## Defining parameters

 Level: $$N$$ = $$450 = 2 \cdot 3^{2} \cdot 5^{2}$$ Weight: $$k$$ = $$6$$ Nonzero newspaces: $$12$$ Sturm bound: $$64800$$ Trace bound: $$3$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{6}(\Gamma_1(450))$$.

Total New Old
Modular forms 27448 6568 20880
Cusp forms 26552 6568 19984
Eisenstein series 896 0 896

## Trace form

 $$6568q + 9q^{3} - 64q^{4} - 85q^{5} - 84q^{6} + 446q^{7} + 192q^{8} + 1075q^{9} + O(q^{10})$$ $$6568q + 9q^{3} - 64q^{4} - 85q^{5} - 84q^{6} + 446q^{7} + 192q^{8} + 1075q^{9} - 1804q^{10} - 4933q^{11} - 128q^{12} + 7192q^{13} + 7880q^{14} + 4344q^{15} - 3072q^{16} - 2264q^{17} - 6520q^{18} - 35886q^{19} - 6784q^{20} - 11070q^{21} + 9948q^{22} + 8702q^{23} + 576q^{24} + 15127q^{25} + 19192q^{26} + 38352q^{27} + 3840q^{28} + 31700q^{29} - 412q^{31} - 5120q^{32} - 131909q^{33} - 8512q^{34} + 17684q^{35} + 41200q^{36} - 167229q^{37} + 85284q^{38} + 88480q^{39} - 192q^{40} + 72729q^{41} - 126368q^{42} + 8897q^{43} - 12704q^{44} - 151280q^{45} + 146000q^{46} + 37934q^{47} - 4352q^{48} + 40377q^{49} + 62628q^{50} + 175941q^{51} - 70016q^{52} + 243723q^{53} + 198660q^{54} - 156020q^{55} + 11392q^{56} - 107197q^{57} + 64688q^{58} - 566087q^{59} - 215552q^{60} + 99078q^{61} + 347664q^{62} + 624516q^{63} + 45056q^{64} + 1217047q^{65} + 181344q^{66} + 96251q^{67} + 65808q^{68} - 198764q^{69} - 412416q^{70} - 499488q^{71} - 59712q^{72} - 460032q^{73} - 808840q^{74} - 1271336q^{75} - 64368q^{76} - 280518q^{77} - 70680q^{78} - 193436q^{79} - 21760q^{80} + 799567q^{81} + 789760q^{82} + 2123680q^{83} + 584352q^{84} + 1657135q^{85} + 683428q^{86} + 492602q^{87} + 8128q^{88} - 1067197q^{89} - 638464q^{90} - 1547576q^{91} - 1596768q^{92} - 3031004q^{93} - 855288q^{94} - 582772q^{95} - 69632q^{96} - 306763q^{97} + 665100q^{98} + 2569762q^{99} + O(q^{100})$$

## Decomposition of $$S_{6}^{\mathrm{new}}(\Gamma_1(450))$$

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
450.6.a $$\chi_{450}(1, \cdot)$$ 450.6.a.a 1 1
450.6.a.b 1
450.6.a.c 1
450.6.a.d 1
450.6.a.e 1
450.6.a.f 1
450.6.a.g 1
450.6.a.h 1
450.6.a.i 1
450.6.a.j 1
450.6.a.k 1
450.6.a.l 1
450.6.a.m 1
450.6.a.n 1
450.6.a.o 1
450.6.a.p 1
450.6.a.q 1
450.6.a.r 1
450.6.a.s 1
450.6.a.t 1
450.6.a.u 1
450.6.a.v 1
450.6.a.w 1
450.6.a.x 1
450.6.a.y 2
450.6.a.z 2
450.6.a.ba 2
450.6.a.bb 2
450.6.a.bc 2
450.6.a.bd 2
450.6.a.be 2
450.6.a.bf 2
450.6.c $$\chi_{450}(199, \cdot)$$ 450.6.c.a 2 1
450.6.c.b 2
450.6.c.c 2
450.6.c.d 2
450.6.c.e 2
450.6.c.f 2
450.6.c.g 2
450.6.c.h 2
450.6.c.i 2
450.6.c.j 2
450.6.c.k 2
450.6.c.l 2
450.6.c.m 2
450.6.c.n 2
450.6.c.o 2
450.6.c.p 4
450.6.c.q 4
450.6.e $$\chi_{450}(151, \cdot)$$ n/a 190 2
450.6.f $$\chi_{450}(107, \cdot)$$ 450.6.f.a 4 2
450.6.f.b 4
450.6.f.c 4
450.6.f.d 4
450.6.f.e 12
450.6.f.f 16
450.6.f.g 16
450.6.h $$\chi_{450}(91, \cdot)$$ n/a 252 4
450.6.j $$\chi_{450}(49, \cdot)$$ n/a 180 2
450.6.l $$\chi_{450}(19, \cdot)$$ n/a 248 4
450.6.p $$\chi_{450}(257, \cdot)$$ n/a 360 4
450.6.q $$\chi_{450}(31, \cdot)$$ n/a 1200 8
450.6.s $$\chi_{450}(17, \cdot)$$ n/a 400 8
450.6.v $$\chi_{450}(79, \cdot)$$ n/a 1200 8
450.6.w $$\chi_{450}(23, \cdot)$$ n/a 2400 16

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

## Decomposition of $$S_{6}^{\mathrm{old}}(\Gamma_1(450))$$ into lower level spaces

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