# Properties

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

## Defining parameters

 Level: $$N$$ = $$450 = 2 \cdot 3^{2} \cdot 5^{2}$$ Weight: $$k$$ = $$2$$ Nonzero newspaces: $$12$$ Newforms: $$63$$ 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

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

## 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

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