# Properties

 Label 324.3 Level 324 Weight 3 Dimension 2632 Nonzero newspaces 8 Newform subspaces 38 Sturm bound 17496 Trace bound 1

## Defining parameters

 Level: $$N$$ = $$324 = 2^{2} \cdot 3^{4}$$ Weight: $$k$$ = $$3$$ Nonzero newspaces: $$8$$ Newform subspaces: $$38$$ Sturm bound: $$17496$$ Trace bound: $$1$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{3}(\Gamma_1(324))$$.

Total New Old
Modular forms 6102 2744 3358
Cusp forms 5562 2632 2930
Eisenstein series 540 112 428

## Trace form

 $$2632 q - 12 q^{2} - 20 q^{4} - 15 q^{5} - 18 q^{6} + 3 q^{7} - 15 q^{8} - 36 q^{9} + O(q^{10})$$ $$2632 q - 12 q^{2} - 20 q^{4} - 15 q^{5} - 18 q^{6} + 3 q^{7} - 15 q^{8} - 36 q^{9} - 37 q^{10} - 36 q^{11} - 18 q^{12} - 79 q^{13} - 57 q^{14} - 32 q^{16} - 30 q^{17} - 18 q^{18} - 72 q^{19} + 51 q^{20} - 171 q^{21} + 48 q^{22} - 63 q^{23} - 18 q^{24} - 63 q^{25} + 165 q^{26} + 27 q^{27} + 87 q^{28} + 3 q^{29} - 18 q^{30} + 81 q^{31} + 78 q^{32} + 153 q^{33} + 26 q^{34} + 486 q^{35} - 18 q^{36} + 152 q^{37} - 90 q^{38} + 65 q^{40} + 570 q^{41} + 477 q^{42} + 270 q^{43} + 849 q^{44} + 396 q^{45} + 351 q^{46} + 567 q^{47} + 477 q^{48} + 205 q^{49} + 498 q^{50} + 126 q^{51} + 11 q^{52} + 6 q^{53} - 144 q^{54} - 126 q^{55} - 495 q^{56} - 252 q^{57} - 187 q^{58} - 630 q^{59} - 837 q^{60} - 307 q^{61} - 1203 q^{62} - 540 q^{63} - 695 q^{64} - 735 q^{65} - 954 q^{66} - 612 q^{67} - 1068 q^{68} + 468 q^{69} - 405 q^{70} + 324 q^{71} - 18 q^{72} - 277 q^{73} - 465 q^{74} + 450 q^{75} - 546 q^{76} - 231 q^{77} - 99 q^{78} - 273 q^{79} - 834 q^{80} - 108 q^{81} - 514 q^{82} - 189 q^{83} - 99 q^{84} + 520 q^{85} - 624 q^{86} - 1008 q^{87} - 708 q^{88} + 321 q^{89} - 1089 q^{90} + 483 q^{91} - 2091 q^{92} + 792 q^{93} - 651 q^{94} + 612 q^{95} - 1305 q^{96} + 1160 q^{97} - 2067 q^{98} - 252 q^{99} + O(q^{100})$$

## Decomposition of $$S_{3}^{\mathrm{new}}(\Gamma_1(324))$$

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
324.3.c $$\chi_{324}(161, \cdot)$$ 324.3.c.a 4 1
324.3.c.b 4
324.3.d $$\chi_{324}(163, \cdot)$$ 324.3.d.a 2 1
324.3.d.b 2
324.3.d.c 2
324.3.d.d 2
324.3.d.e 6
324.3.d.f 6
324.3.d.g 8
324.3.d.h 8
324.3.d.i 8
324.3.f $$\chi_{324}(55, \cdot)$$ 324.3.f.a 2 2
324.3.f.b 2
324.3.f.c 2
324.3.f.d 2
324.3.f.e 2
324.3.f.f 2
324.3.f.g 2
324.3.f.h 2
324.3.f.i 2
324.3.f.j 2
324.3.f.k 4
324.3.f.l 4
324.3.f.m 4
324.3.f.n 4
324.3.f.o 8
324.3.f.p 8
324.3.f.q 12
324.3.f.r 12
324.3.f.s 16
324.3.g $$\chi_{324}(53, \cdot)$$ 324.3.g.a 2 2
324.3.g.b 2
324.3.g.c 4
324.3.g.d 8
324.3.j $$\chi_{324}(19, \cdot)$$ 324.3.j.a 204 6
324.3.k $$\chi_{324}(17, \cdot)$$ 324.3.k.a 36 6
324.3.n $$\chi_{324}(7, \cdot)$$ 324.3.n.a 1908 18
324.3.o $$\chi_{324}(5, \cdot)$$ 324.3.o.a 324 18

## Decomposition of $$S_{3}^{\mathrm{old}}(\Gamma_1(324))$$ into lower level spaces

$$S_{3}^{\mathrm{old}}(\Gamma_1(324)) \cong$$ $$S_{3}^{\mathrm{new}}(\Gamma_1(9))$$$$^{\oplus 9}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(\Gamma_1(12))$$$$^{\oplus 4}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(\Gamma_1(18))$$$$^{\oplus 6}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(\Gamma_1(27))$$$$^{\oplus 6}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(\Gamma_1(36))$$$$^{\oplus 3}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(\Gamma_1(54))$$$$^{\oplus 4}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(\Gamma_1(81))$$$$^{\oplus 3}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(\Gamma_1(108))$$$$^{\oplus 2}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(\Gamma_1(162))$$$$^{\oplus 2}$$