $$\chi_{112}(37, \cdot)$$ 112.2.w.a 4 4
112.2.w.b 4
112.2.w.c 48

## Decomposition of $$S_{2}^{\mathrm{old}}(\Gamma_1(112))$$ into lower level spaces

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