## Defining parameters

 Level: $$N$$ = $$63 = 3^{2} \cdot 7$$ Weight: $$k$$ = $$2$$ Nonzero newspaces: $$10$$ Newform subspaces: $$17$$ Sturm bound: $$576$$ Trace bound: $$4$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(\Gamma_1(63))$$.

Total New Old
Modular forms 192 131 61
Cusp forms 97 87 10
Eisenstein series 95 44 51

## Trace form

 $$87q - 9q^{2} - 12q^{3} - 13q^{4} - 12q^{5} - 12q^{6} - 13q^{7} - 27q^{8} - 12q^{9} + O(q^{10})$$ $$87q - 9q^{2} - 12q^{3} - 13q^{4} - 12q^{5} - 12q^{6} - 13q^{7} - 27q^{8} - 12q^{9} - 42q^{10} - 18q^{11} - 16q^{13} - 3q^{14} - 12q^{15} + 3q^{16} + 12q^{17} + 12q^{18} - 16q^{19} + 36q^{20} + 6q^{21} + 6q^{23} + 24q^{24} - 9q^{25} + 24q^{26} + 6q^{27} - 33q^{28} - 18q^{29} + 30q^{30} - 10q^{31} + 39q^{32} + 12q^{33} + 12q^{34} + 24q^{35} + 36q^{36} - 14q^{37} + 54q^{38} + 24q^{39} + 48q^{40} + 60q^{41} + 54q^{42} - 8q^{43} + 66q^{44} + 36q^{45} + 24q^{46} + 18q^{47} - 24q^{48} + 3q^{49} - 9q^{50} - 12q^{51} + 8q^{52} - 42q^{53} - 42q^{54} - 48q^{55} - 87q^{56} - 72q^{57} - 36q^{58} - 96q^{59} - 114q^{60} - 22q^{61} - 156q^{62} - 96q^{63} - 127q^{64} - 72q^{65} - 54q^{66} - 12q^{67} - 120q^{68} - 36q^{69} + 24q^{70} - 12q^{71} - 66q^{72} + 14q^{73} + 12q^{74} + 38q^{76} + 36q^{77} + 12q^{78} + 36q^{79} + 36q^{80} + 72q^{81} + 18q^{82} + 66q^{83} + 132q^{84} + 24q^{85} + 126q^{86} + 78q^{87} + 78q^{88} + 132q^{89} + 150q^{90} + 38q^{91} + 132q^{92} + 102q^{93} + 90q^{94} + 114q^{95} + 168q^{96} + 56q^{97} + 123q^{98} + 72q^{99} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(\Gamma_1(63))$$

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
63.2.a $$\chi_{63}(1, \cdot)$$ 63.2.a.a 1 1
63.2.a.b 2
63.2.c $$\chi_{63}(62, \cdot)$$ 63.2.c.a 4 1
63.2.e $$\chi_{63}(37, \cdot)$$ 63.2.e.a 2 2
63.2.e.b 2
63.2.f $$\chi_{63}(22, \cdot)$$ 63.2.f.a 6 2
63.2.f.b 6
63.2.g $$\chi_{63}(4, \cdot)$$ 63.2.g.a 2 2
63.2.g.b 10
63.2.h $$\chi_{63}(25, \cdot)$$ 63.2.h.a 2 2
63.2.h.b 10
63.2.i $$\chi_{63}(5, \cdot)$$ 63.2.i.a 2 2
63.2.i.b 10
63.2.o $$\chi_{63}(20, \cdot)$$ 63.2.o.a 12 2
63.2.p $$\chi_{63}(17, \cdot)$$ 63.2.p.a 4 2
63.2.s $$\chi_{63}(47, \cdot)$$ 63.2.s.a 2 2
63.2.s.b 10

## Decomposition of $$S_{2}^{\mathrm{old}}(\Gamma_1(63))$$ into lower level spaces

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