## Defining parameters

 Level: $$N$$ = $$552 = 2^{3} \cdot 3 \cdot 23$$ Weight: $$k$$ = $$2$$ Nonzero newspaces: $$12$$ Newform subspaces: $$32$$ Sturm bound: $$33792$$ Trace bound: $$4$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(\Gamma_1(552))$$.

Total New Old
Modular forms 8976 3634 5342
Cusp forms 7921 3466 4455
Eisenstein series 1055 168 887

## Trace form

 $$3466 q + 4 q^{2} - 16 q^{3} - 36 q^{4} + 4 q^{5} - 26 q^{6} - 36 q^{7} - 8 q^{8} - 38 q^{9} + O(q^{10})$$ $$3466 q + 4 q^{2} - 16 q^{3} - 36 q^{4} + 4 q^{5} - 26 q^{6} - 36 q^{7} - 8 q^{8} - 38 q^{9} - 52 q^{10} - 8 q^{11} - 38 q^{12} + 4 q^{13} - 8 q^{14} - 34 q^{15} - 44 q^{16} + 4 q^{17} - 10 q^{18} - 44 q^{19} + 16 q^{20} - 28 q^{22} - 20 q^{24} - 70 q^{25} + 16 q^{26} - 40 q^{27} - 44 q^{28} - 12 q^{29} - 14 q^{30} - 68 q^{31} - 16 q^{32} - 52 q^{33} - 84 q^{34} + 44 q^{35} - 30 q^{36} + 76 q^{37} - 16 q^{38} + 34 q^{39} - 60 q^{40} + 48 q^{41} - 30 q^{42} + 76 q^{43} + 4 q^{45} - 36 q^{46} + 136 q^{47} - 6 q^{48} + 42 q^{49} + 4 q^{50} + 58 q^{51} - 76 q^{52} + 48 q^{53} - 34 q^{54} + 60 q^{55} + 16 q^{56} - 16 q^{57} - 20 q^{58} + 36 q^{59} - 22 q^{60} + 4 q^{61} + 8 q^{62} - 30 q^{63} - 12 q^{64} - 40 q^{65} - 104 q^{66} - 92 q^{67} - 8 q^{69} - 72 q^{70} - 64 q^{71} - 46 q^{72} - 92 q^{73} - 76 q^{74} - 110 q^{75} - 216 q^{76} - 258 q^{78} - 200 q^{79} - 396 q^{80} - 154 q^{81} - 236 q^{82} - 124 q^{83} - 314 q^{84} - 124 q^{85} - 424 q^{86} - 184 q^{87} - 428 q^{88} - 80 q^{89} - 432 q^{90} - 352 q^{91} - 396 q^{92} + 16 q^{93} - 488 q^{94} - 248 q^{95} - 488 q^{96} - 176 q^{97} - 364 q^{98} - 196 q^{99} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(\Gamma_1(552))$$

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
552.2.a $$\chi_{552}(1, \cdot)$$ 552.2.a.a 1 1
552.2.a.b 1
552.2.a.c 1
552.2.a.d 1
552.2.a.e 1
552.2.a.f 2
552.2.a.g 3
552.2.b $$\chi_{552}(413, \cdot)$$ 552.2.b.a 6 1
552.2.b.b 6
552.2.b.c 80
552.2.e $$\chi_{552}(47, \cdot)$$ None 0 1
552.2.f $$\chi_{552}(277, \cdot)$$ 552.2.f.a 2 1
552.2.f.b 4
552.2.f.c 18
552.2.f.d 20
552.2.i $$\chi_{552}(367, \cdot)$$ None 0 1
552.2.j $$\chi_{552}(323, \cdot)$$ 552.2.j.a 2 1
552.2.j.b 2
552.2.j.c 42
552.2.j.d 42
552.2.m $$\chi_{552}(137, \cdot)$$ 552.2.m.a 8 1
552.2.m.b 16
552.2.n $$\chi_{552}(91, \cdot)$$ 552.2.n.a 24 1
552.2.n.b 24
552.2.q $$\chi_{552}(25, \cdot)$$ 552.2.q.a 30 10
552.2.q.b 30
552.2.q.c 30
552.2.q.d 30
552.2.t $$\chi_{552}(19, \cdot)$$ 552.2.t.a 240 10
552.2.t.b 240
552.2.u $$\chi_{552}(17, \cdot)$$ 552.2.u.a 240 10
552.2.x $$\chi_{552}(35, \cdot)$$ 552.2.x.a 920 10
552.2.y $$\chi_{552}(7, \cdot)$$ None 0 10
552.2.bb $$\chi_{552}(13, \cdot)$$ 552.2.bb.a 480 10
552.2.bc $$\chi_{552}(71, \cdot)$$ None 0 10
552.2.bf $$\chi_{552}(5, \cdot)$$ 552.2.bf.a 920 10

## Decomposition of $$S_{2}^{\mathrm{old}}(\Gamma_1(552))$$ into lower level spaces

$$S_{2}^{\mathrm{old}}(\Gamma_1(552)) \cong$$ $$S_{2}^{\mathrm{new}}(\Gamma_1(23))$$$$^{\oplus 8}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(24))$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(46))$$$$^{\oplus 6}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(69))$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(92))$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(138))$$$$^{\oplus 3}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(184))$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(276))$$$$^{\oplus 2}$$