## Defining parameters

 Level: $$N$$ = $$162 = 2 \cdot 3^{4}$$ Weight: $$k$$ = $$3$$ Nonzero newspaces: $$4$$ Newform subspaces: $$7$$ Sturm bound: $$4374$$ Trace bound: $$1$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{3}(\Gamma_1(162))$$.

Total New Old
Modular forms 1566 384 1182
Cusp forms 1350 384 966
Eisenstein series 216 0 216

## Trace form

 $$384 q - 18 q^{5} - 6 q^{7} + O(q^{10})$$ $$384 q - 18 q^{5} - 6 q^{7} + 12 q^{10} + 18 q^{11} + 6 q^{13} + 36 q^{14} + 72 q^{18} + 180 q^{19} + 180 q^{20} + 270 q^{21} + 84 q^{22} + 342 q^{23} + 90 q^{25} - 54 q^{27} - 48 q^{28} - 306 q^{29} - 216 q^{30} - 234 q^{31} - 378 q^{33} - 228 q^{34} - 972 q^{35} - 180 q^{36} - 336 q^{37} - 288 q^{38} - 24 q^{40} + 306 q^{41} + 234 q^{43} + 432 q^{45} + 24 q^{46} + 702 q^{47} + 210 q^{49} + 126 q^{51} - 12 q^{52} - 180 q^{55} + 72 q^{56} - 216 q^{57} + 12 q^{58} - 630 q^{59} - 414 q^{61} - 540 q^{63} - 48 q^{64} - 1746 q^{65} - 1008 q^{66} + 342 q^{67} - 36 q^{68} - 1872 q^{69} + 564 q^{70} - 648 q^{71} - 576 q^{72} + 138 q^{73} - 360 q^{74} - 900 q^{75} + 168 q^{76} - 522 q^{77} - 288 q^{78} + 6 q^{79} + 144 q^{81} - 96 q^{82} - 162 q^{83} + 144 q^{84} - 720 q^{85} + 324 q^{86} + 2016 q^{87} - 168 q^{88} + 1134 q^{89} + 1440 q^{90} + 330 q^{91} + 468 q^{92} + 2124 q^{93} - 684 q^{94} + 2556 q^{95} + 288 q^{96} - 498 q^{97} + 648 q^{98} + 1800 q^{99} + O(q^{100})$$

## Decomposition of $$S_{3}^{\mathrm{new}}(\Gamma_1(162))$$

We only show spaces with odd 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
162.3.b $$\chi_{162}(161, \cdot)$$ 162.3.b.a 4 1
162.3.b.b 4
162.3.d $$\chi_{162}(53, \cdot)$$ 162.3.d.a 4 2
162.3.d.b 4
162.3.d.c 8
162.3.f $$\chi_{162}(17, \cdot)$$ 162.3.f.a 36 6
162.3.h $$\chi_{162}(5, \cdot)$$ 162.3.h.a 324 18

## Decomposition of $$S_{3}^{\mathrm{old}}(\Gamma_1(162))$$ into lower level spaces

$$S_{3}^{\mathrm{old}}(\Gamma_1(162)) \cong$$ $$S_{3}^{\mathrm{new}}(\Gamma_1(9))$$$$^{\oplus 6}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(\Gamma_1(18))$$$$^{\oplus 3}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(\Gamma_1(27))$$$$^{\oplus 4}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(\Gamma_1(54))$$$$^{\oplus 2}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(\Gamma_1(81))$$$$^{\oplus 2}$$