## Defining parameters

 Level: $$N$$ = $$189 = 3^{3} \cdot 7$$ Weight: $$k$$ = $$2$$ Nonzero newspaces: $$16$$ Newform subspaces: $$37$$ Sturm bound: $$5184$$ Trace bound: $$9$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(\Gamma_1(189))$$.

Total New Old
Modular forms 1476 1062 414
Cusp forms 1117 902 215
Eisenstein series 359 160 199

## Trace form

 $$902q - 12q^{2} - 24q^{3} - 26q^{4} - 18q^{5} - 36q^{6} - 35q^{7} - 60q^{8} - 36q^{9} + O(q^{10})$$ $$902q - 12q^{2} - 24q^{3} - 26q^{4} - 18q^{5} - 36q^{6} - 35q^{7} - 60q^{8} - 36q^{9} - 36q^{10} - 30q^{11} - 60q^{12} - 42q^{13} - 51q^{14} - 90q^{15} - 70q^{16} - 60q^{17} - 54q^{18} - 36q^{19} - 60q^{20} - 24q^{21} - 114q^{22} - 18q^{23} - 52q^{25} - 18q^{27} - 106q^{28} - 48q^{29} - 18q^{30} - 54q^{31} - 78q^{32} - 36q^{33} - 96q^{34} - 72q^{35} - 108q^{36} - 82q^{37} - 168q^{38} - 102q^{39} - 120q^{40} - 108q^{41} - 54q^{42} - 130q^{43} - 108q^{44} - 18q^{45} - 96q^{46} - 30q^{47} + 102q^{48} - 43q^{49} + 78q^{50} + 54q^{51} + 12q^{52} + 138q^{53} + 180q^{54} + 12q^{55} + 267q^{56} + 6q^{57} + 48q^{58} + 150q^{59} + 180q^{60} - 6q^{61} + 270q^{62} + 81q^{63} + 100q^{64} + 150q^{65} + 126q^{66} + 28q^{67} + 276q^{68} + 36q^{69} + 57q^{70} - 30q^{71} + 108q^{72} + 12q^{73} + 6q^{74} - 24q^{75} + 42q^{76} - 21q^{77} - 36q^{78} - 26q^{79} - 6q^{80} - 108q^{81} - 120q^{82} - 180q^{83} - 42q^{84} - 102q^{85} - 258q^{86} - 54q^{87} - 42q^{88} - 156q^{89} - 72q^{90} - 123q^{91} - 126q^{92} + 42q^{93} - 132q^{94} - 78q^{95} - 36q^{96} - 90q^{97} - 96q^{98} - 18q^{99} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(\Gamma_1(189))$$

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
189.2.a $$\chi_{189}(1, \cdot)$$ 189.2.a.a 1 1
189.2.a.b 1
189.2.a.c 1
189.2.a.d 1
189.2.a.e 2
189.2.a.f 2
189.2.c $$\chi_{189}(188, \cdot)$$ 189.2.c.a 2 1
189.2.c.b 4
189.2.c.c 4
189.2.e $$\chi_{189}(109, \cdot)$$ 189.2.e.a 2 2
189.2.e.b 2
189.2.e.c 2
189.2.e.d 4
189.2.e.e 6
189.2.e.f 6
189.2.f $$\chi_{189}(64, \cdot)$$ 189.2.f.a 6 2
189.2.f.b 6
189.2.g $$\chi_{189}(100, \cdot)$$ 189.2.g.a 2 2
189.2.g.b 10
189.2.h $$\chi_{189}(37, \cdot)$$ 189.2.h.a 2 2
189.2.h.b 10
189.2.i $$\chi_{189}(143, \cdot)$$ 189.2.i.a 2 2
189.2.i.b 10
189.2.o $$\chi_{189}(62, \cdot)$$ 189.2.o.a 12 2
189.2.p $$\chi_{189}(26, \cdot)$$ 189.2.p.a 2 2
189.2.p.b 4
189.2.p.c 4
189.2.p.d 12
189.2.s $$\chi_{189}(17, \cdot)$$ 189.2.s.a 2 2
189.2.s.b 10
189.2.u $$\chi_{189}(4, \cdot)$$ 189.2.u.a 132 6
189.2.v $$\chi_{189}(22, \cdot)$$ 189.2.v.a 54 6
189.2.v.b 54
189.2.w $$\chi_{189}(25, \cdot)$$ 189.2.w.a 132 6
189.2.ba $$\chi_{189}(5, \cdot)$$ 189.2.ba.a 132 6
189.2.bd $$\chi_{189}(47, \cdot)$$ 189.2.bd.a 132 6
189.2.be $$\chi_{189}(20, \cdot)$$ 189.2.be.a 132 6

## Decomposition of $$S_{2}^{\mathrm{old}}(\Gamma_1(189))$$ into lower level spaces

$$S_{2}^{\mathrm{old}}(\Gamma_1(189)) \cong$$ $$S_{2}^{\mathrm{new}}(\Gamma_1(21))$$$$^{\oplus 3}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(27))$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(63))$$$$^{\oplus 2}$$