## Defining parameters

 Level: $$N$$ = $$112 = 2^{4} \cdot 7$$ Weight: $$k$$ = $$2$$ Nonzero newspaces: $$8$$ Newform subspaces: $$23$$ 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

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

## 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

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