## Defining parameters

 Level: $$N$$ = $$473 = 11 \cdot 43$$ Weight: $$k$$ = $$2$$ Nonzero newspaces: $$16$$ Newforms: $$33$$ Sturm bound: $$36960$$ Trace bound: $$3$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(\Gamma_1(473))$$.

Total New Old
Modular forms 9660 9445 215
Cusp forms 8821 8705 116
Eisenstein series 839 740 99

## Trace form

 $$8705q - 167q^{2} - 170q^{3} - 179q^{4} - 176q^{5} - 184q^{6} - 172q^{7} - 183q^{8} - 177q^{9} + O(q^{10})$$ $$8705q - 167q^{2} - 170q^{3} - 179q^{4} - 176q^{5} - 184q^{6} - 172q^{7} - 183q^{8} - 177q^{9} - 182q^{10} - 192q^{11} - 402q^{12} - 190q^{13} - 200q^{14} - 190q^{15} - 191q^{16} - 182q^{17} - 215q^{18} - 188q^{19} - 214q^{20} - 204q^{21} - 188q^{22} - 400q^{23} - 228q^{24} - 191q^{25} - 194q^{26} - 218q^{27} - 216q^{28} - 198q^{29} - 244q^{30} - 200q^{31} - 163q^{32} - 149q^{33} - 314q^{34} - 128q^{35} - 27q^{36} - 128q^{37} - 60q^{38} - 118q^{39} + 78q^{40} - 152q^{41} - 46q^{42} + 5q^{43} - 305q^{44} - 242q^{45} - 76q^{46} - 190q^{47} + 36q^{48} - 121q^{49} - 109q^{50} - 160q^{51} - 58q^{52} - 126q^{53} - 142q^{54} - 155q^{55} - 414q^{56} - 234q^{57} - 258q^{58} - 238q^{59} - 332q^{60} - 254q^{61} - 236q^{62} - 280q^{63} - 279q^{64} - 260q^{65} - 205q^{66} - 432q^{67} - 286q^{68} - 182q^{69} - 152q^{70} - 150q^{71} + 57q^{72} - 166q^{73} + 24q^{74} - 6q^{75} - 14q^{76} - 67q^{77} - 16q^{78} - 60q^{79} - 10q^{80} + 45q^{81} + 94q^{82} - 72q^{83} + 440q^{84} - 62q^{85} + 27q^{86} - 36q^{87} + 6q^{88} - 330q^{89} + 346q^{90} - 96q^{91} + 88q^{92} + 54q^{93} + 56q^{94} - 120q^{95} + 142q^{96} - 57q^{98} - 51q^{99} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(\Gamma_1(473))$$

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
473.2.a $$\chi_{473}(1, \cdot)$$ 473.2.a.a 1 1
473.2.a.b 2
473.2.a.c 2
473.2.a.d 5
473.2.a.e 5
473.2.a.f 9
473.2.a.g 11
473.2.d $$\chi_{473}(472, \cdot)$$ 473.2.d.a 2 1
473.2.d.b 4
473.2.d.c 4
473.2.d.d 4
473.2.d.e 28
473.2.e $$\chi_{473}(122, \cdot)$$ 473.2.e.a 2 2
473.2.e.b 2
473.2.e.c 34
473.2.e.d 34
473.2.f $$\chi_{473}(130, \cdot)$$ 473.2.f.a 76 4
473.2.f.b 92
473.2.i $$\chi_{473}(252, \cdot)$$ 473.2.i.a 84 2
473.2.j $$\chi_{473}(78, \cdot)$$ 473.2.j.a 114 6
473.2.j.b 114
473.2.k $$\chi_{473}(85, \cdot)$$ 473.2.k.a 8 4
473.2.k.b 160
473.2.n $$\chi_{473}(32, \cdot)$$ 473.2.n.a 252 6
473.2.q $$\chi_{473}(36, \cdot)$$ 473.2.q.a 336 8
473.2.r $$\chi_{473}(23, \cdot)$$ 473.2.r.a 216 12
473.2.r.b 216
473.2.s $$\chi_{473}(7, \cdot)$$ 473.2.s.a 336 8
473.2.v $$\chi_{473}(4, \cdot)$$ 473.2.v.a 1008 24
473.2.w $$\chi_{473}(76, \cdot)$$ 473.2.w.a 504 12
473.2.bb $$\chi_{473}(2, \cdot)$$ 473.2.bb.a 1008 24
473.2.bc $$\chi_{473}(9, \cdot)$$ 473.2.bc.a 2016 48
473.2.bf $$\chi_{473}(18, \cdot)$$ 473.2.bf.a 2016 48

## Decomposition of $$S_{2}^{\mathrm{old}}(\Gamma_1(473))$$ into lower level spaces

$$S_{2}^{\mathrm{old}}(\Gamma_1(473)) \cong$$ $$S_{2}^{\mathrm{new}}(\Gamma_1(11))$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(43))$$$$^{\oplus 2}$$