# Properties

 Label 450.2 Level 450 Weight 2 Dimension 1313 Nonzero newspaces 12 Newform subspaces 63 Sturm bound 21600 Trace bound 3

## Defining parameters

 Level: $$N$$ = $$450 = 2 \cdot 3^{2} \cdot 5^{2}$$ Weight: $$k$$ = $$2$$ Nonzero newspaces: $$12$$ Newform subspaces: $$63$$ Sturm bound: $$21600$$ Trace bound: $$3$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(\Gamma_1(450))$$.

Total New Old
Modular forms 5848 1313 4535
Cusp forms 4953 1313 3640
Eisenstein series 895 0 895

## Trace form

 $$1313 q - 3 q^{2} - 3 q^{3} - 3 q^{4} - 5 q^{5} + 3 q^{6} - 18 q^{7} + 19 q^{9} + O(q^{10})$$ $$1313 q - 3 q^{2} - 3 q^{3} - 3 q^{4} - 5 q^{5} + 3 q^{6} - 18 q^{7} + 19 q^{9} + 11 q^{10} + 43 q^{11} + 16 q^{12} + 34 q^{13} + 46 q^{14} + 24 q^{15} - 3 q^{16} + 74 q^{17} + 26 q^{18} + 62 q^{19} + 16 q^{20} + 54 q^{21} + 51 q^{22} + 94 q^{23} - 3 q^{24} + 67 q^{25} - 4 q^{26} + 48 q^{27} + 8 q^{28} + 70 q^{29} + 44 q^{31} + 2 q^{32} + 7 q^{33} + 8 q^{34} + 4 q^{35} - 29 q^{36} + 81 q^{37} - 87 q^{38} - 128 q^{39} + 3 q^{40} - 105 q^{41} - 128 q^{42} + 17 q^{43} - 46 q^{44} - 80 q^{45} + 36 q^{46} - 98 q^{47} - 29 q^{48} + 30 q^{49} - 57 q^{50} - 87 q^{51} - 14 q^{52} - 87 q^{53} - 87 q^{54} + 60 q^{55} - 34 q^{56} - 61 q^{57} - 50 q^{58} - 31 q^{59} - 32 q^{60} - 104 q^{61} - 156 q^{62} - 84 q^{63} - 253 q^{65} - 24 q^{66} - 189 q^{67} - 117 q^{68} - 152 q^{69} - 216 q^{70} - 24 q^{71} + 3 q^{72} - 190 q^{73} - 212 q^{74} - 296 q^{75} - 31 q^{76} - 474 q^{77} - 126 q^{78} - 188 q^{79} - 5 q^{80} - 137 q^{81} - 178 q^{82} - 436 q^{83} - 78 q^{84} - 185 q^{85} - 43 q^{86} - 214 q^{87} + 11 q^{88} - 239 q^{89} - 64 q^{90} + 24 q^{91} - 6 q^{92} - 104 q^{93} + 38 q^{94} + 28 q^{95} + 16 q^{96} + 123 q^{97} + 72 q^{98} - 14 q^{99} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(\Gamma_1(450))$$

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
450.2.a $$\chi_{450}(1, \cdot)$$ 450.2.a.a 1 1
450.2.a.b 1
450.2.a.c 1
450.2.a.d 1
450.2.a.e 1
450.2.a.f 1
450.2.a.g 1
450.2.c $$\chi_{450}(199, \cdot)$$ 450.2.c.a 2 1
450.2.c.b 2
450.2.c.c 2
450.2.c.d 2
450.2.e $$\chi_{450}(151, \cdot)$$ 450.2.e.a 2 2
450.2.e.b 2
450.2.e.c 2
450.2.e.d 2
450.2.e.e 2
450.2.e.f 2
450.2.e.g 2
450.2.e.h 2
450.2.e.i 2
450.2.e.j 4
450.2.e.k 4
450.2.e.l 4
450.2.e.m 4
450.2.e.n 4
450.2.f $$\chi_{450}(107, \cdot)$$ 450.2.f.a 4 2
450.2.f.b 4
450.2.f.c 4
450.2.h $$\chi_{450}(91, \cdot)$$ 450.2.h.a 4 4
450.2.h.b 4
450.2.h.c 4
450.2.h.d 8
450.2.h.e 8
450.2.h.f 12
450.2.h.g 12
450.2.j $$\chi_{450}(49, \cdot)$$ 450.2.j.a 4 2
450.2.j.b 4
450.2.j.c 4
450.2.j.d 4
450.2.j.e 4
450.2.j.f 8
450.2.j.g 8
450.2.l $$\chi_{450}(19, \cdot)$$ 450.2.l.a 8 4
450.2.l.b 8
450.2.l.c 16
450.2.l.d 16
450.2.p $$\chi_{450}(257, \cdot)$$ 450.2.p.a 8 4
450.2.p.b 8
450.2.p.c 8
450.2.p.d 8
450.2.p.e 8
450.2.p.f 8
450.2.p.g 8
450.2.p.h 16
450.2.q $$\chi_{450}(31, \cdot)$$ 450.2.q.a 8 8
450.2.q.b 112
450.2.q.c 120
450.2.s $$\chi_{450}(17, \cdot)$$ 450.2.s.a 16 8
450.2.s.b 16
450.2.s.c 16
450.2.s.d 32
450.2.v $$\chi_{450}(79, \cdot)$$ 450.2.v.a 240 8
450.2.w $$\chi_{450}(23, \cdot)$$ 450.2.w.a 480 16

## Decomposition of $$S_{2}^{\mathrm{old}}(\Gamma_1(450))$$ into lower level spaces

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