## Defining parameters

 Level: $$N$$ = $$18 = 2 \cdot 3^{2}$$ Weight: $$k$$ = $$4$$ Nonzero newspaces: $$2$$ Newform subspaces: $$3$$ 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

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

## 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

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