## 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

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

## 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

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