## Defining parameters

 Level: $$N$$ = $$162 = 2 \cdot 3^{4}$$ Weight: $$k$$ = $$2$$ Nonzero newspaces: $$4$$ Newform subspaces: $$12$$ Sturm bound: $$2916$$ Trace bound: $$1$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(\Gamma_1(162))$$.

Total New Old
Modular forms 837 192 645
Cusp forms 622 192 430
Eisenstein series 215 0 215

## Trace form

 $$192 q + 6 q^{5} + 6 q^{7} + 3 q^{8} + O(q^{10})$$ $$192 q + 6 q^{5} + 6 q^{7} + 3 q^{8} + 6 q^{10} + 15 q^{11} + 12 q^{13} + 6 q^{14} + 12 q^{17} - 9 q^{18} - 18 q^{19} - 30 q^{20} - 54 q^{21} - 21 q^{22} - 78 q^{23} - 36 q^{25} - 90 q^{26} - 54 q^{27} - 24 q^{28} - 84 q^{29} - 54 q^{30} - 36 q^{31} - 54 q^{33} - 15 q^{34} - 60 q^{35} - 18 q^{36} - 6 q^{37} + 3 q^{38} + 6 q^{40} - 3 q^{41} + 9 q^{43} + 6 q^{44} - 54 q^{45} + 12 q^{46} - 54 q^{47} - 12 q^{49} + 36 q^{50} - 63 q^{51} + 12 q^{52} - 30 q^{53} - 36 q^{55} + 6 q^{56} - 54 q^{57} + 24 q^{58} - 69 q^{59} - 18 q^{61} + 24 q^{62} - 54 q^{63} + 3 q^{64} + 12 q^{65} + 72 q^{66} - 45 q^{67} + 30 q^{68} + 126 q^{69} - 24 q^{70} + 120 q^{71} + 72 q^{72} + 42 q^{73} + 114 q^{74} + 180 q^{75} - 15 q^{76} + 246 q^{77} + 144 q^{78} - 60 q^{79} + 24 q^{80} + 144 q^{81} + 6 q^{82} + 144 q^{83} + 36 q^{84} - 36 q^{85} + 117 q^{86} + 288 q^{87} - 21 q^{88} + 174 q^{89} + 144 q^{90} + 30 q^{91} + 66 q^{92} + 126 q^{93} - 18 q^{94} + 84 q^{95} + 18 q^{96} - 15 q^{97} + 87 q^{98} + 18 q^{99} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(\Gamma_1(162))$$

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
162.2.a $$\chi_{162}(1, \cdot)$$ 162.2.a.a 1 1
162.2.a.b 1
162.2.a.c 1
162.2.a.d 1
162.2.c $$\chi_{162}(55, \cdot)$$ 162.2.c.a 2 2
162.2.c.b 2
162.2.c.c 2
162.2.c.d 2
162.2.e $$\chi_{162}(19, \cdot)$$ 162.2.e.a 6 6
162.2.e.b 12
162.2.g $$\chi_{162}(7, \cdot)$$ 162.2.g.a 72 18
162.2.g.b 90

## Decomposition of $$S_{2}^{\mathrm{old}}(\Gamma_1(162))$$ into lower level spaces

$$S_{2}^{\mathrm{old}}(\Gamma_1(162)) \cong$$ $$S_{2}^{\mathrm{new}}(\Gamma_1(18))$$$$^{\oplus 3}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(27))$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(54))$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(81))$$$$^{\oplus 2}$$