# Properties

 Label 550.2 Level 550 Weight 2 Dimension 2717 Nonzero newspaces 21 Newform subspaces 82 Sturm bound 36000 Trace bound 13

## Defining parameters

 Level: $$N$$ = $$550 = 2 \cdot 5^{2} \cdot 11$$ Weight: $$k$$ = $$2$$ Nonzero newspaces: $$21$$ Newform subspaces: $$82$$ Sturm bound: $$36000$$ Trace bound: $$13$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(\Gamma_1(550))$$.

Total New Old
Modular forms 9560 2717 6843
Cusp forms 8441 2717 5724
Eisenstein series 1119 0 1119

## Trace form

 $$2717 q + 2 q^{2} + 8 q^{3} + 2 q^{4} + 10 q^{5} + 13 q^{6} + 26 q^{7} + 2 q^{8} + 46 q^{9} + O(q^{10})$$ $$2717 q + 2 q^{2} + 8 q^{3} + 2 q^{4} + 10 q^{5} + 13 q^{6} + 26 q^{7} + 2 q^{8} + 46 q^{9} + 10 q^{10} + 22 q^{11} + 18 q^{12} + 38 q^{13} + 26 q^{14} + 40 q^{15} + 2 q^{16} + 16 q^{17} - 19 q^{18} - 25 q^{19} + 4 q^{21} - 28 q^{22} - 12 q^{23} - 27 q^{24} - 70 q^{25} + 8 q^{26} + 5 q^{27} - 14 q^{28} + 20 q^{29} - 40 q^{30} + 14 q^{31} - 3 q^{32} + 63 q^{33} + 6 q^{34} + 40 q^{35} + 31 q^{36} + 86 q^{37} + 70 q^{38} + 82 q^{39} + 10 q^{40} + 84 q^{41} + 14 q^{42} - 102 q^{43} - 53 q^{44} - 150 q^{45} - 92 q^{46} - 184 q^{47} + 8 q^{48} - 316 q^{49} - 30 q^{50} - 231 q^{51} - 112 q^{52} - 172 q^{53} - 200 q^{54} - 140 q^{55} + 16 q^{56} - 235 q^{57} - 220 q^{58} - 175 q^{59} - 80 q^{60} - 246 q^{61} - 186 q^{62} - 352 q^{63} + 2 q^{64} - 190 q^{65} - 112 q^{66} - 174 q^{67} - 144 q^{68} - 138 q^{69} - 120 q^{70} + 64 q^{71} + 46 q^{72} + 48 q^{73} - 104 q^{74} - 120 q^{75} - 30 q^{76} + 16 q^{77} - 48 q^{78} + 50 q^{79} + 10 q^{80} + 217 q^{81} - 41 q^{82} - 87 q^{83} - 16 q^{84} - 30 q^{85} + 33 q^{86} + 60 q^{87} + 22 q^{88} + 50 q^{89} + 10 q^{90} + 84 q^{91} + 38 q^{92} + 26 q^{93} + 156 q^{94} + 40 q^{95} + 8 q^{96} + 11 q^{97} + 159 q^{98} - 174 q^{99} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(\Gamma_1(550))$$

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
550.2.a $$\chi_{550}(1, \cdot)$$ 550.2.a.a 1 1
550.2.a.b 1
550.2.a.c 1
550.2.a.d 1
550.2.a.e 1
550.2.a.f 1
550.2.a.g 1
550.2.a.h 1
550.2.a.i 1
550.2.a.j 1
550.2.a.k 1
550.2.a.l 1
550.2.a.m 1
550.2.a.n 2
550.2.b $$\chi_{550}(199, \cdot)$$ 550.2.b.a 2 1
550.2.b.b 2
550.2.b.c 2
550.2.b.d 2
550.2.b.e 2
550.2.b.f 4
550.2.f $$\chi_{550}(43, \cdot)$$ 550.2.f.a 4 2
550.2.f.b 4
550.2.f.c 4
550.2.f.d 8
550.2.f.e 16
550.2.g $$\chi_{550}(291, \cdot)$$ 550.2.g.a 12 4
550.2.g.b 48
550.2.g.c 60
550.2.h $$\chi_{550}(201, \cdot)$$ 550.2.h.a 4 4
550.2.h.b 4
550.2.h.c 4
550.2.h.d 4
550.2.h.e 4
550.2.h.f 4
550.2.h.g 4
550.2.h.h 4
550.2.h.i 4
550.2.h.j 8
550.2.h.k 8
550.2.h.l 8
550.2.h.m 8
550.2.h.n 8
550.2.i $$\chi_{550}(31, \cdot)$$ 550.2.i.a 4 4
550.2.i.b 4
550.2.i.c 4
550.2.i.d 48
550.2.i.e 60
550.2.j $$\chi_{550}(81, \cdot)$$ 550.2.j.a 12 4
550.2.j.b 48
550.2.j.c 60
550.2.k $$\chi_{550}(111, \cdot)$$ 550.2.k.a 4 4
550.2.k.b 20
550.2.k.c 20
550.2.k.d 24
550.2.k.e 28
550.2.l $$\chi_{550}(181, \cdot)$$ 550.2.l.a 4 4
550.2.l.b 4
550.2.l.c 4
550.2.l.d 48
550.2.l.e 60
550.2.n $$\chi_{550}(59, \cdot)$$ 550.2.n.a 120 4
550.2.t $$\chi_{550}(89, \cdot)$$ 550.2.t.a 8 4
550.2.t.b 40
550.2.t.c 56
550.2.y $$\chi_{550}(9, \cdot)$$ 550.2.y.a 120 4
550.2.z $$\chi_{550}(69, \cdot)$$ 550.2.z.a 120 4
550.2.ba $$\chi_{550}(49, \cdot)$$ 550.2.ba.a 8 4
550.2.ba.b 8
550.2.ba.c 8
550.2.ba.d 8
550.2.ba.e 8
550.2.ba.f 16
550.2.ba.g 16
550.2.bb $$\chi_{550}(119, \cdot)$$ 550.2.bb.a 120 4
550.2.be $$\chi_{550}(17, \cdot)$$ 550.2.be.a 240 8
550.2.bh $$\chi_{550}(7, \cdot)$$ 550.2.bh.a 32 8
550.2.bh.b 48
550.2.bh.c 64
550.2.bi $$\chi_{550}(87, \cdot)$$ 550.2.bi.a 240 8
550.2.bj $$\chi_{550}(13, \cdot)$$ 550.2.bj.a 240 8
550.2.bk $$\chi_{550}(123, \cdot)$$ 550.2.bk.a 240 8
550.2.bp $$\chi_{550}(63, \cdot)$$ 550.2.bp.a 240 8

## Decomposition of $$S_{2}^{\mathrm{old}}(\Gamma_1(550))$$ into lower level spaces

$$S_{2}^{\mathrm{old}}(\Gamma_1(550)) \cong$$ $$S_{2}^{\mathrm{new}}(\Gamma_1(1))$$$$^{\oplus 12}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(2))$$$$^{\oplus 6}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(5))$$$$^{\oplus 8}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(10))$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(11))$$$$^{\oplus 6}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(22))$$$$^{\oplus 3}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(25))$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(50))$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(55))$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(110))$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(275))$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(550))$$$$^{\oplus 1}$$