## Defining parameters

 Level: $$N$$ = $$168 = 2^{3} \cdot 3 \cdot 7$$ Weight: $$k$$ = $$2$$ Nonzero newspaces: $$12$$ Newform subspaces: $$25$$ Sturm bound: $$3072$$ Trace bound: $$3$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(\Gamma_1(168))$$.

Total New Old
Modular forms 912 330 582
Cusp forms 625 290 335
Eisenstein series 287 40 247

## Trace form

 $$290 q + 4 q^{2} - 4 q^{4} + 4 q^{5} - 10 q^{6} - 8 q^{7} - 8 q^{8} + O(q^{10})$$ $$290 q + 4 q^{2} - 4 q^{4} + 4 q^{5} - 10 q^{6} - 8 q^{7} - 8 q^{8} - 20 q^{10} + 4 q^{11} - 22 q^{12} + 16 q^{13} - 4 q^{14} - 12 q^{15} - 12 q^{16} + 16 q^{17} + 6 q^{18} - 12 q^{19} - 20 q^{20} - 68 q^{22} - 36 q^{23} - 18 q^{24} - 42 q^{25} - 44 q^{26} - 36 q^{27} - 108 q^{28} - 12 q^{29} - 46 q^{30} - 84 q^{31} - 76 q^{32} - 50 q^{33} - 124 q^{34} - 36 q^{35} - 32 q^{36} - 24 q^{37} - 52 q^{38} - 36 q^{39} - 88 q^{40} + 4 q^{41} + 26 q^{42} - 16 q^{43} + 48 q^{44} - 38 q^{45} + 64 q^{46} - 24 q^{47} + 70 q^{48} - 54 q^{49} + 88 q^{50} - 48 q^{51} + 64 q^{52} - 32 q^{53} + 78 q^{54} - 68 q^{55} + 92 q^{56} - 64 q^{57} + 72 q^{58} - 80 q^{59} + 96 q^{60} + 4 q^{61} + 116 q^{62} - 64 q^{63} + 92 q^{64} - 40 q^{65} + 74 q^{66} - 84 q^{67} + 60 q^{68} + 8 q^{69} + 44 q^{70} - 64 q^{71} + 84 q^{72} + 20 q^{73} - 32 q^{74} + 20 q^{75} + 36 q^{76} + 36 q^{77} + 56 q^{78} + 12 q^{79} + 40 q^{81} + 60 q^{82} + 56 q^{83} + 110 q^{84} + 80 q^{85} + 16 q^{86} + 96 q^{87} - 44 q^{88} + 76 q^{89} + 116 q^{90} + 96 q^{91} + 12 q^{92} + 82 q^{93} + 12 q^{94} + 136 q^{95} + 90 q^{96} - 64 q^{97} + 64 q^{98} + 120 q^{99} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(\Gamma_1(168))$$

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
168.2.a $$\chi_{168}(1, \cdot)$$ 168.2.a.a 1 1
168.2.a.b 1
168.2.b $$\chi_{168}(55, \cdot)$$ None 0 1
168.2.c $$\chi_{168}(85, \cdot)$$ 168.2.c.a 4 1
168.2.c.b 8
168.2.h $$\chi_{168}(71, \cdot)$$ None 0 1
168.2.i $$\chi_{168}(125, \cdot)$$ 168.2.i.a 4 1
168.2.i.b 4
168.2.i.c 4
168.2.i.d 8
168.2.i.e 8
168.2.j $$\chi_{168}(155, \cdot)$$ 168.2.j.a 4 1
168.2.j.b 4
168.2.j.c 4
168.2.j.d 12
168.2.k $$\chi_{168}(41, \cdot)$$ 168.2.k.a 8 1
168.2.p $$\chi_{168}(139, \cdot)$$ 168.2.p.a 16 1
168.2.q $$\chi_{168}(25, \cdot)$$ 168.2.q.a 2 2
168.2.q.b 2
168.2.q.c 4
168.2.t $$\chi_{168}(19, \cdot)$$ 168.2.t.a 32 2
168.2.u $$\chi_{168}(17, \cdot)$$ 168.2.u.a 16 2
168.2.v $$\chi_{168}(11, \cdot)$$ 168.2.v.a 56 2
168.2.ba $$\chi_{168}(5, \cdot)$$ 168.2.ba.a 4 2
168.2.ba.b 4
168.2.ba.c 48
168.2.bb $$\chi_{168}(23, \cdot)$$ None 0 2
168.2.bc $$\chi_{168}(37, \cdot)$$ 168.2.bc.a 32 2
168.2.bd $$\chi_{168}(31, \cdot)$$ None 0 2

## Decomposition of $$S_{2}^{\mathrm{old}}(\Gamma_1(168))$$ into lower level spaces

$$S_{2}^{\mathrm{old}}(\Gamma_1(168)) \cong$$ $$S_{2}^{\mathrm{new}}(\Gamma_1(14))$$$$^{\oplus 6}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(21))$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(24))$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(28))$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(42))$$$$^{\oplus 3}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(56))$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(84))$$$$^{\oplus 2}$$