## Defining parameters

 Level: $$N$$ = $$222 = 2 \cdot 3 \cdot 37$$ Weight: $$k$$ = $$2$$ Nonzero newspaces: $$9$$ Newform subspaces: $$22$$ Sturm bound: $$5472$$ Trace bound: $$1$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(\Gamma_1(222))$$.

Total New Old
Modular forms 1512 343 1169
Cusp forms 1225 343 882
Eisenstein series 287 0 287

## Trace form

 $$343q + q^{2} + q^{3} + q^{4} + 6q^{5} + q^{6} + 8q^{7} + q^{8} + q^{9} + O(q^{10})$$ $$343q + q^{2} + q^{3} + q^{4} + 6q^{5} + q^{6} + 8q^{7} + q^{8} + q^{9} + 6q^{10} + 12q^{11} + q^{12} + 14q^{13} + 8q^{14} + 6q^{15} + q^{16} + 18q^{17} + q^{18} + 20q^{19} + 6q^{20} + 8q^{21} + 12q^{22} + 24q^{23} + q^{24} + 31q^{25} - 4q^{26} - 11q^{27} - 16q^{28} - 42q^{29} - 66q^{30} - 184q^{31} + q^{32} - 60q^{33} - 126q^{34} - 168q^{35} - 17q^{36} - 131q^{37} - 52q^{38} - 70q^{39} - 84q^{40} - 174q^{41} - 64q^{42} - 100q^{43} + 12q^{44} + 6q^{45} - 120q^{46} - 24q^{47} - 11q^{48} - 39q^{49} + 13q^{50} + 18q^{51} + 14q^{52} + 54q^{53} + q^{54} + 72q^{55} + 8q^{56} + 20q^{57} + 30q^{58} - 12q^{59} + 6q^{60} - 28q^{61} + 32q^{62} - 100q^{63} + q^{64} - 78q^{65} + 12q^{66} - 4q^{67} + 18q^{68} - 120q^{69} + 48q^{70} - 72q^{71} + q^{72} - 70q^{73} + 37q^{74} - 149q^{75} + 20q^{76} - 48q^{77} + 14q^{78} - 64q^{79} + 6q^{80} - 143q^{81} + 42q^{82} + 12q^{83} + 8q^{84} - 54q^{85} + 44q^{86} - 78q^{87} + 12q^{88} + 6q^{90} + 40q^{91} + 24q^{92} + 140q^{93} + 48q^{94} + 120q^{95} + q^{96} + 98q^{97} + 57q^{98} + 192q^{99} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(\Gamma_1(222))$$

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
222.2.a $$\chi_{222}(1, \cdot)$$ 222.2.a.a 1 1
222.2.a.b 1
222.2.a.c 1
222.2.a.d 1
222.2.a.e 1
222.2.c $$\chi_{222}(73, \cdot)$$ 222.2.c.a 2 1
222.2.c.b 4
222.2.e $$\chi_{222}(121, \cdot)$$ 222.2.e.a 2 2
222.2.e.b 2
222.2.e.c 4
222.2.g $$\chi_{222}(179, \cdot)$$ 222.2.g.a 28 2
222.2.j $$\chi_{222}(85, \cdot)$$ 222.2.j.a 4 2
222.2.j.b 8
222.2.k $$\chi_{222}(7, \cdot)$$ 222.2.k.a 6 6
222.2.k.b 6
222.2.k.c 6
222.2.k.d 6
222.2.k.e 12
222.2.m $$\chi_{222}(23, \cdot)$$ 222.2.m.a 56 4
222.2.n $$\chi_{222}(25, \cdot)$$ 222.2.n.a 24 6
222.2.n.b 24
222.2.q $$\chi_{222}(5, \cdot)$$ 222.2.q.a 144 12

## Decomposition of $$S_{2}^{\mathrm{old}}(\Gamma_1(222))$$ into lower level spaces

$$S_{2}^{\mathrm{old}}(\Gamma_1(222)) \cong$$ $$S_{2}^{\mathrm{new}}(\Gamma_1(37))$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(74))$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(111))$$$$^{\oplus 2}$$