## Defining parameters

 Level: $$N$$ = $$126 = 2 \cdot 3^{2} \cdot 7$$ Weight: $$k$$ = $$2$$ Nonzero newspaces: $$9$$ Newform subspaces: $$22$$ Sturm bound: $$1728$$ Trace bound: $$9$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(\Gamma_1(126))$$.

Total New Old
Modular forms 528 110 418
Cusp forms 337 110 227
Eisenstein series 191 0 191

## Trace form

 $$110 q + 2 q^{2} + 6 q^{3} + 4 q^{4} + 6 q^{5} - 6 q^{6} + 10 q^{7} - 4 q^{8} - 6 q^{9} + O(q^{10})$$ $$110 q + 2 q^{2} + 6 q^{3} + 4 q^{4} + 6 q^{5} - 6 q^{6} + 10 q^{7} - 4 q^{8} - 6 q^{9} + 6 q^{10} + 6 q^{11} + 6 q^{13} - 16 q^{14} - 24 q^{15} - 4 q^{16} - 36 q^{17} - 12 q^{18} - 30 q^{19} - 18 q^{20} - 42 q^{21} - 42 q^{22} - 60 q^{23} - 6 q^{24} - 40 q^{25} - 38 q^{26} - 36 q^{27} - 4 q^{28} - 12 q^{29} - 12 q^{30} - 12 q^{31} + 2 q^{32} - 6 q^{33} + 6 q^{34} - 30 q^{35} - 6 q^{36} + 8 q^{37} + 16 q^{38} - 36 q^{39} + 6 q^{40} - 30 q^{41} + 2 q^{43} + 24 q^{44} - 48 q^{45} + 48 q^{46} - 48 q^{47} - 6 q^{48} - 28 q^{49} + 50 q^{50} + 18 q^{51} - 6 q^{52} + 24 q^{53} + 54 q^{54} + 8 q^{56} + 90 q^{57} - 12 q^{58} + 120 q^{59} + 72 q^{60} - 6 q^{61} + 100 q^{62} + 180 q^{63} - 2 q^{64} + 168 q^{65} + 96 q^{66} - 26 q^{67} + 54 q^{68} + 108 q^{69} - 18 q^{70} + 120 q^{71} - 6 q^{72} - 24 q^{73} + 64 q^{74} + 114 q^{75} + 66 q^{77} + 60 q^{78} + 4 q^{79} + 6 q^{80} + 6 q^{81} + 30 q^{83} + 6 q^{84} + 60 q^{85} + 10 q^{86} - 108 q^{87} - 18 q^{88} - 84 q^{89} - 48 q^{90} - 66 q^{91} - 36 q^{92} - 108 q^{93} - 48 q^{94} - 144 q^{95} - 12 q^{96} - 54 q^{97} - 76 q^{98} - 156 q^{99} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(\Gamma_1(126))$$

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
126.2.a $$\chi_{126}(1, \cdot)$$ 126.2.a.a 1 1
126.2.a.b 1
126.2.d $$\chi_{126}(125, \cdot)$$ None 0 1
126.2.e $$\chi_{126}(25, \cdot)$$ 126.2.e.a 2 2
126.2.e.b 2
126.2.e.c 6
126.2.e.d 6
126.2.f $$\chi_{126}(43, \cdot)$$ 126.2.f.a 2 2
126.2.f.b 2
126.2.f.c 4
126.2.f.d 4
126.2.g $$\chi_{126}(37, \cdot)$$ 126.2.g.a 2 2
126.2.g.b 2
126.2.g.c 2
126.2.g.d 2
126.2.h $$\chi_{126}(67, \cdot)$$ 126.2.h.a 2 2
126.2.h.b 2
126.2.h.c 6
126.2.h.d 6
126.2.k $$\chi_{126}(17, \cdot)$$ 126.2.k.a 8 2
126.2.l $$\chi_{126}(5, \cdot)$$ 126.2.l.a 16 2
126.2.m $$\chi_{126}(41, \cdot)$$ 126.2.m.a 16 2
126.2.t $$\chi_{126}(47, \cdot)$$ 126.2.t.a 16 2

## Decomposition of $$S_{2}^{\mathrm{old}}(\Gamma_1(126))$$ into lower level spaces

$$S_{2}^{\mathrm{old}}(\Gamma_1(126)) \cong$$ $$S_{2}^{\mathrm{new}}(\Gamma_1(14))$$$$^{\oplus 3}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(18))$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(21))$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(42))$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(63))$$$$^{\oplus 2}$$