## Defining parameters

 Level: $$N$$ = $$63 = 3^{2} \cdot 7$$ Weight: $$k$$ = $$4$$ Nonzero newspaces: $$10$$ Newform subspaces: $$20$$ Sturm bound: $$1152$$ Trace bound: $$2$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{4}(\Gamma_1(63))$$.

Total New Old
Modular forms 480 349 131
Cusp forms 384 305 79
Eisenstein series 96 44 52

## Trace form

 $$305q - 3q^{2} - 6q^{3} + 17q^{4} + 33q^{5} - 30q^{6} + 2q^{7} - 177q^{8} - 102q^{9} + O(q^{10})$$ $$305q - 3q^{2} - 6q^{3} + 17q^{4} + 33q^{5} - 30q^{6} + 2q^{7} - 177q^{8} - 102q^{9} - 78q^{10} + 123q^{11} + 348q^{12} + 256q^{13} + 603q^{14} + 42q^{15} - 91q^{16} - 615q^{17} - 780q^{18} - 491q^{19} - 1626q^{20} - 393q^{21} - 954q^{22} - 237q^{23} - 102q^{24} + 209q^{25} + 1626q^{26} + 1248q^{27} + 1433q^{28} + 1140q^{29} + 1002q^{30} + 1345q^{31} + 2535q^{32} + 336q^{33} + 1530q^{34} - 3q^{35} - 234q^{36} - 1517q^{37} - 4596q^{38} - 2790q^{39} - 3654q^{40} - 2784q^{41} - 2232q^{42} - 788q^{43} - 3570q^{44} - 54q^{45} - 1218q^{46} - 201q^{47} - 2502q^{48} + 1148q^{49} + 159q^{50} - 1362q^{51} + 3970q^{52} + 2841q^{53} + 336q^{54} + 510q^{55} + 4599q^{56} + 3954q^{57} - 2652q^{58} + 4005q^{59} + 8670q^{60} - 2327q^{61} + 3084q^{62} + 4593q^{63} + 1733q^{64} + 4722q^{65} + 8226q^{66} + 4651q^{67} + 12738q^{68} + 3978q^{69} + 9078q^{70} - 348q^{71} - 1920q^{72} + 3355q^{73} + 90q^{74} - 5718q^{75} - 878q^{76} - 8508q^{77} - 11040q^{78} - 7793q^{79} - 18114q^{80} - 5346q^{81} - 19734q^{82} - 9438q^{83} - 8322q^{84} - 9918q^{85} - 10644q^{86} - 1182q^{87} + 5988q^{88} + 1893q^{89} + 1914q^{90} + 3070q^{91} + 4848q^{92} + 2094q^{93} + 7284q^{94} + 3339q^{95} + 1320q^{96} + 7912q^{97} + 11763q^{98} + 3366q^{99} + O(q^{100})$$

## Decomposition of $$S_{4}^{\mathrm{new}}(\Gamma_1(63))$$

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
63.4.a $$\chi_{63}(1, \cdot)$$ 63.4.a.a 1 1
63.4.a.b 1
63.4.a.c 1
63.4.a.d 2
63.4.a.e 2
63.4.c $$\chi_{63}(62, \cdot)$$ 63.4.c.a 4 1
63.4.c.b 4
63.4.e $$\chi_{63}(37, \cdot)$$ 63.4.e.a 2 2
63.4.e.b 2
63.4.e.c 6
63.4.e.d 8
63.4.f $$\chi_{63}(22, \cdot)$$ 63.4.f.a 2 2
63.4.f.b 16
63.4.f.c 18
63.4.g $$\chi_{63}(4, \cdot)$$ 63.4.g.a 44 2
63.4.h $$\chi_{63}(25, \cdot)$$ 63.4.h.a 44 2
63.4.i $$\chi_{63}(5, \cdot)$$ 63.4.i.a 44 2
63.4.o $$\chi_{63}(20, \cdot)$$ 63.4.o.a 44 2
63.4.p $$\chi_{63}(17, \cdot)$$ 63.4.p.a 16 2
63.4.s $$\chi_{63}(47, \cdot)$$ 63.4.s.a 44 2

## Decomposition of $$S_{4}^{\mathrm{old}}(\Gamma_1(63))$$ into lower level spaces

$$S_{4}^{\mathrm{old}}(\Gamma_1(63)) \cong$$ $$S_{4}^{\mathrm{new}}(\Gamma_1(7))$$$$^{\oplus 3}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(9))$$$$^{\oplus 2}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(21))$$$$^{\oplus 2}$$