## Defining parameters

 Level: $$N$$ = $$324 = 2^{2} \cdot 3^{4}$$ Weight: $$k$$ = $$2$$ Nonzero newspaces: $$8$$ Newform subspaces: $$21$$ Sturm bound: $$11664$$ Trace bound: $$1$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(\Gamma_1(324))$$.

Total New Old
Modular forms 3186 1400 1786
Cusp forms 2647 1288 1359
Eisenstein series 539 112 427

## Trace form

 $$1288 q - 12 q^{2} - 20 q^{4} - 27 q^{5} - 18 q^{6} - 3 q^{7} - 9 q^{8} - 36 q^{9} + O(q^{10})$$ $$1288 q - 12 q^{2} - 20 q^{4} - 27 q^{5} - 18 q^{6} - 3 q^{7} - 9 q^{8} - 36 q^{9} - 25 q^{10} - 3 q^{11} - 18 q^{12} - 37 q^{13} + 3 q^{14} - 8 q^{16} - 6 q^{17} - 18 q^{18} + 18 q^{19} + 9 q^{20} - 9 q^{21} - 12 q^{22} + 57 q^{23} - 18 q^{24} - 3 q^{25} - 27 q^{26} + 27 q^{27} - 33 q^{28} + 39 q^{29} - 18 q^{30} + 27 q^{31} - 42 q^{32} - 9 q^{33} - 52 q^{34} + 48 q^{35} - 18 q^{36} - 34 q^{37} - 36 q^{38} - 73 q^{40} - 33 q^{41} - 63 q^{42} - 27 q^{43} - 153 q^{44} - 90 q^{45} - 117 q^{46} - 99 q^{47} - 117 q^{48} - 83 q^{49} - 222 q^{50} - 63 q^{51} - 91 q^{52} - 174 q^{53} - 144 q^{54} - 90 q^{55} - 243 q^{56} - 90 q^{57} - 91 q^{58} - 123 q^{59} - 135 q^{60} - 97 q^{61} - 207 q^{62} - 54 q^{63} - 119 q^{64} - 171 q^{65} - 90 q^{66} - 9 q^{67} - 114 q^{68} - 126 q^{69} - 81 q^{70} - 60 q^{71} - 18 q^{72} - 73 q^{73} - 39 q^{74} - 90 q^{75} - 66 q^{76} - 117 q^{77} + 9 q^{78} + 57 q^{79} - 108 q^{81} - 34 q^{82} - 27 q^{83} - 45 q^{84} - 68 q^{85} + 12 q^{86} - 144 q^{87} + 60 q^{88} - 105 q^{89} + 45 q^{90} - 15 q^{91} + 195 q^{92} - 234 q^{93} + 75 q^{94} - 132 q^{95} + 99 q^{96} - 79 q^{97} + 333 q^{98} - 90 q^{99} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(\Gamma_1(324))$$

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
324.2.a $$\chi_{324}(1, \cdot)$$ 324.2.a.a 1 1
324.2.a.b 1
324.2.a.c 1
324.2.a.d 1
324.2.b $$\chi_{324}(323, \cdot)$$ 324.2.b.a 4 1
324.2.b.b 8
324.2.b.c 8
324.2.e $$\chi_{324}(109, \cdot)$$ 324.2.e.a 2 2
324.2.e.b 2
324.2.e.c 2
324.2.e.d 2
324.2.h $$\chi_{324}(107, \cdot)$$ 324.2.h.a 4 2
324.2.h.b 4
324.2.h.c 4
324.2.h.d 8
324.2.h.e 8
324.2.h.f 16
324.2.i $$\chi_{324}(37, \cdot)$$ 324.2.i.a 18 6
324.2.l $$\chi_{324}(35, \cdot)$$ 324.2.l.a 96 6
324.2.m $$\chi_{324}(13, \cdot)$$ 324.2.m.a 162 18
324.2.p $$\chi_{324}(11, \cdot)$$ 324.2.p.a 936 18

## Decomposition of $$S_{2}^{\mathrm{old}}(\Gamma_1(324))$$ into lower level spaces

$$S_{2}^{\mathrm{old}}(\Gamma_1(324)) \cong$$ $$S_{2}^{\mathrm{new}}(\Gamma_1(18))$$$$^{\oplus 6}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(27))$$$$^{\oplus 6}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(36))$$$$^{\oplus 3}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(54))$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(81))$$$$^{\oplus 3}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(108))$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(162))$$$$^{\oplus 2}$$