## Defining parameters

 Level: $$N$$ = $$186 = 2 \cdot 3 \cdot 31$$ Weight: $$k$$ = $$2$$ Nonzero newspaces: $$8$$ Newform subspaces: $$21$$ Sturm bound: $$3840$$ Trace bound: $$4$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(\Gamma_1(186))$$.

Total New Old
Modular forms 1080 241 839
Cusp forms 841 241 600
Eisenstein series 239 0 239

## Trace form

 $$241q + q^{2} + q^{3} + q^{4} + 6q^{5} + q^{6} + 8q^{7} + q^{8} + q^{9} + O(q^{10})$$ $$241q + q^{2} + q^{3} + q^{4} + 6q^{5} + q^{6} + 8q^{7} + q^{8} + q^{9} + 6q^{10} + 12q^{11} + q^{12} + 14q^{13} + 8q^{14} + 6q^{15} + q^{16} + 18q^{17} + q^{18} + 20q^{19} + 6q^{20} - 2q^{21} - 48q^{22} - 36q^{23} + q^{24} - 109q^{25} - 46q^{26} + q^{27} - 72q^{28} - 90q^{29} - 54q^{30} - 89q^{31} + q^{32} - 48q^{33} - 102q^{34} - 72q^{35} - 9q^{36} - 142q^{37} - 40q^{38} - 56q^{39} + 6q^{40} - 18q^{41} - 22q^{42} + 24q^{43} + 12q^{44} + 6q^{45} + 24q^{46} + 48q^{47} + q^{48} - 3q^{49} + 31q^{50} - 72q^{51} + 14q^{52} - 6q^{53} + q^{54} - 48q^{55} + 8q^{56} - 70q^{57} + 30q^{58} + 6q^{60} - 118q^{61} + 31q^{62} - 112q^{63} + q^{64} - 96q^{65} + 12q^{66} + 8q^{67} + 18q^{68} - 66q^{69} + 48q^{70} - 48q^{71} + q^{72} + 14q^{73} + 38q^{74} - 29q^{75} + 20q^{76} + 36q^{77} + 104q^{78} + 80q^{79} + 6q^{80} + 121q^{81} + 42q^{82} + 84q^{83} + 68q^{84} + 108q^{85} + 44q^{86} + 150q^{87} + 12q^{88} + 90q^{89} + 186q^{90} + 112q^{91} + 24q^{92} + 271q^{93} + 48q^{94} + 120q^{95} + q^{96} + 98q^{97} + 57q^{98} + 162q^{99} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(\Gamma_1(186))$$

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
186.2.a $$\chi_{186}(1, \cdot)$$ 186.2.a.a 1 1
186.2.a.b 1
186.2.a.c 1
186.2.a.d 2
186.2.c $$\chi_{186}(185, \cdot)$$ 186.2.c.a 4 1
186.2.c.b 8
186.2.e $$\chi_{186}(25, \cdot)$$ 186.2.e.a 2 2
186.2.e.b 2
186.2.e.c 4
186.2.e.d 4
186.2.f $$\chi_{186}(97, \cdot)$$ 186.2.f.a 4 4
186.2.f.b 4
186.2.f.c 8
186.2.h $$\chi_{186}(119, \cdot)$$ 186.2.h.a 20 2
186.2.j $$\chi_{186}(23, \cdot)$$ 186.2.j.a 48 4
186.2.m $$\chi_{186}(7, \cdot)$$ 186.2.m.a 8 8
186.2.m.b 8
186.2.m.c 8
186.2.m.d 8
186.2.m.e 16
186.2.p $$\chi_{186}(11, \cdot)$$ 186.2.p.a 80 8

## Decomposition of $$S_{2}^{\mathrm{old}}(\Gamma_1(186))$$ into lower level spaces

$$S_{2}^{\mathrm{old}}(\Gamma_1(186)) \cong$$ $$S_{2}^{\mathrm{new}}(\Gamma_1(31))$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(62))$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(93))$$$$^{\oplus 2}$$