# Properties

 Label 600.2 Level 600 Weight 2 Dimension 3651 Nonzero newspaces 18 Newform subspaces 81 Sturm bound 38400 Trace bound 8

## Defining parameters

 Level: $$N$$ = $$600 = 2^{3} \cdot 3 \cdot 5^{2}$$ Weight: $$k$$ = $$2$$ Nonzero newspaces: $$18$$ Newform subspaces: $$81$$ Sturm bound: $$38400$$ Trace bound: $$8$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(\Gamma_1(600))$$.

Total New Old
Modular forms 10272 3815 6457
Cusp forms 8929 3651 5278
Eisenstein series 1343 164 1179

## Trace form

 $$3651 q - 2 q^{2} - 15 q^{3} - 28 q^{4} - 2 q^{5} - 18 q^{6} - 44 q^{7} + 4 q^{8} - 31 q^{9} + O(q^{10})$$ $$3651 q - 2 q^{2} - 15 q^{3} - 28 q^{4} - 2 q^{5} - 18 q^{6} - 44 q^{7} + 4 q^{8} - 31 q^{9} - 32 q^{10} - 12 q^{11} + 20 q^{12} - 10 q^{13} + 52 q^{14} - 8 q^{15} + 24 q^{16} + 6 q^{17} + 38 q^{18} + 40 q^{19} + 40 q^{20} + 32 q^{21} + 48 q^{22} + 48 q^{23} + 32 q^{24} - 58 q^{25} + 56 q^{26} + 45 q^{27} - 40 q^{28} + 30 q^{29} - 12 q^{30} + 36 q^{31} - 72 q^{32} + 28 q^{33} - 132 q^{34} + 48 q^{35} - 32 q^{36} + 72 q^{37} - 104 q^{38} + 78 q^{39} - 112 q^{40} + 46 q^{41} - 96 q^{42} + 136 q^{43} - 112 q^{44} + 70 q^{45} - 176 q^{46} + 120 q^{47} - 140 q^{48} + 55 q^{49} - 40 q^{50} - 62 q^{51} - 72 q^{52} + 64 q^{53} - 110 q^{54} + 64 q^{55} - 8 q^{56} + 16 q^{57} - 36 q^{58} + 20 q^{59} - 92 q^{60} - 50 q^{61} - 4 q^{62} - 120 q^{63} - 88 q^{64} + 6 q^{65} - 188 q^{66} - 160 q^{67} - 112 q^{68} - 24 q^{69} - 104 q^{70} - 128 q^{71} - 200 q^{72} + 74 q^{73} - 160 q^{74} - 88 q^{75} - 232 q^{76} - 188 q^{78} - 204 q^{79} - 80 q^{80} + 57 q^{81} - 348 q^{82} - 268 q^{83} - 300 q^{84} - 82 q^{85} - 184 q^{86} - 206 q^{87} - 472 q^{88} + 128 q^{89} - 172 q^{90} - 184 q^{91} - 384 q^{92} - 32 q^{93} - 496 q^{94} - 256 q^{95} - 84 q^{96} - 86 q^{97} - 378 q^{98} - 180 q^{99} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(\Gamma_1(600))$$

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 available newforms together with their dimension.

Label $$\chi$$ Newforms Dimension $$\chi$$ degree
600.2.a $$\chi_{600}(1, \cdot)$$ 600.2.a.a 1 1
600.2.a.b 1
600.2.a.c 1
600.2.a.d 1
600.2.a.e 1
600.2.a.f 1
600.2.a.g 1
600.2.a.h 1
600.2.a.i 1
600.2.b $$\chi_{600}(251, \cdot)$$ 600.2.b.a 2 1
600.2.b.b 4
600.2.b.c 4
600.2.b.d 4
600.2.b.e 8
600.2.b.f 8
600.2.b.g 12
600.2.b.h 12
600.2.b.i 16
600.2.d $$\chi_{600}(349, \cdot)$$ 600.2.d.a 2 1
600.2.d.b 2
600.2.d.c 2
600.2.d.d 2
600.2.d.e 6
600.2.d.f 6
600.2.d.g 8
600.2.d.h 8
600.2.f $$\chi_{600}(49, \cdot)$$ 600.2.f.a 2 1
600.2.f.b 2
600.2.f.c 2
600.2.f.d 2
600.2.f.e 2
600.2.h $$\chi_{600}(551, \cdot)$$ None 0 1
600.2.k $$\chi_{600}(301, \cdot)$$ 600.2.k.a 2 1
600.2.k.b 2
600.2.k.c 6
600.2.k.d 8
600.2.k.e 8
600.2.k.f 12
600.2.m $$\chi_{600}(299, \cdot)$$ 600.2.m.a 4 1
600.2.m.b 8
600.2.m.c 16
600.2.m.d 16
600.2.m.e 24
600.2.o $$\chi_{600}(599, \cdot)$$ None 0 1
600.2.r $$\chi_{600}(257, \cdot)$$ 600.2.r.a 4 2
600.2.r.b 4
600.2.r.c 4
600.2.r.d 4
600.2.r.e 4
600.2.r.f 16
600.2.s $$\chi_{600}(7, \cdot)$$ None 0 2
600.2.v $$\chi_{600}(43, \cdot)$$ 600.2.v.a 16 2
600.2.v.b 24
600.2.v.c 32
600.2.w $$\chi_{600}(293, \cdot)$$ 600.2.w.a 4 2
600.2.w.b 4
600.2.w.c 4
600.2.w.d 4
600.2.w.e 4
600.2.w.f 4
600.2.w.g 4
600.2.w.h 4
600.2.w.i 8
600.2.w.j 32
600.2.w.k 64
600.2.y $$\chi_{600}(121, \cdot)$$ 600.2.y.a 4 4
600.2.y.b 4
600.2.y.c 12
600.2.y.d 12
600.2.y.e 16
600.2.y.f 16
600.2.ba $$\chi_{600}(71, \cdot)$$ None 0 4
600.2.bc $$\chi_{600}(169, \cdot)$$ 600.2.bc.a 8 4
600.2.bc.b 24
600.2.bc.c 24
600.2.be $$\chi_{600}(109, \cdot)$$ 600.2.be.a 120 4
600.2.be.b 120
600.2.bg $$\chi_{600}(11, \cdot)$$ 600.2.bg.a 464 4
600.2.bi $$\chi_{600}(119, \cdot)$$ None 0 4
600.2.bk $$\chi_{600}(59, \cdot)$$ 600.2.bk.a 464 4
600.2.bm $$\chi_{600}(61, \cdot)$$ 600.2.bm.a 240 4
600.2.bp $$\chi_{600}(53, \cdot)$$ 600.2.bp.a 16 8
600.2.bp.b 16
600.2.bp.c 896
600.2.bq $$\chi_{600}(67, \cdot)$$ 600.2.bq.a 480 8
600.2.bt $$\chi_{600}(103, \cdot)$$ None 0 8
600.2.bu $$\chi_{600}(17, \cdot)$$ 600.2.bu.a 240 8

## Decomposition of $$S_{2}^{\mathrm{old}}(\Gamma_1(600))$$ into lower level spaces

$$S_{2}^{\mathrm{old}}(\Gamma_1(600)) \cong$$ $$S_{2}^{\mathrm{new}}(\Gamma_1(1))$$$$^{\oplus 24}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(2))$$$$^{\oplus 18}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(3))$$$$^{\oplus 12}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(4))$$$$^{\oplus 12}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(5))$$$$^{\oplus 16}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(6))$$$$^{\oplus 9}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(8))$$$$^{\oplus 6}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(10))$$$$^{\oplus 12}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(12))$$$$^{\oplus 6}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(15))$$$$^{\oplus 8}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(20))$$$$^{\oplus 8}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(24))$$$$^{\oplus 3}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(25))$$$$^{\oplus 8}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(30))$$$$^{\oplus 6}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(40))$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(50))$$$$^{\oplus 6}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(60))$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(75))$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(100))$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(120))$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(150))$$$$^{\oplus 3}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(200))$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(300))$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(600))$$$$^{\oplus 1}$$