## Defining parameters

 Level: $$N$$ = $$18 = 2 \cdot 3^{2}$$ Weight: $$k$$ = $$4$$ Nonzero newspaces: $$2$$ Newforms: $$3$$ Sturm bound: $$72$$ Trace bound: $$1$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{4}(\Gamma_1(18))$$.

Total New Old
Modular forms 35 7 28
Cusp forms 19 7 12
Eisenstein series 16 0 16

## Trace form

 $$7q + 4q^{2} + 3q^{3} - 8q^{4} + 12q^{5} - 18q^{6} - 4q^{7} - 8q^{8} - 105q^{9} + O(q^{10})$$ $$7q + 4q^{2} + 3q^{3} - 8q^{4} + 12q^{5} - 18q^{6} - 4q^{7} - 8q^{8} - 105q^{9} - 12q^{10} + 27q^{11} + 24q^{12} + 14q^{13} + 68q^{14} + 90q^{15} - 32q^{16} + 48q^{17} + 156q^{18} + 230q^{19} + 48q^{20} - 36q^{21} - 42q^{22} - 432q^{23} - 24q^{24} - 308q^{25} - 316q^{26} - 160q^{28} - 378q^{29} - 288q^{30} - 94q^{31} + 64q^{32} + 765q^{33} + 342q^{34} + 1428q^{35} + 516q^{36} + 446q^{37} + 362q^{38} - 582q^{39} - 48q^{40} - 249q^{41} - 816q^{42} + 77q^{43} - 360q^{44} - 702q^{45} + 168q^{46} - 564q^{47} - 144q^{48} - 672q^{49} + 436q^{50} - 153q^{51} + 56q^{52} + 330q^{53} + 954q^{54} - 1332q^{55} + 272q^{56} + 987q^{57} + 192q^{58} + 333q^{59} - 936q^{60} + 320q^{61} - 1840q^{62} - 1794q^{63} + 448q^{64} + 186q^{65} + 288q^{66} + 2471q^{67} + 660q^{68} + 1494q^{69} + 408q^{70} - 1104q^{71} - 24q^{72} - 40q^{73} + 1364q^{74} + 3231q^{75} - 340q^{76} + 900q^{77} + 132q^{78} - 2002q^{79} - 672q^{80} + 315q^{81} - 3000q^{82} + 354q^{83} + 744q^{84} - 648q^{85} - 190q^{86} - 3204q^{87} - 168q^{88} - 3894q^{89} - 432q^{90} + 1900q^{91} - 1728q^{92} - 2634q^{93} + 804q^{94} - 2136q^{95} + 384q^{96} + 2183q^{97} + 2430q^{98} + 1152q^{99} + O(q^{100})$$

## Decomposition of $$S_{4}^{\mathrm{new}}(\Gamma_1(18))$$

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
18.4.a $$\chi_{18}(1, \cdot)$$ 18.4.a.a 1 1
18.4.c $$\chi_{18}(7, \cdot)$$ 18.4.c.a 2 2
18.4.c.b 4

## Decomposition of $$S_{4}^{\mathrm{old}}(\Gamma_1(18))$$ into lower level spaces

$$S_{4}^{\mathrm{old}}(\Gamma_1(18)) \cong$$ $$S_{4}^{\mathrm{new}}(\Gamma_1(6))$$$$^{\oplus 2}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(9))$$$$^{\oplus 2}$$