# Properties

 Label 450.4 Level 450 Weight 4 Dimension 3940 Nonzero newspaces 12 Sturm bound 43200 Trace bound 3

## Defining parameters

 Level: $$N$$ = $$450 = 2 \cdot 3^{2} \cdot 5^{2}$$ Weight: $$k$$ = $$4$$ Nonzero newspaces: $$12$$ Sturm bound: $$43200$$ Trace bound: $$3$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{4}(\Gamma_1(450))$$.

Total New Old
Modular forms 16648 3940 12708
Cusp forms 15752 3940 11812
Eisenstein series 896 0 896

## Trace form

 $$3940 q + 3 q^{3} + 5 q^{5} - 18 q^{6} + 4 q^{7} - 24 q^{8} + 151 q^{9} + O(q^{10})$$ $$3940 q + 3 q^{3} + 5 q^{5} - 18 q^{6} + 4 q^{7} - 24 q^{8} + 151 q^{9} + 46 q^{10} - 13 q^{11} - 104 q^{12} - 638 q^{13} - 604 q^{14} - 336 q^{15} - 128 q^{16} - 212 q^{17} + 92 q^{18} + 814 q^{19} + 176 q^{20} + 1548 q^{21} + 534 q^{22} + 1112 q^{23} - 24 q^{24} - 1603 q^{25} - 620 q^{26} - 1104 q^{27} - 368 q^{28} - 2782 q^{29} - 894 q^{31} + 160 q^{32} - 2867 q^{33} + 2272 q^{34} + 164 q^{35} - 380 q^{36} + 3929 q^{37} + 858 q^{38} + 3610 q^{39} - 72 q^{40} + 5463 q^{41} + 4048 q^{42} - 2955 q^{43} - 200 q^{44} + 3520 q^{45} - 3928 q^{46} + 4268 q^{47} + 880 q^{48} - 795 q^{49} + 798 q^{50} + 519 q^{51} + 712 q^{52} - 4563 q^{53} - 3462 q^{54} + 820 q^{55} - 1904 q^{56} - 11461 q^{57} + 2760 q^{58} - 11915 q^{59} - 4352 q^{60} - 1128 q^{61} - 9336 q^{62} - 11010 q^{63} + 576 q^{64} - 2963 q^{65} + 336 q^{66} + 6343 q^{67} + 2820 q^{68} + 9382 q^{69} + 9024 q^{70} + 2256 q^{71} - 24 q^{72} + 14164 q^{73} + 13748 q^{74} + 14584 q^{75} + 844 q^{76} + 11940 q^{77} + 7572 q^{78} + 3166 q^{79} + 80 q^{80} + 2491 q^{81} + 768 q^{82} + 7822 q^{83} - 408 q^{84} - 6475 q^{85} + 5098 q^{86} + 9932 q^{87} + 5016 q^{88} + 3845 q^{89} - 12544 q^{90} + 5108 q^{91} - 5952 q^{92} + 8854 q^{93} + 4196 q^{94} - 1492 q^{95} - 128 q^{96} - 14801 q^{97} - 8166 q^{98} - 8480 q^{99} + O(q^{100})$$

## Decomposition of $$S_{4}^{\mathrm{new}}(\Gamma_1(450))$$

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
450.4.a $$\chi_{450}(1, \cdot)$$ 450.4.a.a 1 1
450.4.a.b 1
450.4.a.c 1
450.4.a.d 1
450.4.a.e 1
450.4.a.f 1
450.4.a.g 1
450.4.a.h 1
450.4.a.i 1
450.4.a.j 1
450.4.a.k 1
450.4.a.l 1
450.4.a.m 1
450.4.a.n 1
450.4.a.o 1
450.4.a.p 1
450.4.a.q 1
450.4.a.r 1
450.4.a.s 1
450.4.a.t 1
450.4.a.u 2
450.4.a.v 2
450.4.c $$\chi_{450}(199, \cdot)$$ 450.4.c.a 2 1
450.4.c.b 2
450.4.c.c 2
450.4.c.d 2
450.4.c.e 2
450.4.c.f 2
450.4.c.g 2
450.4.c.h 2
450.4.c.i 2
450.4.c.j 2
450.4.c.k 2
450.4.e $$\chi_{450}(151, \cdot)$$ n/a 114 2
450.4.f $$\chi_{450}(107, \cdot)$$ 450.4.f.a 4 2
450.4.f.b 4
450.4.f.c 4
450.4.f.d 8
450.4.f.e 8
450.4.f.f 8
450.4.h $$\chi_{450}(91, \cdot)$$ n/a 148 4
450.4.j $$\chi_{450}(49, \cdot)$$ n/a 108 2
450.4.l $$\chi_{450}(19, \cdot)$$ n/a 152 4
450.4.p $$\chi_{450}(257, \cdot)$$ n/a 216 4
450.4.q $$\chi_{450}(31, \cdot)$$ n/a 720 8
450.4.s $$\chi_{450}(17, \cdot)$$ n/a 240 8
450.4.v $$\chi_{450}(79, \cdot)$$ n/a 720 8
450.4.w $$\chi_{450}(23, \cdot)$$ n/a 1440 16

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

## Decomposition of $$S_{4}^{\mathrm{old}}(\Gamma_1(450))$$ into lower level spaces

$$S_{4}^{\mathrm{old}}(\Gamma_1(450)) \cong$$ $$S_{4}^{\mathrm{new}}(\Gamma_1(5))$$$$^{\oplus 12}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(6))$$$$^{\oplus 6}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(9))$$$$^{\oplus 6}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(10))$$$$^{\oplus 6}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(15))$$$$^{\oplus 8}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(18))$$$$^{\oplus 3}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(25))$$$$^{\oplus 6}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(30))$$$$^{\oplus 4}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(45))$$$$^{\oplus 4}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(50))$$$$^{\oplus 3}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(75))$$$$^{\oplus 4}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(90))$$$$^{\oplus 2}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(150))$$$$^{\oplus 2}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(225))$$$$^{\oplus 2}$$