## Defining parameters

 Level: $$N$$ = $$228 = 2^{2} \cdot 3 \cdot 19$$ Weight: $$k$$ = $$2$$ Nonzero newspaces: $$12$$ Newform subspaces: $$24$$ Sturm bound: $$5760$$ Trace bound: $$1$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(\Gamma_1(228))$$.

Total New Old
Modular forms 1620 658 962
Cusp forms 1261 594 667
Eisenstein series 359 64 295

## Trace form

 $$594 q - 18 q^{4} - 9 q^{6} - 18 q^{9} + O(q^{10})$$ $$594 q - 18 q^{4} - 9 q^{6} - 18 q^{9} - 18 q^{10} - 9 q^{12} - 12 q^{13} + 18 q^{15} - 18 q^{16} + 18 q^{17} - 18 q^{18} + 42 q^{19} + 3 q^{21} - 18 q^{22} + 18 q^{23} - 9 q^{24} - 6 q^{27} - 72 q^{28} - 36 q^{29} - 72 q^{30} - 36 q^{31} - 90 q^{32} - 72 q^{33} - 108 q^{34} - 72 q^{35} - 99 q^{36} - 108 q^{37} - 126 q^{38} - 54 q^{39} - 144 q^{40} - 36 q^{41} - 99 q^{42} - 72 q^{43} - 90 q^{44} - 99 q^{45} - 108 q^{46} - 36 q^{47} - 72 q^{48} - 72 q^{49} - 54 q^{50} - 54 q^{51} - 18 q^{52} - 9 q^{54} - 63 q^{57} - 36 q^{58} - 18 q^{60} - 120 q^{61} + 90 q^{62} - 42 q^{63} + 126 q^{64} - 108 q^{65} + 63 q^{66} - 72 q^{67} + 126 q^{68} - 189 q^{69} + 198 q^{70} - 36 q^{71} + 45 q^{72} - 198 q^{73} + 144 q^{74} - 42 q^{75} + 162 q^{76} - 234 q^{77} + 63 q^{78} - 48 q^{79} + 144 q^{80} + 18 q^{81} + 252 q^{82} + 18 q^{83} + 99 q^{84} - 252 q^{85} + 126 q^{86} + 36 q^{87} + 126 q^{88} - 54 q^{89} + 54 q^{90} + 24 q^{91} + 90 q^{92} + 12 q^{93} + 54 q^{95} + 54 q^{96} + 18 q^{97} + 135 q^{99} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(\Gamma_1(228))$$

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
228.2.a $$\chi_{228}(1, \cdot)$$ 228.2.a.a 1 1
228.2.a.b 1
228.2.a.c 2
228.2.c $$\chi_{228}(191, \cdot)$$ 228.2.c.a 36 1
228.2.d $$\chi_{228}(113, \cdot)$$ 228.2.d.a 2 1
228.2.d.b 4
228.2.f $$\chi_{228}(151, \cdot)$$ 228.2.f.a 10 1
228.2.f.b 10
228.2.i $$\chi_{228}(49, \cdot)$$ 228.2.i.a 4 2
228.2.i.b 4
228.2.k $$\chi_{228}(31, \cdot)$$ 228.2.k.a 20 2
228.2.k.b 20
228.2.m $$\chi_{228}(11, \cdot)$$ 228.2.m.a 72 2
228.2.p $$\chi_{228}(65, \cdot)$$ 228.2.p.a 2 2
228.2.p.b 2
228.2.p.c 4
228.2.p.d 4
228.2.q $$\chi_{228}(25, \cdot)$$ 228.2.q.a 6 6
228.2.q.b 12
228.2.t $$\chi_{228}(29, \cdot)$$ 228.2.t.a 6 6
228.2.t.b 36
228.2.v $$\chi_{228}(23, \cdot)$$ 228.2.v.a 216 6
228.2.w $$\chi_{228}(67, \cdot)$$ 228.2.w.a 60 6
228.2.w.b 60

## Decomposition of $$S_{2}^{\mathrm{old}}(\Gamma_1(228))$$ into lower level spaces

$$S_{2}^{\mathrm{old}}(\Gamma_1(228)) \cong$$ $$S_{2}^{\mathrm{new}}(\Gamma_1(19))$$$$^{\oplus 6}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(38))$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(57))$$$$^{\oplus 3}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(76))$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(114))$$$$^{\oplus 2}$$