# Properties

 Label 324.4 Level 324 Weight 4 Dimension 3976 Nonzero newspaces 8 Sturm bound 23328 Trace bound 1

## Defining parameters

 Level: $$N$$ = $$324 = 2^{2} \cdot 3^{4}$$ Weight: $$k$$ = $$4$$ Nonzero newspaces: $$8$$ Sturm bound: $$23328$$ Trace bound: $$1$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{4}(\Gamma_1(324))$$.

Total New Old
Modular forms 9018 4088 4930
Cusp forms 8478 3976 4502
Eisenstein series 540 112 428

## Trace form

 $$3976 q - 12 q^{2} - 20 q^{4} - 36 q^{5} - 18 q^{6} + 18 q^{7} - 9 q^{8} - 36 q^{9} + O(q^{10})$$ $$3976 q - 12 q^{2} - 20 q^{4} - 36 q^{5} - 18 q^{6} + 18 q^{7} - 9 q^{8} - 36 q^{9} - 13 q^{10} + 87 q^{11} - 18 q^{12} - 58 q^{13} - 87 q^{14} - 200 q^{16} - 222 q^{17} - 18 q^{18} + 270 q^{19} - 243 q^{20} + 72 q^{21} - 474 q^{23} - 18 q^{24} - 60 q^{25} - 27 q^{26} - 702 q^{27} - 153 q^{28} - 798 q^{29} - 18 q^{30} - 216 q^{31} + 678 q^{32} + 396 q^{33} + 518 q^{34} + 1236 q^{35} - 18 q^{36} + 284 q^{37} + 882 q^{38} - 625 q^{40} - 2895 q^{41} - 3573 q^{42} - 567 q^{43} - 5337 q^{44} - 1062 q^{45} - 1161 q^{46} + 2142 q^{47} + 2259 q^{48} + 1960 q^{49} + 7608 q^{50} + 2961 q^{51} + 3065 q^{52} + 3876 q^{53} + 6660 q^{54} + 3348 q^{55} + 8829 q^{56} + 2178 q^{57} + 743 q^{58} + 2667 q^{59} + 1269 q^{60} + 140 q^{61} - 3771 q^{62} - 1998 q^{63} - 3863 q^{64} - 9450 q^{65} - 7434 q^{66} - 189 q^{67} - 9294 q^{68} - 7146 q^{69} + 369 q^{70} - 600 q^{71} - 18 q^{72} + 950 q^{73} + 5865 q^{74} + 4500 q^{75} + 3042 q^{76} + 7272 q^{77} + 225 q^{78} - 3528 q^{79} + 5724 q^{81} - 1930 q^{82} + 864 q^{83} - 261 q^{84} - 7226 q^{85} - 8340 q^{86} + 5472 q^{87} - 1980 q^{88} - 13902 q^{89} + 12519 q^{90} - 5634 q^{91} + 15207 q^{92} - 18486 q^{93} + 2205 q^{94} - 10356 q^{95} + 2907 q^{96} + 3569 q^{97} - 5823 q^{98} - 7542 q^{99} + O(q^{100})$$

## Decomposition of $$S_{4}^{\mathrm{new}}(\Gamma_1(324))$$

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
324.4.a $$\chi_{324}(1, \cdot)$$ 324.4.a.a 1 1
324.4.a.b 1
324.4.a.c 3
324.4.a.d 3
324.4.a.e 4
324.4.b $$\chi_{324}(323, \cdot)$$ 324.4.b.a 4 1
324.4.b.b 8
324.4.b.c 24
324.4.b.d 32
324.4.e $$\chi_{324}(109, \cdot)$$ 324.4.e.a 2 2
324.4.e.b 2
324.4.e.c 2
324.4.e.d 2
324.4.e.e 2
324.4.e.f 2
324.4.e.g 2
324.4.e.h 2
324.4.e.i 8
324.4.h $$\chi_{324}(107, \cdot)$$ n/a 140 2
324.4.i $$\chi_{324}(37, \cdot)$$ 324.4.i.a 54 6
324.4.l $$\chi_{324}(35, \cdot)$$ n/a 312 6
324.4.m $$\chi_{324}(13, \cdot)$$ n/a 486 18
324.4.p $$\chi_{324}(11, \cdot)$$ n/a 2880 18

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

## Decomposition of $$S_{4}^{\mathrm{old}}(\Gamma_1(324))$$ into lower level spaces

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