## 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

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

## 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

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