# Properties

 Label 324.9 Level 324 Weight 9 Dimension 10696 Nonzero newspaces 8 Sturm bound 52488 Trace bound 1

## Defining parameters

 Level: $$N$$ = $$324 = 2^{2} \cdot 3^{4}$$ Weight: $$k$$ = $$9$$ Nonzero newspaces: $$8$$ Sturm bound: $$52488$$ Trace bound: $$1$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{9}(\Gamma_1(324))$$.

Total New Old
Modular forms 23598 10808 12790
Cusp forms 23058 10696 12362
Eisenstein series 540 112 428

## Trace form

 $$10696 q - 12 q^{2} - 20 q^{4} + 417 q^{5} - 18 q^{6} - 2769 q^{7} - 15 q^{8} - 36 q^{9} + O(q^{10})$$ $$10696 q - 12 q^{2} - 20 q^{4} + 417 q^{5} - 18 q^{6} - 2769 q^{7} - 15 q^{8} - 36 q^{9} - 541 q^{10} - 5490 q^{11} - 18 q^{12} - 62215 q^{13} - 26517 q^{14} + 167008 q^{16} - 30 q^{17} - 18 q^{18} + 378252 q^{19} - 314445 q^{20} - 1360971 q^{21} + 692472 q^{22} + 2243205 q^{23} - 18 q^{24} - 305199 q^{25} - 2400675 q^{26} - 1933281 q^{27} + 1024215 q^{28} - 7602657 q^{29} - 18 q^{30} + 2362581 q^{31} - 4836162 q^{32} + 6069321 q^{33} - 2623690 q^{34} - 12450834 q^{35} - 18 q^{36} - 6511564 q^{37} + 9654786 q^{38} + 1868105 q^{40} + 16937508 q^{41} - 39542103 q^{42} - 10499436 q^{43} + 23121489 q^{44} + 37896732 q^{45} + 64625607 q^{46} + 36843255 q^{47} - 15217803 q^{48} - 11977859 q^{49} - 175835922 q^{50} - 54355896 q^{51} - 33880381 q^{52} + 6 q^{53} + 75728376 q^{54} + 95397354 q^{55} + 298722321 q^{56} + 65747772 q^{57} + 12113921 q^{58} - 58027680 q^{59} - 125543709 q^{60} - 78127255 q^{61} - 298279491 q^{62} - 101958480 q^{63} - 25724567 q^{64} + 189047001 q^{65} + 254302758 q^{66} + 181901682 q^{67} + 30424476 q^{68} - 25284492 q^{69} - 88779753 q^{70} + 125718804 q^{71} - 18 q^{72} - 29594041 q^{73} + 268406751 q^{74} - 246093750 q^{75} + 148975662 q^{76} - 333783303 q^{77} - 59067 q^{78} - 46223121 q^{79} - 419710434 q^{80} + 148365972 q^{81} - 105216994 q^{82} + 236810763 q^{83} - 59067 q^{84} - 59662172 q^{85} + 596778000 q^{86} + 61362000 q^{87} + 40600956 q^{88} + 351477789 q^{89} + 504195723 q^{90} - 482661921 q^{91} + 996653013 q^{92} - 574019964 q^{93} - 530413959 q^{94} + 866815848 q^{95} - 1223629425 q^{96} + 338570018 q^{97} - 1411868355 q^{98} - 681481116 q^{99} + O(q^{100})$$

## Decomposition of $$S_{9}^{\mathrm{new}}(\Gamma_1(324))$$

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 available newforms together with their dimension.

Label $$\chi$$ Newforms Dimension $$\chi$$ degree
324.9.c $$\chi_{324}(161, \cdot)$$ 324.9.c.a 16 1
324.9.c.b 16
324.9.d $$\chi_{324}(163, \cdot)$$ n/a 188 1
324.9.f $$\chi_{324}(55, \cdot)$$ n/a 380 2
324.9.g $$\chi_{324}(53, \cdot)$$ 324.9.g.a 2 2
324.9.g.b 2
324.9.g.c 4
324.9.g.d 4
324.9.g.e 4
324.9.g.f 4
324.9.g.g 12
324.9.g.h 32
324.9.j $$\chi_{324}(19, \cdot)$$ n/a 852 6
324.9.k $$\chi_{324}(17, \cdot)$$ n/a 144 6
324.9.n $$\chi_{324}(7, \cdot)$$ n/a 7740 18
324.9.o $$\chi_{324}(5, \cdot)$$ n/a 1296 18

"n/a" means that newforms for that character have not been added to the database yet

## Decomposition of $$S_{9}^{\mathrm{old}}(\Gamma_1(324))$$ into lower level spaces

$$S_{9}^{\mathrm{old}}(\Gamma_1(324)) \cong$$ $$S_{9}^{\mathrm{new}}(\Gamma_1(1))$$$$^{\oplus 15}$$$$\oplus$$$$S_{9}^{\mathrm{new}}(\Gamma_1(2))$$$$^{\oplus 10}$$$$\oplus$$$$S_{9}^{\mathrm{new}}(\Gamma_1(3))$$$$^{\oplus 12}$$$$\oplus$$$$S_{9}^{\mathrm{new}}(\Gamma_1(4))$$$$^{\oplus 5}$$$$\oplus$$$$S_{9}^{\mathrm{new}}(\Gamma_1(6))$$$$^{\oplus 8}$$$$\oplus$$$$S_{9}^{\mathrm{new}}(\Gamma_1(9))$$$$^{\oplus 9}$$$$\oplus$$$$S_{9}^{\mathrm{new}}(\Gamma_1(12))$$$$^{\oplus 4}$$$$\oplus$$$$S_{9}^{\mathrm{new}}(\Gamma_1(18))$$$$^{\oplus 6}$$$$\oplus$$$$S_{9}^{\mathrm{new}}(\Gamma_1(27))$$$$^{\oplus 6}$$$$\oplus$$$$S_{9}^{\mathrm{new}}(\Gamma_1(36))$$$$^{\oplus 3}$$$$\oplus$$$$S_{9}^{\mathrm{new}}(\Gamma_1(54))$$$$^{\oplus 4}$$$$\oplus$$$$S_{9}^{\mathrm{new}}(\Gamma_1(81))$$$$^{\oplus 3}$$$$\oplus$$$$S_{9}^{\mathrm{new}}(\Gamma_1(108))$$$$^{\oplus 2}$$$$\oplus$$$$S_{9}^{\mathrm{new}}(\Gamma_1(162))$$$$^{\oplus 2}$$$$\oplus$$$$S_{9}^{\mathrm{new}}(\Gamma_1(324))$$$$^{\oplus 1}$$