# Properties

 Label 168.4 Level 168 Weight 4 Dimension 914 Nonzero newspaces 12 Newform subspaces 29 Sturm bound 6144 Trace bound 3

## Defining parameters

 Level: $$N$$ = $$168 = 2^{3} \cdot 3 \cdot 7$$ Weight: $$k$$ = $$4$$ Nonzero newspaces: $$12$$ Newform subspaces: $$29$$ Sturm bound: $$6144$$ Trace bound: $$3$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{4}(\Gamma_1(168))$$.

Total New Old
Modular forms 2448 954 1494
Cusp forms 2160 914 1246
Eisenstein series 288 40 248

## Trace form

 $$914q - 4q^{2} - 8q^{3} - 52q^{4} - 28q^{5} + 22q^{6} - 16q^{7} + 152q^{8} + 124q^{9} + O(q^{10})$$ $$914q - 4q^{2} - 8q^{3} - 52q^{4} - 28q^{5} + 22q^{6} - 16q^{7} + 152q^{8} + 124q^{9} - 84q^{10} - 28q^{11} + 106q^{12} - 8q^{13} + 100q^{14} - 108q^{15} + 180q^{16} - 280q^{17} - 338q^{18} - 476q^{19} - 1156q^{20} + 240q^{21} - 1100q^{22} - 420q^{23} - 530q^{24} + 710q^{25} + 1388q^{26} + 364q^{27} + 2404q^{28} + 276q^{29} + 1506q^{30} + 1820q^{31} + 916q^{32} - 142q^{33} + 1572q^{34} - 708q^{35} + 1504q^{36} - 816q^{37} - 1436q^{38} - 348q^{39} + 520q^{40} - 1036q^{41} + 534q^{42} - 3312q^{43} + 1392q^{44} - 546q^{45} - 1448q^{46} - 888q^{47} - 4538q^{48} - 1206q^{49} - 6040q^{50} - 784q^{51} - 8144q^{52} + 2024q^{53} - 182q^{54} + 5708q^{55} - 2756q^{56} + 2480q^{57} + 728q^{58} + 3920q^{59} + 4800q^{60} + 1444q^{61} + 2812q^{62} + 240q^{63} + 4508q^{64} + 2536q^{65} + 1886q^{66} - 4436q^{67} + 108q^{68} - 1704q^{69} + 2380q^{70} - 4352q^{71} - 1628q^{72} - 3100q^{73} - 3136q^{74} + 1204q^{75} - 3740q^{76} - 1140q^{77} - 288q^{78} + 1148q^{79} + 4224q^{80} - 740q^{81} + 10732q^{82} + 8440q^{83} - 2754q^{84} + 4640q^{85} + 1520q^{86} - 3528q^{87} + 5140q^{88} + 332q^{89} - 11808q^{90} - 2160q^{91} - 228q^{92} - 4386q^{93} + 252q^{94} - 5560q^{95} + 1018q^{96} - 9784q^{97} + 3296q^{98} - 5560q^{99} + O(q^{100})$$

## Decomposition of $$S_{4}^{\mathrm{new}}(\Gamma_1(168))$$

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
168.4.a $$\chi_{168}(1, \cdot)$$ 168.4.a.a 1 1
168.4.a.b 1
168.4.a.c 1
168.4.a.d 1
168.4.a.e 1
168.4.a.f 1
168.4.a.g 2
168.4.a.h 2
168.4.b $$\chi_{168}(55, \cdot)$$ None 0 1
168.4.c $$\chi_{168}(85, \cdot)$$ 168.4.c.a 16 1
168.4.c.b 20
168.4.h $$\chi_{168}(71, \cdot)$$ None 0 1
168.4.i $$\chi_{168}(125, \cdot)$$ 168.4.i.a 4 1
168.4.i.b 8
168.4.i.c 80
168.4.j $$\chi_{168}(155, \cdot)$$ 168.4.j.a 72 1
168.4.k $$\chi_{168}(41, \cdot)$$ 168.4.k.a 24 1
168.4.p $$\chi_{168}(139, \cdot)$$ 168.4.p.a 48 1
168.4.q $$\chi_{168}(25, \cdot)$$ 168.4.q.a 2 2
168.4.q.b 2
168.4.q.c 2
168.4.q.d 4
168.4.q.e 6
168.4.q.f 8
168.4.t $$\chi_{168}(19, \cdot)$$ 168.4.t.a 96 2
168.4.u $$\chi_{168}(17, \cdot)$$ 168.4.u.a 48 2
168.4.v $$\chi_{168}(11, \cdot)$$ 168.4.v.a 184 2
168.4.ba $$\chi_{168}(5, \cdot)$$ 168.4.ba.a 4 2
168.4.ba.b 4
168.4.ba.c 176
168.4.bb $$\chi_{168}(23, \cdot)$$ None 0 2
168.4.bc $$\chi_{168}(37, \cdot)$$ 168.4.bc.a 96 2
168.4.bd $$\chi_{168}(31, \cdot)$$ None 0 2

## Decomposition of $$S_{4}^{\mathrm{old}}(\Gamma_1(168))$$ into lower level spaces

$$S_{4}^{\mathrm{old}}(\Gamma_1(168)) \cong$$ $$S_{4}^{\mathrm{new}}(\Gamma_1(6))$$$$^{\oplus 6}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(7))$$$$^{\oplus 8}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(8))$$$$^{\oplus 4}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(12))$$$$^{\oplus 4}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(14))$$$$^{\oplus 6}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(21))$$$$^{\oplus 4}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(24))$$$$^{\oplus 2}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(28))$$$$^{\oplus 4}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(42))$$$$^{\oplus 3}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(56))$$$$^{\oplus 2}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(84))$$$$^{\oplus 2}$$